Packet-Based Control for Networked Control Systems

Packet-Based Control for Networked Control Systems

Yun-Bo Zhao

A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy

University of Glamorgan Prifysgol Morgannwg

June 2008 c

Yun-Bo Zhao 2008. All Rights Reserved.

Abstract Networked control systems (NCSs) are such control systems where the control loop is closed via some form of communication networks. These control systems are widely applicable in remote and distributed control applications. The inserted network however presents great challenges to conventional control theory as far as the design and analysis of NCSs are concerned. These challenges are caused primarily by the communication constraints in NCSs, e.g., network-induced delay, data packet dropout, data packet disorder, network access constraint, etc., which significantly degrade the system performance or even destabilize the system. When applying conventional control approaches to NCSs, considerable conservativeness is inevitable due to the failure to exploit network characteristics. Therefore, the co-design approach to NCSs in which control approaches and characteristics of NCSs are both fully considered, is believed to be the best way forward for the design of NCSs. In this thesis, we investigate the packet-based transmission of the network being used in NCSs, and propose a packet-based control (PB-control) approach to NCSs. In this approach, the “packet” structure of data transmission in NCSs which is distinct from conventional control systems, is taken advantage of where, the control signals are first “packed” and then sent as a sequence instead of one at a time as done in conventional control systems. With the efficient use of the “packet” structure, we can then actively compensate for the communication constraints in NCSs including the network-induced delay, data packet dropout and data packet disorder simultaneously. After determining the PB-control structure, we then extend its application to several categories of problems as follows.

• The first application is to two types of special nonlinear systems described by a Hammerstein model and a Wiener model respectively. A “two-step” approach is adopted in this situation to separate the nonlinear process from the whole system which then enables the PB-control approach to be implemented. • It is observed that the communication constraints in NCSs are stochastic in nature, and thus a stochastic analysis of the PB-control approach is presented

iii under the Markov jump system framework, by modeling the network-induced delay and data packet dropout as a homogeneous ergodic Markov chain. The sufficient and necessary conditions for stochastic stability and stabilization in this situation are also obtained. • Continuous-time plant and continuous network-induced delay are observed to be more difficult to handle when implementing the PB-control approach. For this challenge, a discretization technique is introduced for the continuous network-induced delay and as a result, a novel model for NCSs is derived which is different to that obtained by conventional analysis from time delay system theory. A stabilized controller is also obtained in this situation by using delay-dependent analysis. • The last application is to deal with the situation where a set of NCSs share the network and thus the network access constraint has to be considered. For this situation, a PB-control and scheduling co-design approach is proposed where, PB-control is still applied to each subsystem while scheduling algorithms are applied to schedule the network resources among the subsystems to guarantee the stability of the whole system.

We also point out in the thesis that further research on the PB-control approach is still needed as far as nonlinear, continuous-time systems and stochastic analysis are concerned.

Acknowledgements I would like to express my deep gratitude to my Director of Studies, Dr. David Rees. He has been a great advisor and supporter of my study. I admire and am indebted to him for his broad knowledge, patient guidance, and trust in me. I am also deeply grateful to my supervisor, Prof. Guo-Ping Liu. His consistent support and detailed direction have always been helpful and indispensable in my study. I am also grateful to them for providing opportunities for me to learn and develop as a researcher in such a pleasant environment. My warm thanks are due to my colleagues Prof. Peng Shi, Dr. Yuanqing Xia, Prof. Yong He, Dr. Ximing Sun, Dr. Rui Wang, Dr. Hua Ouyang, Dr. Senchun Chai, Mr. Bo Wang and Dr. Wenshan Hu, who work (worked) at the University of Glamorgan. Their helpful discussion and kind support are greatly appreciated. I am particulary grateful to Dr. Wenshan Hu for his help in implementing the experimental examples reported in Chapters 3, 4 and 5. I also thank the assistance of Dr. Yu Kang from the University of Science and Technology of China, who collaborated with me on the study reported in Chapter 6. Research studentship award and financial support from the Faculty of Advanced Technology, University of Glamorgan are gratefully acknowledged. My life and research at the University of Glamorgan have been influenced by, and have benefited from, a number of colleagues and friends. I would like to take this opportunity to thank all my colleagues and friends that have given me support and encouragement in my study. Thank you! Finally, but most importantly, this thesis is dedicated to my family for their continued love, understanding and support that have enabled me to complete my PhD degree.

Yun-Bo Zhao Pontypridd, UK June 2008 iv

Contents Abstract

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Acknowledgements

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List of Figures

viii

Abbreviations

x

Symbols

xii

1 Introduction 1.1 Motivation of research . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the thesis and publications . . . . . . . . . . . . . . . . . 2 A Brief Survey of Networked Control Systems 2.1 Characteristics of NCSs . . . . . . . . . . . . . . . . . 2.1.1 Two structures . . . . . . . . . . . . . . . . . . 2.1.2 Network topology . . . . . . . . . . . . . . . . . 2.1.3 PB-transmission . . . . . . . . . . . . . . . . . . 2.1.4 Limited network resources . . . . . . . . . . . . 2.2 Research background . . . . . . . . . . . . . . . . . . . 2.2.1 Conventional research on NCSs . . . . . . . . . 2.2.2 Co-design approaches to NCSs: A new direction 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 3 PB-Control for NCSs 3.1 PB-control for NCSs . . . . . . . . . . . . . . 3.1.1 PB-control for NCSs: A unified model 3.1.2 Design of PB-Control for NCSs . . . . 3.2 Stability of PB-NCSs . . . . . . . . . . . . . . 3.2.1 A switched system theory approach . . 3.2.2 A delay dependent analysis approach . v

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3.3 Controller design: A GPC-based approach . . . . . . . . . . . . . . 33 3.4 Numerical & experimental examples . . . . . . . . . . . . . . . . . . 35 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 PB-Control for Networked Hammerstein Systems 4.1 System descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 PB-Control for networked Hammerstein systems . . . . . . . . . . 4.2.1 Intermediate FCS (FCIS) . . . . . . . . . . . . . . . . . . 4.2.2 The nonlinear input process . . . . . . . . . . . . . . . . . 4.2.3 PB-Control for networked Hammerstein systems . . . . . . 4.3 Stability analysis of packet-based networked Hammerstein systems 4.3.1 Stability criterion in input-output description . . . . . . . 4.3.2 Stability criterion in state-space description . . . . . . . . 4.4 Numerical & experimental examples . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 PB-Control for Networked Wiener Systems 5.1 System description . . . . . . . . . . . . . . . . . . 5.2 PB-Control for networked Wiener systems . . . . . 5.3 Stability analysis of packet-based networked Wiener 5.3.1 Observer error . . . . . . . . . . . . . . . . . 5.3.2 Closed-loop stability . . . . . . . . . . . . . 5.4 Numerical & experimental examples . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 6 Stochastic Stabilization of PB-NCSs 6.1 Stochastic analysis of PB-NCSs . . . . . . . 6.1.1 Stochastic model of PB-NCSs . . . . 6.1.2 Stochastic stability and stabilization 6.2 A numerical example . . . . . . . . . . . . . 6.3 Conclusion . . . . . . . . . . . . . . . . . . .

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7 Continuous-Time PB-NCSs 7.1 PB-control: Modeling of NCSs . . . . . . . . . . 7.1.1 PB-control for NCSs in continuous time 7.1.2 A novel model for NCSs . . . . . . . . . 7.2 Stability and stabilization . . . . . . . . . . . . 7.3 A numerical example . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . .

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NCSs . . . . . . . . . . . . . . . .

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8 PB-Control and Scheduling Co-Design for 8.1 Problem statement . . . . . . . . . . . . . 8.2 PB-control for subsystems . . . . . . . . . 8.3 Scheduling . . . . . . . . . . . . . . . . . . 8.3.1 Static scheduling . . . . . . . . . .

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Contents

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8.3.2 Dynamic feedback scheduling . . . . . . . . . . . . . . . . . 105 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9 Conclusions and Future Work 113 9.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Bibliography

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Published & Finished Papers

126

List of Figures 1.1

Ubiquitous network, ubiquitous control. . . . . . . . . . . . . . . . .

2.1 Networked control systems in the direct structure. . . . . 2.2 Networked control systems in the hierarchical structure. . 2.3 Relationship between the sampling period, network loads tem performance in NCSs. . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

2

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The block diagram of networked control systems in discrete time. Packet-based networked control systems in discrete time (with time synchronization). . . . . . . . . . . . . . . . . . . . . . . . . . . . Timeline in the PB-NCSs . . . . . . . . . . . . . . . . . . . . . . Packet-based networked control systems in discrete time (without time synchronization). . . . . . . . . . . . . . . . . . . . . . . . . Example 3.1. System is unstable using LQR controller. . . . . . . Example 3.1. System is stable using PB-controller. . . . . . . . . Example 3.2. PB-control, unstable, τ¯sc = 2, τ¯ca = 1. . . . . . . . . Example 3.2. VFG scheme, stable, τ¯sc = 1, τ¯ca = 1. . . . . . . . . Example 3.2. FFG scheme, unstable, τ¯sc = 1, τ¯ca = 1. . . . . . . . Example 3.3. PB-control, stable, τ¯sc = 3, τ¯ca = 2. . . . . . . . . . The Internet-based test rig. . . . . . . . . . . . . . . . . . . . . . Example 3.4. Comparison between simulation and experimental results of linear packet-based control system. . . . . . . . . . . . .

. 23

The block diagram of networked Hammerstein systems. . . . . . . Two-step approach to networked Hammerstein systems. . . . . . . PB-control for networked Hammerstein systems. . . . . . . . . . . Popov criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The simplified block diagram of PB-control for networked Hammerstein systems in (4.1). . . . . . . . . . . . . . . . . . . . . . . . . Example 4.1. i) (τca , τsc ) = (0, 0); ii) (τca , τsc ) = (2, 3); . . . . . . . Example 4.1. iii) (τca , τsc ) = (3, 7) . . . . . . . . . . . . . . . . . . Example 4.2. Arbitrary delays in the forward channel. . . . . . . Example 4.2. The effectiveness of PB-control for networked Hammerstein systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4.3. Comparison between simulation and experimental results of packet-based control for Hammerstein systems. . . . . .

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List of Figures 5.1 The block diagram of networked Wiener Systems. . . . . . . . . . 5.2 Two-step approach to networked Wiener systems. . . . . . . . . . 5.3 PB-control for networked Wiener systems. . . . . . . . . . . . . . 5.4 Example 5.1. A comparison between with/without compensation for network constraints. . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Example 5.1. A comparison between with/without compensation for output nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Example 5.2. Comparison between simulation and experimental results of packet-based control for Wiener systems. . . . . . . . . 6.1

ix . 61 . 62 . 64 . 68 . 68 . 69

Example 6.1. States evolution of the PB-control approach to NCSs.

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7.1 The block diagram of networked control systems in continuous time. 7.2 Timeline of packet-based networked control systems. . . . . . . . . 7.3 Packet-based networked control systems in continuous time. . . . . 7.4 Example 7.1. State response and communication constraints. The discretized delay & dropout is obtained by ik ϑ. . . . . . . . . . . .

82 84 86 95

8.1 Multiple networked control systems share the communication channel. 98 8.2 Dynamic feedback scheduling of multiple systems. . . . . . . . . . . 105 8.3 Example 8.1. State evolution using RM algorithm. Only the first state is illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.4 Example 8.2. State evolution using DFS algorithm. Only the first state is illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Abbreviations CARIMA

Controlled Auto Regressive Integrated Moving Average

CAS

Control Action Selector

CCS(s)

Conventional Control System(s)

DFS

Dynamic Feedback Scheduling

FCIS

Forward Control Increment Sequence

FCS

Forward Control Sequence

FFG

Fixed Feedback Gain

GPC

Generalized Predictive Control

LGPC

Linear Generalized Predictive Control

LQR

Linear Quadratic Optimal

LTI

Linear Time-Invariant

MPC

Model Predictive Control

NCS(s)

Networked Control System(s)

PB-control

Packet-Based control

PB-controller

Packet-Based controller

PB-transmission

Packet-Based transmission

PB-NCS(s)

Packet-Based Networked Control System(s)

QoP

Quality of Performance

RM

Rate Monotonic

RHC

Receding Horizon Control

SISO

Single-Input-Single-Output

SSRTD

Stability-guaranteed Supremum of Round Trip Delay

TCP

Transmission Control Protocol

TDS(s)

Time Delay System(s) x

Abbreviations

xi

UDP

User Datagram Protocol

VFG

Varying Feedback Gain

Symbols h

Sampling period

u

System input

v

Intermediate control input

x

System state

y

System output

A B

System matrix

C I

Identity matrix

J

Objective function

U

System input in augmented form

X

System state in augmented form

Y

System output in augmented form

Bc

The bits required to encode a single step control signal

Bp

The size of the effective load of the data packet

Nu

Control horizon in model predictive control

Np

Predictive horizon in model predictive control

Sc

Linear time-invariant control system in continuous time

Sd

Linear time-invariant control system in discrete time

SI1

Hammerstein model in input-output description

SI2

Hammerstein model in state-space description xii

Symbols

xiii

So

Wiener model

τ

Network-induced delay

τsc,k

Backward channel delay at time k

τca,k

Forward channel delay at time k

τk

Round trip delay at time k

∗ τsc,k

Backward channel delay of the control signal that is actually applied at time k

∗ τca,k

Forward channel delay of the control signal that is actually applied at time k

τk∗

Round trip delay of the control signal that is actually applied at time k

τ¯

Upper bound of delay and dropout in round trip

τ¯sc

Upper bound of delay and dropout in the backward channel

τ¯ca

Upper bound of delay and dropout in the forward channel

ω

Reference input

N

Set of integers

Rn

Real space with dimension n

Rn×m

Matrix space with dimension n × m

|| · ||

Norm of ·

dxe

min{ς|ς ∈ N, ς ≥ x}

bxc

max{ς|ς ∈ N, ς ≤ x}

E{·}

Expectation of ·

P {·}

Probability of event ·

Re(·)

Real part of complex number ·

AT P

Transpose of matrix A Sum

◦

Composite function operator

∞

Infinity

∆

Increment operator

max

Maximum

Symbols

xiv

min

Minimum

inf

Infimum

sup

Supremum

lim

Limit

To my family

xv

Chapter 1 Introduction Networked Control Systems (NCSs) are such systems where the control components such as sensors, actuators, and controllers, etc. are interconnected via some form of communication network instead of connected directly as assumed in Conventional Control Systems (CCSs). In NCSs, instantaneous and perfect data transmissions between these control components are not achievable due to the inserted network. The study of NCSs therefore requires multi-field knowledge with the integration of communication, computing and control technologies. On the basis of this observation, the goal of this thesis is to investigate a class of “co-design” approach in NCSs which takes advantage of the packet-based transmission (PBtransmission) of the network – we call it the “packet-based control” (PB-control) approach to NCSs in the thesis – its design issues and the theoretical analysis of the corresponding closed-loop system. In this introductory chapter, we first present the motivation of our research, our aims and objectives, and the outline of the thesis and the resulting publications.

1.1

Motivation of research

The Internet, as the convergence of communication and computation during the last two decades, has brought a lot of interesting and important applications such as remote communications, E-commerce, distribute computing, etc.. It can be regarded as one of the most important and intelligent inventions in history. However, in the rapid development of information technology, an important aspect – the role of the control technology – has had a much lower priorities for decades, 1

Chapter 1. Introduction

2

yet it should have been given a greater importance in this information-rich world. The recognition of the control aspect, which is believed to be the key point of the next phase in information technology, will inevitably provide the ability of a large number of sensors, actuators and controllers, all interconnected over the network, to interact with the physical environment, that is, provide remote, distribute and ubiquitous control all over the world (Murray et al., 2003; Murray, 2003; Graham and Kumar, 2003), see Fig. 1.1.

Figure 1.1: Ubiquitous network, ubiquitous control.

The new area of research activity – networked control systems – which is the topic of this thesis, is the convergence of control, communication and computation (Walsh et al., 1999; Hespanha et al., 2007; Nair et al., 2007; Yang, 2006). Distinct from CCSs where the data exchange between sensors, controllers, actuators, etc. is assumed to be seamless, NCSs can contain a large number of control devices interconnected and data is exchanged through communication networks which inevitably introduces communication constraints to the control system, e.g., network-induced delay, data packet dropout, data packet disorder, data rate constraint, etc.. These communication constraints in NCSs present great challenges for conventional control theory (Hespanha et al., 2007; Baillieul and Antsaklis, 2007; Nair et al., 2007; Yang, 2006; Tipsuwan and Chow, 2003; Savkin, 2006; Walsh et al., 2001), and therefore are areas explored in the thesis.


3

Though the theoretical foundation of NCSs has been improved considerably during the last decade, it is still in its infancy. An interesting phenomenon of the early research work on NCSs is that the potential of optimizing the system performance by taking advantage of the network characteristics is somewhat neglected. The network is simply treated as a negative parameter (mostly a delay parameter) to the system and as a result a CCS instead of an NCS is actually considered, see e.g., Yue et al. (2005); Montestruque and Antsaklis (2003) and the survey paper in Tipsuwan and Chow (2003). While this kind of approach enables standard design and analysis tools in CCSs to be applied to NCSs, it has not made full use of the characteristics of the network, especially those which may be positive to the system performance, which therefore inevitably results in considerable conservativeness. On the basis of this observation, preliminary work has been done recently under the so called “co-design” framework, where the characteristics of the network in NCSs are analyzed and utilized further, see, e.g., in Branicky et al. (2003); Montestruque and Antsaklis (2003); Goodwin et al. (2004); Tipsuwan and Chow (2004); Zhang and Hristu-Varsakelis (2006); Liu et al. (2006); Georgiev and Tilbury (2006); Zhao et al. (2007a); Baillieul and Antsaklis (2007); Goodwin et al. (2008). Most of the work in the co-design area is motivated by the observation of PBtransmission in NCSs (Hespanha et al., 2007). This characteristic can mean that whether an NCS sends one single bit or several hundreds bits of data, consumes the same amount of network resource. More specifically, it can be concluded that the same network resource is required to send either a single step control signal or multiple steps of forward control signals with a certain length. This observation motivates the study on the so called PB-control for NCSs in this thesis, where by designing a special packet-based controller (PB-controller) and a corresponding comparison rule at the actuator side, the PB-control approach can explicitly compensate for the communication constraints including the network-induced delay, data packet dropout and data packet disorder simultaneously in both forward and backward channels. This merit can not be achieved using conventional control approaches as in, e.g., Zhang et al. (2005); Gao et al. (2007) and thus results in a better performance than that in previously reported results (Zhao et al., 2008e,f,h,i).


1.2

4

Aims and objectives

In this thesis, we aim at improving the system performance of NCSs by taking advantage of the characteristics of the network, that is, by using co-design approach. Specifically, our main objectives are focused on the following two aspects: • A novel design approach to NCSs which can efficiently deal with the communication constraints in NCSs; • The corresponding theoretical analysis on the stability of the closed-loop system. Based on the aforementioned objectives, our ultimate objective is to provide a unified design & analysis framework for NCSs which can result in a better performance than that in previously reported results.

1.3

Outline of the thesis and publications

The remaining chapters of the thesis and the candidate’s publications are outlined as follows. Chapter 2 A Brief Survey of Networked Control Systems In this chapter, we discuss the characteristics of NCSs and provide the state-of-art research background of our work that will be presented in the following chapters. Chapter 3 PB-Control for Networked Control Systems In this chapter, the design and stability analysis of the PB-control approach to NCSs is presented, with a predictive-based controller designed as an example. This chapter is the basis of the remaining chapters where various applications of the PB-control approach are considered. The presentation of this chapter is based on all the candidate’s publications and especially on: (Zhao et al., 2008d) Y.-B. Zhao, G. P. Liu, and D. Rees. An Improved Predictive Control Approach to Networked Control Systems. Accepted.

IET Control Theory Appl..


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(Zhao et al., 2008g) Y.-B. Zhao, G. P. Liu, and D. Rees. A Packet-Based Control Approach to Networked Control Systems. Ready for submission. Chapter 4 PB-Control for Networked Hammerstein Systems This chapter extends the application of the PB-control approach to a class of input nonlinear systems described by a Hammerstein model. Two descriptions of the Hammerstein model, in input-output description and state-space description respectively, are considered, and stability criteria are also obtained for both descriptions. The presentation of this chapter is mainly based on: (Zhao et al., 2007b) Y.-B. Zhao, G. P. Liu, and D. Rees. Time delay compensation and stability analysis of networked predictive control systems based on Hammerstein model. In Proc. 2007 IEEE Int. Conf. Netowking, Sensing and Control, pages 808–811, London, UK, April 2007. (Zhao et al., 2007a) Y.-B. Zhao, G. P. Liu, and D. Rees. A predictive control based approach to networked control systems with input nonlinearity: Design and stability analysis. In Proc. 13th Int. Conf. Autom. Comp., pages 7–12, Stafford, UK, 15 Sept. 2007. (Zhao et al., 2008f) Y.-B. Zhao, G. P. Liu, and D. Rees. Networked predictive control systems based on Hammerstein model. IEEE Trans. Circuits Syst. IIExpress Briefs 55(5), 469-473 (Zhao et al., 2008h) Y.-B. Zhao, G. P. Liu, and D. Rees.

A Predictive Con-

trol Based Approach to Networked Hammerstein Systems: Design and Stability Analysis. IEEE Trans. Syst. Man Cybern. Part B-Cybern. 38(3), 700-708 Chapter 5 PB-Control for Networked Wiener Systems This chapter extends the application of the PB-control approach to a class of output nonlinear system described by a Wiener model. A special state observer is design for the implementation of the PB-control approach in this case and the stability criterion of the closed-loop system is also obtained. The presentation of this chapter is mainly based on:


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(Zhao et al., 2008i) Y.-B. Zhao, G. P. Liu, and D. Rees. A Predictive Control Based Approach to Networked Wiener Systems.

Int. J. Innov. Comp. Inf.

Control 4(10), to appear. Chapter 6 Stochastic Stabilization of PB-NCSs In this chapter, the PB-control approach is modeled in a stochastic fashion and then analyzed under the Markov jump system framework. Stochastic stability conditions are obtained and stochastic stabilization problem is also solved. The presentation of this chapter is mainly based on: (Zhao et al., 2008a) Y.-B. Zhao, Y. Kang, G. P. Liu, and D. Rees. Packet-Based Networked Control Systems. Automatica. Submitted. Chapter 7 Continuous-Time PB-NCSs In this chapter, the PB-control approach is considered for a continuous-time system with a continuous network-induced delay. A discretization technique is proposed to enable the PB-control approach to be implemented in this situation, and stability criterion is obtained based on the average dwell time analysis. The presentation of this chapter is mainly based on: (Zhao et al., 2008b) Y.-B. Zhao, G. P. Liu, and D. Rees. Design and Stability Analysis of Packet-Based Networked Control Systems in Continuous Time. 2008 IEEE Int. Conf. Syst. Man Cybern.. Accepted. (Zhao et al., 2008c) Y.-B. Zhao, G. P. Liu, and D. Rees. Modeling and Stabilization of Continuous-Time Packet-Based Networked Control Systems. Automatica. Submitted. Chapter 8 PB-Control and Scheduling Co-Design for NCSs This chapter considers a different system setup where a set of NCSs share the network resources. The PB-control approach still works for each subsystem and the whole system is scheduled by two different scheduling algorithms with the guarantee of the stability of each subsystem. The presentation of this chapter is mainly based on:


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(Zhao et al., 2008e) Y.-B. Zhao, G. P. Liu, and D. Rees, 2008. Integrated predictive control and scheduling co-design for networked control systems.

IET Control

Theory Appl. 2 (1), 7-15 Chapter 9 Conclusions The conclusions are given in this chapter, focusing both on the main contributions of the thesis and suggestions of future work. Appendix Published & Finished Papers The papers that have been published, accepted, submitted, or finished, during the period of the candidate’s doctoral study, are listed in this part. The full text of the published & accepted journal papers is also attached as appendix.

Chapter 2 A Brief Survey of Networked Control Systems As an introduction to the thesis, a brief survey of NCSs is given in this chapter. The survey covers both the unique characteristics of NCSs and reviews the research in this area. The characteristics of NCSs are explored from a perspective that emphasizes the differences between NCSs and CCSs, and thus the role played by the network in NCSs is extensively examined, including, e.g., the characteristics of the network topology, the PB-transmission and the limited resource of the network, etc.. Among all these characteristics, the PB-transmission will be further investigated in the following chapters to derive the so called “packet-based control” for NCSs which is the main subject of the thesis. The second part of the survey covers the state-of-art research on NCSs, which specifically makes the comparison between conventional studies and the so called “co-design” approach to NCSs. The co-design approach is in the author’s view the direction of future research on NCSs and the PB-control approach proposed in the thesis falls into this category of research.

8

Chapter 2. Survey of NCSs

2.1

9

Characteristics of NCSs

The term “networked control system” which is well known today, is believed to be first introduced by Walsh et al. (1999). It is used to describe the configuration where “a control loop is closed via a serial communication network” (Walsh et al., 1999), see Fig. 2.1 and Fig. 2.2. From a historical viewpoint, interest in such a configuration dates back to as early as 1980s’, when the so called “Integrated Communication and Control Networks” (ICCS) was first introduced by Halevi and Ray (1988). There were also several other alias in history such as “Networkbased Control Systems” (NBCS) and “Control over (through) Networks”, which were used to describe almost the same configuration as NCSs but are seldom used today. As indicated by its name, the most important and distinct feature of NCSs is the use of communication networks in the control loop. Due to the limitations of communication networks, perfect data exchanges among the control components can not be achieved in NCSs which, however, is a fundamental basis of CCSs. The communication constraints of the network in NCSs greatly degrade the system performance or even destabilize the system at certain conditions, and simple extensions of conventional control approaches can not be obtained directly in a networked control environment. Therefore, in the following investigation on the characteristics of NCSs, the emphasis will be on the differences between NCSs and CCSs, that is, the distinct and unique characteristics of NCSs that are brought by the inserted network.

2.1.1

Two structures

In Tipsuwan and Chow (2003) the authors argued that there are two different structures of NCSs: the “direct structure” (Fig. 2.1) and the “hierarchical structure” (Fig. 2.2), where the difference between these two structures is in whether a local controller is present or not. However, it is noticed that the aforementioned difference between the two structures has nothing to do with the network. That is, from the perspective of NCSs, these two structures do not significantly differ, and most research work done to date has been concerned with the so called direct structure as shown in Fig. 2.1.


10

Figure 2.1: Networked control systems in the direct structure.

Figure 2.2: Networked control systems in the hierarchical structure.

2.1.2

Network topology

In the presence of the network in NCSs, the conventional control components including the sensor, the controller and the actuator work as network nodes from the perspective of network topology. From this perspective, two issues need to be addressed as follows.

A. Time-synchronization Time-synchronization, or clock synchronization, is a fundamental basis of the implementation of distributed communication networks (Stallings, 2004, 2000). It is important to note that time-synchronization among the control components in NCSs may not be a necessary condition if the network-induced delay in the backward channel (“backward channel delay”) is not required for the calculation of the control signals and/or the network-induced delay in the forward channel (“forward channel delay”) is not required for the implementation of the control actions, which


11

is typically true in conventional Time Delay Systems (TDSs). However, as discussed in Zhang et al. (2001b); Zhao et al. (2008e), time-synchronization together with the use of time stamps in NCSs can offer an advantage over conventional TDSs in that the backward channel delay is known by the controller and the forward channel delay (round trip delay as well) is known by the actuator. This advantage can then be used to derive a better control structure for NCSs as done in Zhang et al. (2001b) and the PB-control structure in this thesis.

B. Drive mechanism Two drive mechanism for the control components in NCSs are usually used, that is, time-driven and event-driven. The difference between the two drive mechanisms lies in the trigger method that initiates the control components. For the timedriven mechanism, the control components are trigged to work at regular intervals, while for event-driven mechanism the control components are only trigged by a predefined “event”. Generally speaking, the sensor is always time-driven, while the controller and the actuator can either be time-driven or event-driven. For more information on the drive mechanism for the control components, the reader is referred to Hu et al. (2007b) and the references therein. It is worth mentioning though, with different drive mechanisms different system models for NCSs are obtained and event-driven control components generally lead to a better system performance.

2.1.3

PB-transmission

PB-transmission is one of the most important characteristics of NCSs that are distinct from CCSs (Baillieul and Antsaklis, 2007; Murray, 2003; Murray et al., 2003). This characteristic can mean that the perfect data transmission as assumed in CCSs is absent in NCSs, which is also the most challenging aspect presented by NCSs. The communication constraints caused by the PB-transmission in NCSs include the network-induced delay, data packet dropout, data packet disorder, etc., which are detailed as follows.


12

A. Network-induced delay Due to the network being inserted into the control loop in NCSs, network-induced delays are introduced in both the forward and backward channels, which are well known to degrade the performance of the control systems. There are two types of network-induced delays according to where they occur: • τsc : Network-induced delay from the sensor to the controller, i.e., backward channel delay; • τca : Network-induced delay from the controller to the actuator, i.e., forward channel delay. The aforementioned two types of network-induced delays may have different characteristics (Nilsson et al., 1998). In most cases, however, these delays are not treated separately and only the round trip delay is of interest (Hespanha et al., 2007; Hu et al., 2007a; Fan et al., 2006; Yang et al., 2006). According to the types of the communication networks being used in NCSs, the characteristics of the network-induced delay vary as follows (Tipsuwan and Chow, 2003; Lian, 2001; Lian et al., 2001). • Cyclic service networks (e.g., Toking-Ring, Toking-Bus): Bounded delays which can be regarded as constant for some occasions; • Random access networks (e.g., Ethernet, CAN): Random and unbounded delays; • Priority order networks(e.g., DeviceNet): Bounded delays for the data packets with higher priority and unbounded delays for those with lower priority. Network-induced delay is one of the most important characteristics of NCSs which has been widely addressed in the literature to date, see, e.g., in Hu and Zhu (2003); Kim et al. (2003); Lian et al. (2003); Montestruque and Antsaklis (2003); He (2004); Yue et al. (2004); Sala (2005); Yue and Han (2005); Zhang et al. (2005); Fan and Arcak (2006); Yang et al. (2006); Liu et al. (2006); Liu and Shen (2006); Tang and de Silva (2006); Garc´ıa-Rivera and Barreiro (2007); Hespanha et al. (2007); Hu et al. (2007a); Schenato et al. (2007).


13

B. Data packet dropout It is well known that the transmission error in communication networks is inevitable, which in the case of NCSs then produces a situation called “data packet dropout”. Data packet dropout can occur either in the backward or forward channel, and it makes either the sensing data or the control signals unavailable to NCSs, thus significantly degrading the performance of NCSs. In communication networks, two different strategies are applied when a data packet is lost, that is, either to send the packet again or simply discard it. Using the terms from communication networks, these two strategies are called Transmission Control Protocol (TCP) and User Datagram Protocol (UDP) respectively (Stallings, 2000). It is readily seen that with TCP, all the data packets will be received successfully, although it may take a considerably long time for some data packets; while with UDP, some data packets will be lost forever. As far as NCSs is concerned, UDP is used in most applications due to the realtime requirement and the robustness of control systems. As a result, the effect of data packet dropout in NCSs has to be explicitly considered, as done in, e.g., Azimi-Sadjadi (2003); Georgiev and Tilbury (2006); Imer et al. (2006); Gupta et al. (2007); Hu and Yan (2007); Xiong and Lam (2007) and the PB-control approach in this thesis.

C. Data packet disorder In most communication networks, different data packets suffer different delays, see Section 2.1.3; it therefore produces a situation where a data packet sent earlier may arrive at the destination later or vice versa, that is, data packet disorder. This characteristic in NCSs can mean that a newly arrived control signal in NCSs may not be the latest, which never occurs in CCSs. Therefore, the effect of data packet disorder has to be specially dealt with which, however, is unfortunately absent in literature on NCSs. It is worth mentioning that this effect can be effectively overcome by using the comparison process in the PB-control approach.


14

D. Single & Multi-packet When the sensing data and the control signals are sent via data packets of the network, another situation occurs: in a case where, for example, multiple sensors are used and distributed spatially in NCSs and thus they send their sensing data separately to the controller over the network, the controller may have to wait for the arrival of all the sensing data packets before it is able to calculate the control actions, and if only one sensing data packet is lost, all the other sensing data packets have to be discarded due to incompleteness. We call this situation the “multi-packet” transmission of the data in NCSs. Another situation in NCSs is where the sensing data or the control signals of multiple steps are sent via a single data packet over the network, since the packet size used in NCSs can be very large compared with the data size required to encode a single step of sensing data or control signal. This “single-packet” transmission of the data in NCSs is the fundamental basis of the PB-control approach adopted in this thesis.

2.1.4

Limited network resources

The limitation of the network resources in NCSs is primarily caused by the limited bandwidth of the communication network, which results in the following three situations in NCSs that are distinct from CCSs 1 .

A. Sampling period, Network loads and System Performance It is noticed that NCSs is a special class of sampled data systems due to the digital transmission of the data in communication networks. However, in NCSs, the limited bandwidth of the network produces a situation where, a smaller sampling period may not result in a better system performance which, however, is normally true for sampled data systems. This situation happens because, with too small a sampling period, too much sensing data will be produced; thus overloading the network and causing congestion, 1 For the data rate constraint in NCSs, the reader is referred to the recent survey in Nair et al. (2007) and the references therein.


15

Figure 2.3: Relationship between the sampling period, network loads and system performance in NCSs.

which will result in more data packet dropouts and longer delays, and thus degrade the system performance. The relationship between the sampling period, network loads and system performance in NCSs is illustrated in Fig. 2.3 where, for example, when the sampling period decreases from the value corresponding to point “a” to “b”, the system performance is getting better as in conventional sampled data systems since the network congestion does not appear until point “b”; However, the system performance is likely to deteriorate due to the network congestion when the sampling period is getting even smaller from the value corresponding to point “b” to “c”. Therefore, The relationship shown in Fig. 2.3 implies that there is a trade-off between the period of sampling the plant data and the system performance in NCSs, that is, in NCSs an optimal sampling period exists which offers the best system performance (point “b” in Fig. 2.3).

B. Quantization Due to the use of data networks with limited bandwidth, signal quantization is inevitable in NCSs, which has a significant impact on the system performance of NCSs. Quantization in the meantime is also a potential method to reduce the


16

bandwidth usage which enables it to be an effective tool to avoid the network congestion in NCSs and thus improve the system performance of NCSs. For more information on the quantization effects in NCSs, the reader is referred to Elia and Mitter (2001); Liberzon (2006); Peng and Tian (2007); Tian et al. (2007) and the references therein.

C. Network access constraint and scheduling In Chapter 8 of the thesis, a system setup is considered where a set of control systems share the communication network to transmit the control signals, see Fig. 8.1. In such a case, the limited bandwidth of the network produces a situation where all the subsystems can not access the network resource at the same time. A scheduling algorithm is therefore needed to schedule the timeline of when and how long a specific subsystem can occupy the network resource.

2.2

Research background

In this section, we briefly survey the state-of-art research on NCSs. It covers first the dominant area of control theory in which the characteristics of NCSs are taken sufficiently account of, and then the new research area on the co-design approach to NCSs. A conclusion is drawn from the discussion that the co-design approach is the future direction of research on NCSs.

2.2.1

Conventional research on NCSs

Since the renewed interest in NCSs by Walsh et al. (1999) (ICCS proposed in Halevi and Ray (1988) did not receive much attention), the research on NCSs has been primarily done within the control theory community, see, e.g., the survey paper in Tipsuwan and Chow (2003). There are, generally speaking, two types of problems in NCSs that interest researchers from the control theory community. One is concerned with the theoretical analysis on the control performance of NCSs where the network in NCSs is modeled by predetermined parameters to the control system. In this kind of research, the aforementioned characteristics of NCSs, mostly the network-induced


17

delay, may be carefully described and formulated, and incorporated into the description of the system, to form a special class of CCSs, mostly TDSs, and then a CCS instead of an NCS, is actually considered. This kind of research inevitably simplify the modeling and analysis of NCSs and, what is more important, all the previous results in CCSs can now be readily applied to NCSs, thus enabling it to be the mainstream research activity on NCSs for a significant period (Richard, 2003; Tipsuwan and Chow, 2003). Corresponding to the aforementioned theoretical analysis of NCSs from the control theory perspective, the synthesis issue is also a main concern. Despite considerable work on this area, the limitations are obvious: most work concentrates on the extension of existing control approaches to NCSs without full use of the characteristics of NCSs. As a result, the system performance obtained for NCSs can never be better than that of CCSs. However, it is worth mentioning that recent studies actually show that the network in NCSs may not be a disadvantage, and by making efficient use of the network characteristics, one can expect a better system performance of NCSs than that of the corresponding CCSs (Zhang, 2005; Goodwin et al., 2008; Colandairaj et al., 2007; Liu et al., 2007a). The conventional control approaches and theories that have been applied to NCSs are briefly surveyed as follows.

• Time Delay Systems. As far as the network-induced delay is concerned, it is natural to model NCSs as a special class of TDSs. This kind of research covers a vast range of research on NCSs, see, e.g., in Mazenc and Niculescu (2001); Zhang et al. (2001a); Hoo et al. (2002); Lin et al. (2003); So (2003); Guan et al. (2005a); Wu et al. (2005); Zhang and Fang (2006), and the survey in Tipsuwan and Chow (2003); Antsaklis and Baillieul (2004); Hespanha et al. (2007). With the development of the analysis technique in TDSs, recently a delay-dependent analysis approach to NCSs is also reported in, e.g., Yue et al. (2004); Yue and Han (2005); Gao et al. (2007). As mentioned above, though modeling NCSs as TDSs simplifies the analysis and enables the existing approaches in TDSs to be applied to NCSs, it has not taken full advantage of the unique characteristics of NCSs and thus is inevitably conservative. An interesting issue in this kind of research is to determine the Maximum Allowable Delay Bound (MADB) of NCSs, which is the upper bound of the


18

transfer interval that ensures the stability or other performance targets of NCSs (Branicky et al., 2000). The determination of MADB is important in theory and can also play a guiding role for practical applications. One can refer to the survey paper in Kim et al. (2003) for more information on this issue. • Stochastic control. As mentioned above, the communication constraints in NCSs are stochastic in nature, thus enabling the application of conventional stochastic control approaches to be applied to NCSs. For example, an early study in this area can be found in Nilsson et al. (1998), where the characteristics of the network-induced delay were explicitly formulated and preliminary stochastic stability criteria of NCSs were obtained; Hu and Zhu (2003) extended the work in Halevi and Ray (1988) to a stochastic optimal control framework and gave the stochastic optimal state feedback and output feedback controllers respectively; In Zhang et al. (2005), the sufficient and necessary conditions of the stochastic stability of NCSs were obtained based on the Markov jump system framework. For further information, the reader is referred to the recent survey in Antsaklis and Baillieul (2007). • Optimal control. As a very successful idea both in theory and practical applications, optimal control can also find its position in the research on NCSs. Undoubtedly, conventional optimal control approaches can be used in the networked control environment to design the controller for NCSs, see, e.g., in Seiler and Sengupta (2005); Basin and Rodriguez-Gonzalez (2006); Georgiev and Tilbury (2006); Imer et al. (2006); Gupta et al. (2007); Schenato et al. (2007); Sahebsara et al. (2008); and as a special class of optimal control approaches, Model Predictive Control (MPC, or Receding Horizon Control (RHC)) seems to be more suitable for the networked control environment and “a major extension required to apply model predictive control in networked environments would be the distributed solution of the underlying optimization problem” (Murray, 2003). Examples of the application of MPC to NCSs can be seen, e.g., in Goodwin et al. (2004); Liu et al. (2006); Zhao et al. (2008e,f,h,i). • Switched system theory. Another important tool in the study on NCSs is switched system theory, which is typically used by modeling different network conditions in NCSs as different system modes. This approach can readily deal with network-induced delay as well as data packet dropout in NCSs, and


19

the limitation of the approach is caused mainly by how well we understand the properties of the changes of the network conditions, which is generally difficult. For the research in this area, the reader is referred to Lin et al. (2003); Nael H. El-Farra (2003); Zhivoglyadov and Middleton (2003); Guan et al. (2005b); Kawka and Alleyne (2005); Wu et al. (2005); Yu et al. (2006); Xia et al. (2007) and the references therein.

As mentioned above, though considerable work on NCSs has been done within the control theory community, the drawbacks of this kind of research is obvious: it has not taken full use of the advantages that the network has brought to the system. Take the classic state feedback control of a control system as an example. The conventional state feedback law is generally obtained as follows without the consideration of the communication constraints in NCSs, u(k) = Kx(k)

(2.1)

where the feedback gain K is time-invariant. However, when the network-induced delay is considered, the state feedback law can not be simply defined as in (2.1) due to the unavailability of the current state information. The resulting control law using conventional approaches in TDSs would have the following form u(k) = Kx(k − τk )

(2.2)

where the effect of the delay is not been specially treated in the design. Furthermore, when data packet dropout is also present, it can be seen from Fig. 3.1 that no matter where data packet dropout occurs, a certain control input will be unavailable to the actuator. In conventional TDS theory, there are mainly two ways to deal with this situation, either use the previous control input or adopt zero control (Richard, 2003). For example, in Wu and Chen (2007), the last step of the control signal is used as the the control strategy in the case of an unsuccessful transmission, as follows,  u¯(k) if transmitted successfully; u(k) = u(k − 1) otherwise.

(2.3)


20

where u¯(k) is the newly arrived control signal at time k. It can be seen that though the conventional control strategies in (2.2) and (2.3) are simple to implement, however they are conservative in that they overlook the potential of providing an active prediction for the unavailable control input using available information of the system dynamics and previous system trajectory as done in Liu et al. (2007b); Zhao et al. (2008e). It is worth mentioning that this drawback in CCSs can be readily dealt with by using the PB-control approach proposed in the thesis, see Section 3.1.

2.2.2

Co-design approaches to NCSs: A new direction

As has been pointed out earlier, it is the communication network which replaces the direct connections among the control components in CCSs that makes NCSs distinct from CCSs. Therefore it is natural and necessary to highlight the effects of the network when investigating NCSs, that is, by using the so called co-design approach to NCSs, which has been an emerging trend in recent years. In this kind of research, the communication constraints are no longer assumed to be predetermined parameters but act as designable factors, and by their efficient use a better performance can be therefore expected, see, e.g., in Branicky et al. (2003); Hartman (2004); Gaid et al. (2006); Baillieul and Antsaklis (2007); Colandairaj et al. (2007); Liu et al. (2007b); Goodwin et al. (2008); Zhao et al. (2008e,h). In the following discussion, two types of work in this area which are related to the theme of the thesis are briefly surveyed. • Packet-based control approach. As discussed in Section 2.1.3, the PBtransmission is one of the most distinct characteristics of NCSs, which can be used to derive a novel control structure for NCSs, as done in Zhang (2001); Georgiev and Tilbury (2006) and in this thesis. It is worth mentioning that the PB-control approach was inspired by the idea reported in Liu et al. (2006, 2007b), where with the use of Generalized Predictive Control (GPC) the packet-based structure of data transmission was efficiently used to actively compensate for the communication constraints in NCSs. • Control and scheduling co-design. In NCSs, a situation may occur where multiple control components share a network whose resources are limited. In such a situation, network resources scheduling among the control


21

components is necessary, see Section 2.1.4. As far as the scheduling algorithms are concerned, Walsh et al. (1999) proposed a dynamic scheduling algorithm called “Try-Once-Discard” (TOD) which allocates the network resources in a way that the node with the greatest error in the last reported period has access to the network resource. Nesic and Teel (2004) proposed a Lyapunov Uniformly Globally Asymptotically Stable (UGAS) protocol based on TOD, which is further improved in Tabbara et al. (2007). In Hristu-Varsakelis and Kumar (2002), the authors used the technique of “communication sequence” (see also in Brochett (1995)) to deal with the network access constraint for such a system configuration and modeled the subsystems as switched systems with two modes “open loop” and “closed loop” which switch according to whether the current subsystem has access to the medium or not. In Branicky et al. (2002), the authors considered a special case of the configuration shown in Fig. 8.1 in Chapter 8 where the channel from controller to actuator is linked directly, and the rate monotonic scheduling algorithm is applied to schedule the transmissions of the sensing data of the subsystems. In Chapter 8 of the thesis, a PB-control and scheduling co-design approach is also proposed to deal with such a problem.

2.3

Conclusion

A brief survey on NCSs was given in this chapter, covering both the characteristics of NCSs to current state-of-art research. It is observed that as an emerging research area, NCSs is distinct from CCSs and therefore a co-design approach to NCSs is required, which is the theme of this thesis.

Chapter 3 PB-Control for Networked Control Systems In this chapter, we exploit the fact that in most communication networks, data is transmitted in a “packet” and within its effective load sending a single bit or several hundred bits of data consumes the same amount of network resources (Hespanha et al., 2007). This makes it possible in NCSs to actively compensate for the communication constraints by sending a sequence of control predictions in one data packet and then selecting the appropriate one corresponding to the current network condition. This PB-transmission characteristic motivates us to design the PB-control approach to NCSs. Due to the active compensation process in the PB-control approach, a better performance can be expected compared to an implementation where no characteristics of the network are specifically considered in the design. This chapter is organized as follows. The design of the PB-control for NCSs is presented in detail in Section 3.1, which leads to a novel controller that can compensate for network-induced delay, data packet dropout and data packet disorder simultaneously. The stability criteria for the corresponding closed-loop system are then investigated in Section 3.2, from the points of view of switched system theory and delay-dependent analysis, respectively. As an example, a GPC-based controller under the PB-control framework is designed in Section 3.3, which is more feasible in practice compared with previous results. Numerical and experimental examples to illustrate the effectiveness of the proposed approach are presented in Section 3.4 and Section 3.5 concludes the chapter. 22

Chapter 3. PB-Control for NCSs

3.1

23

PB-control for NCSs

The NCS setup considered in this chapter is shown in Fig. 3.11 , where τsc,k and τca,k are the backward and forward channel delays respectively and the plant is linear in discrete-time which can be represented by ( Sd :

x(k + 1) = Ax(k) + Bu(k)

(3.1a)

y(k) = Cx(k)

(3.1b)

with x(k) ∈ Rn , u(k) ∈ Rm , A ∈ Rn×n , B ∈ Rn×m and C ∈ Rr×n .

Figure 3.1: The block diagram of networked control systems in discrete time.

3.1.1

PB-control for NCSs: A unified model

It is necessary to point out that in an NCS as shown in Fig. 3.1, the forward channel delay τca,k is not available for the controller when the control action is calculated at time k, since τca,k occurs after the determination of the control action. For this reason, when applying conventional design techniques as in TDSs to NCSs, the active compensation for the forward channel delay can not be provided, see (2.2) and (2.3) and the discussion in Section 2.2.1. This fact can mean that the conventional design techniques are conservative in the networked control environment, which however can be dealt with using the PB-control approach in this thesis. As discussed earlier in Section 2.1.3, the presence of the network in NCSs brings to the system network-induced delay, data packet dropout, data packet disorder, 1

It is worth mentioning that though it has not been explicitly shown in Fig. 3.1, the effects of data packet dropout and data packet disorder are also considered in the PB-control approach.


24

etc.. These communication constraints degrade the system performance significantly whereas the PB-transmission of the network also offers the potential of transmitting a sequence of control signals simultaneously instead of one at a time as typically done in CCSs. This observation is the motivation of the PB-control approach to NCSs proposed in this thesis. The control law based on the PB-control approach is obtained as follows with explicit compensation for the communication constraint (see Algorithm 3.6 which will be given later), ∗ ∗ ∗ ∗ u(k) = K(τsc,k , τca,k )x(k − τsc,k − τca,k )

(3.2)

or simply (see Algorithm 3.9 which will be given later), u(k) = K(τk∗ )x(k − τk∗ )

(3.3)

∗ ∗ where τsc,k and τca,k are the network-induced delays of the control action that is ∗ ∗ actually applied to the plant at time k and τk∗ = τsc,k + τca,k .

It is seen from the control laws in (3.2) and (3.3) that in the PB-control approach, different feedback gains apply for different network conditions. This is why we call it a “Varying Feedback Gain” (VFG) scheme for NCSs. As will be presented later, these PB-control laws can actively deal with the network-induced delay, data packet dropout and data packet disorder simultaneously, and therefore can be regarded as a unified model for NCSs. This control strategy can be compared with the conventional approach as in (2.2) and (2.3) where no active compensation is available. Remark 3.1. In Zhang et al. (2005), the authors noticed the unavailability of the forward channel delay τca,k and a controller was designed with the following form u(k) = K(τsc,k , τca,k−1 )x(k − τsc,k − τca,k )

(3.4)

where the forward channel delay of the last step τca,k−1 was used instead. However, actually even τca,k−1 is generally unavailable for the controller in NCSs since in the case of a arbitrary forward channel delay, τca,k−1 can not be known to the controller until the controller receives information of τca,k−1 from the actuator. Therefore, it is seen that τca,k−1 can not be available for the controller earlier than time k − 1 + τca,k−1 even if an additional delay-free channel exists to send the


25

information of τca,k−1 from the actuator to the controller. As a result, the above model in (3.4) is inappropriate in practice unless a special control structure is designed for the networked control environment as done in this chapter.

3.1.2

Design of PB-Control for NCSs

For the design of the PB-control approach, the following assumptions are required. Assumption 3.2. The control components in the considered NCS including the sensor, the controller and the actuator, are time-synchronized and the data packets sent from both the sensor and the controller are time-stamped. Assumption 3.3. The sum of the maximum forward (backward) channel delay and the maximum number of continuous data packet dropout is upper bounded by τ¯ca (¯ τsc accordingly) and τ¯ca ≤

Bp −1 Bc

(3.5)

where Bp is the size of the effective load of the data packet and Bc is the bits required to encode a single step control signal. Remark 3.4. From Assumption 3.2, the network-induced delay that each data packet experiences is known by the controller and the actuator on its arrival. Remark 3.5. Assumption 3.3 is required due to the need of packing the forward control signals and compensating for the network-induced delay in the PB-control approach, which will be detailed later. The constraint in (3.5) is easy to be satisfied, e.g., Bp = 368 bit for an Ethernet IEEE 802.3 frame which is often used (Stallings, 2000), while an 8-bit data (i.e., Bc = 8 bit) can encode 28 = 256 different control actions which is ample for most control implementations; In this case, 45 steps of forward channel delay is allowed by (3.5) which can actually meet the requirements of most practical control systems. The block diagram of the packet-based control structure is illustrated in Fig. 3.2. It is distinct from a conventional control structure in two aspects: the specially designed PB-controller and the corresponding Control Action Selector (CAS) at the actuator side. In order to implement the control law in (3.2) and (3.3), we take advantage of the PB-transmission of the network to design a PB-controller instead of trying to


26

Figure 3.2: Packet-based networked control systems in discrete time (with time synchronization).

obtain directly the current forward channel delay as this is actually impossible in practice. As for the control law in (3.2), the PB-controller determines a sequence of forward control actions (called “Forward Control Sequence” (FCS)) as follows and sends them together in one data packet to the actuator, U1 (k|k − τsc,k ) = [u(k|k − τsc,k ) . . . u(k + τ¯ca |k − τsc,k ]T

(3.6)

where u(k + i|k − τsc,k ), i = 0, 1, . . . , τca,k are the forward control action predictions based on information up to time k − τsc,k . When a data packet arrives at the actuator, the designed CAS compares its time stamp with the one already in CAS and only the one with the latest time stamp is saved. Denote the forward control sequence already in CAS and the one just arrived by U1 (k1 − τca,k1 |k1 − τk1 ) and U1 (k2 − τca,k2 |k2 − τk2 ) respectively, then the chosen sequence is determined by the following comparison rule,  U1 (k2 − τca,k |k2 − τk ), if k1 − τk < k2 − τk ; 2 2 1 2 ∗ ∗ U1 (k − τca,k |k − τk ) = U (k − τ 1 1 ca,k1 |k1 − τk1 ), otherwise.

(3.7)

The comparison process is introduced due to the fact that different data packets may experience different delays thus producing such a situation where a packet sent earlier may arrive at the actuator later or vice versa, that is, data packet disorder. After the comparison process, only the latest available information is used and the effect of data packet disorder is effectively overcome. ∗ CAS also determines the appropriate control action from the FCS U1 (k − τca,k |k −

τk∗ ) at each time instant as follows u(k) = u(k|k − τk∗ )

(3.8)


27

The timeline of the PB-control approach is illustrated in Fig. 3.3. It is necessary to point out that the appropriate control action determined by (3.8) is always available provided Assumption 3.3 holds.

Figure 3.3: Timeline in the PB-NCSs

The PB-control algorithm under Assumptions 3.2 and 3.3 can now be summarized as follows. Algorithm 3.6 (PB-control with the control law in (3.2)). S1. At time k, if the PB-controller receives the delayed state data x(k − τsc,k ), it then: S1a. Reads the current backward channel delay τsc,k ; S1b. Calculates the FCS as in (3.6); S1c. Packs U1 (k|k − τsc,k ) and sends it to the actuator in one data packet with time stamps k and τsc,k . If no data packet is received at time k, then let k = k + 1 and wait for the next time instant. S2. CAS updates its FCS by (3.7) once a data packet arrives; S3. The control action in (3.8) is picked out from CAS and applied to the plant. In practice, it is often the case that we do not need to identify separately the forward and backward channel delays since it is normally the round trip delay that affects the system performance. In such a case, the simpler control law in (3.3) instead of that in (3.2) is applied, for which the following assumption is required instead of Assumption 3.3.


28

Assumption 3.7. The sum of the maximum network-induced delay and the maximum number of continuous data packet dropout in round trip is upper bounded by τ¯ and τ¯ ≤

Bp −1 Bc

(3.9)

With the above assumption, the PB-controller is modified as follows U2 (k − τsc,k |k − τsc,k ) = [u(k − τsc,k |k − τsc,k ) . . . u(k − τsc,k + τ¯|k − τsc,k ]T (3.10) It is noticed that in such a case the backward channel delay τsc,k is not required for the controller, since the controller simply produces (¯ τ + 1) step forward control actions whenever a data packet containing sensing data arrives. This relaxation implies that the time-synchronization between the controller and the actuator (plant) is not required any more and thus Assumption 3.2 can then be modified as follows. Assumption 3.8. The data packets sent from the sensor are time-stamped. The comparison rule in (3.7) and the determination of the actual control action in (3.8) remain unchanged since both of them are based on the round trip delay τk . The PB-control algorithm with the control law in (3.3) can now be summarized as follows based on Assumptions 3.7 and 3.8. Algorithm 3.9 (PB-control with the control law in (3.3)). S1. At time k, if the PB-controller receives the delayed state data x(k − τsc,k ), then, S1a. Calculate the FCS as in (3.10); S1b. Pack U2 (k − τsc,k |k − τsc,k ) and send it to the actuator in one data packet. If no data packet is received at time k, then let k = k + 1 and wait for the next time instant. S2-S3 remain the same as in Algorithm 3.6. The block diagram of the PB-control approach in Algorithm 3.9 is illustrated in Fig. 3.4.


29

Figure 3.4: Packet-based networked control systems in discrete time (without time synchronization).

3.2

Stability of PB-NCSs

In this section the stability criteria for the system in (3.1) using the aforementioned PB-control approach with the control laws in (3.2) and (3.3) are investigated. Two stability analysis approaches, i.e., results from switched system theory and delaydependent analysis, are applied, by modeling the closed-loop system into different forms. Unless otherwise specified, all the stability related notions in this thesis are under the Lyapunov framework.

3.2.1

A switched system theory approach

An intuitive observation on the PB-control approach is that, at every execution time, a specific control action is determined by the CAS according to the current network condition. Thus, regarding this selection process as “switches” among different subsystems, then yields the following analysis from the viewpoint of switched system theory. Let X(k) = [x(k) x(k − 1) · · · x(k − τ¯)]. The closed-loop formula for the system in (3.1) using the PB-controllers in (3.2) and (3.3) can then be represented in augmented forms as ∗ ,τ ∗ X(k + 1) = Ξτsc,k X(k) ca,k

(3.11)

X(k + 1) = Ξτk∗ X(k)

(3.12)

and


30

respectively, where 

∗ ,τ ∗ Ξτsc,k ca,k

∗ ,τ ∗ A · · · BKτsc,k ··· ca,k

  In   =   

In ..

. In



Ξτk∗

A · · · BKτk∗ · · ·

  In   =   

In ... In

···

···



 0    0 , ..  .   0



 0    0 , ..  .   0

and In is the identity matrix with rank n. With the closed-loop system model in (3.11), we then obtain the following stability criterion. Theorem 3.10. The closed-loop system in (3.11) is stable if there exists a positive definite solution P = P T > 0 for the following (¯ τsc + 1) × (¯ τca + 1) LMIs ∗ ,τ ∗ ΞTτsc,k P Ξτsc,k −P 0, i = 1, 2, 3, X =

∗ τsc,k

It is

≤ τ¯sc . !

X11 X12

≥ 0, ∗ X22 Ni , i = 1, 2 with appropriate dimensions and γ > 0 satisfying the following two LMIs, 

X11 X12 N1

  ∗  ∗



Φ11

  ∗   ∗  ∗



 X22 N2  ≥0 ∗ P3

Φ12

(3.15)

(A − I)T H P1 Bm

Φ22 + γI

0

0

∗

−H

HBm

∗

∗

−γI

    0 where Ej = 1 + ej,1 z −1 + ... + ej,j+τsc,k −1 z −(j+τsc,k −1) , Fj = fj,0 + fj,1 z −1 + · · · + ej,n z −n , Ej0 = e0j,0 + e0j,1 z −1 + ... + e0j,m−1 z −(m−1) , and Gj = gj,0 + gj,1 z −1 + · · · + gj,j−1 z −(j+τsc,k −1) . 0 0 T 0 · · · EN ] , if m > 0; 0(Np +τsc,k )×1 , otherwise; E−τ Define E = [E−τ p sc,k +1 sc,k +2

G ∈ R(Np +τsc,k )×(Nu +τsc,k ) (z −1 ) with all the entries 0 but G(j, j) = Gj if m > 0; G(j, j) = Ej bI1 , otherwise, for j = −τsc,k + 1, −τsc,k + 2, · · · , Nu , and F = [F−τsc,k +1 F−τsc,k +2 · · · FNp ]T , T = (GT QG + R)−1 GT Q, Y0 (k|k − τsc,k ) = E∆v(k − τsc,k − 1) + F y(k  1 0 ···   1 1 ···  S =  . . .  .. .. . .  1 1 ···

− τsc,k ), M = [1 1 · · · 1]T(Nu +τsc,k )×1 , P = [0Nu ×τsc,k INu ×Nu ],  0  0   . The FCS from k to k + Nu − 1 using objective ..  .   1 Nu +τsc,k

Chapter 4. Networked Hammerstein Systems

46

function in (4.4) based on the information up to time k − τsc,k is then obtained as V (k|k − τsc,k ) = P (M v(k − τsc,k − 1) + ST ($ − Y0 (k|k − τsc,k ))

(4.5)

B. FCIS for the state-space description For the state-space description, the following objective function is adopted, I2 Jk,τ sc,k

=

N2 X

2

qj (ˆ y (k + j|k − τsc,k ) − ω(k + j)) +

Nu X

rj (∆v(k + j − 1))2 (4.6)

j=1

j=N1

0 Let x¯(k) = [xT (k) v(k − 1)]T , and then system SI2 can be transformed to SI2 as

follows, ( 0 SI2 :

where A¯ =

¯x(k) + ¯b∆v(k) x¯(k + 1) = A¯ y(k) = c¯x¯(k)

A bI2 0

1

! , ¯b =

bI2 1

! , c¯ =

c 0

.

The forward output predictions at time k based on the information of the state on time k − τsc,k and control signals from time k − τsc,k − 1 is yˆ(k + j|k − τsc,k ) = c¯A¯j+τsc,k x¯(k − τsc,k ) +

j−1 X

c¯A¯j−l−1¯b∆v(k + l|k − τsc,k )

l=−τsc,k

Let Yˆ (k|k −τsc,k ) = [ˆ y (k +N1 |k −τsc,k ) · · · yˆ(k +N2 |k −τsc,k )]T , ∆V 0 (k|k −τsc,k ) = [∆v(k − τsc,k |k − τsc,k ) · · · ∆v(k + Nu − 1|k − τsc,k )]T . Then we obtain Yˆ (k|k − τsc,k ) = Eτsc,k x¯(k − τsc,k ) + Fτsc,k ∆V 0 (k|k − τsc,k ) where Eτsc,k = [(¯ cA¯N1 +τsc,k )T (¯ cA¯N1 +τsc,k +1 )T · · · (¯ cA¯N2 +τsc,k )T ]T and Fτsc,k is a (N2 − N1 + 1) × (Nu + τsc,k ) matrix with the non-null entries defined by (Fτsc,k )ij = c¯A¯N1 +τsc,k +i−j−1¯b, j − i ≤ N1 + τsc,k − 1. Note here that Eτ and Fτ vary with sc,k

sc,k

different τsc,k s. Let $k = [ω(k + N1 ) · · · ω(k + N2 )]T , and then the optimal predictive control increments from k to k +Nu −1 can be calculated by following standard techniques


47

in GPC approach, ∆V (k|k − τsc,k ) = Mτsc,k ($k − Eτsc,k x¯(k − τsc,k ))

(4.8)

where Mτsc,k = Hτsc,k (FτTsc,k QFτsc,k + R)−1 FτTsc,k Q, Q, R are diagonal matrices with Qi,i = qi , Ri,i = ri respectively and Hτsc,k = [0Nu ×τsc,k INu ×Nu ], INu ×Nu is the identity matrix with rank Nu . Remark 4.2. Normally, the minimum prediction horizon can be set as 1. Rewrite the maximum prediction horizon N2 as Np . The following constraint between Nu and Np needs to be always held in order to implement the LGPC method successfully, Nu ≤ Np

4.2.2

(4.9)

The nonlinear input process

With the designed intermediate FCIS in (4.8), the nonlinear input process in SI2 is first considered as follows. Assume the nonlinear function f (·) in (4.2c) is invertible and denote its inverse by fˆ−1 (·). Then we obtain ∆u(k|k − τsc,k ) = fˆ−1 (∆v(k|k − τsc,k ))

(4.10)

Thus, at every time instant k, the intermediate control increments ∆v(k|k − τsc,k ), k = 1, 2, · · · , Nu can be obtained from (4.8), and then the real control increments ∆u(k|k −τsc,k ), k = 1, 2, · · · , Nu can be calculated from (4.10) thus enabling 0 the control law to be defined for system SI2 .

If ∆u(k|k − τsc,k ) can be calculated accurately using (4.10), thus enabling the function fˆ−1 (·) to be exactly known, then the system with compensation for the nonlinear input process is equivalent to LGPC and the system is stable if and only if the linear part of system SI2 with LGPC is stable. However, in practice, it is usually impossible to calculate ∆u(k|k−τsc,k ) that accurately, i.e., fˆ−1 (f (·)) 6≡ 1(·). This inaccuracy introduces to the LGPC a nonlinear disturbance, which makes the stability analysis difficult.


48

ˆ ˆ For simplicity of notation, let f~−1 (·) : RNu → RNu with f~−1 (∆V (k|k − τsc,k )) = [fˆ−1 (∆v(k|k −τsc,k )) · · · fˆ−1 (∆v(k +Nu −1|k −τsc,k ))]T . Then from the discussion above, the real FCIS for system SI2 can be obtained as ˆ ∆U (k|k − τsc,k ) = f~−1 (∆V (k|k − τsc,k )

(4.11)

where ∆U (k|k − τsc,k ) = [∆u(k|k − τsc,k ) · · · ∆u(k + Nu − 1|k − τsc,k )]T . Remark 4.3. Note that the control increment instead of the control signal itself is used in the compensation for the nonlinear input process in (4.10). Though the use of control increments complicates the problem in that the past control increments are also needed to determine the current control increment, it is inevitable since the objective function to be optimized in (4.6) takes the form of control increments. In order to implement the predictive controller in this chapter, the past control increments are sent to the controller as well as the state information, which is different from both CCSs and standard PB-control approach. Note that for a system without a nonlinear input process in (4.2c), it makes no difference whether the intermediate control increment or the intermediate control signal itself is used to calculate the real control signal, while for system SI2 , generally, these two methods give different control input at time k, i.e., f (∆v(k)) 6= f (v(k)) − f (v(k − 1)). Remark 4.4. For system SI1 , the real FCS can be obtained analogously as follows using the similar inverse compensation scheme as aforementioned ˆ U (k|k − τsc,k ) = f~−1 (V (k|k − τsc,k )

(4.12)

where U (k|k − τsc,k ) = [u(k|k − τsc,k ) · · · u(k + Nu − 1|k − τsc,k )]T .

4.2.3

PB-Control for networked Hammerstein systems

With the aforementioned discussion, the PB-controllers for networked Hammerstein systems have been successfully obtained for both descriptions, which enables the PB-control structure proposed in Section 3.1 to be implemented. Since the PB-control structure for networked Hammerstein systems is exactly the same as the linear system case in Section 3.1, we therefore will not address the design details but only present the PB-control algorithm as follows, where we take the


49

Figure 4.3: PB-control for networked Hammerstein systems.

state-space description as an example; the reader is referred to Section 3.1 for more details of the PB-control structure. Algorithm 4.5 (PB-control for networked Hammerstein systems in state-space model description in (4.2)). S1. Calculation. The PB-controller calculates the intermediate FCIS ∆V (k|k − τsc,k ) using (4.8) and then obtains the real FCIS ∆U (k|k − τsc,k ) by compensating for the nonlinear input process using (4.11); S2. Forward-transmission. ∆U (k|k − τsc,k ) is packed and sent to the actuator simultaneously with time stamps k and τsc,k ; S3. Comparison. CAS updates its FCIS according to the time stamps once a data packet arrives; S4. Execution. An appropriate control increment signal is picked out from CAS and applied to the plant; S5. Backward-transmission. The information of the applied control increment with the sensing state is sent to the controller. The block diagram of the PB-control structure for networked Hammerstein systems in state-space description is shown in Fig. 4.3.


4.3

50

Stability analysis of packet-based networked Hammerstein systems

In this section, the stability conditions of networked Hammerstein systems using the PB-control approach are investigated, for both descriptions in (4.1) and (4.2). For the input-output description, the Popov criterion is adopted from which a stability criterion is derived only for a constant network-induced delay whereas for the state-space description, switched system theory is applied which yields a stability criterion that is valid for arbitrary network-induced delays.

4.3.1

Stability criterion in input-output description

From the design of the CAS in Section 3.1, the control action adopted by the actuator at time k is readily obtained as ∗ u(k) = dTτca,k U (k − τca,k |k − τk∗ ) ∗

(4.13)

∗ ∗ where dτca,k is a Nu × 1 column vector with all entries 0 but the τca,k th being 1, ∗ ∗ ∗ and τk∗ is the round trip delay with respect to τca,k , i.e. τk∗ = τca,k + τsc,k .

Combining (4.1),(4.5),(4.12),(4.13), the PB-control approach applied to the networked Hammerstein system in (4.1) can then be fully described by the following ∗ system SI1 (ω is set to 0 without loss of generality),

∗ SI1

 ay(k) = bI1 v(k − 1)       v(k) = f (u(k)) : ˆ ∗  u(k) = dTτca,k f~−1 (V (k − τca,k |k − τk∗ )) ∗      ∗ V (k − τca,k |k − τk∗ ) = Lτ (z −1 )y(k − τk∗ ) ∗

∗

(4.14a) (4.14b) (4.14c) (4.14d)

where Lτ (z −1 ) = (z −τk −1 P M dTτca,k −z −τk −1 P CDE∆dTτca,k −I)−1 P ST F and (4.14d) ∗ ∗ is obtained by noticing ∗ V (k − τca,k |k − τk∗ ) =P M v(k − τk∗ − 1)

− P ST E∆v(k − τk∗ − 1) − P ST F y(k − τk∗ ),


51

∗

∗ |k − τk∗ ), V (k − τca,k v(k − τk∗ − 1) = z −τk −1 dTτca,k ∗

and substituting the latter to the former. ∗ In order to derive the stability criterion for system SI1 , the following Popov crite-

rion is required. Lemma 4.6 (Popov criterion, see Ding et al. (2003)). Suppose that H(z −1 ) in Fig. 4.4 is stable and 0 ≤ Φ(θ) ≤ Kθ. Then the closed-loop system is stable if 1/K + Re(H(z −1 ) > 0, ∀|z| = 1.

Figure 4.4: Popov criterion. ∗ ∗ ∗ ∗ In the case of constant delays, we have that τ ∗ = τk∗ , τca = τca,k , τsc = τsc,k , ∀k, are ∗ and denote the characteristic polyall constant. Apply Lemma 1 to system SI1

nomial of a transfer function H(z −1 ) by δ(H(z −1 )), we then obtain the following theorem. Theorem 4.7. Suppose the roots of δ(Aτ (z −1 )) = 0 are all located in the unit circle. Then the system in (4.14) is stable if there exists a positive constant K such that 1. the input nonlinearity of the plant satisfies 0 ≤ v ≤ K¯ v,

(4.15a)

2. the network-induced delay satisfies 1 + Re{Aτ (z −1 )} > 0, ∀|z| = 1, K where Aτ (z −1 ) =

z −τ

∗ −1 T dτ ∗ Lτ (z −1 )bI1 ca

a

input value to the CARIMA model.

(4.15b)

, and v¯(k) = Aτ (z −1 v(k)) is the theoretical


52

Proof. Without loss of generality assume ω = 0. Notice that for any column ~ˆ−1 (P )) = f · fˆ−1 (dT∗ P ) from the vector P with an appropriate dimension, f (dTτca ∗ f τca ˆ definition of f~−1 (·). Then from (4.14) we obtain v(k) =f (u(k)) ~ˆ−1 (V (k − τ ∗ |k − τ ∗ )) =f (dTτca ∗ f ca −1 =f · fˆ−1 (dTτca )y(k − τ ∗ )) ∗ Lτ (z

=f · fˆ−1 (Aτ (z −1 )v(k)) =f · fˆ−1 (¯ v (k))

This is equivalent to the block diagram shown in Fig. 4.5. Thus the theorem can be easily obtained by applying Lemma 1 to Fig. 4.5.

Figure 4.5: The simplified block diagram of PB-control for networked Hammerstein systems in (4.1).

4.3.2

Stability criterion in state-space description

Similar to (4.13), from the design of the CAS in Section 3.1, the incremental control action adopted by the actuator at time k is readily obtained as ∗ ∆U (k − τca,k |k − τk∗ ) ∆u(k) = ∆u(k|k − τk∗ ) = dTτca,k ∗

(4.16)

∗ ∗ ∗ ∗ where dτca,k , τk∗ , τca,k and τk∗ = τca,k + τsc,k are defined in (4.13).

From (4.8), (4.11), (4.16) and noticing for any vector V with an appropriate diˆ mension, dTτca,k f~−1 (V ) = fˆ−1 (dTτca,k V ), we then obtain (assume the set point ω = 0 ∗ ∗


53

w.l.o.g.) ∗ ∆u(k) =dTτca,k ∆U (k − τca,k |k − τk∗ ) ∗

ˆ ∗ =dTτca,k f~−1 (∆V (k − τca,k |k − τk∗ ) ∗ ∗ =fˆ−1 (dTτca,k ∆V (k − τca,k |k − τk∗ ) ∗ ∗ =fˆ−1 (−Kτ,k x¯(k − τk∗ ))

(4.17)

∗ Mτsc,k Eτsc,k 1 . The real incremental control action for the linear = dTτca,k where Kτ,k ∗

system in (4.2a) and (4.2b) at time k can then be obtained as ∗ x¯(k − τk∗ )) ∆v(k) = f (∆u(k)) = f ◦ fˆ−1 (−Kτ,k

(4.18)

where f ◦ fˆ−1 (·) = f (fˆ−1 (·)) is the composite function of f (·) and fˆ−1 (·). Let X(k) = [¯ xT (k − τ¯) · · · x¯T (k)]T , w(k) = ∆v(k). The closed-loop formula for the system in (4.7) with the controller in (4.18) can then be represented by ( ∗ SI2

:

e X(k + 1) = AX(k) + ebw(k)

(4.19a)

w(k) = f ◦ fˆ−1 (−Kτ¯∗,k X(k))

(4.19b)

where eb = [0n+1,1 · · · 0n+1,1 ¯bTn+1,1 ]T , Kτ¯∗,k is a 1×(¯ τ +1) block matrix with block size ∗ of 1×(n+1) and all its blocks 0 except the (¯ τ +1−τk∗ )th being Kτ,k (the set of all the   0n+1 In+1     0n+1 In+1     . . ∗ e= . .. .. possible Kτ¯,k will be denoted by K), and A       0 I n+1 n+1   A¯

0

0

As has been pointed out in Section 4.2.2, the compensation for the nonlinear input process using (4.10) is generally not accurate, and this inaccuracy introduces to the linear part of the Hammerstein system in (4.2a) and (4.2b) a nonlinear disturbance, which appears in the form of f ◦ fˆ−1 (·). Though generally f ◦ fˆ−1 (·) 6≡ 1(·), it is reasonable to assume that the calculation error meets some accuracy requirement ∗ Note that the value of Kτ,k varies with the delays in both channels, and thus it has (¯ τca + 1)(¯ τsc + 1) different values in total. 1


54

to a certain extent, which results in a sector constraint for the term f ◦ fˆ−1 (·), as described in Assumption 4.8 as follows2 . Assumption 4.8. The nonlinearity due to the calculation inaccuracy is supposed to satisfy the following sector constraint: ∃0 < ε ≤ ε¯ < ∞, s.t. εα ≤ f ◦ fˆ−1 (α) ≤ ε¯α, ∀α ∈ R

(4.20a)

or denoted by f ◦ fˆ−1 (·) ∈ [ε, ε¯]

(4.20b)

Notice here generally 0 < ε ≤ 1 ≤ ε¯ < ∞. Assumption 4.8 implies that for any specific α ∈ R, there exists a real number εα , ε ≤ εα ≤ ε¯ such that f ◦ fˆ−1 (α) = εα α. With this observation, (4.19b) can then be rewritten as w(k) = f ◦ fˆ−1 (−Kτ¯∗,k X(k)) = −εk Kτ¯∗,k X(k)

(4.21)

where εk ∈ [ε, ε¯] represents the compensation for the specific nonlinearity for the term Kτ¯∗,k X(k) at time k. ∗ can then be written as With (4.19a) and (4.21), the closed-loop system in SI2

e X(k + 1) =AX(k) + ebw(k) e − εkebKτ¯∗,k )X(k) =(A =Λ(εk , Kτ¯∗,k )X(k)

(4.22)

e − εkebK ∗ has the following form where the closed loop matrix Λ(εk , Kτ¯∗,k ) = A τ¯,k      Λ(εk , Kτ¯∗,k ) =     

0n+1



In+1 0n+1

In+1 .. .

0 ···

∗ −εk¯bKτ,k

···

0 ..

.

0n+1 In+1 A¯

    .    

2 Note that though it is reasonable to place a sector constraint as in Assumption 4.8 to f ◦ fˆ−1 (·), it is somewhat conservative since the calculation of some strongly nonlinear function may not be that accurate and thus does not satisfy Assumption 4.8.


55

∗ depend on specific delays in where the position and value of the term −εk¯bKτ,k ∗ ∗ both channels at time k, i.e., (Λ(εk , Kτ¯,k ))τ¯+1,j = −εk¯bKτ,k , j = τk∗ = 1, 2, · · · , τ¯, and (Λ(εk , K ∗ ))τ¯+1,¯τ +1 = A¯ − εk¯bK ∗ , if τ ∗ = τ¯ + 1. τ¯,k

τ,k

k

∗ Theorem 4.9. The closed-loop system SI2 is stable if Assumption 4.8 holds and

there exists a positive definite solution P = P T > 0 for the following 2(¯ τca +1)(¯ τsc + 1) LMIs ΛT (ε, Kτ¯∗,k )P Λ(ε, Kτ¯∗,k ) − P ≤ 0

(4.23a)

ΛT (¯ ε, Kτ¯∗,k )P Λ(¯ ε, Kτ¯∗,k ) − P ≤ 0

(4.23b)

where Kτ¯∗,k ∈ K. Proof. Let V (k) = X T (k)P X(k) be a Lyapunov function candidate for system ∗ ∗ SI2 . The increment of V (k) along the trajectory of system SI2 can be obtained

using (4.22) as ∆V (k) =X T (k)(Λ(εk , Kτ¯∗,k )T P Λ(εk , Kτ¯∗,k ) − P )X(k) eT P A e − P − εk A eT P ebK ∗ − εk K ∗T ebT P A e =X T (k)(A τ¯,k τ¯,k + ε2k Kτ¯∗T,kebT P ebKτ¯∗,k )X(k) ,X T (k)A (εk , Kτ¯∗,k )X(k)

(4.24)

where εk ∈ [ε, ε¯], Kτ¯∗,k ∈ K. Notice that for any εk ∈ [ε, ε¯], there exists 0 ≤ λk ≤ 1 s.t. εk = λk ε + (1 − λk )¯ ε, and thus we obtain by substituting this into (4.24) that A (εk , Kτ¯∗,k ) = λk A (ε, Kτ¯∗,k )+(1−λk )A (¯ ε, Kτ¯∗,k )−λk (1−λk )(ε− ε¯)2 Kτ¯∗T,kebT P ebKτ¯∗,k From (4.23) and (4.24) it is seen that A (ε, Kτ¯∗,k ) and A (¯ ε, Kτ¯∗,k ) are semi-negative definite for all Kτ¯∗,k ∈ K. Notice that P is symmetric positive definite, and then K ∗T ebT P ebK ∗ is semi-positive definite as a symmetric matrix, thus enabling A

τ¯,k (εk , Kτ¯∗,k )

τ¯,k

to be semi-negative definite for any εk ∈ [ε, ε¯] and Kτ¯∗,k ∈ K, which

completes the proof. Remark 4.10. It is necessary to point out that according to Theorem 4.9, what is required for the stability of the system is to satisfactorily meet the sector constraint in Assumption 4.8, no matter how the inverse function fˆ−1 (·) is calculated. It


56

implies that the function f (·) does not need to be theoretically invertible as long as its inverse can be obtained by a numerical method and satisfies the sector constraint. The reader is referred to Tao and Kokotovic (1996) and the references therein for more information of the calculation of fˆ−1 (·). The following two special cases are also considered for system SI2 . Case 4.11. The network-induced delays in both channels are constant (noted by 0 0 τsc and τca respectively).

Case 4.12. The calculation of the inverse of the nonlinear function is accurate. The following corollary readily follows from Theorem 4.9. ∗ Corollary 4.13. The closed loop system SI2 is stable if any one of the following

three conditions holds. 1. Assumption 4.8 and Case 4.11 hold and there exists a positive definite solution P = P T > 0 for the following two LMIs ΛT (ε, Kτ¯∗,k )P Λ(ε, Kτ¯∗,k ) − P ≤ 0

(4.25a)

ΛT (¯ ε, Kτ¯∗,k )P Λ(¯ ε, Kτ¯∗,k ) − P ≤ 0

(4.25b)

0 0 where τsc,k ≡ τsc , τca,k ≡ τca and Kτ¯∗,k is therefore fixed.

2. Case 4.12 holds and there exists a positive definite solution P = P T > 0 for the following (¯ τca + 1)(¯ τsc + 1) LMIs ΛT (1, Kτ¯∗,k )P Λ(1, Kτ¯∗,k ) − P ≤ 0

(4.26)

where Kτ¯∗,k ∈ K. 3. Both of Case 4.11 and Case 4.12 hold and there exists a positive definite solution P = P T > 0 for the following LMI ΛT (1, Kτ¯∗,k )P Λ(1, Kτ¯∗,k ) − P ≤ 0 0 0 where τsc,k ≡ τsc , τca,k ≡ τca and Kτ¯∗,k is therefore fixed.

(4.27)


4.4

57

Numerical & experimental examples

In this section, numerical and experimental examples are considered to illustrate the effectiveness of the proposed PB-control approach to networked Hammerstein systems. 0.06

3000 (0,0) (2,3)

0.04

(3,7) 2000

0.02

1000

0 0 −0.02 −1000 −0.04 −2000

−0.06

−3000

−0.08 −0.1

0

50

100

150

200

250

Figure 4.6: Example 4.1. i) (τca , τsc ) = (0, 0); ii) (τca , τsc ) = (2, 3);

−4000

0

100

200

300

400

Figure 4.7: Example 4.1. (τca , τsc ) = (3, 7)

500

iii)

Example 4.1. This numerical example is used to illustrate the effectiveness of the PB-control approach to system SI1 in input-output description. The linear part in (4.1a) is adopted as y(k) − 0.8y(k − 1) = 2v(k − 1) + 3v(k − 2) The nonlinear input process in (4.1b) is chosen as v = f (u) = u2 and the practical √ inverse of f (·) is fˆ−1 = v × , where is a random number with a uniform distribution in [0 1]. is introduced to represent the uncertainty in a practical implementation. From (4.15a) in Theorem 4.7 it is seen that K = 1. The predictive horizon and control horizon are chosen as Np = Nu = 12 in the simulation. It is seen that the system is stable only for the first two cases according to Theorem 4.7 since for too large a time delay the system will not satisfy (4.15b) in Theorem 4.7. The simulation results of three cases: i) (τca , τsc ) = (0, 0); ii) (τca , τsc ) = (2, 3); and iii) (τca , τsc ) = (3, 7) are shown in Fig. 4.6 and Fig. 4.7 and illustrate the validity of the theoretical analysis. Example 4.2. This numerical example is used to illustrate the effectiveness of the PB-control approach to system SI2 in state-space description.


58

0.4

3 Random delays in the forward channel

0.2 2.5 0 2

x1(k)

τca,k

−0.2 1.5

−0.4 −0.6

1 −0.8 0.5 −1 0

0

20

40

60

80

−1.2

100

System without input nonlinear process System with input nonlinear process 0

20

40

60

k

Figure 4.8: Example 4.2. Arbitrary delays in the forward channel.

80 k

100

120

140

160

Figure 4.9: Example 4.2. The effectiveness of PB-control for networked Hammerstein systems.

The linear part of system SI2 is defined as follows which is open-loop unstable, A=

0.98 0.1 0

1

! ,b =

0.04 0.1

! ,c =

1 0

.

Note the fact that with an inverse process to compensate for the static nonlinear input process in system SI2 , from (4.19b) we know that the system performance depends only on the accuracy of the compensation process, i.e., the size of the sector constraint [ε, ε¯] for f ◦ fˆ−1 (·) (see Assumption 4.8). In this simulation, we set [ε, ε¯] = [0.5, 1.5] which means there is approximately 50% error in the compensation for the input nonlinearity while the input nonlinear function f (·) can be of any form provided this compensation accuracy is satisfied. All the other parameters are set the same as above. Such a system with those parameters can be proved to be stable using Theorem 4.9. The compensation for the nonlinear input process is shown in Fig. 4.9, from which it is seen that this compensation strategy is effective for networked Hammerstein systems, where the parameters are set as Nu = 8, Np = 10, τ¯ = 3, τ¯ca = 2, τ¯sc = 1 and x0 = [−1 − 1]T . Example 4.3. In this example, we use the same experiment setup as in Example 3.4 only that a compensation scheme for an input nonlinear process is present with [ε, ε¯] = [0.8, 1.2]. Since the linear part of the system remains the same, the same PB-controller is designed here as in Example 3.4. The comparison between the simulation and experimental results is illustrated in Fig. 4.10, where it is seen that the compensation scheme is effective in practice.


59

30

Deflection angle (Degree)

20

10

0

−10

−20

−30

Reference signal Simulation of PB−control for Hammerstein systems Internet−based PB−control for Hammerstein systems 0

1

2

3 4 Time (Second)

5

6

7

Figure 4.10: Example 4.3. Comparison between simulation and experimental results of packet-based control for Hammerstein systems.

4.5

Conclusion

In this chapter, the PB-control approach proposed in Chapter 3 was extended to networked Hammerstein systems. In order to deal with the nonlinear input process in the Hammerstein system, a two-step approach was applied to separate the nonlinear input process from the whole system, which proved to be effective for both descriptions of the Hammerstein system, i.e., the input-output description and the state-space description. For the input-output description, a stability criterion was obtained using Popov criterion, which is valid for a constant delay, while for the state-space description, stability conditions were obtained for arbitrary delays by using switched system theory. Numerical and experimental examples illustrated the effectiveness of the proposed approaches.

Chapter 5 PB-Control for Networked Wiener Systems Following the extension of the PB-control approach to networked Hammerstein systems in Chapter 4, another extension readily follows which is the category of output nonlinear systems described by the Wiener model, where a static nonlinear output process is present in the system. For this type of nonlinear systems, the two-step approach proposed in Chapter 4 can still be applied to separate the nonlinear process from the system, thus enabling the PB-control approach to be implemented in this case. Different from the input nonlinearity case, a specially designed observer is proposed for the implementation of the two-step approach to networked Wiener systems, and as a result, the stability criterion of the corresponding closed-loop system depends not only on the communication conditions but the error of the observer. This chapter is organized as follows. Section 5.2 presents the design details of the PB-control approach to networked Wiener systems; Section 5.3 analyzes the stability of the closed-loop system; Section 5.4 presents numerical and experimental examples to illustrate the validity of the proposed approach and Section 5.5 concludes the chapter.

60

Chapter 5. Networked Wiener Systems

5.1

61

System description

The following SISO Wiener system So is considered in this chapter,  x(k + 1) = Ax(k) + bu(k)    So : y(k) = cx(k)    z(k) = f (y(k))

(5.1a) (5.1b) (5.1c)

where x ∈ Rn , u, y, z ∈ R, A ∈ Rn×n , b ∈ Rn×1 , c ∈ R1×n , f (·) is a memoryless static nonlinear function and u(k) is to be determined (see Section 5.2). In this chapter the Wiener system is assumed to be controlled over the network, see Fig. 5.1 for its configuration.

Figure 5.1: The block diagram of networked Wiener Systems.

As in Chapter 4, the memoryless static nonlinear function f (·) in this chapter is assumed to be invertible with its inverse denoted by fˆ−1 (·). Notice that fˆ−1 (·) can not be obtained accurately in practice which means ϕ(·) , fˆ−1 (f (·)) 6≡ 1(·). The approximate intermediate output y˜(k) (Fig.5.2) can thus be obtained as follows, y˜(k) = fˆ−1 (z(k)) = ϕ(y(k))

(5.2)

With this inverse process, the PB-controller for networked Wiener systems in (5.1) can then be obtained using a LGPC method and a specially designed state observer as follows.


62

Figure 5.2: Two-step approach to networked Wiener systems.

5.2

PB-Control for networked Wiener systems

Let the objective function for system So be defined by o Jk,τ sc,k

=

N2 X

2

qj (ˆ y (k + j|k − τsc,k ) − ω(y; k + j)) +

j=N1

Nu X

rj (∆u(k + j − 1))2 (5.3)

j=1

where N1 and N2 are the minimum and maximum prediction horizons, Nu is the control horizon, qj , N1 ≤ j ≤ N2 and rj , 1 ≤ j ≤ Nu are weighting factors, ∆u(k) = u(k) − u(k − 1) is the control increment, yˆ(k + j|k − τsc,k ), j = N1 , ..., N2 are the forward predictions of the system outputs, which are obtained on data up to time k − τsc,k ; ω(y; k + j) is the set point with respect to y and can be obtained approximately by inverting corresponding set point ω(z; k + j) with respect to z, i.e., ω(y; k + j) = fˆ−1 (ω(z; k + j)), j = N1 , ..., N2

(5.4)

Letting x¯(k) = [xT (k) u(k − 1)]T , then the linear part of system So (i.e. (5.1a) and (5.1b)) can be rewritten as follows, ( So0

:

¯x(k) + ¯b∆u(k) x¯(k + 1) = A¯

(5.5a)

y(k) = c¯x¯(k)

(5.5b)


where A¯ =

A b 0 1

! , ¯b =

b

!

1

, c¯ =

63

c 0

.

Following the same procedure as in Section 3.3, the optimal FCIS from k to k + Nu − 1 can then be obtained as ∆U (k|k − τsc,k ) = Mτsc,k ($k (y; ·) − Eτsc,k x¯(k − τsc,k ))

where ∆U (k|k − τsc,k ) = [∆u(k|k − τsc,k ) · · · ∆u(k + Nu − 1|k − τsc,k )]T , $k (y; ·) = cA¯N2 +τsc,k )T ]T , Fτ [ω(y; k + N1 ) · · · ω(y; k + N2 )]T , Eτ = [(¯ cA¯N1 +τsc,k )T · · · (¯ sc,k

sc,k

is a (N2 − N1 + 1) × (Nu + τsc,k ) matrix with the non-null entries defined by (Fτsc,k )ij = c¯A¯N1 +τsc,k +i−j−1¯b, j − i ≤ N1 + τsc,k − 1, Mτsc,k = Hτsc,k (FτTsc,k QFτsc,k + R)−1 FτTsc,k Q, Q, R are diagonal matrices with Qi,i = qi , Ri,i = ri respectively, Hτsc,k = [0Nu ×τsc,k INu ×Nu ], and INu ×Nu is the identity matrix with rank Nu . Since the system states are normally unavailable for the controller, the following observed system is then constructed: (

xˆ(k + 1) = Aˆ x(k) + bu(k)

(5.6a)

yˆ(k) = ϕ(cˆ x(k))

(5.6b)

to observe the system states, xˆ(k + 1) = Aˆ x(k) + bu(k) + L(˜ y (k) − yˆ(k))

(5.7)

where xˆ(k) is the observed state at time k. Letting xˆ¯(k) = [ˆ xT (k) uT (k − 1)]T , the real FCIS can then be obtained as follows when the state observer in (5.6) is present, ∆U (k|k − τsc,k ) = Mτsc,k ($k (y; ·) − Eτsc,k xˆ¯(k − τsc,k ))

(5.8)

With the FCIS obtained in (5.8), the PB-control approach can then be implemented to networked Wiener systems in (5.1). Since the whole PB-control structure here is exactly the same as in Section 3.1, we therefore only illustrate its block diagram in Fig. 5.3 without further discussion; the reader is referred to Section 3.1 for more information on the design of the PB-control approach.


64

Figure 5.3: PB-control for networked Wiener systems.

5.3

Stability analysis of packet-based networked Wiener systems

In this section, we first prove the proposed state observer in (5.6) is stable under certain conditions. This fact enables us to construct the stability criterion for the closed-loop system.

5.3.1

Observer error

Let the observer error e(k) = x(k) − xˆ(k). From (5.1a), (5.6a) we obtain e(k + 1) =x(k + 1) − xˆ(k + 1) =Ae(k) − L(˜ y (k) − yˆ(k))

(5.9)

Assume ϕ(·) ∈ C 1 , then by mean value theorem, y˜(k) − yˆ(k) =ϕ(cx(k)) − ϕ(cˆ x(k)) =cϕ0 (ξk ))e(k)

(5.10)

where ξk ∈ [min{cx(k), cˆ x(k)} max{cx(k), cˆ x(k)}]. Combining equations (5.9) and (5.10) yields e(k + 1) = (A − Lcϕ0 (ξk ))e(k)

(5.11)


65

Notice that though ϕ(·) 6≡ 1(·), it is reasonable to assume that the compensation for the nonlinear function f (·) is smooth, which means there exists ε > 0 s.t. |ϕ0 (α) − 1| ≤ ε, ∀α ∈ R. Thus the dynamics of the observer error can be obtained as e(k + 1) =(A − Lc − ζk Lc)e(k) =Aζk e(k)

(5.12)

where Aζk = A − Lc − ζk Lc, |ζk | ≤ ε. Theorem 5.1 (Observer Error). The observer error converges to 0 if there exists a positive definite solution Pe = PeT > 0 for the following two LMIs ATε Pe Aε − Pe < 0 AT−ε Pe A−ε − Pe < 0

(5.13)

where Aε = A − Lc − εLc and A−ε = A − Lc + εLc. Proof. Let V (k) = eT (k)Pe e(k) be a Lyapunov function candidate for the system in (5.12). Notice the fact that for any ζk , there exists 0 ≤ λk ≤ 1 such that ζk = λk ε + (1 − λk )(−ε). Thus by simple calculation, the incremental V (k) for the system in (5.12) can be obtained as ∆V (k + 1) =eT (k)Γζk e(k) =eT (k)(λk Γε + (1 − λk )Γ−ε − 4λk (1 − λk )(Lc)T Pe Lc)e(k) (5.14) where Γζk = ATζk Pe Aζk − Pe . Noticing λk (1 − λk ) ≥ 0 and (Lc)T Pe Lc is semi positive definite, then yields that ∆V (k) is decreasing which completes the proof.


5.3.2

66

Closed-loop stability

From (5.2) and the design of the CAS, the incremental control action adopted by the actuator at time k is readily obtained as

∗ ∆u(k) = dTτca,k ∆U (k − τca,k |k − τk∗ ) ∗

= −dTτca,k Mτk∗ Eτk∗ xˆ¯(k − τk∗ ) ∗ = −Στk xˆ¯(k − τk∗ )

(5.15)

∗ ∗ where dτca,k is a Nu × 1 matrix with all entries 0 except the (τca,k + 1)th being ∗ ∗ are defined in (3.3), Στk = dTτca,k and τsc,k Mτk∗ Eτk∗ and the set point is 1, τk∗ , τca,k ∗

assumed to be 0 without loss of generality. Let e¯(k) = x¯(k) − xˆ¯(k) = [e(k) 0]T . Then e¯(k + 1) = A¯ξk e¯(k)

where A¯ξk =

(5.16)

A − Lcϕ0 (ξk ) 0 0

!

0

.

Let Z(k) = [¯ xT (k − τ¯) · · · x¯T (k) e¯(k − τ¯) · · · e¯(k)]T . The closed-loop system can then be obtained as Z(k + 1) = Λξk ,τk Z(k)

(5.17) 

where Λξk ,τk =

12 Λ11 τk Λτk

0

Λ22 ξk

! , Λ11 τk

0n+1



In+1

    =   

In+1 ... ···

−Στk

···

In+1 A¯

    ,   

Chapter 5. Networked Wiener Systems 

Λ22 ξk



0n+1 In+1

    =    

67

0n+1 In+1 ... .. . ... In+1 A¯ξ

    , and Λ12 τk is a block matrix with all its   

k

entries (blocks) 0 except (Λ12 τ −1)×(¯ τ −τk∗ +1) = −Στk . τk )(¯ Theorem 5.2 (Closed-loop stability). The closed-loop system in (5.17) is stable if (5.1) holds and there exists a positive definite solution Pc = PcT > 0 for the following (¯ τca + 1)(¯ τsc + 1) LMIs T 11 (Λ11 τk ) Pc Λτk − Pc ≤ 0

(5.18)

Proof. By noticing the block-triangular structure of the system matrix Λξk ,τk for the closed-loop system, it is seen that the state observer in (5.6) can be designed separately without affecting the stability of the system and the closed-loop system is stable if we can guarantee the stability of the state observer (Theorem 5.1) and the following system, X(k + 1) = Λ11 τk X(k)

(5.19)

where X(k) = [¯ xT (k − τ¯) · · · x¯T (k)]. Let V (k) = X T (k)Pc X(k) be a Lyapunov function candidate for the system in (5.19). The incremental V (k) along the trajectory of the system in (5.19) is then obtained as T 11 ∆V (k) = X T (k)((Λ11 τk ) Pc Λτk − Pc )X(k)

which completes the proof using (5.18). Remark 5.3. It is worth mentioning that the two conditions ((5.13) and (5.18)) that guarantee the stability of the closed-loop system are with respect to the compensation accuracy for the nonlinearity and the effect of the network constraints respectively.


5.4

68

Numerical & experimental examples

In this section, a numerical example is considered to illustrate the effectiveness of the proposed PB-control approach for networked Wiener systems. Example 5.1. The linear system in Example 4.2 with a static nonlinear output process and random delays in both channels and data packet dropout in the forward channel, is adopted, with other parameters of the simulation chosen as τ¯ = 8, τ¯ca = 4, τ¯sc + χ¯ = 4, Nu = 8, Np = 10, ε = 0.5 and the initial state x(0) = x0 = [−0.1 0.2]T . The delays in both channels are set to vary randomly within their upper bounds. Such a system using the PB-control approach can be proved to be stable under Theorem 5.2. Two cases which illustrate the validity of the compensation for the communication constraints and the compensation for the output nonlinearity , are shown in Fig. 5.4 and Fig. 5.5 respectively. In both cases, all the other parameters remain the same and only the evolution of the first state of the system is illustrated. The simulation results show that the system is stable with the compensation scheme while unstable without it, which illustrate the validity of the proposed approach in this chapter.

0.25

15 State evolution without compensation for network constraints State evolution with compensation for network constraints

0.2

State evolution without compensation for the nonlinearity State evolution with compensation for the nonlinearity 10

0.15 0.1

x1(k)

x1(k)

5 0.05 0 0 −0.05 −0.1

−5

−0.15 −0.2

0

20

40

60

80

100

120

140

k

Figure 5.4: Example 5.1. A comparison between with/without compensation for network constraints.

−10

0

20

40

60 k

80

100

120

Figure 5.5: Example 5.1. A comparison between with/without compensation for output nonlinearity.

Example 5.2. In this example, we use the same experiment setup as in Example 3.4 only that a compensation scheme for an output nonlinear process is present with [ε, ε¯] = [0.8, 1.2]. Since the linear part of the system remains the same, the same PB-controller is designed here as in Example 3.4. The comparison between


69

30

Deflection angle (Degree)

20

10

0

−10

−20

−30

Reference signal Simulation of PB−control for Wiener systems Internet−based PB−control for Wiener systems 0

1

2

3 4 Time (Second)

5

6

7

Figure 5.6: Example 5.2. Comparison between simulation and experimental results of packet-based control for Wiener systems.

the simulation and experimental results is illustrated in Fig. 5.6, where it is seen that the compensation scheme is effective in practice.

5.5

Conclusion

In this chapter, the PB-control approach was extended to networked Wiener systems. The idea of the two-step approach proposed for networked Hammerstein systems in Chapter 4 was still adopted, which together with a specially designed state observer enabled the PB-control approach to be implemented in this case. Closed-loop stability was obtained by using the separate principle and switched system theory, the validity of which was illustrated by numerical and experimental examples.

Chapter 6 Stochastic Stabilization of PB-NCSs In the previous chapters (Chapter 3, 4 and 5) the communication constraints including network-induced delay, data packet dropout and data packet disorder, are all assumed to be deterministic which, however, are actually stochastic in nature. This observation motivates the study in this chapter on the stochastic stabilization of PB-NCSs under the Markov jump system framework, where the network-induced delay (data packet dropout as well) in round trip is modeled as a homogeneous ergodic Markov chain. Under this framework, the sufficient and necessary conditions for stochastic stability and stabilization of PB-NCSs are obtained, which can be compared with the deterministic analysis in Chapter 3, 4 and 5 where only sufficient conditions to guarantee the closed-loop stability are obtained and no stabilization analysis is given. This chapter is organized as follows. The stochastic analysis of PB-NCSs is presented in Section 6.1, covering the stochastic model of PB-NCSs and the corresponding stochastic stability and stabilization analysis. A numerical example is then given in Section 6.2 to illustrate the validity of the theoretical analysis and Section 6.3 concludes the chapter.

70

Chapter 6. Stochastic PB-NCSs

6.1

71

Stochastic analysis of PB-NCSs

Note that all the analysis in this chapter is based on the PB-control approach designed in Section 3.1; the reader is referred to Section 3.1 for more information on the design details of the PB-control approach and this chapter only focuses on the corresponding stochastic analysis. It is noticed that the control law in (3.2) ∗ ∗ , τca,k ) which is generally true in practice. equals that in (3.3) if K(τk∗ ) = K(τsc,k

Thus for simplicity only the closed-loop system with the control law in (3.3) (i.e., Algorithm 3.9) is analyzed in this chapter. The augmented closed-loop system of system Sd in (3.1) with the control law in (3.3) was shown in (3.12) as follows, X(k + 1) = Ξ(τk∗ )X(k) where X(k) = [xT (k) xT (k − 1) · · · xT (k − τ¯)]T , 

A · · · BK(τk∗ ) · · ·

  In   ∗ Ξ(τk ) =    

···

In ... In



 0    0 , ..  .   0

and In is the identity matrix with rank n. In this section, the stochastic model of the closed-loop system in (3.12) is first obtained, which is then further analyzed to derive the stochastic stability and stabilization conditions as follows.

6.1.1

Stochastic model of PB-NCSs

In NCSs, it is reasonable to model the round trip delay {τk ; k = 0, 1, . . .} as a homogeneous ergodic Markov chain (Zhang et al., 2005). Here in order to take explicit account of data packet dropout, Markov chain {τk ; k = 0, 1, . . .} is assumed to take values from M = {0, 1, 2, . . . , τ¯, ∞} where τk = 0 means no delay in round trip while τk = ∞ implies a data packet dropout in either the backward or the forward channel. Let the transition probability matrix of {τk ; k = 0, 1, . . .} be


72

denoted by Λ = [λij ] where λij = P {τk+1 = j|τk = i}, i, j ∈ M P {τk+1 = j|τk = i} is the probability of τk jumping from state i to j, λij ≥ 0 and X

λij = 1, ∀i, j ∈ M

j∈M

The initial distribution of {τk ; k = 0, 1, . . .} is defined by P {τ0 = i} = pi , i ∈ M According to the comparison rule in (3.7), the round trip delay of the control actions that are actually applied to the plant can be determined by the following formula,

∗ τk+1

 τ ∗ + 1, if τk+1 > τ ∗ ; k k = τ ∗ − r, if τ ∗ − r = τ ∗ k+1 ≤ τk . k k

(6.1)

The following lemma shows that the delay τk∗ in (6.1) that is derived from the comparison rule in (3.7) is a Markov chain. Lemma 6.1. {τk∗ ; k = 0, 1, . . .} is a non-homogeneous Markov chain with state space M∗ = {0, 1, 2, . . . , τ¯} whose transition probability matrix Λ∗ (k) = [λ∗ij (k)] is defined by

λ∗ij (k) =

 P πl1 (k)λl1 j   l1 ∈M,l1 ≥i  P , j ≤ i;  πl1 (k)    l1 P ∈M,l1 ≥i P  π (k)λ l1 ∈M,l1 ≥i l2 ∈M,l2 >i

l1

P  πl1 (k)   l1 ∈M,l1 ≥i     0, otherwise.

where πj (k) =

P

(k)

l1 l2

, j = i + 1;

(6.2)

(k)

pi λij and λij is the k-step transition probability of τk from

i∈M

state i to j. Proof. The comparison rule in (6.1) implies that the probability event {τk∗ = i} ∈ σ(τk , τk−1 , . . . , τ1 , τ0 ). Thus it is readily concluded that τk∗ is also a Markov chain since τk as a Markov chain evolves independently. It is obvious that τk∗ can


73

not be ∞ and thus its state space is M∗ = {0, 1, 2, . . . , τ¯}. Furthermore, since ∗ ∗ {τk∗ = i} = {τk−1 = i − 1, τk > i − 1} ∪ {τk−1 ≥ i, τk = i} we have

1. If j ≤ i, then ∗ P {τk+1 = j|τk∗ = i} =P {τk+1 = j|τk∗ = i}

=P {τk+1 = j|τk ≥ i} P πl1 (k)λl1 j l1 ∈M,l1 ≥i P = πl1 (k) l1 ∈M,l1 ≥i

2. If j = i + 1, then ∗ = j|τk∗ = i} =P {τk+1 > i|τk∗ = i} P {τk+1

=P {τk+1 > i|τk ≥ i} X P {τk+1 = l2 |τk ≥ i} = l2 ∈M,l2 >i

P =

P

πl1 (k)λl1 l2

l1 ∈M,l1 ≥i l2 ∈M,l2 >i

P

πl1 (k)

l1 ∈M,l1 ≥i

which completes the proof. With Lemma 6.1, the closed-loop system in (3.12) can now be regarded as a Markov jump system where the system matrix Ξ(τk∗ ) evolves with the Markov chain {τk∗ ; k = 0, 1, . . .} whose transition probability matrix is defined in (6.2). Remark 6.2. The data packet dropout is explicitly considered by including the state τk = ∞ into the state space Λ; The data packet disorder is also considered by (6.1): In our stochastic model the network-induced delay, data packet dropout and data packet disorder are all considered simultaneously. To the best knowledge of the authors, there is no analogous stochastic analysis available in literature to date. The following well-known result for homogeneous ergodic Markov chains is required for the stochastic stability analysis in this chapter.


74

Lemma 6.3 (Billingsley (1995)). For the homogeneous ergodic Markov chain {τk ; k = 0, 1, . . .} with any initial distribution, there exists a limit probability distribution π = {πi ; πi > 0, i ∈ M} such that for each j ∈ M, X

λij πi = πj ,

i∈M

X

πi = 1

(6.3)

i∈M

and |πi (k) − πi | ≤ ηξ k

(6.4)

for some η ≥ 0 and 0 < ξ < 1. Proposition 6.4. For N1 that is large enough and some nonzero η ∗ the following inequality holds |λ∗ij (k) − λ∗ij | ≤ η ∗ ξ k , k > N1

(6.5)

where Λ∗ = [λ∗ij ] with

λ∗ij =

 P πl1 λl1 j   l1 ∈M,l1 ≥i  P , if j ≤ i;    l1 ∈M,l1 ≥i πl1  P  P π λ l1 ∈M,l1 ≥i l2 ∈M,l2 >i P

 πl1   l1 ∈M,l1 ≥i     0, otherwise.

l1 l1 l2

, if j = i + 1;

(6.6)

Proof. It can be readily obtained from (6.2), (6.4) and (6.6).

6.1.2

Stochastic stability and stabilization

The following definition of stochastic stability is used in this chapter. Definition 6.5 (Stochastic stability, see Zhang et al. (2005).). The closed-loop system in (3.12) is said to be stochastically stable if for every finite X0 = X(0) and initial state τ0∗ = τ ∗ (0) ∈ M, there exists a finite W > 0 such that the following inequality holds, E{

∞ X k=0

||X(k)||2 |X0 , τ0∗ } < X0T W X0

(6.7)


75

where E{X} is the expectation of the random variable X. Theorem 6.6 (Stochastic stability). The closed-loop system in (3.12) is stochastically stable if and only if there exists P (i) > 0, i ∈ M∗ such that the following (¯ τ + 1) LMIs hold L(i) =

X

λ∗ij ΞT (j)P (j)Ξ(j) − P (i) < 0, ∀i ∈ M∗

(6.8)

j∈M∗

Proof. Sufficiency. For the closed-loop system in (3.12), consider the following quadratic function given by V (X(k), k) = X T (k)P (τk∗ )X(k) We have E{∆V (X(k), k)} ∗ =E{X T (k + 1)P (τk+1 )X(k + 1)|X(k), τk∗ = i} − X T (k)P (i)X(k) X = λ∗ij (k + 1)X T (k)ΞT (j)P (j)Ξ(j)X(k) − X T (k)P (i)X(k) j∈M∗

X

=X T (k)[

λ∗ij (k + 1)ΞT (j)P (j)Ξ(j) − P (i)]X(k)

j∈M∗

From (6.8) we obtain X T (k)[

X

λ∗ij ΞT (j)P (j)Ξ(j) − P (i)]X(k)

j∈M∗

≤ − λmin (−L(i))X T (k)X(k) ≤ − β||X(k)||2

(6.9)


76

where β = inf{λmin (−L(i)); i ∈ M∗ } > 0. Thus for k > N1 , E{∆V (X(k), k)} X =X T (k)[ λ∗ij (k + 1)ΞT (j)P (j)Ξ(j) − P (i)]X(k) j∈M∗

≤X T (k)[

X

λ∗ij ΞT (j)P (j)Ξ(j) − P (i)]X(k)

j∈M∗

+ X T (k)

X

|λ∗ij (k + 1) − λ∗ij |ΞT (j)P (j)Ξ(j)X(k)

j∈M∗

≤ − β||X(k)||2 + η ∗ ξ k+1 X T (k)

X

ΞT (j)P (j)Ξ(j)X(k)

j∈M∗

≤(αη ∗ ξ k+1 − β)||X(k)||2 where α = sup{λmax (ΞT (j)P (j)Ξ(j)); j ∈ M∗ } > 0. Let N2 = inf{M ; M ∈ N+ , M > max{N1 , logξ

β αη ∗

− 1}}. Then we have for k ≥ N2

E{∆V (X(k), k)} ≤ −β ∗ ||X(k)||2

(6.10)

where β ∗ = β − αη ∗ ξ N2 +1 > 0. Summing from N2 to N > N2 we obtain E{

N X

||X(k)||2 }

k=N2

1 ≤ ∗ (E{V (X(N2 ), N2 )} − E{V (X(N + 1), N + 1)}) β 1 ≤ ∗ E{V (X(N2 ), N2 )} β which implies that N 2 −1 X 1 ||X(k)|| } ≤ ∗ E{V (X(N2 ), N2 )} + E{ ||X(k)||2 } E{ β k=0 k=0 ∞ X

2

(6.11)

This proves the stochastic stability of the closed-loop system in (3.12) by Definition 6.5. Necessity. Suppose the closed-loop system in (3.12) is stochastically stable, that is, E{

∞ X k=0

||X(k)||2 |X0 , τ0∗ } < X0T W X0

(6.12)


77

Define N X X T (n)P¯ (N − n, τn∗ )X(n) = E{ X T (k)Q(τk∗ )X(k)|Xn , τn∗ }

(6.13)

k=n

with Q(τk∗ ) > 0. It is noticed that X T (n)P¯ (N − n, τn∗ )X(n) is upper bounded from (6.12) and monotonically non-decreasing as N increases since Q(τk∗ ) > 0. Therefore its limit exists which is denoted by X T (n)P (i)X(n) = lim X T (n)P¯ (N − n, τn∗ = i)X(n) N →∞

(6.14)

Since (6.14) is valid for any X(n), we obtain P (i) = lim P¯ (N − n, τn∗ = i) > 0 N →∞

(6.15)

Now consider E{X T (n)P¯ (N − n, τn∗ )X(n) ∗ − X T (n + 1)P¯ (N − n − 1, τn+1 )X(n + 1)|Xn , τn∗ = i} X =X T (n)[P¯ (N − n, i) − λ∗ij (n + 1)ΞT (j)P¯ (N − n − 1, j)Ξ(j)]X(n) j∈M∗

=X T (n)Q(i)X(n)

(6.16)

Since (6.16) is valid for any X(n), we obtain P¯ (N − n, i) −

X

λ∗ij (n + 1)ΞT (j)P¯ (N − n − 1, j)Ξ(j)) = Q(i) > 0 (6.17)

j∈M∗

Let N → ∞, P (i) −

X

λ∗ij (n + 1)ΞT (j)P (j)Ξ(j) > 0, ∀n

j∈M∗

Let n → ∞, P (i) −

X

λ∗ij ΞT (j)P (i)Ξ(j) > 0

j∈M∗

which completes the proof.


78

The stochastic stabilization result in Corollary 6.7 readily follows using the Schur complement. Corollary 6.7 (Stochastic stabilization). System Sd is stochastically stabilizable using the PB-control approach with the control law in (3.3) if and only if there exist P (i) > 0, Z(i) > 0, K(i), i ∈ M∗ such that the following (¯ τ + 1) LMIs hold P (i)

R(i)

T

R (i)

! > 0, i ∈ M∗

Q

(6.18)

with the equation constraints P (i)Z(i) = I, ∀i ∈ M∗ 1

(6.19) 1

where R(i) = [(λ∗i0 ) 2 ΞT (0) . . . (λ∗i¯τ ) 2 ΞT (¯ τ )], Q = diag{Z(0) . . . Z(¯ τ )} and Ξ(i) (consequently K(i)) is defined in (3.12). The LMIs in Corollary 6.7 with the matrix inverse constraints in (6.19) can be solved using the Cone Complementarity Linearization (CCL) algorithm (Ghaoui et al., 1997).

6.2

A numerical example

In this section, a numerical example is considered to illustrate the validity of Theorem 6.6 and Corollary 6.7. Example 6.1. Consider the example in Zhang et al. (2005) where the system matrices are as follows,    A=  

1.0000 0.1000 −0.0166 −0.0005 0 0 0





0.0045

   1.0000 −0.3374 −0.0166   , B =  0.0896   −0.0068 0 1.0996 0.1033   0 2.0247 1.0996 −0.1377

   .  

This system is open-loop unstable with the eigenvalues at 1, 1, 1.5569 and 0.6423. In the simulation, the random round trip delay is bounded by 4, i.e., τk ∈ M =


79

0.1 x (k) 1

0.08

x2(k) x3(k)

0.06

x (k) 4

0.04

x(k)

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1

0

50

100 k

150

200

Figure 6.1: Example 6.1. States evolution of the PB-control approach to NCSs.

{0, 1, 2, 3, 4, ∞}, with the transition probability matrix as follows, 

0.1

0.2

0.2

0.3

0.2

     Λ=     

0.2

0.2

0.2

0.2

0.1

0.24 0.06 0.48 0.12 0.1 0.15 0.25

0.3

0.15 0.1

0.3

0.3

0.2

0.1

0.3

0.3

0.15 0.15 0.1

0.1

0



 0.1    0  . 0.05    0   0

The limit distribution of the above ergodic Markov chain can be simply obtained as in Lemma 6.3, π=

0.1982 0.1814 0.3000 0.1738 0.1198 0.0268

Λ∗ in Proposition 6.4 can then be calculated by (6.6) as 

0.1982 0.8018

0

0

0



   0.2224 0.1767 0.6008  0 0     ∗ Λ =  0.2290 0.1699 0.3612 0.2398 . 0    0.2186 0.2729 0.2501 0.1313 0.1271    0.3000 0.3000 0.1909 0.1091 0.1000

.


80

From Corollary 6.7, the packet-based controller is obtained as follows,

K(0) =

K(1) =

K(2) =

K(3) =

K(4) =

0.5292 0.6489 22.4115 2.8205 0.3792 0.8912 20.2425 5.3681 0.0499 0.4266 15.6574 5.7322

,

,

,

−0.4400 −0.3003 9.2976 5.0540

,

−0.8400 −1.3422 2.7723 2.9173

.

The state trajectories of the closed-loop system under the packet-based controller are shown in Fig. 6.1 with the initial states x(−3) = x(−2) = x(−1) = x(0) = [0 0.1 0 − 0.1]T , which illustrates the stochastic stability of the closed-loop system.

6.3

Conclusion

It is observed that the communication constraints in NCSs including networkinduced delay, data packet dropout and data packet disorder, are stochastic in nature. Based on this observation, a stochastic analysis was presented for the PBcontrol approach proposed in Chapter 3. Both stochastic stability and stabilization conditions were obtained, which was then validated by a numerical example.

Chapter 7 Continuous-Time PB-NCSs In all the previous chapters (Chapter 3, 4, 5 and 6) the PB-control approach is considered for plants in discrete time and discrete network-induced delay. Based on this observation, the PB-control approach is extended to the continuous time case in this chapter, with the use of a discretization technique for the continuous network-induced delay. The derived approach leads to a novel model for NCSs in continuous time. This model, as in the discrete time case, offers the designer the freedom of designing different controllers with respect to specific network conditions, which is distinct from previously reported results and results in a better performance. By applying switched system theory, the stability criterion for the derived model is obtained, which is then used to obtain an LMI-based stabilized controller for the continuous-time PB-NCSs. This chapter is organized as follows. The design details of the PB-control approach to NCSs in continuous time is first presented in Section 7.1, which leads to a novel model for NCSs. This model is then further analyzed in Section 7.2 to obtain the stability criterion and a stabilized controller by using the results from switched system theory. A numerical example is given in Section 7.3 to illustrate the effectiveness of the proposed approach and Section 7.4 concludes the chapter.

81

Chapter 7. Continuous-Time PB-NCSs

7.1

82

PB-control: Modeling of NCSs

The following linear plant in continuous time is considered in this chapter, which is assumed to be controlled over the network as shown in Fig. 7.1: Sc : x(t) ˙ = Ax(t) + Bu(t)

(7.1)

where x ∈ Rn , u ∈ Rm , A ∈ Rn×n and B ∈ Rn×m .

Figure 7.1: The block diagram of networked control systems in continuous time.

In this section the PB-control approach proposed in Chapter 3 in the discrete time fashion is extended to the continuous time case, with a discretization technique to merge the gap between discrete and continuous time. The reader is referred to Section 3.1 for more information on the PB-control approach in the discrete time case.

7.1.1

PB-control for NCSs in continuous time

A fundamental basis of the implementation of the PB-control for NCSs in discrete time is the construction of the FCS in (3.6) and (3.10), which can be readily obtained in discrete time. However, a time delay system in continuous time is of infinite dimension, thus making it difficult to implement readily the PB-control approach in the continuous time case because of the difficulty in determining the FCS as in (3.6) and (3.10). To deal with this difficulty, the continuous networkinduced delay is discretized as follows. Let τ¯d = dh + τ¯ and ϑ =

τ¯d , N

where h is the sampling period. A different FCS

structure compared to (3.6) and (3.10) can then be constructed as follows, which


83

uses N discrete levels to approach the real network-induced delay,

U (t − τsc (t)|t − τsc (t)) = [u(t − τsc (t)|t − τsc (t)) . . . u(t − τsc (t) + (N − 1)ϑ|t − τsc (t))]

(7.2)

where τsc (t) is the continuous backward channel delay of the data packet received by the controller at time t. The FCS in (7.2) can now be transmitted in one data packet by the network, provided the data size required for encoding a single step of the control signal is the same as the discrete time case. This is generally true since, both single step control signals are the specific values at one time instant, and therefore the data sizes of encoding both signals depend only on the range of the signals and the corresponding quantization levels, which can be assumed to be the same in both cases. In order to implement the PB-control approach, a similar CAS is also designed at the actuator side. The designed CAS consists of a simple comparison logic and a memory which can store only a single forward control sequence. When a FCS arrives at the actuator, it will first be compared using the comparison logic of the CAS with the one already in the memory of the CAS and only the latest is stored and applied to the plant. This comparison process is also introduced to overcome the effect of data packet disorder as done in Section 3.1. For clarity, a FCS is called an “effective” one if it is actually stored after the comparison process. Note that the kth effective FCS is U (t∗k − τk∗ |t∗k − τk∗ ), where t∗k is the time when this sequence is received by the actuator and τk∗ the corresponding round trip delay. The control law during the time period [t∗ik , t∗ik +1 ) can then be defined by u(t) = u(t∗k − τk∗ + ik ϑ|t∗k − τk∗ ), t ∈ [t∗ik , t∗ik +1 )

(7.3)

where [t∗ik , t∗ik +1 ) = [t∗k − τk∗ + ik ϑ, t∗k − τk∗ + (ik + 1)ϑ) with ik ∈ N satisfying t∗k ≤ t∗k −τk∗ +ik ϑ < t∗k+1 , and u(t∗k −τk∗ +ik ϑ|t∗k −τk∗ ) is selected from U (t∗k −τk∗ |t∗k −τk∗ ) which in this chapter is of the form of state feedback as follows, u(t∗k − τk∗ + ik ϑ|t∗k − τk∗ ) = K(ik )x(t∗k − τk∗ )

(7.4)


84

Note here that the value of the feedback gain K(ik ) is dependent on the current range of the network-induced delay which, for t ∈ [t∗ik , t∗ik +1 ), is τk∗ (t) = t − (t∗k − τk∗ ) ∈ [ik ϑ, (ik + 1)ϑ)

(7.5)

Figure 7.2: Timeline of packet-based networked control systems.

Remark 7.1. Note that t∗k − τk∗ is the time when the sensing data packet is sent from the sensor from which the kth effective forward control sequence is calculated, and the sum of the continuous data packet dropout and network-induced delay is upper bounded by τ¯d , see Fig. 7.2. Therefore the time when the (k + 1)th effective FCS arrives at the actuator is not later than t∗k − τk∗ + τ¯d , that is, t∗k+1 ≤ t∗k − τk∗ + τ¯d , ∀k ≥ 1 The definition of ik yields τk∗ ≤ ik ϑ < t∗k+1 − t∗k + τk∗ ≤ τ¯d , ∀k ≥ 1 Thus d

τk∗ τ¯d e ≤ ik < = N, ∀k ≥ 1 ϑ ϑ τ∗

where d ϑk e = min{ς|ς ∈ N, ς ≥

τk∗ }. ϑ

Noticing the structure of U (t∗k − τk∗ |t∗k − τk∗ ) it

is seen that the control action in (7.3) is always available from U (t∗k − τk∗ |t∗k − τk∗ ).


85

Remark 7.2. It is necessary to point out that there exists a situation where, for some k ≥ 1 and ik , the following relationship holds, t∗k ≤ t∗ik < t∗k+1 < t∗ik +1 By the control law in (7.3), in this situation the (k + 1)th effective FCS is not applied to the plant immediately but waits until t∗ik +1 , and during the time period [t∗k+1 , t∗ik +1 ) the kth effective FCS is still in action. It is seen that this strategy artificially increases the delay (less than ϑ) however it provides the advantage that it produces a constant switch interval between two subsequent switches of control actions. A constant switch interval undoubtedly simplifies the modeling and analysis, and what is more important, it avoids a situation where, the switch interval is too short which may affect the stability of the system according to switched system theory (Liberzon, 2003). Based on the above analysis, the algorithm of the packet-based control for NCSs in continuous time under Assumptions 3.7 and 3.8 can now be summarized as follows. Algorithm 7.3 (PB-control in continuous time). S1. As soon as the PB-controller receives a delayed sensing data packet containing the state information x(t − τsc (t)), it then,

S1a. Calculates the FCS as in (7.2); S1b. Packs U (t−τsc (t)|t−τsc (t)) into one data packet and sends it to the actuator.

S2. CAS updates its FCS once a data packet arrives; S3. The effective FCS is applied to the plant by the control law in (7.3). The block diagram of the continuous packet-based network control system is shown in Fig. 7.3.


86

Figure 7.3: Packet-based networked control systems in continuous time.

7.1.2

A novel model for NCSs

Under Algorithm 7.3, and assuming u(t) = 0, t ∈ [t∗0 , t∗1 ], t∗0 = t∗1 − τ¯d , a novel model for NCSs can now be obtained as ( x(t) ˙ = Ax(t) + Bu(t), t ∈ [t∗ik , t∗ik +1 ), k ≥ 1 ∗ Sc : u(t) = K(ik )x(t∗k − τk∗ ), k ≥ 1

(7.6a) (7.6b)

∗

with initial state evolving as x(t) = x(t∗0 )eA(t−t0 ) , φ(t), t ∈ [t∗0 , t∗1 ], where ik and K(ik ) are defined in (7.3) and (7.4) respectively. It is noticed that the derived model for NCSs in (7.6) is distinct from the previous models as in, e.g., Gao et al. (2007) in that the network-induced delay is considered more precisely and the effects of the data packet dropout and disorder are also included in the same model. As shown in (7.6b), the discretization of the networkinduced delay and the implementation of the packet-based control approach offer us the advantage of designing different control actions as in (7.3) for different network conditions. It is obvious that this advantage results in at least the same system performance as previous approaches (by designing the same control action for all network conditions), whereas a better performance is expected since more freedom is given to designers. Remark 7.4. If Assumption 3.2 holds, then the network-induced delay in the backward channel can be known to the controller by using time stamps as done in Section 3.1. Thus a different FCS compared with (7.2) can be used, which is defined by

U 0 (t|t − τsc (t)) = [u(t|t − τsc (t)) . . . u(t + (N − 1)ϑ|t − τsc (t))]


87

That is, the control signals from time t − τsc (t) to t − 1 which are obviously useless, are discarded from FCS. As a result, the network-induced delay (data packet dropout as well) in the backward channel will not affect the delay range that the PB-control approach can handle. One can see that, both cases, with or without the time synchronization, have very similar models (simply replace the round trip delay related parameters in the aforementioned model to the forward channel delay related ones in the presence of the time synchronization). Therefore without loss of generality we will focus only on the system model in (7.6) in the following stability and stabilization analysis.

7.2

Stability and stabilization

In this section, switched system theory is applied to system Sc∗ to derive a stability criterion. To this end, we first evaluate the growth of the following Lyapunov function candidate Vik (x(t)), t ∈ [t∗ik , t∗ik +1 ), k ≥ 1 defined by Vik (x(t)) = xT (t)Pik x(t) Z t + xT (s)eα(s−t) Rik x(s)ds t−ik ϑ Z t 0

Z +

x˙ T (s)eα(s−t) Q1ik x(s)dsdθ ˙

−ik ϑ t+θ Z t −ik ϑ

Z +

−(ik +1)ϑ 0

Z

t+θ t

Z

+ −(ik +1)ϑ

t+θ

x˙ T (s)eα(s−t) Q2ik x(s)dsdθ ˙ x˙ T (s)eα(s−t) Q3ik x(s)dsdθ. ˙

Note that ik = 0 is a special case where R0 = 0, Q10 = 0, Q20 = 0. For simplicity this case will not be specially addressed in the following analysis. Lemma 7.5. For a given constant α > 0 and given feedback gain matrices K(ik ), if the following LMI-based problems are feasible,

Pik (α) :

 i i    ∃Pik > 0, Rik > 0, Qik > 0, Nik , i = 1, 2, 3, s.t.    Ξ (α) < 0. ik


88

where Ξ0ik (α) Ξ5ik

Ξik (α) =

! ,

Ξ6ik

∗

(7.7)

Ξ1ik (α) + Ξ2ik + (Ξ2ik )T

Ξ0ik (α) =

∗ 

 Ξ1ik (α) =  

Ξ11 ik (α)

0

∗

Ξ22 ik (α)

∗

∗

Ξ13 ik

!

Ξ3ik

,

Ξ4ik (α) 

 0  , 0

T Ξ11 ik (α) = Pik A + A Pik + αPik + Rik ,

Ξ13 ik = Pik BK(ik ), −αik ϑ Ξ22 Rik , ik (α) = −e

Ξ2ik = [Ni1k + Ni3k

− Ni1k + Ni2k

− Ni2k − Ni3k ],

Ξ3ik = [Ni1k Ni2k Ni3k ],

Ξ4ik (α) = −diag{(ik ϑ)−1 e−αik ϑ Q1ik , ϑ−1 e−α(ik +1)ϑ Q2ik , ((ik + 1)ϑ)−1 e−α(ik +1)ϑ Q3ik }, ¯ i BK(ik ) 0 0 0]T , ¯i A 0 Q Ξ5ik = [Q k k ¯i , Ξ6ik = −Q k ¯ i = ik ϑQ1 + ϑQ2 + (ik + 1)ϑQ3 , Q ik ik ik k then along the trajectory of the system in (7.6), the following inequality holds ∗

Vik (x(t)) ≤ e−α(t−tik ) Vik (x(t∗ik )), t ∈ [t∗ik , t∗ik +1 ), k ≥ 1

(7.8)

Proof. Note that for any Niik , i = 1, 2, 3, with appropriate dimensions we have Γ1ik

Γ2ik

=ξ

=ξ

T

T

(t)Ni1k [x(t)

(t)Ni2k [x(t

Z

t

− x(t − ik ϑ) −

− ik ϑ) −

x(s)ds] ˙ =0

(7.9a)

t−ik ϑ

x(t∗k

−

τk∗ )

Z

t−ik ϑ

−

x(s)ds] ˙ =0 t∗k −τk∗

(7.9b)


Γ3ik

=ξ

T

(t)Ni3k [x(t)

−

x(t∗k

−

τk∗ )

89

Z

t

−

x(s)ds] ˙ =0

(7.9c)

t∗k −τk∗

where ξ(t) = [xT (t), xT (t − ik ϑ), xT (t∗k − τk∗ )]T . Using (7.9) and noticing t − (ik + 1)ϑ < t∗k − τk∗ ≤ t − ik ϑ for t ∈ [t∗ik , t∗ik +1 ) we then obtain

V˙ ik (x(t)) + αVik (x(t)) =2xT (t)Pik x(t) ˙ + xT (t)(αPik + Rik )x(t) − xT (t − ik ϑ)e−αik ϑ Rik x(t − ik ϑ) Z t 3 2 T 1 ˙ ˙ − x˙ T (s)eα(s−t) Q1ik x(s)ds + x˙ (t)(ik ϑQik + ϑQik + (ik + 1)ϑQik )x(t) t−ik ϑ

Z

t−ik ϑ

− t−(ik +1)ϑ

x˙ T (s)eα(s−t) Q2ik x(s)ds ˙ −

t

Z

t−(ik +1)ϑ

x˙ T (s)eα(s−t) Q3ik x(s)ds ˙

≤2x (t)Pik x(t) ˙ + x (t)(αPik + Rik )x(t) − xT (t − ik ϑ)e−αik ϑ Rik x(t − ik ϑ) Z t T 1 2 3 + x˙ (t)(ik ϑQik + ϑQik + (ik + 1)ϑQik )x(t) ˙ − x˙ T (s)e−αik ϑ Q1ik x(s)ds ˙ T

T

t−ik ϑ

Z

t−ik ϑ

− t∗k −τk∗

x˙ T (s)e−α(ik +1)ϑ Q2ik x(s)ds ˙ −

Z

t

t∗k −τk∗

x˙ T (s)e−α(ik +1)ϑ Q3ik x(s)ds ˙

+ 2Γ1ik + 2Γ2ik + 2Γ3ik =ξ

T

(t)(Ξ1ik

+

Ξ2ik

+

(Ξ2ik )T

+

Ξ7ik

+

Ξ8ik )ξ(t)

−

11 X

Ξiik

i=9

where 

¯i A 0 AT Q k

 Ξ7ik =  

∗ ∗

¯ i BK(ik ) AT Q k



 , 0  T ¯ ∗ (BK(ik )) Qik BK(ik ) 0

Ξ8ik =ik ϑNi1k eαik ϑ (Q1ik )−1 Ni1k + ϑNi2k eα(ik +1)ϑ (Q2ik )−1 Ni2k + (ik + 1)ϑNi3k eα(ik +1)ϑ (Q3ik )−1 Ni3k , Ξ9ik

Z

t

= t−ik ϑ

T αik ϑ 91 1 T −αik ϑ 1 ˙ (Ξ91 (Q1ik )−1 Ξ91 Qik x(s), ik ) e ik , Ξik = (Nik ) ξ(t) + e

Chapter 7. Continuous-Time PB-NCSs Ξ10 ik Ξ11 ik

t−ik ϑ

Z =

t∗k −τk∗

Z

90

T α(ik +1)ϑ 101 2 T −α(ik +1)ϑ 2 (Q2ik )−1 Ξ101 Qik x(s), ˙ (Ξ101 ik ) e ik , Ξik = (Nik ) ξ(t) + e

t

= t∗k −τk∗

T α(ik +1)ϑ 111 3 T −α(ik +1)ϑ 3 (Ξ111 (Q3ik )−1 Ξ111 Qik x(s). ˙ ik ) e ik , Ξik = (Nik ) ξ(t) + e

Notice that Qiik > 0, i = 1, 2, 3 implies Ξ4ik < 0, Ξ6ik < 0 and Ξiik ≥ 0, i = 9, 10, 11. Then by Schur complements, Ξik (α) < 0 guarantees Ξ1ik + Ξ2ik + (Ξ2ik )T + Ξ7ik Ξ3ik ∗

Ξ4ik

! < 0,

which furthermore guarantees Ξ1ik + Ξ2ik + (Ξ2ik )T + Ξ7ik + Ξ8ik < 0. Thus we obtain V˙ ik (x(t)) + αVik (x(t)) ≤ 0, t ∈ [t∗ik , t∗ik +1 ), ∀k ≥ 1

Integrating this inequality then completes the proof. Using Lemma 7.5, we then obtain the following stability criterion for the system in (7.6), based on the average dwell time analysis (Sun et al., 2006). Theorem 7.6. Suppose for the system in (7.6) the following inequality holds ϑ > ϑ∗

(7.10)

where ϑ∗ = inf { lnαµα } with µα = inf{µ|µ ≥ 1, Pik ≤ µPjk , Rik ≤ µRjk , Qiik ≤ α∈Ω

µQijk , i = 1, 2, 3, ∀ik , jk ∈ M}, Ω = {α|α > 0, Pik (α) feasible, ∀ik ∈ M} and M = {0, 1, 2, . . . , N − 1}. Then the system in (7.6) is exponentially stable. Proof. For any given α ∈ Ω, define for the system in (7.6) the following piecewise Lyapunov functional V (x(t)) = Vik (x(t)), t ∈ [t∗ik , t∗ik +1 ), k ≥ 1

(7.11)

From Lemma 7.5 the following inequality holds for t ∈ [t∗ik , t∗ik +1 ), k ≥ 1, ∗

∗

V (x(t)) = Vik (x(t)) ≤ e−α(t−tik ) Vik (x(t∗ik )) = e−α(t−tik ) V (x(t∗ik ))


91

The definition of µα implies that Vik (x(t∗ik )) ≤ µα Vik −1 (x(t∗− ik )), ∀k ≥ 1, if ik ≥ d

τk∗ e + 1, ϑ

or Vik (x(t∗ik ))

≤

µα V¯ik−1 (x(t∗− ik )), ∀k

τk∗ ≥ 2, if ik = d e, ϑ

where ¯ik−1 = max{ik−1 |ik−1 satisfying (7.3)}. Thus by iteration we obtain ∗

V (x(t)) ≤ e−α(t−t1 ) µIαk −1 V1 (x(t∗1 )), t ∈ [t¯∗ik , t¯∗ik +1 ) where V1 (x(t∗1 )) is defined over [t∗1 , t∗1 + ϑ) and Ik =

PP k

is readily seen that Ik − 1 = b V (x(t)) ≤ e−(α−

(t−t∗1 ) c, ϑ

ln µα )(t−t∗1 ) ϑ

1. From Remark 7.2, it

ik

and thus

V1 (x(t∗1 )), t ∈ [t¯∗ik , t¯∗ik +1 )

The definition of ϑ∗ implies that ∀ε > 0, ∃αε ∈ Ω and correspondingly µαε such that ϑ∗ + ε >

ln µαε αε

Choosing a sufficiently small ε = ε0 such that ϑ > ϑ∗ + ε0 yields ϑ > ϑ∗ + ε0 >

ln µαε0 αε0

which implies αε0 −

ln µαε0 >0 ϑ

Correspondingly, we obtain V (x(t)) ≤ e−(αε0 −

ln µαε 0 ϑ

)(t−t∗1 )

V1 (x(t∗1 )), t ∈ [t¯∗ik , t¯∗ik +1 )

which completes the proof following the same procedure as in, e.g., Sun et al. (2006).


92

The following proposition solves the synthesis problem of the PB-control approach to NCSs based on Theorem 7.6. Proposition 7.7. Suppose for the system in (7.6) the following inequality holds, ϑ > ϑ∗

0

(7.12)

0

where ϑ∗ = inf 0 { β∈Ω

ln µ0β } β

with µ0β = inf{µ0 |µ0 ≥ 1, Lik ≤ µ0 Ljk , Zik ≤ µ0 Zjk , Yiik ≤

µ0 Yjik , i = 1, 2, 3, ∀ik , jk ∈ M}, Ω0 = {β|β > 0, Lik (β) feasible, ∀ik ∈ M}, and

Lik (β) :

 i i    ∃Lik > 0, Zik > 0, Yik > 0, Mik , i = 1, 2, 3, Vik , s.t.    Π (β) < 0. ik

where Π0ik (β) Π5ik

Πik (β) =

∗

! ,

Π6ik

(7.13)

Π1ik (β) + Π2ik + (Π2ik )T

Π0ik (β) =

∗ 

 Π1ik (β) =  

Π11 ik (β)

0

∗

Π22 ik (β)

∗

∗

Π13 ik

Π3ik Π4ik (β) 

! ,

 0  , 0

T Π11 ik (β) = ALik + Lik A + βLik + Zik ,

Π13 ik = BVik , −βik ϑ Π22 Zik , ik (β) = −e

Π2ik = [Mi1k + Mi3k

− Mi1k + Mi2k

− Mi2k − Mi3k ],

Π3ik = [Mi1k Mi2k Mi3k ],

Π4ik (β) = −diag{(ik ϑ)−1 e−βik ϑ Yi1k , ϑ−1 e−β(ik +1)ϑ Yi2k , ((ik + 1)ϑ)−1 e−β(ik +1)ϑ Yi3k }, Π5ik = [ALik 0 BVik 0 0 0]T , Π6ik = −Lik (Y¯ik )−1 Lik ,


93

Y¯ik = ik ϑYi1k + ϑYi2k + (ik + 1)ϑYi3k . Then, the system in (7.6) is exponentially stabilizable by the control law K(ik ) = Vik L−1 ik , ik ∈ M. ¯ −1 } to (7.7) , Pi−1 , Pi−1 , Pi−1 , Pi−1 , Pi−1 ,Q Proof. Pre- and post-multiply diag{Pi−1 ik k k k k k k ik −1 ¯ ¯ and let Lik = Pik , Yik = Lik Qik Lik , Zik = Lik Rik Lik , Mi = Lik Niik Lik , Yiik = Lik Qiik Lik , i = 1, 2, 3, and Vik = K(ik )Lik . Then we complete the proof by using Theorem 7.6. It is noticed that the feasibility problem of Lik (β) is no longer LMI conditions because of the term Π6ik . There are several techniques available to deal with this difficulty, among which the cone complementarity technique is one of the most commonly used (Ghaoui et al., 1997; Moon et al., 2001). In the following theorem, this technique is used to derive a suboptimal solution for Lik (β) by transforming the feasibility problem of Lik (β) to a nonlinear minimization problem involving LMI conditions. Theorem 7.8. Suppose (7.12) holds for the system in (7.6), where the feasibility problem of Lik (β) is redefined to the following nonlinear minimization problem involving LMI conditions,

Li0k (β) :

where Ψ1ik = Lik

I

 ¯    Minimize Tr(Sik Tik + Lik Jik + Yik Uik )

Subject to Lik > 0, Zik > 0, Yiik > 0, i = 1, 2, 3,    Ψ1 ≤ 0, Ψ2 ≥ 0, Ψ3 ≥ 0, Ψ4 ≥ 0, Ψ5 ≥ 0. ik ik ik ik ik Π0ik (β)

Π5ik

∗

−Sik Y¯i

! , Ψ5ik =

k

! , Ψ2ik = I

Tik Jik ∗

Uik

! , Ψ3ik =

Sik

I

∗

Tik

! , Ψ4ik =

! . If the solution of Li0k (β) = 6n, ∀ik ∈ M,

∗ Jik ∗ Uik then the system in (7.6) is exponentially stabilizable by the control law defined in Proposition 7.7. Proof. Applying the cone complementarity technique proposed in Moon et al. (2001) to (7.13) in Proposition 7.7 then we complete the proof.


7.3

94

A numerical example

In this section, a numerical example is considered to illustrate the effectiveness of the proposed approach in this chapter. Example 7.1. Consider the system in (7.1) with the following system matrices borrowed from Xiong and Lam (2007), 

−1

 A=  1

0

0 −0.5 0

−0.5





0



   ,B =  0 .    0.5 1 0

When the plant is sampled with h = 0.1s, it yields the following discretized system 

0.9048

0

−0.04881





−0.00246



     x(k) +  −8.13 × e−5  u(k). x(k + 1) =  0.09278 0.9512 −0.002419     0.1025 0 0 1.051

Let d = 3, τ¯ = 0.6s, and thus τ¯d = 0.9s. Assume N = 3 which means one data packet of the network can contain three steps of control signals. Applying Theorem 7.8 we then obtain the following PB-controllers with respect to different network conditions, K(0) =

0.0200 0.0004 −1.3267

K(1) =

−0.0004 −0.0001 −1.0088

,

K(2) =

−0.0002 −0.0001 −1.0098

.

,

Notice here that the continuous network-induced delay (data packet dropout as well) is discretized into three levels by ϑ = 0.3s, corresponding to the above three different PB-controllers. That is, for different delays, different controllers apply. The system response and the network-induced delay (data packet dropout as well) is shown in Fig. 7.4 with the initial state x(t) = [−5 0 5]T , t ∈ [−0.9 0), which illustrates the effectiveness of the PB-control approach for NCSs in the presence


95

of network-induced delay, data packet dropout and data packet disorder simultaneously. This can be compared with the example in Xiong and Lam (2007) where only data packet dropout and an unit step delay is considered.

5 x1

4

x2

3

x3

System states

2 1 0 −1 −2 −3 −4 −5

0

2

4

6

8

10 Time(s)

12

14

16

18

20

0.9 Discretized delay & dropout Continuous delay & dropout

0.8

Delay & dropout

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10 Time(s)

12

14

16

18

20

Figure 7.4: Example 7.1. State response and communication constraints. The discretized delay & dropout is obtained by ik ϑ.

7.4

Conclusion

By applying the discretization technique to the continuous network-induced delay, the PB-control approach was extended to the continuous time case in this chapter, from which a novel model for NCSs was derived. The proposed approach and the derived model can deal with network-induced delay, data packet dropout and data


96

packet disorder simultaneously as in the discrete time case in Chapter 3 and offer the designer the freedom of designing different controllers for different network conditions. The stability criterion was obtained using switched system theory and the stabilization problem was also solved, the effectiveness of which was illustrated by a numerical example.

Chapter 8 PB-Control and Scheduling Co-Design for NCSs Different from the previous chapters where only one NCS occupies the network resource, in this chapter, the design and analysis of a situation where a set of linear NCSs share the limited network resources to transmit their control signals, is considered, See Fig. 8.1 for the general configuration 1 , where each NCS is regarded to be a subsystem to the whole system. In this situation, the PB-control approach is still applied to each subsystem and two scheduling algorithms, the existing static Rate Monotonic (RM) algorithm and a newly proposed Dynamic Feedback Scheduling (DFS) algorithm, are considered to schedule the network resource allocations among those subsystems, to achieve the target that all the subsystems are stable under the limited network resources. This chapter is organized as follows. The problem being studied is first described in Section 8.1, and then the PB-controller for each subsystem is obtained in Section 8.2, from which an important definition for the subsystems is derived which is the supremum of round trip delay under which the stability of the subsystems is guaranteed. With this definition, two scheduling algorithms are presented in Section 8.3, and a numerical example is presented in Section 8.4. Section 8.5 concludes the chapter.

1

Note that there are also random delays in the backward channel which is not shown in Fig.

8.1.

97

Chapter 8. PB-Control and Scheduling Co-Design

8.1

98

Problem statement

In this chapter, a set of N continuous-time LTI systems (Sic )1≤i≤N are considered which share the network resource as shown in Fig. 8.1, ( Sci

:

x˙ ci (t) = Aci xci (t) + Bic uci (t)

(8.1a)

yic (t) = Cic xci (t)

(8.1b)

where xci (t) ∈ Rni , uci (t) ∈ Rmi , and yic (t) ∈ Rri . System N

System 1

Plant

Plant

Controller

Controller

Shared Netw ork Figure 8.1: Multiple networked control systems share the communication channel.

In a digital control environment, a discrete-time representation Sdi of system Sci is obtained using a sampling period Ti , ( Sdi :

xi (k + 1) = Ai xi (k) + Bi ui (k)

(8.2a)

yi (k) = Ci xi (k)

(8.2b) c

where xi (k) = xci (kTi ), ui (k) = uci (kTi ), yi (k) = yic (kTi ), Ai = eAi Ti , and Bi = R Ti Ac s e i dsBic . 0 Suppose that the backward channel delays of all the subsystems are random but bounded and the transmissions from the controllers to the actuators share a communication network with limited resource. The communication resource is limited in the sense that, at each time instant, only one controller can access the network for transmission. Therefore the forward channel delay for each subsystem depends on not only the time during which the data is transmitted over the network but the time taken for waiting for the permission of network access which is determined by the used scheduling algorithm, see Fig. 8.1.


99

Thus the problem here is not only to design a controller for each subsystem Sdi but also to design the scheduling scheme for the network resource allocations for all the subsystems (Sdi )1≤i≤N , in an environment of network-induced delay, data packet dropout and data packet disorder. To this end, a co-design approach is proposed with the integration of the PB-control approach and the scheduling algorithm. In the following section, the PB-controller for each subsystem is first determined, and two different scheduling algorithms, the existing static RM algorithm and a novel DFS algorithm, are then adopted to schedule the transmissions of FCSs, with the guarantee of the stability of all the subsystems.

8.2

PB-control for subsystems

For each subsystem (Sdi )1≤i≤N , exactly the same PB-control approach proposed in Section 3.1 is applied, with a similar objective function as defined in (4.6), i

i Jk,τ sc,k

=

N2 X

i

i

2

qj (ˆ y (k + j|k − τsc,k ) − ω(k + j)) +

j=N1i

Nu X

rj ∆u2 (k + j − 1) (8.3)

j=1

where the definitions of the parameters are referred to (4.6). Following the same procedure as in Section 4.2.1, FCIS for system Sdi is then obtained as ∆U (k|k − τsc,k ) = Mτsc,k ($k − Eτsc,k x¯(k − τsc,k )) where ∆U (k|k − τsc,k ), Mτsc,k , $k and Eτsc,k can be similarly defined as in Section 4.2.1. With this FCIS, the following FCS from k to k + Nu − 1 is readily obtained as U (k|k − τsc,k ) = Gu(k − τsc,k − 1) + H∆U (k|k − τsc,k )

(8.4)


100

where G = [Im · · · Im ]TmNu ×m , Im is the identity matrix with rank m and     H=  

···

0



Im · · · Im Im · · · .. . .. . . . . · · · .. .

0 .. .

     

Im · · · Im

0

Im · · · Im Im · · · Im

. mNu ×m(Nu +τsc,k )

Thus, for each subsystem with the PB-control approach and the aforementioned FCS, following the same procedure as in Section 3.2.1, a stability theory similar to Theorem 3.10 can then be obtained using switched system theory. We now explore a little further on Theorem 3.10. As a matter of fact, Theorem 3.10 implies that system Sdi with the PB-control approach and the aforementioned FCS in (8.4), is stable under certain conditions if the round trip delay is less than a fixed value. In other words, given a linear system, the least upper bound, or the supremum of the round trip delay, under which the system is stable can be found from Theorem 3.10. We call this supremum of round trip delay that guarantees the stability of the system the “Stability-guaranteed Supremum of Round Trip Delay (SSRTD)”, which is an inherent characteristic of a given system. The techniques such as the LMI tool-box, are useful to find the SSRTD for a given system. It is also necessary to point out that if other performance constraints besides stability are considered, a smaller supremum of round trip delay than SSRTD is needed. For ˆ i > 0, 1 ≤ i ≤ N . convenience, denote the SSRTDs of the systems (S i )1≤i≤N by D d

Remark 8.1. Note that the notion of SSRTD here is similar to MADB (see Section 2.2.1) which has been used in a number of publications, see Kim et al. (2003) for an overview. We prefer SSRTD to MADB in this thesis since the former can better express the particular requirement of the round trip delay in Theorem 3.10 for the stability of the system.

8.3

Scheduling

In this section, scheduling theory is applied to allocate the limited network resources for the transmission tasks of the PB-NCSs that are derived from the subsystems (Sdi )1≤i≤N . The static, priority-based scheduling algorithm RM is applied to the set of subsystems under a private network environment first, and then the


101

dynamic, feedback-based scheduling algorithm DFS is presented to extend the application to the public network. When a scheduling algorithm is applied to schedule the transmission tasks of a set of PB-NCSs, stability of the subsystems have to be guaranteed as a precondition. To ensure this, we define “Stable Schedulability” as follows. Definition 8.2 (Stable Schedulability). A set of PB-NCSs sharing the network resources is said to be stable schedulable by a scheduling algorithm if the transmissions of all the subsystems can be scheduled so that all the subsystems are stable.

8.3.1

Static scheduling

In the static scheduling case, the network is assumed to be used only by the subsystems (Sdi )1≤i≤N , i.e. is private to the set of subsystems. In the analysis, the transmissions of the FCSs for the subsystem (Sdi )1≤i≤N are regarded as real-time tasks in scheduling theory, which are defined by analyzing the formation of the network-induced delay. The RM algorithm is then adopted over these transmission tasks and the feasibility theorem is obtained as well.

A. The transmission tasks of the PB-NCSs As shown in Lian et al. (2001), the forward channel delay τca is mainly composed of the following three parts.

1. The propagation delay, which is the time from when a packet is put onto the network till it successfully arrives at its destination. Since the network is private to the subsystems (Sdi )1≤i≤N , the propagation delay depends merely on the speed of signal transmission and the distance between the source and the destination, which are assumed to be fixed for all the subsystems. 0 Therefore this delay is assumed to be known as a constant τca in the static

scheduling case. 2. The frame time delay, which is the time for the source to place a packet on the network. Suppose that the size of the packet which contains the FCS is Bc Nu , where Bc is data size required for encoding a single step control signal


102

as defined in Assumption 3.3, which can be assumed to be the same for all the subsystems. The frame time delay is then obtained as Bc Nui ei = BN

(8.5)

where BN is the bandwidth of the network, Nui is the control horizon of subsystem Sdi . It is natural to assume that all the subsystems use the same control horizon Nu , since the selection of Nu mainly depends on the round trip delay of the network and all the subsystems endure similar network-induced delays and data packet dropouts by sharing the network. Hence, ei = e =

Bc Nu , i = 1, 2, ..., N BN

(8.6)

The frame network-induced delay e serves as the execution time in the transmission tasks of the PB-NCSs. 3. The waiting network-induced delay, is defined as the time a FCS has to wait for queuing and network availability before actually being sent. From earlier ˆ i , therefore, to ensure the stability discussion, the SSRTD for system S i is D d

of all the subsystems, the waiting delay for each system should not be larger than 0 ˆ i − τca Di = D − τ¯sc − e

(8.7)

Note that the upper bound of the backward channel delay τ¯sc is used since the RM algorithm assigns the priority of each task statically and the stability of the subsystems needs to be guaranteed under the worst case. It is only the stability of the system that we care about in the RM algorithm in this chapter, and therefore the transmission period of subsystem Sdi needs to be no longer than Di . As a result, the transmission period hi of subsystem Sdi is assumed to be equal to Di in the static RM scheduling algorithm, i.e. hi = Di , i = 1, 2, ..., N

(8.8)


103

hi is chosen by (8.8) so that the FCS of subsystem Sdi is sent every hi seconds no matter what the sampling period is or how fast the controller can generate FCSs.

Thus from the analysis of τca , the transmission tasks of the set of PB-NCSs can now be described as follows: All the tasks have the same execution time e, the deadline of each task equals its period hi , and the first release time ϑi of task i is the time when subsystem Sdi first operates. We denote the tasks by Ti = T (ϑi , e, hi ), i = 1, 2, · · · , N

(8.9)

B. Scheduling of PB-NCSs by RM RM is a widely used scheduling algorithm, where tasks with shorter periods have higher priorities. It is a fixed-priority assignment: priorities are assigned to tasks before execution and do not change over time. Liu and Layland (1973) have shown that RM is superior to other fixed-priority assignments in the sense that no other fixed-priority algorithm can schedule a task set that cannot be scheduled by RM . Consider the set of real-time transmission tasks Ti , 1 ≤ i ≤ N defined in (8.9). These tasks are periodic, independent, non-preemptive, and the period of each task equals its deadline. These characteristics are just what the operation of the RM algorithm needs. Therefore, the RM scheduling algorithm can be applied to schedule the set of transmission tasks in PB-NCSs, by which the transmission with shorter deadlines (or periods) are assigned higher priorities, and thus the corresponding FCSs can be transmitted first if the network is idle, i.e. if hi < hj , thenΥi > Υj , i, j = 1, 2, · · · , N

(8.10)

where Υi represents the priority of the transmission task of subsystem Sdi . Theorem 8.3. A set of N PB-NCSs sharing the network resource in their forward channel (indexed by the increasing order of their transmission periods, i.e. hi ≤ hi+1 , i = 1, 2, ...N − 1) are stable schedulable if for all i = 1, ..., N U (i) ≤ f (i)

(8.11)


104

where f (i) = i(21/i − 1) and

U (i) =

 i X  1 1   e( + )    h hi j=1 j

i = 1, 2, ..., N − 1

(8.12a)

i  X  1   e( )   h j j=1

i=N

(8.12b)

Proof. From Theorem 16 in Sha et al. (1990), a set of nonpreemptive periodic real-time tasks are schedulable if e2 ei ¯bl,i e1 + + ··· + + ≤ i(21/i − 1) h1 h2 hi hi

(8.13)

where ei is the frame time, hi is the transmission period, each for the ith task, and ¯bl,i is task i’s worst-case blocking time by the lower priority tasks, i.e., ¯bl,i =

max

j=i+1,...,N

ej .

(8.14)

As has been pointed out earlier, for the transmission tasks Ti , 1 ≤ i ≤ N , (8.6) holds, and therefore ¯bl,i = e, i = 1, 2, ..., N − 1 and ¯bl,N = 0 from (8.14). Hence the theorem holds. 1

Corollary 8.4. If 1) hi+1 ≤ 2hi , i = 1, 2, ...N − 2, and 2)

hN hN −1

≤

N −1 2 N −1 −1 , 1 N 2 N −1

then the set of tasks of PB-NCSs is stable schedulable if N X 1 e ≤ N (21/N − 1) h i=1 i

(8.15)

Proof. From 1) we obtain for i = 1, 2, ..., N − 2 that U (i + 1) − U (i) = e(

2 hi+1

−

1 )≥0 hi

It is obvious that U (N ) ≥ U (N − 1) from 2). Thus we obtain U (N ) = max U (i) 1≤i≤N


105

F e e d b a c k S c h e d u le r

S y s tem 1 P

S y s tem n P

C

C

S h ar e d N etw o r k

Figure 8.2: Dynamic feedback scheduling of multiple systems.

On the other hand, it is easy to show that function f (·) is nonincreasing, and therefore U (N ) ≤ f (N ) implies U (i) ≤ f (i), i = 1, 2, ..., N − 1, which completes the proof by Theorem 8.3. Corollary 8.5. If the transmission periods of all the subsystems are the same, i.e., hi = h, i = 1, 2, ..., N , then (Sid )1≤i≤N are stable schedulable if 1 e ≤ 2N − 1 h

(8.16)

Proof. It can be obtained directly from Corollary 8.4.

8.3.2

Dynamic feedback scheduling

In the static RM scheduling scheme presented earlier, the transmission periods hi for all the subsystems are assigned a priori to ensure the stability of the subsystems and do not change any more. In the case of the network being shared only by (Sdi )1≤i≤N , this method works though the performance of the subsystems may not be optimum because the network is not fully used. However, if the network is not private to these subsystems, i.e., there are other components occupying the network, it can not be assumed that the propagation delay is constant due to the change of the network loads. Based on this reality, DFS scheme is designed. In this scheme, a higher level feedback scheduler is proposed, which gets the information of the network utilization from the network and the control performances from all the


106

subsystems as well, and then regularly calculates and reassigns the transmission period for each subsystem. During the interval of two successive reassignments of periods, the RM algorithm still works. The framework of the DFS scheme is depicted in Fig. 8.2. In order to implement DFS, such issues as the selection of the period of DFS, the measurement of the network utilization and the reassignment of the transmission periods of the PB-NCSs, need to be dealt with first.

A. The period of DFS This period, noted by TDF S , has to be chosen carefully. Generally, its value depends on the speed at which the condition of the network changes. A small TDF S is needed if the network condition changes rapidly, while a larger one can still guarantee the performance of the system without overloading the network if the parameters of the network do not change much over a long time. However, in any case, TDF S should be always not less than the transmission periods of all the subsystems, i.e. TDF S ≥ maxN i=1 hi .

B. The measurement of the network utilization To obtain the utilization information of the network, a packet containing this information is sent to the feedback scheduler using the period of TDF S . This information is mainly reflected by the propagation delays of the subsystems. The propagation time will increase to a certain extent with the increase of the network load. Another factor affecting the stability of the subsystems is the change of the backward channel delay. In order to take this factor into account and for simplicity, we assume that the upper bound of the backward channel delay during i the kth period of DFS (denoted by τ¯sc (k) for subsystem Sdi ) can be obtained from

the network and the network does not change too much during this period thus i enabling us to use τ¯sc (k) to estimate its value during the (k + 1)th period. Then

the deadline hi of the task Ti will be recalculated by updating the propagation o time τca and the upper bound of the backward channel delay every TDF S seconds

as follows ˆ i − µ(τ 0 (k) + τ¯i (k)) − e Di (k + 1) = D sc ca

(8.17)


107

where µ close to 1 is a smoothing factor satisfying 0 i 0 i µ(τca (k) + τ¯sc (k)) ≥ τca (k + 1) + τ¯sc (k + 1), ∀k

C. The reassignment of the transmission periods for all subsystems In order to obtain the control performance of the subsystems, an obvious idea is to use the predictive Quality of Performance (QoP) during the next DFS period. This QoP during the kth period of DFS can be defined for subsystem Sdi as d(k+1)TDF S /hi e

X

Pî (k) =

(ˆ yi (j|j − τsc,j ) − ωi (j))2

(8.18)

j=dkTDF S /hi e

The calculation of the new transmission periods for all the subsystems can then be modeled as an optimization problem P as follows:

P:

                  

Select hi , i = 1, 2, ...N , s.t. N P min Pî , hi

i=1

subject to U (i) ≤ f (i), i = 1, 2, ..., N , hi ≥ Ti , i = 1, 2, ..., N .

where U (i) and f (i) are defined in Theorem 8.3 and Ti is the sampling period for system Sdi defined in (8.2). In practice, the predictive outputs yˆ(j|j − τsc,j ) of the subsystems can be obtained using the open-loop prediction, whereas the online operation of the optimization problem P is not a simple one. Therefore, not the predictive QoP but the previous QoP, J¯i (k), i for subsystem S i and k for the kth period of DFS, is used to d

represent the performance of the system, which is defined as follows and can be easily obtained, i J¯k,τ = sc,k

X i Jk,τ ∈Πk sc,k

i Jk,τ sc,k

(8.19)


108

i is the objective function of subsystem Sdi defined in (8.3), Πk is the set where Jk,τ sc,k

of objective functions during the kth period of DFS, or from the d(k−1)TDF S /hi eth transmission period of subsystem Sdi to the dkTDF S /hi eth. Let the new transmission periods chosen in this way be i J¯k,τ 1 = κ(k + 1) PN sc,k ¯j hi (k + 1) j=1 Jk,τ

= κ(k + 1)θi (k)

(8.20)

sc,k

where κ is a proportion factor and can be chosen as follows to include the constraints of stable schedulability in Theorem 8.3, κ(k + 1) =

f (i) f (N ) } max { Pi , i=1,...,N −1 e( e θ (k) + θ (k)) j i j=1

Considering the fact that the network load may change greatly between two periods of DFS, a smoothing factor ρ(0 < ρ ≤ 1) is introduced to avoid network overload. Also taking account of the fact that the transmission period hi can never exceed Di for the stability of the system, then the transmission periods are obtained as hi (k + 1) = min{

ρ , Di (k + 1)} κ(k + 1)θi (k)

(8.21)

The algorithm of DFS can then be summarized as follows. Algorithm 8.6 (Dynamic feedback scheduling). ˆ i − e, 1 ≤ i ≤ N ; S1. Initialize TDF S , k = 1, t = 0, hi = D S2. During time period t ∈ [(k − 1)TDF S kTDF S ], S2a. Calculate the FCSs Ui (k|k − τsc,k ), 1 ≤ i ≤ N using (8.4) for all the subsystems; S2b. Apply RM algorithm in Section 8.3.1 to determine the order of the transmission tasks of the PB-NCSs; S2c. Transmit the FCS as S2b. determines. S3. When t = kTDF S , the DFS module checks the utilization information of network. If the network is in full use, set k = k + 1, return to S2; else go to S4;


109

S4. The DFS module calculates the new transmission periods using (8.20) and reassigns the priorities for all the subsystems; set k = k + 1, return to S2.

D. Stability of DFS Theorem 8.7. (Sic )1≤i≤N with the PB-control approach are stable under DFS if the transmission tasks of the PB-NCSs are always stable schedulable. Proof. It is noticed that the use of DFS to the set of PB-NCSs does not change the backward channel delay, since the DFS module is at the controller side, while it does change the forward channel delay by reassigning the transmission period hi , i = 1, 2, · · · , N for the subsystems. However, from (8.17) and (8.21), we obtain

ˆ i −τ 0 (k +1)− τ¯i (k +1)−e, ∀k, i = 1, 2, · · · , N (8.22) hi (k +1) ≤ Di (k +1) ≤ D ca sc which implies, 0 i ˆ i , i = 1, 2, · · · , N sup{hi (k) + τca (k) + τ¯sc (k) + e} ≤ D

(8.23)

k

Note that the left side of (8.23) is the effective maximum of round trip delay for subsystem Sdi , which is always no more than SSRTD. Thus the theorem is valid by Theorem 8.3.

8.4

Numerical examples

Numerical examples are given in this section to illustrate the validity of the proposed co-design approach. Three second order linear subsystems in (Sic )1≤i≤N are considered in the examples, whose system matrices are as follows Ac1 =

−11.1572

−106.0132

−110.3637

−5.2680

! , Ac2 =

−23.7783

−48.9313

−107.2959 −34.0550

! ,


Ac3 =

B3c =

−91.6291 −160.9438 −69.3147 ! 9.1471 4.0444

! , B1c =

−35.6675

110

−0.1295 2.6890

!

4.3801

, B2c =

2.3278

! ,

, C1c = C2c = C3c = 1,

The sampling periods are set as T1 = 0.02s, T2 = 0.015s, T3 = 0.01s, respectively. The corresponding discrete-time subsystems (Sid )1≤i≤N can then be obtained with the following system matrices

A1 =

B1 =

0.8

0.12

0.11

0.9

0.02 0.05

! , A2 =

! , B2 =

0.08 0.06

0.7 0.48 0.2

0.6

! , B3 =

! , A3 =

0.08 0.1

0.4 0.2

!

0.5 0.7

,

! ,

C1 = C2 = C3 = 1. For the simplicity of simulation, assume for all the three subsystems that the set point ω = 0, weighting factors W1 = I, W2 = I, and the state vector can be obtained directly. Other parameters of the simulation are shown in Table 1. A Gaussian white noise with standard deviation 0.1 is also introduced as the disturbance of the state. Table 8.1: Simulation parameters

System 1 System 2 System 3 T 0.02 0.015 0.01 ϑ 0 0.05 0.06 x0 [−1 − 1]T [−1 − 1]T [−1 − 1]T P1 [1 30 20 0 I I] P2 [0.008 0.002 0.01 10] T is the sampling period. ϑ is the first release time. x0 = [x01 x02 ]T is the initial state. P1 = [N1 N2 Nu ω W1 W2 ] is the predictive parameters 0 P2 = [e τca τ¯sc Tsim ], Tsim is the simulation time.

î Example 8.1 (RM algorithm). Using the LMI toolbox in Matlab, the SSRTD D for the subsystem (Sid )1≤i≤N can be obtained by Theorem 3.10, thus enabling the


111

transmission periods hi to be calculated according to (8.7) and (8.8), as shown in Table 2. Table 8.2: SSRTD and Transmission periods

î D hi

System 1 System 2 System 3 5 7 8 4 5 6

Note that the execution time of each job is e = 0.008s. The value of the utilization function U (·) can then be obtained as U (i) = 0.2, 0.3133, 0.34, i = 1, 2, 3 while f (i) = 1, 0.8284, 0.7798, i = 1, 2, 3 respectively. It is readily seen that in this case (8.11) holds and by Theorem 8.3, the set of NCSs is stable schedulable under RM. The state evolution of the first state of the three subsystems under RM is shown in Fig. 8.3. Simulation of DFS algorithm 0.2

0

0

−0.2

−0.2

−0.4

−0.4

State evolution

State evolution

Simulation of RM algorithm 0.2

−0.6 −0.8 −1

−0.8 −1

System 1 System 2 System 3

−1.2 −1.4

−0.6

0

10

20

30 Time

40

50

System 1 System 2 System 3

−1.2

60

Figure 8.3: Example 8.1. State evolution using RM algorithm. Only the first state is illustrated.

−1.4

0

10

20

30 Time

40

50

60

Figure 8.4: Example 8.2. State evolution using DFS algorithm. Only the first state is illustrated.

Example 8.2 (DFS algorithm). It is noted that the SSRTD obtained in Theorem 3.10 is conservative. In the simulation of DFS, the deadlines of the three subsystems are set to be 8, 10 and 12 steps respectively, and the propagation delays of the subsystems in the forward channel are set to be randomly changing under the constraint that the real round trip delay are no more than the new SSRTD, in order to simulate the changes of the network loads. All the other parameters remain the same as in RM algorithm. The simulation result (Fig. 8.4) shows that the subsystems are still stable under this larger SSRTD and with fluctuating propagation delays.


8.5

112

Conclusion

Different from the previous chapters where only one NCS occupies the network resources, in this chapter a situation where multiple NCSs share the network resources to transmit the FCS was considered. The PB-control approach was still applied to the subsystems, and scheduling theory was also considered to schedule the network resources to guarantee the stability of all the subsystems. Two scheduling algorithms, the existing RM algorithm and a novel designed DFS algorithm were discussed, the validity of which were also illustrated by numerical examples.

Chapter 9 Conclusions and Future Work A PB-control approach to NCSs was proposed in this thesis, which takes advantage of the PB-transmission of the network being used in NCSs and as a result, it can efficiently deal with the communication constraints in NCSs including the network-induced delay, data packet dropout and data packet disorder simultaneously. Besides the basic application of the PB-control approach to linear NCSs, applications were also considered with respect to networked Hammerstein and Wiener systems, stochastic stabilization of the PB-control approach, continuoustime PB-control, and a co-design approach with scheduling theory. In the following sections, the main contributions of the thesis are summarized and suggestions for future work are also addressed.

9.1

Main contributions

In the thesis, we focus on both the design and the theoretical analysis of the PBcontrol approach in various implementations. The main contributions of the thesis are as follows.

1. The PB-control approach to NCSs, which provides a unified framework under which the communication constraints in NCSs including the network-induced delay, data packet dropout and data packet disorder can be efficiently dealt with simultaneously, which has not been achieved by previously reported results to date. It is worth mentioning that under the 113

Chapter 9. Conclusions and Future Work

114

PB-control framework, any conventional control methods are eligible to be applied to determine the forward control (increment) sequence provided it can result in a satisfactory system performance. This merit significantly increases the applicability of the PB-control approach. 2. Application of PB-control to networked Hammerstein systems, which effectively solves the control problem of a class of input nonlinear systems in the networked control environment, by using PB-control and a two-step approach to separate the nonlinear input process from the system. Two descriptions of the Hammerstein system are considered, in the input-output and state-space form respectively. For both descriptions, PBcontrollers are obtained and the stability criteria are also presented. 3. Application of PB-control to networked Wiener systems, which effectively solves the control problem of a class of output nonlinear systems in the networked control environment, by using PB-control and a similar twostep approach to separate the nonlinear output process from the system. The PB-controller in this situation is designed with a specially designed state observer and the stability criterion of the closed-loop system is obtained by using a separation principle. 4. Stochastic stabilization of PB-NCSs, which solves the synthesis problem of PB-NCSs in the stochastic fashion, under the PB-control and Markov jump system frameworks. This stochastic model takes account of networkinduced delay, data packet dropout and data packet disorder simultaneously which has not been achieved in previously reported results, and sufficient and necessary conditions for the closed-loop stability are obtained which are then used to solve the stochastic stabilization problem in this situation. 5. PB-control for continuous-time NCSs, which solves the application of PB-control to the continuous time case, and the corresponding synthesis problem. This is achieved mainly by the use of the discretization technique proposed for the continuous network-induced delay. Closed-loop stability is obtained by using delay-dependent analysis and stabilization problem is also solved. 6. PB-control and scheduling co-design, which solves the control problem where a set of subsystems share the network resources, by integrating PBcontrol and scheduling theory. Besides the analysis of the existing static

Chapter 9. Conclusions and Future Work

115

rate monotonic scheduling algorithm, a novel dynamic feedback scheduling algorithm is also designed, both of which are stability-guaranteed.

9.2

Future work

Based on the research work done in the thesis on the PB-control approach, the following research areas are worth investigating in the near future.

1. Nonlinear PB-NCSs. Though two classes of nonlinear systems have been addressed in Chapter 4 and 5, these nonlinear systems are too particular to represent general ones. Moreover, GPC-based controllers have been designed in Chapter 4 and 5 which, however, can not guarantee the stability of the closed-loop system a prior due to the constraints of GPC itself. Therefore, stabilization design of more general nonlinear systems is necessary. 2. Continuous-time PB-NCSs. It is observed in Chapter 7 that the theoretical analysis in the continuous time case is far more difficult than its counterpart in discrete time. Therefore, better stabilized controller design and stability criteria either from delay-dependent analysis as done in Chapter 7 or other techniques are important. 3. Stochastic PB-NCSs. The stochastic analysis in Chapter 6 is in discrete time. However, with the continuous-time model proposed in Chapter 7, further stochastic analysis in continuous time is necessary. 4. Co-design in PB-NCSs. In Chapter 8 we have addressed the co-design problem of PB-control and scheduling theory, by which the network access constraint is also considered besides the communication constraints of the network-induced delay, data packet dropout and data packet disorder. Then a natural question arises: How should we redesign PB-control when it encounters other communication constraints in NCSs as presented in Section 2.1?

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Published & Finished Papers Journal Papers: [1] Y.-B. Zhao, G. P. Liu, and D. Rees, 2008. Integrated predictive control and scheduling co-design for networked control systems. IET Control Theory Appl. 2 (1), 7-15 [2] Y.-B. Zhao, G. P. Liu, and D. Rees. Networked predictive control systems based on hammerstein model.

IEEE Trans. Circuits Syst. II-Express Briefs

55(5), 469-473 [3] Y.-B. Zhao, G. P. Liu, and D. Rees. A Predictive Control Based Approach to Networked Hammerstein Systems: Design and Stability Analysis. IEEE Trans. Syst. Man Cybern. Part B-Cybern. 38(3), 700-708 [4] Y.-B. Zhao, G. P. Liu, and D. Rees. A Predictive Control Based Approach to Networked Wiener Systems. Int. J. Innov. Comp. Inf. Control 4(10), to appear. [5] Y.-B. Zhao, G. P. Liu, and D. Rees. An Improved Predictive Control Approach to Networked Control Systems. IET Control Theory Appl.. Accepted. [6] Y.-B. Zhao, G. P. Liu, and D. Rees. A Packet-Based Control Approach to Networked Control Systems. Ready for submission. [7] Y.-B. Zhao, Y. Kang, G. P. Liu, and D. Rees. Packet-Based Networked Control Systems. Automatica. Submitted. [8] Y.-B. Zhao, G. P. Liu, and D. Rees. Modeling and Stabilization of ContinuousTime Packet-Based Networked Control Systems. Automatica. Submitted. Conference Papers:

126

Published & Finished Papers

127

[1] Y.-B. Zhao, G. P. Liu, and D. Rees. A predictive control based approach to networked control systems with input nonlinearity: Design and stability analysis. In Proc. 13th Int. Conf. Autom. Comp., pages 7–12, Stafford, UK, 15 Sept. 2007. [2] Y.-B. Zhao, G. P. Liu, and D. Rees. Time delay compensation and stability analysis of networked predictive control systems based on Hammerstein model. In Proc. 2007 IEEE Int. Conf. Networking, Sensing and Control, pages 808–811, London, UK, April 2007. [3] Y.-B. Zhao, G. P. Liu, and D. Rees. Design and Stability Analysis of PacketBased Networked Control Systems in Continuous Time. 2008 IEEE Int. Conf. Syst. Man Cybern.. Accepted.

Integrated predictive control and scheduling co-design for networked control systems Y.B. Zhao, G.P. Liu and D. Rees Abstract: A predictive control and scheduling co-design approach is proposed to deal with the controller and scheduler design for a set of networked control systems which are connected to a shared communication network. In the proposed approach, a predictive controller is applied to generate the control predictions for each system using delayed sensing data and previous control information, and a time delay compensator is designed at the actuator side to actively compensate for the network-induced delay in the forward channel when the control action is taken. Two different scheduling algorithms, the existing static rate monotonic (RM) scheduling algorithm and a new dynamic scheduling algorithm called dynamic feedback scheduling (DFS), are considered to schedule the transmissions of the control signals generated by the predictive controller, which are packed and transmitted to the actuator in one packet simultaneously. Both the scheduling algorithms are designed with the guarantee of the stability of all the systems, which is achieved by ensuring that the time delay of the systems do not exceed the upper bound under which the systems are stable. It is also pointed out that the RM algorithm is a special case of the proposed DFS algorithm, in the sense that the former can work only in a private network environment, whereas the latter extends its application to such networks where other components occupying the network. Simulations for both the RM and the DFS algorithms, illustrate the validity of the proposed approach.

1

Introduction

With the rapid development of control theory and communication technology, a new area of research activity called ‘networked control systems’ (NCSs) has received much attention in recent years. Although the notion of NCSs is quite new and the theory is still in its infancy, fruitful research work can be found in the most popular journals in both fields of control theory and communication networks, considering different issues in NCSs, such as the network-induced delay, data packet dropout, and so on. [1 – 4]. So far, most research work on NCSs, especially in dealing with the network-induced delay, which is introduced by the inserted network and greatly degrades the performance of the system at certain conditions, has been done by the control theory community. Various methodologies in conventional control theory, such as the theories of timedelay systems, switched systems, stochastic control, optimal control and so on, have found their applications in NCSs [5, 6]. In this kind of research, the characteristics of the network are assumed to be given in advance, and thus a conventional time-delay system, rather than an NCS is considered. However, it is the communication network which replaces the direct connections between sensors, controllers and actuators in conventional control systems that makes NCSs distinct from the latter. Thus, it is necessary to take # The Institution of Engineering and Technology 2008 doi:10.1049/iet-cta:20070005 Paper first received 3rd January and in revised form 21st April 2007 The authors are with Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, UK E-mail: [email protected] IET Control Theory Appl., 2008, 2, (1), pp. 7– 15

the characteristics of the network into account in the study of NCSs, and actually, this kind of research (the so called ‘co-design’ approach) has been an emerging trend in recent years. The co-design approach to NCSs generally considers NCSs with such communication constraints as network-induced delays, data packet dropout, medium access constraints and so on, which are not assumed to be given as parameters or constraints for conventional control systems in advance, but act as designable factors using techniques of communications and networks. It is therefore reasonable to expect that a better performance can be obtained using the co-design approach. For further information of the co-design approach to NCSs, the reader is referred to recent papers [7– 15] and the references therein. In this paper, the design and analysis of a set of linear NCSs which share the network with limited resource to transmit their control signals, are considered. (See Fig. 1 for the general configuration. Note that there are also random delays in the backward channel which is not shown in Fig. 1.) A similar problem setup can be found in [16, 17]. Hristu Varsakelis and Kumar [16] use the technique of ‘communication sequence’ (see also in [18, 19]) to deal with medium access constraint for such a system configuration and model the subsystems as switched systems with two modes ‘open loop’ and ‘closed loop’ which switch according to whether the current subsystem has access to the medium or not. Branicky et al. [17] consider a special case of Fig. 1 where the channel from controller to actuator is linked directly, and the rate monotonic (RM) scheduling algorithm is applied to schedule the transmissions of the sensing data of the subsystems. Both of the papers do not explicitly take the network-induced delay nor data packet dropout into consideration. In this paper, however, we will consider all the communication constraints, network-induced delay, data 7

system Sci is obtained using a sampling period Ti xi (k þ 1) ¼ Ai xi (k) þ Bi ui (k) Si : yi (k) ¼ Ci xi (k)

Fig. 1 Multiple networked control systems share the communication channel

packet dropout and medium access constraint, for the system configuration shown in Fig. 1. To this end, a co-design approach is proposed with the integration of the model predictive control (MPC) method and the scheduling algorithm. In conventional time-delay systems (TDS), there are mainly two ways to deal with the case when there is no current control signal available at the plant side due to the delay. This is to use either the last control signal or zero control. In both methods, the previous information of the system, including the system states, outputs and inputs and the structure information of the system have not been considered. However, if this information is well organised to derive a predictive control signal, it is reasonable to expect that a better performance can be obtained. On the basis of this insight, a modified MPC method is applied to design the predictive controller and a time-delay compensator is used at the actuator side to compensate for the network-induced delay in NCSs [20]. This method is validated using both simulation and a practical experiment. A similar idea is also used to deal with the network-induced delays and medium access constraints in [21]. In this paper, different from the input – output form in [20], a modified predictive control method in state-space form is applied and a delay compensation scheme is designed at both the controller and the actuator sides to compensate for both the network-induced delay and data packet dropout. In order to optimise the resource allocations of the shared network, scheduling theory is applied in this paper to allocate the medium access of the transmissions of the predictive control sequence [22, 23]. Two different scheduling algorithms, the existing static RM algorithm and a novel dynamic feedback scheduling (DFS) algorithm, are adopted to schedule the transmissions of the control predictions under different environments by defining carefully the transmission task for each system, with the guarantee of the stability of all the systems by using the notion of ‘Stable Supremum of Round Trip Time (SSRTT)’. The remainder of the paper is organised as follows. The problem being studied is described in Section 2; then the design of the predictive controller and the time delay compensator is given in Section 3. After that the scheduling algorithms are presented in Section 4, and the simulation results are illustrated in Section 5. Section 6 concludes the paper. 2

Problem description

Consider a set of N continuous-time LTI systems (Sic)1iN described by the state equations ( Sic : xci (t)

ni

x_ ci (t) ¼ Aci xci (t) þ Bci uci (t) yci (t) ¼ Cic xci (t)

uci (t)

mi

3 Predictive controller and time delay compensation scheme design In this section, the predictive controller with a delay compensation scheme at the controller side and a modified delay compensator at the actuator side for each system are first presented and then the stability theorem of the closed-loop system is obtained, from which the important notion SSRTT is derived. This notion will be used in the scheduling algorithm design covered in the next section. Note that though the design is for a separate system Si , the subscript i of all the parameters for system Si is ignored in this section for the simplicity of notation due to the fact that the design method is exactly the same for all the systems (Si )1iN . 3.1

Design of the predictive controller

Assume that the objective function has the form J (N1 , N2 , Nu ) ¼

N2 X

dj [^y(k þ jjk tsc,k ) v(k þ j)]2

j¼N1

þ

Nu X

lj [Du(k þ j 1)]2

(1)

j¼1

where N1 and N2 are the minimum and maximum prediction horizons, Nu is the control horizon, dj , j ¼ N1 , . . . , N2 , lj , j ¼ 1, . . . , Nu the weighting factors, v(k þ j), j ¼ N1 , . . . , N2 the set points, y^ (k þ jjk tsc,k ), j ¼ N1 , . . . , N2 the predictive outputs based on previous information up to time k 2 tsc,k and tsc,k the time delay in the backward channel at time k. Let x (k) ¼ [x(k) u(k 1)]T , then system Si (any system chosen from (Si )1iN ) can be represented by S 0 x (k þ 1) ¼ M x (k) þ N Du(k) 0 S: y(k) ¼ Qx(k)

yci (t)

ri

[R , [ R and [ R . In a digital where control environment, a discrete-time representation Si of 8

with xi (k) ¼ xci (kTi ), ui (k) ¼ uci (kTi ), yi (k) ¼ yci (kTi ), Ai ¼ ÐT c c eAi Ti and Bi ¼ 0 i eAi s dsBci : Suppose that the time delays in the backward channel of all the systems are random but bounded and the transmissions from the controllers to the actuators share a communication network with limited resource. The communication resource is limited in the sense that, at each time instant, only one controller can access the network for transmission. Thus, the time delay in the forward channel depends not only on the delay when the data are transmitted through the network but also on the scheduling algorithm used for the medium access control of the transmissions of all the systems (Fig. 1). Thus the problem here is not only to design a controller for each system Si but also to design the scheduling scheme for the communication resource allocation for all the systems (Si )1iN , in an environment of network-induced delays and data packet dropouts.

A B B , N¼ , Q¼ C 0 I I Du(k) ¼ u(k) u(k 1). where M ¼

0

and

IET Control Theory Appl., Vol. 2, No. 1, January 2008

Thus the j0 step forward output prediction at time k 0 is 0

0

0

0

j0

0

y^ (k þ j jk ) ¼ QM x (k ) þ

j 1 X

QM

j0 l0 1

0

0

N Du(k þ l )

l0 ¼0

Let j0 ¼ j þ tsc,k , k 0 ¼ k tsc,k and l0 ¼ l þ tsc,k , then the predictive outputs at time k based on the information of the state up to time k 2 tsc,k and the control sequence from ktsc,k is y^ (k þ jjk tsc,k ) ¼ QM jþtsc,k x (k tsc,k ) þ

j1 X

QM jl1 NDu(k þ l) (2)

l¼tsc,k

If the state vector x is not available, an observer must be included x^ (k þ 1jk) ¼ A^x(kjk 1) þ Bu(k) þ L(ym (k) C x^ (kjk 1)) (3) where ym(k) is the measured output. If the plant is subject to white noise disturbances affecting the process and the output with known covariance matrices, the observer becomes a Kalman filter and the gain L is calculated solving a Riccati equation. In [20], the previous control sequence u(k 1), . . . , u(k tsc,k ) is used to calculate the predictive control sequence at the controller side at time k. However, the previous control signals from u(k tsc,k ) to u(k 2 1) are not available for the controller due to the random time delay in the forward channel. As will be discussed further in Section 3.2, in a networked predictive control environment, a sequence of future control signals is packed to send to the actuator, and the actuator only picks out one from the sequence of the data corresponding to the specific time delay in the forward channel. It can therefore be seen that the controller does not know the real control signal adopted by the actuator unless it receives the information about the previous control signals applied to the actuator. Only in the special case where there is no delay in the forward channel, the previous control sequence is known immediately by the controller. Therefore in this paper, we develop a new method to deal with this problem, in which only the control and output information before k 2 tsc,k are used to generate the predictive control sequence by including the control sequence from time k 2 tsc,k to k 2 1 as part of the predictive control sequence. Let Y^ (kjk tsc,k ) ¼ [^y(k þ N1 jk tsc,k ) y^ (k þ N2 jk tsc,k )]T , DU (kjk tsc,k ) ¼ [Du(ktsc,k jk tsc,k ) Du(kþ Nu 1jk tsc,k )]T , then Y^ (kjk tsc,k ) ¼ Ex(k tsc,k ) þ FDU (kjk tsc,k ) N1 þtsc,k T

N1 þtsc,k þ1 T

(4)

N2 þtsc,k T T

where E ¼ [(QM ) (QM ) (QM ) ] and F is a block lower triangular matrix with its non-null elements defined by (F)ij ¼ QM N1 þtsc,k þij1 N , j i N1 þ tsc,k 1. Let 4k ¼ [v(k þ N1 ) v(k þ N2 )]T , then the optimal control increment sequence from k 2 tsc,k to k þ Nu 2 1 can be calculated by letting @J ()=@DU ¼ 0 using equations (1), (2) and (4) DU (kjk tsc,k ) ¼ (F T W1 F þ W2 )1 F T W1 (4k Ex(k tsc,k ))

(5)

where W1 and W2 are the weighting matrices, which are diagonal with the entries (W1 )ii ¼ di , i ¼ 1, 2, . . . , N2 N1 þ 1 and (W2 )ii ¼ li , i ¼ 1, 2, . . . , Nu : The optimal IET Control Theory Appl., Vol. 2, No. 1, January 2008

predictive control sequence from k to k þ Nu 2 1 is U (kjk tsc,k ) ¼ Gu(k tsc,k 1) þ HDU (kjk tsc,k )

(6)

where U (kjk tsc,k ) ¼ [u(kjk tsc,k ) u(k þ Nu 1jk tsc,k )]TNu 1 , G ¼ [Im Im ]TmNu m , Im is the identity matrix with rank m and 0 1 Im Im 0 0 B Im Im Im 0 C B C H ¼B . C @ .. ... ... . . . ... A Im Im Im Im mNu m(Nu þtsc,k ) Remark 1: Although we have not specially pointed this out earlier, it is a fact that the complexity of the calculation of the control predictions [(5) and (6)] seriously depends on the backward channel delay tsc since the matrices E, F and H vary with tsc . Thus, for the online implementation, it is a great burden for the controller to calculate the control predictions if tsc varies over a large range. However, all these matrices, actually, can be calculated offline for a given tsc given the nature of these matrices. This advantage enables us to calculate offline all the matrices relating to the specific tscs, store them in the controller and just pick out the appropriate ones when calculating online the control predictions, according to the current value of tsc , which can be known to the controller by using a time stamp for each sensing data packet as described in the following time-delay compensator design. 3.2

Design of the time-delay compensator

The network introduces to the NCSs delays which greatly degrade the performance of the system, even making the system unstable under certain conditions, while at the same time, the network also brings an advantage to the system in that a sequence of signals can be packed and transmitted simultaneously [6, 20]. The time-delay compensator adopted in this paper takes advantage of this property of NCSs. The following assumptions are made in the design of the time delay compensator. 1. For the sake of the calculation of the predictive control sequence, the time delay of the backward channel, tsc , is known to the controller, which can be easily done by issuing a time stamp on each data packet from the sensor side to the controller side. 2. The round trip time (RTT, noted by t, the total time delay of a packet, i.e. t ¼ tsc þ tca ) is known to the actuator, which can also be done by using the time stamps. 3. The predictive control sequences are packed and transmitted to the actuator simultaneously. 4. The sum of the forward time delay tca and the maximum of continuous data packet dropout is less than the control horizon Nu . The time delay compensator works as follows. At every time instant k, the predictive controller calculates a sequence of predictive control signals based on the outputs and control sequence up to time k 2 tsc,k , the future control sequence part of which U (kjk 2 tsc,k) (6) is then transmitted in one packet to the actuator with a time stamp k and its backward channel delay tsc,k . When a packet of a control sequence arrives at the actuator side (different packets may experience different time delays), it is compared with the one already in the cache of the actuator according to the time stamp and only the later one is saved. 9

As for the actuator, it picks out the control action u(k þ tca,kjk 2 tsc,k) from the control sequence in its cache if the time stamp of the control sequence in its cache is k. Note that the time instant k in the time-delay compensator described above is based on the controller. Let tca,k denote the time delay in the forward channel of the control sequence which is applied to the actuator at time instant k (the time at the plant side), then the time stamp of this sequence (the time when it is sent at the controller side) is

the state vector is not available, then the state observer (3) is also included in the enhanced system. The theory of switched systems is applied to derive the following stability theorem. A similar theorem can also be found in [20].

k ¼ k tca,k ¼ max{jjU (jjj tsc,j ) [ Gk }

LT (tsc,k , tca,k )PL(tsc,k , tca,k ) P 0

j

(7)

where Gk is the set of the control sequences that are available at time interval (k 2 1, k] at the actuator side. From (5), (6) and (7) and the definition of x¯(k), the control signal adopted by the actuator at time k is obtained as u(k) ¼ dt U (k tca,k jk tk ) ca,k

¼ dt Gu(k tk 1) þ dt HDU (k tca,k jk tk ) ca,k

ca,k

¼ u(k

tk

T

1) þ dt H(F W1 F þ W2 )

1

ca,k

T

F W1 (4kt Ex(k tk )) ca,k

¼ (Im Kt E2 )u(k tk 1) k

þ Kt (4kt E1 x(k tk )) k

(8)

ca,k

where dtca,k is a 1 Nu block matrix with all entries are 0 except the tca,k th is Im , tk the RTT with respect to tca,k , that is, tk ¼ tca,k þ tsc,k , tsc,k the delay in the backward channel corresponding to tca,k , Ktk ¼ dtca,k H(F T W1 F þ W2 )1 F T W1 , and E can be written as E ¼ [E1 E2] with appropriate dimensions such that x(k tk ) Ex(k tk ) ¼ [E1 E2 ] ¼ E1 x(k tk ) u(k tk 1) þ E2 u(k tk 1) 3.3

Stability analysis

It is assumed in this section that the RTT is bounded by a finite value t¯ , t¯ ¼ t¯ ca þ t¯ sc , where t¯ ca and t¯ sc are the upper bounds of the delay in the forward and backward channels respectively, and v ¼ 0 without loss of generality. Let U (k) ¼ [u(k 1) u(k t¯ 1)]T , X (k) ¼ [x(k) x (k t¯ )]T and Z(k) ¼ [X T (k)U T (k)]T , then an enhanced system can be obtained from (8) and the system description of Si [any system chosen from (Si )1iN ] Z(k þ 1) ¼ L(tsc,k , tca,k )Z(k)

(9)

where 0

A BI B n B B B B B L(tsc,k , tca,k ) ¼ B B B B B B @

BKt1 ..

.

0 k

Kt2 k

Im ..

(10)

for all tca,k [ {0, 1, . . . , t¯ ca } and tsc,k [ {0, 1, . . . , t¯ sc }: Proof: Let V (k) ¼ Z T (k)PZ(k) be a Lyapunov candidate. Then the incremental V for system (9) is DV (k þ 1) ¼ Z T (k)(LT (tsc,k , tca,k )PL(tsc,k , tca,k ) P) Z(k) 0, thus enabling the theorem to be proved given the assumption above. A Remark 2: Theorem 1 implies that the linear system Si which uses MPC method and the delay compensation scheme described above to compensate for the network-induced delay and data packet dropout in NCSs (called a ‘network predictive control system (NPCS)’ hereafter), is stable under certain conditions if the RTT is less than a fixed value (Fig. 2). In other words, given a linear system, the least upper bound, or the supremum of the RTT, under which the system is stable can be found from Theorem 1. We call this supremum of RTT that guarantees the stability of the system the SSRTT, which is an inherent characteristic of a given system (Note that the notion of SSRTT here is similar to ‘maximum allowable delay bound’ (MADB) which has been used in a number of papers; see [24] for an overview. We prefer SSRTT to MADB in this paper since the former can better express the particular requirement of the RTT in Theorem 1 for the stability of the system.). Techniques such as the LMI tool-box are useful in the process of finding the SSRTT of a given system. It is also necessary to point out that if other performance constraints besides stability are considered, a smaller supremum of RTT than SSRTT is needed. For convenience, denote the SSRTTs of the ^ i . 0, 1 i N . systems (Si )1iN by D 4

Scheduling

In this section, the scheduling theory is applied to allocate the limited network resource for the transmission of the control information for the systems (Si )1iN in the forward channel. The static, priority-based scheduling algorithm RM is applied to the set of systems to schedule their transmissions under a private network environment first, and then a dynamic, feedback-based scheduling algorithm DFS is presented to extend the application to the public network.

C C C C C C C C C C C C C A

k

In Kt1 0

1

BKt2

k

Theorem 1: The closed-loop system (9) is stable if there exists a positive definite matrix P such that

. Im

which varies with tca,k and tsc,k , Kt1 ¼ Kt E1 , Kt2 ¼ k k k Im Kt E2 , and In is the identity matrix with rank n. If k

10

Fig. 2 NPCS with a time-delay compensator IET Control Theory Appl., Vol. 2, No. 1, January 2008

When a scheduling algorithm is applied to schedule the transmission tasks of a set of NPCSs, the stability of the systems need to be guaranteed as a precondition. To ensure this, the concept of ‘stable schedulability’ is defined. Stable schedulability: A set of NPCSs sharing the network resource is said to be stable schedulable by a scheduling algorithm if the transmissions of all the systems can be scheduled so that all the systems are stable. 4.1

Static scheduling

In the static scheduling case, the network is assumed to be used only by the systems (Si)1i N , that is, it is private to the set of systems. Since the transmissions of the predictive control sequences for each system are viewed as different real-time tasks in scheduling theory, in the following the transmission tasks of the NPCSs will be defined by analysing the formation of the network-induced delays of NPCSs first, and then the RM algorithm is described over these transmission tasks and the feasibility theorem is obtained as well. 4.1.1 Transmission tasks of the NPCSs: As shown in [25], the time delay of the forward channel tca is mainly composed of the following three parts. 1. The propagation delay, which is the time from when a packet is put onto the network till it successfully arrives at its destination. Since the network is private to the systems (Si)1i N , the propagation delay depends merely on the speed of signal transmission and the distance between the source and the destination, which are fixed in the system model of this paper, and hence this delay is assumed to be known as a constant t0ca in the static scheduling case. 2. The frame time delay, which is the time for the source to place a packet on the network. Suppose that the size of the packet which contains the predictive control sequence is expressed as gNu , where g is a constant to the number of bytes contained in the one step ahead predictive control signal. This can be viewed as the same for all the systems, then the frame time delay for a transmission of system Si is ei ¼

gNui B

(11)

where B is the bandwidth of the network and Niu the control horizon of system Si . In the networked predictive control scheme proposed in Section 3, it is normal to assume that all the systems use the same control horizon Nu , since the selection of Nu mainly depends on the RTT of the network and all the systems endure similar network-induced delays and data packet dropouts by sharing the network. Hence ei ¼ e ¼

gNu , B

i ¼ 1, 2, . . . , N

(12)

The frame time delay e serves as the execution time in the transmission tasks of the NPCSs. 3. The waiting time delay is defined as the time a predictive control sequence has to wait for queuing and network availability before actually being sent. From Remark 2, the ^ i , and therefore to ensure the stabSSRTT for system Si is D ility of all the systems, the waiting delay for the transmission of the control signals of each system should not IET Control Theory Appl., Vol. 2, No. 1, January 2008

be larger than ^ i t0ca t¯ sc e Di ¼ D

(13)

Note here that the upper bound of the delay in the backward channel tˆsc is used since the RM algorithm assigns the priority of each task statically and the stability of the systems needs to be guaranteed under the worst case. It is only the stability of the system that we care about in the RM algorithm, and thus the transmission period of system Si needs to be no longer than Di . Therefore the transmission period hi of system Si is assumed to be equal to Di in the static RM scheduling algorithm, that is hi ¼ Di ,

i ¼ 1, 2, . . . , N

(14)

hi is chosen by (14) so that the predictive control sequence of system Si is sent every hi seconds no matter what the sampling period is or how fast the controller can generate the predictive control sequence. Thus, from the analysis of tca , the transmission tasks of the set of NPCSs can be described as follows. All the tasks have the same execution time e, the deadline of each task equals its period hi and the first release time qi of task i is the time when system Si first operates. We denote the tasks by Ti ¼ T(qi , e, hi ),

i ¼ 1, 2, . . . , N

(15)

4.1.2 Scheduling of NPCSs by RM: RM is a widely used scheduling algorithm, where tasks with shorter periods have higher priorities. It is a fixed-priority assignment: priorities are assigned to tasks before execution and do not change over time. Liu and Layland [26] have shown that RM is superior to other fixed-priority assignments in the sense that no other fixed-priority algorithm can schedule a task set that cannot be scheduled by RM. Now consider the set of real-time transmission tasks Ti , 1 i N defined in (15). These tasks are periodic, independent, non-pre-emptive and the period of each task equals its deadline. These characteristics are just what the operation of the RM algorithm needs. Therefore the RM scheduling algorithm can be applied to schedule the set of transmission tasks in NPCSs, by which the transmission with shorter deadlines (or periods) are assigned higher priorities. Thus, the predictive control sequence can be transmitted first if the network is idle, that is if hi , hj , then Yi . Yj ,

i, j ¼ 1, 2, . . . , N

(16)

where Yi represents the priority of the transmission task of system Si . Theorem 2: A set of N NPCSs sharing the network resource in their forward channel (indexed by the increasing order of their transmission periods, that is, hi hiþ1 , i ¼ 1, 2, . . . , N 2 1) are stable schedulable if for all i ¼ 1, . . . , N U(i) f (i)

(17)

where f(i) ¼ i(21/i 2 1) and ! 8 i 1 P 1 > > > > < e j¼1 hj þ hi ! U(i) ¼ > i 1 P > > > :e j¼1 hj

i ¼ 1, 2, . . . , N 1 i¼N 11

Proof: From [27, Theorem 16], a set of non-pre-emptive periodic real-time tasks are schedulable if e1 e 2 e b þ þ þ i þ l,i i(21=i 1) h1 h2 hi hi

(18)

where ei is the frame time, hi the transmission period, each for the ith task, and b¯l,i is task i’s worst-case blocking time by the lower priority tasks, that is b l,i ¼

max j¼iþ1,...,N

ej

(19)

As has been pointed out above, in the proposed networked predictive control systems, (12) holds, and therefore b l,i ¼ e, i ¼ 1, 2, . . . , N 1 and b l,N ¼ 0 from (19). Hence, the theorem holds. Corollary 1: If (1) hiþ1 2 hi , i ¼ 1, 2, . . . , N 2 2 and (2) hN/(hN21) ((N 2 1)/N) ((21/(N21) 2 1)/(21/N 2 1)), then the set of tasks of NPCSs is stable schedulable if e

N X 1 i¼1

hi

N (21=N 1)

Proof: From assumption i ¼ 1, 2, . . . , N 2

(1),

we

Fig. 3 DFS of multiple systems

algorithm still works. The framework of the DFS scheme is depicted in Fig. 3. In order to implement DFS, issues such as the selection of the period of DFS, the measurement of the network utilisation and the reassignment of the transmission periods of the NPCSs, need to be dealt with first.

(20) obtain

for

1 U(i þ 1) U(i) ¼ e 0 hiþ1 hi 2

and it is apparent that U(N ) U(N 1) from assumption (2). Thus U(N ) ¼ max U(i)

4.2.1 Period of DFS: This period, noted by TDFS , needs to be chosen carefully. Generally, its value depends on the speed at which the condition of the network changes. A small TDFS is needed if the network condition changes rapidly, whereas a larger one can still guarantee the performance of the system without overloading the network if the parameters of the network do not change much over a long time. However, in any case, TDFS should be always not less than the transmission periods of all the systems, that is, TDFS maxN i¼1 hi .

1iN

On the other hand, it is easy to show that the function f (.) is non-increasing, and therefore U(N ) f (N ) implies U(i) f (i), i ¼ 1, 2, . . . , N 1, which completes the proof recalling Theorem 2. Corollary 2: If the transmission periods of all the systems are the same, that is, hi ¼ h, i ¼ 1, 2, . . . , N , then the collection of systems are stable schedulable if e 21=N 1 h Proof: It can be obtained directly from Corollary 1. 4.2

(21) A

Dynamic feedback scheduling

In the static RM scheduling scheme presented above, the transmission periods hi for all the systems are assigned a priori to ensure the stability of the systems and do not change in the process of their operation. In the case of the network being shared only by systems (Si )1iN , this method works though the performance of the systems may not be optimum because the network is not fully used. However, if the network is not private to these systems, that is, there are other components occupying the network, it cannot be assumed that the propagation delay is constant due to the change of the network loads. On the basis of this reality, a DFS scheme is designed. In this scheme, a higher-level feedback scheduler is proposed, which obtains the information of the network utilisation from the network and the control performances from all the systems as well, and regularly calculates and reassigns the transmission period for each system. During the interval of two successive reassignments of periods, the RM 12

4.2.2 Measurement of the network utilisation: To obtain the utilisation information of the network, a packet containing this information is sent to the feedback scheduler using the period of TDFS . This information is mainly reflected by the propagation time delay in the forward channel of each system. The propagation time will increase to a certain extent with the increase of the network load. Another factor affecting the stability of each system is the change of the delay in the backward channel. In order to take this factor into account and for simplicity, we assume that the upper bound of the delay in the backward channel during the kth period of DFS (noted by t¯isc(k) for system Si) can be obtained from the network and the network does not change too much during this period, thus enabling us to use t¯isc(k) to estimate its value during the (k þ 1)th period. Then the deadline hi of each task for each system will be recalculated by updating the propagation time toca and the upper bound of the delay in the backward channel every TDFS seconds as follows ^ i m(t0ca (k) þ t¯ isc (k)) e Di (k þ 1) ¼ D

(22)

where m close to 1 is a smoothing factor satisfying

m(t0ca (k) þ t¯ isc (k)) t0ca (k þ 1) þ t¯ isc (k þ 1), 8k

4.2.3 Reassignment of the transmission periods of all the systems: In order to obtain the control performance of each system, an obvious idea is to use the predictive quality of performance (QoP) during the next DFS period . This QoP during the kth period of DFS can be defined for IET Control Theory Appl., Vol. 2, No. 1, January 2008

system Si as P^ i (k) ¼

d(kþ1)T DFS =hi e X

(^yi (jjj tsc,j ) vi (j))2

(23)

j¼dkTDFS =hi e

where dxe is the nearest integer to x satisfying dxe x. The calculation of the new transmission periods for all the systems can then be modelled as an optimisation problem P 8 Select hi , i ¼ 1, 2, . . . ; N > > > > N > P < s.t. min P^ i , P: hi i¼1 > > > subject to U(i) f (i), i ¼ 1, 2, . . . , N > > : hi Ti , i ¼ 1, 2, . . . , N where U(i) and f(i) are defined in Theorem 2 and Ti is the sampling period for system Si as defined in Section 2. In practice, the predictive outputs y^ (jjj tsc,j ) of the systems can be obtained using the open-loop prediction as shown in (2), whereas the online operation of the optimisation problem P is not a simple one. Therefore not the predictive QoP but the previous QoP, J¯i(k), i for system Si and k for the kth period of DFS, is used to represent the performance of the system, which is defined as follows and can be easily obtained X J i (k) ¼ Ji (k; N1j , N2j , Nuj ) (24)

S3. When t ¼ kTDFS , the DFS module checks the utilisation information of network. If the network is in full use, set k ¼ k þ 1, return to S2; else go to S4. S4. The DFS module calculates the new transmission periods using (25) and reassign the priorities for all the systems; set k ¼ k þ 1, return to S2. 4.2.5 Stability of DFS Theorem 3: A set of NPCSs is stable under DFS if the transmission tasks of the NPCSs are always stable schedulable.

Proof: The application of DFS to the set of NPCSs in this paper does not change the delay in the backward channel, since the DFS module is on the controller side, whereas it does change the delay in the forward channel by reassigning the transmission period hi , i ¼ 1, 2, . . . , N to the systems. However, by taking account of (22) and (27), we obtain ^ i t0ca (k þ 1) t¯ isc (k þ 1) e, hi (k þ 1) Di (k þ 1) D 8k, i ¼ 1, 2, . . . , N which implies ^ i , i ¼ 1, 2, . . . , N sup{hi (k) þ t0ca (k) þ t isc (k) þ e} D k

Ji ()[Pk

where Jj (k; N1j , N2j , Nuj ) is the objective function of system Si in the predictive controller defined in (1) and Pk is the set of objective functions during the kth period of DFS, or from the d(k 2 1)TDFS/hieth transmission period of system Si to the dkTDFS/hieth. Let the new transmission periods chosen in this way be 1 J (k) ¼ k(k þ 1) PN i ¼ k(k þ 1)ui (k) hi (k þ 1) j¼1 J j (k)

(28)

(29) Note that the left-hand side of (29) is the effective maximum of RTT for system Si , which is always no more than the required SSRTT. Thus, the theorem is valid recalling Theorem 1. A

(25) 5

Simulation

where k is a proportion factor and can be chosen as follows to include the constraints of stable schedulability of Theorem 2 ( ) f (i) f (N ) (26) k(k þ 1) ¼ max , Pi e i¼1,...,N 1 e( j¼1 uj (k) þ ui (k))

An example of the co-design method proposed above by using simulation is given in this section.

Considering the fact that the network load may change greatly between two periods of DFS, a smoothing factor r(0 , r 1)is introduced to avoid network overload. Also taking account of the fact that the transmission period hi can never exceed Di for the stability of the system, then the transmission periods are obtained as r hi (k þ 1) ¼ min , Di (k þ 1) (27) k(k þ 1)ui (k)

Three second-order linear systems are considered in the simulation, with the expressions in continuous form

4.2.4 Algorithm of DFS scheme: The algorithm of DFS can then be described as follows. ^ i e, 1 i N . S1. Initialise TDFS , k ¼ 1, t ¼ 0, hi ¼ D S2. During time t [ [(k 1)TDFS kTDFS ], do the following: S2a. Generate the predictive control sequences Ui (), 1 i N using (6) for all the systems. S2b. Apply RM algorithm described in Section 4.1 to determine the order of the transmission tasks of the NPCSs. S2c. Transmit the predictive control sequence as S2b determines. IET Control Theory Appl., Vol. 2, No. 1, January 2008

5.1

Simulation parameters

Ac1 ¼

106:0132

110:3637

5:2680

,

48:9313 , 34:0550 91:6291 160:9438 , Ac3 ¼ 69:3147 35:6675 0:1295 4:3801 Bc1 ¼ , Bc2 ¼ , 2:6890 2:3278 9:1471 , C1c ¼ C2c ¼ C3c ¼ 1 Bc3 ¼ 4:0444 Ac2 ¼

11:1572 23:7783 107:2959

and sampling periods T1 ¼ 0.02 s, T2 ¼ 0.015 s, T3 ¼ 0.01 s, respectively. The corresponding discrete 13

Table 1:

Simulation parameters System 1

System 2

System 3

T

0.02

0.015

0.01

q

0

0.05

0.06

x0

[21 21]T

[21 21]T

[21 21]T

P1

[1 30

P2

20 0 I I]

[0.008 0.002 0.01 10]

T is the sampling period

q is the first release time x02]T is the initial state

x0 ¼ [x01 P1 ¼ [N1

N2

Nu v W1 W2] is the predictive parameters

t0ca t¯sc Tsim], Tsim is the simulation time

P2 ¼ [e

Table 2:

SSRTT and transmission periods

System 1

System 2

System 3

D^ i

5

7

8

hi

4

5

6

systems can be obtained as 0:8 0:12 0:7 0:48 , A2 ¼ , A1 ¼ 0:11 0:9 0:2 0:6 0:4 0:2 0:02 0:08 A3 ¼ , B1 ¼ , B2 ¼ , 0:5 0:7 0:05 0:06 0:08 , C1 ¼ C2 ¼ C3 ¼ 1: B3 ¼ 0:1 For the simplicity of simulation, assume for all the three systems that the set point v ¼ 0, weighting factors W1 ¼ I and W2 ¼ I and the state vector can be obtained directly so that the state observer (3) is not required. Under these assumptions, the closed-loop system can be obtained from (9). Other parameters of the simulation are shown in Table 1. A Gaussian white noise with standard deviation 0.1 is also introduced as the disturbance of the state.

Fig. 5 State evolution using DFS algorithm Only the first state is illustrated

5.2

Using the LMI toolbox in Matlab, the SSRTT of each ^ i can be obtained by Theorem 1, thus enabling system D the transmission periods hi to be calculated according to (13) and (14), as shown in Table 2. Note that the execution time of each job is e ¼ 0.008 s, then the value of the utilization function U() can be obtained as follows, U(i) ¼ 0:2, 0:3133, 0:34, i ¼ 1, 2, 3 whereas f(i) ¼ 1, 0.8284, 0.7798, i ¼ 1, 2, 3, thus (17) holds, and by Theorem 2, the set of NCSs is stable schedulable under RM. The state evolution of the first state of the three systems under RM is illustrated in Fig. 4. 5.3

Only the first state is illustrated 14

Simulation of DFS algorithm

It is noted that the SSRTT obtained in Theorem 2 is conservative. In the simulation of DFS, the deadlines of the three systems are set to be 8, 10 and 12 steps, respectively, and the propagation delays of the systems in the forward channel are set to be randomly changing under the constraint that the real RTT are no more than the new SSRTT, in order to simulate the changes of the network loads. All the other parameters remain the same as in RM algorithm. The simulation result (Fig. 5) shows that the systems are still stable under this larger SSRTT and with fluctuating propagation delays. 6

Fig. 4 State evolution using RM algorithm

Simulation of RM algorithm

Conclusion

In this paper, a co-design approach is proposed to deal with the communication constraints for a set of NCSs which are connected to a shared network. To reduce the negative effect of the network-induced delay and data packet dropout, predictive control theory is applied to produce future control inputs to the systems, whereas the scheduling theorem, both the static algorithm RM and a dynamic DFS which takes advantage of the feedback information of the system performances, is applied to schedule the transmissions of the predictive control sequences of all the systems. Simulation results illustrate the validity of the integration of both predictive control and scheduling theories. IET Control Theory Appl., Vol. 2, No. 1, January 2008

7

References

1 Tipsuwan, Y., and Chow, M.-Y.: ‘Control methodologies in networked control systems’, Control Eng. Pract., 2003, 11, (10), pp. 1099– 1111 2 Savkin, A.V.: ‘Analysis and synthesis of networked control systems, topological entropy, observability, robustness and optimal control’, Automatica, 2005, 42, (1), pp. 51–62 3 Walsh, G.C., Beldiman, O., and Bushnell, L.: ‘Asymptotic behaviour of nonlinear networked control systems’, IEEE Trans. Autom. Control, 2001, 46, (7), pp. 1093– 1097 4 Yang, T.C.: ‘Networked control systems: a brief survey’, IET Proc., Control Theory Appl., 2006, 153, (4), pp. 403– 412 5 Shousong, H., and Qixin, Z.: ‘Stochastic optimal control and analysis of stability of networked control systems with long delay’, Automatica, 2003, 39, (11), pp. 1877–1884 6 Georgiev, D., and Tilbury, D.M.: ‘Packet-based control: the H2-optimal solution’, Automatica, 2006, 42, (1), pp. 137– 144 7 Tipsuwan, Y., and Chow, M.-Y.: ‘Gain scheduler middleware: a methodology to enable existing controllers for networked control and teleoperation – part I: networked control’, IEEE Trans. Ind. Electron., 2004, 51, (6), pp. 1218–1227 8 Colandairaj, J., Irwin, G.W., and Scanlon, W.G.: ‘Wireless networked control systems with qos-based sampling’, IEE Proc., Control Theory Appl., 2007, 1, (1), pp. 430– 438 9 Branicky, M.S., Liberatore, V., and Phillips, S.M.: ‘Networked control system co-simulation for co-design’. Proc. American Control Conf., (Denver, USA), June 2003, vol. 4, pp. 3341–3346 10 Zhang, L., and Hua-Jing, F.: ‘A novel controller design and evaluation for networked control systems with time-variant delays’, J. Frankl. Inst., Eng. Appl. Math., 2006, 343, pp. 161– 167 11 Lian, F.L., Yook, J.K., Tilbury, D.M., and Moyne, J.: ‘Network architecture and communication modules for guaranteeing acceptable control and communication performance for networked multi-agent systems’, IEEE Trans. Ind. Inform., 2006, 2, (1), pp. 12–24 12 Imer, O.C., Yu¨ksel, S., and Basar, T.: ‘Optimal control of LTI systems over unreliable communication links’, Automatica, 2006, 42, (9), pp. 1429– 1439 13 Ling, Q., and Lemmon, M.D.: ‘Optimal dropout compensation in networked control systems’. Proc. 42nd IEEE Conf. Decision and Control, Maul, Hawali, USA, December 2003, pp. 670– 675

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14 Tatikonda, S., and Mitter, S.: ‘Control under communication constraints’, IEEE Trans. Autom. Control, 2004, 49, (7), pp. 1056–1068 15 Ben Gaid, M.E.M., Cela, A., and Hamam, Y.: ‘Optimal integrated control and scheduling of networked control systems with communication constraints: application to a car suspension system’, IEEE Trans. Control Syst. Technol., 2006, 14, (4), pp. 776– 787 16 Hristu Varsakelis, D., and Kumar, P.R.: ‘Interrupt-based feedback control over a shared communication medium’. Proc. 41st IEEE Conf. Decision and Control, December 2002, vol. 3, pp. 3223–3228 17 Branicky, M.S., Phillips, S.M., and Zhang, W.: ‘Scheduling and feedback co-design for networked control systems’. Proc. IEEE Conf. Decision and Control, Las Vegas, USA, December 2002 18 Brochett, R.W.: ‘Stabilization of motor networks’. Proc. 34th IEEE Conf. Decision and Control, Phoenix, USA, 1995, pp. 1484–1488 19 Zhang, L., and Hristu-Varsakelis, D.: ‘Communication and control co-design for networked control systems’, Automatica, 2006, 42, (6), pp. 953–958 20 Liu, G.P., Mu, J.X., Rees, D., and Chai, S.C.: ‘Design and stability analysis of networked control systems with random communication time delay using the modified MPC’, Int. J. Control, 2006, 79, (4), pp. 288–297 21 Hristu-Varsakelis, D.: ‘Stabilization of networked control systems with access constraints and delays’. Proc. 45th IEEE Conf. Decision and Control, December 2006 22 Cervin, A.: ‘Integrated control and realtime scheduling’, PhD thesis, Department of Automatic Control, Lund Institute of Technology Sweden, April 2003 23 Walsh, G.C., and Ye, H.: ‘Scheduling of networked control systems’, IEEE Control Syst. Mag., 2001, 21, (1), pp. 57–65 24 Kim, D.S., Lee, Y.S., Kwon, W.H., and Park, H.S.: ‘Maximum allowable delay bounds of networked control systems’, Control Eng. Pract., 2003, 11, pp. 1301– 1313 25 Lian, F.L., Moyne, J.R., and Tilbury, D.M.: ‘Performance evaluation of control networks: Ethernet, controlnet, and devicenet’, IEEE Control Syst. Mag., 2001, 21, (1), pp. 66–83 26 Liu, C.L., and Layland, J.W.: ‘Scheduling algorithm for multiprogramming in a hard-real-time environment’, J. ACM, 1973, 20, (1), pp. 46–61 27 Sha, L., Rajkumar, R., and Lehoczky, J.: ‘Priority inheritance protocols: an approach to real-time synchronization’, IEEE Trans. Comput., 1990, 39, (9), pp. 1175– 1185

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 5, MAY 2008

469

Networked Predictive Control Systems Based on the Hammerstein Model Yun-Bo Zhao, Guo-Ping Liu, Senior Member, IEEE, and David Rees

Abstract—In this paper, a novel predictive control-based approach is proposed for a networked control system with random delays containing an input nonlinear process based on a Hammerstein model. The method uses a time-delay two-step generalized predictive control scheme, which consists of two parts: one is to deal with the input nonlinearity of the Hammerstein model and the other is to compensate for the network-induced delay in the networked control system. A theoretical result using the Popov criterion is presented for the closed-loop stability of the system in the case of a constant delay. Simulation examples illustrating the validity of the approach are also presented. Index Terms—Hammerstein model, networked control systems, predictive control, time-delay compensator, two-step approach.

I. INTRODUCTION CONTROL system is called a “networked control system” (NCS) when the link from sensor to controller and/or the link from controller to actuator are/is connected not directly as is assumed in conventional control systems but via a serial communication network with limited resources [1], [2]. This configuration, due to the advantages the network introduces, brings to the system lower cost, flexibility, and the ability of remote control while at the same time greatly degrades the performance of the control system or even makes the system unstable under certain conditions, due to the disadvantages the network introduces, such as the time delay (so called “network-induced delay”), data packet dropout, and quantization. Such an implementation presents a new challenge to conventional control theory. A large number of papers have addressed NCSs to date, considering different issues, mainly on the treatment of the network-induced delay. The majority of papers published have largely been restricted to linear systems [3], [4]. In this paper, we will consider a particular category of nonlinear system represented by a Hammerstein model [5], which consists of a cascade connection of a static nonlinearity followed by a dynamic linear time-invariant (LTI) system. This nonlinear model is important in theory and applies to a number of practical applications, see, e.g., [6] and [7]. To deal with the control problem when such a nonlinear system is implemented in a networked control environment, a time-delay two-step generalized predictive control (TDTSGPC) approach is proposed in this paper. In this approach, the nonlinear input process is dealt with using the two-step design approach developed in [8], and a predictive-based compensation scheme is also designed to

A

Manuscript received February 10, 2007; revised October 16, 2007. This paper was recommended by Associate Editor M. di Bernardo. The authors are with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCSII.2007.914423

compensate for the network-induced delay. It works as follows. The predictive control method is first applied to the linear part of the Hammerstein model to generate the intermediate control predictions. It is assumed that, for the nonlinear part, numerical methods can be used to obtain the real control predictions from the intermediate control predictions, and then a sequence of the predictions is packed and sent to the actuator through the network simultaneously. At every execution time, the specially designed time-delay compensator will select the appropriate control signal to compensate for the network-induced delay. Using this compensation scheme, the network-induced delay can be exactly compensated for in an active way. The stability criterion of the proposed TDTSGPC approach is obtained using the Popov theorem. Simulations are also done to illustrate the validity of the approach. The remainder of this paper is organized as follows. The design of TDTSGPC based on a Hammerstein model is presented in Section II. Then, the theoretical results for the system stability and the simulation results of TDTSGPC are presented in Sections III and IV, respectively. The paper gives the conclusions in Section V. II. DESIGN OF TDTSGPC BASED ON THE HAMMERSTEIN MODEL The following Hammerstein system is considered in this paper, with the combination of the CARIMA model and a static as input nonlinear function (1a) (1b) where

is the Gaussian white noise with zero mean value, is the input, intermediate input, and output at time , respectively, with , is memoryless, static with and the input nonlinear function . The network-induced delay is one of the key problems when a control system is implemented in a networked control environment [1], which will make the current control input unavailable to the actuator. It is also an important problem in conventional time-delay systems (TDSs), in which there are mainly two ways to deal with this situation. This is to use either the last available control signal or to use zero control [9]. In both methods, the previous information of the system, including the system states, outputs and inputs, and the structure information of the system is not considered. However, with the use of the network in NCSs, it is possible to send a sequence of the control signals together due to the packet-based transmission of the network. Thus, in order to compensate for the network-induced delay, a sequence of forward control predictions can be calculated and

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sent to the actuator simultaneously, from which the appropriate control input can be picked out to actively compensate for the network-induced delay [12]. Since more information is used in this compensation approach, a better performance can be expected than that obtained in conventional TDSs. Following this idea, in this paper, the Linear Generalized Predictive Control (LGPC) approach is applied to generate the control predictions of the linear part of the Hammerstein model while the nonlinear part remains to be solved using a numerical method. This is why it is called a “two-step” approach for a Hammerstein model [8]. The two parts of TDTSGPC, the design of the two-step generalized predictive control (TSGPC) and the time-delay compensator, will now be considered. A. Design of TSGPC The key idea of TSGPC is to first design the intermediate control sequence of the linear part of Hammerstein model (1a) with the LGPC method and then obtain the real control sequence from the relationship , is the control horizon [8]. Hence, we will first present where the design of LGPC and then describe how to obtain the real inputs from the nonlinear relationship in the following subsections. 1) Design of LGPC: Without consideration of the input nonlinearity of the Hammerstein model, the LGPC problem for (1a) is solved for the following objective function: (2) where weight matrixes

is the set-point, are diagonal, are the outputs,

is the delay of the feedback channel, is the predictive means . horizon, and In [12], the previous control sequence is used to generate the control predictions at the controller side at time , while in reality this information is hard to obtain for the controller due to the time delay in both channels. In this paper, we propose a new method to deal with this problem, in which are only the control and output information before time used to calculate the predictive control sequence by including to as part of the the control sequence from time predictive control sequence. This is obtained using the objective function above and the equation of the control predictions below. Introduce the following Diophantine equations:

, otherwise; with all the entries 0 but , otherwise, for

if

.

.. .

, then the

predictive control sequence from to using objective function (2) at time based on the information before is (5) where the control increment sequence

. This can be obtained by (6) are all in terms Note that the matrixes and vectors , and we have omitted only for simplicity of notaof tion. The predictive control sequence obtained is different from a conventional GPC method, since the network-induced delay is also considered. For more details of the calculation of the typical predictive control sequences, the reader is referred to [10] and the references therein. Remark 1: Though we have not specifically pointed this out earlier, it is a fact that the complexity of the calculation of the control predictions seriously depends on the feedsince all of the matrixes and vectors back channel delay vary with . Thus, for the online implementation, it is a great burden for the controller to calvaries over a large range. culate the control predictions if However, all of these matrixes can be calculated offline for a . This advantage enables us to calculate offline all given s,1 store them in the of the matrixes relating to specific controller, and just choose the appropriate matrixes when calculating online the control predictions, according to the current , which can be known to the controller by using a value of time stamp for each data packet as described in the following time-delay compensator design. is invertible, and then 2) Input Nonlinearity: Assume exists such that its inverse (7)

; .

1Denotes the upper bound of the delay in the backward channel by , then the number of the choices with different delays (from 0 to ) for each matrix above is + 1.

where

, if

..

. ;

(4)

Define

.. .

Thus, at every time instant , the intermediate input is obtained from (5), and then the real can be calculated from (7), thus input enabling the control law to be derived. can be calculated accurately using (7), If the real input to be exactly known, then thus enabling the function TSGPC is equivalent to LGPC and the system is stable if and only if the linear system (1a) with LGPC is stable. However, that accuin practice, it is usually impossible to calculate rately. Hence, in this paper, we denote the practical inverse of

(3) when

.. .

ZHAO et al.: NETWORKED PREDICTIVE CONTROL SYSTEMS BASED ON THE HAMMERSTEIN MODEL

by and assume that this nonlinearity due to the inaccuracy of calculation satisfies for some (8) From the discussion above, the real predictive control sequence of TSGPC can be obtained as (9) where

. Remark 2: It is necessary to point out that what is required in implementing TSGPC is to satisfactorily meet the sector conis calstraint in (8), no matter how the inverse function does not need to be culated. It implies that the function invertible as long as its inverse can be obtained by a numerical method and satisfies the sector constraint. One can refer to [11] and the references therein for more information of the calculation of .

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III. STABILITY ANALYSIS Here, we first give the explicit expression of the closed-loop system using the two-step predictive control approach and the delay compensator and then obtain the stability criterion of the closed-loop system using a Popov criterion. A. Closed-Loop System Note that the time instant in the time compensator described denote above is based on the time at the controller side. Let the time delay in the forward channel of the control sequence which is applied by the actuator at time instant (the time at the plant side), and then the time stamp of this sequence (the time when it is sent at the controller side) is (10) is the delay in the feedback channel corresponding where to time instant at the controller side, and is the set of control at the sequences that are available at time interval actuator side, which includes the one in the cache of the actuator and any one that arrives at the actuator between this interval. From (9) and (10), the control signal adopted by the actuator at time is obtained as (11)

B. Design of Time-Delay Compensator The network introduces to the NCSs not only delays but also an advantage to the system in that a sequence of signals can be packed and transmitted simultaneously [12], [13]. Our time-delay compensator takes advantage of this characteristic of NCSs. The following assumptions are first made in the time delay compensator design. 1) For the sake of the calculation of the predictive control sequence, the time delay of the feedback channel needs to be known to the controller, which can be easily done by issuing a time stamp on each data packet from the sensor side to the controller side. 2) The round-trip time (RTT, noted by , the total time delay of feedback channel and forward channel, i.e., ) is known to the actuator, which can also be done by using the time stamps. 3) The predictive control sequences are packed and transmitted to the actuator simultaneously. . 4) The forward time delay is less than the control horizon The time-delay compensator works as follows: at every time instant , the predictive controller calculates a sequence of future control signals based on the outputs and control information be(the time stamp of the packet received at time fore of the controller side). The future control signals are then transmitted to the actuator side with a time stamp all in one packet. When a packet of a control sequence arrives at the actuator side (different packets may experience different time delays), it is compared with the one already in the cache of the actuator according to the time stamp, and only the latest is saved. The acif tuator then chooses the control action the time stamp of the control sequence in its cache is and the . forward time delay is Using this compensation scheme, the network-induced delays can be exactly compensated for.

where the

is a column vector with all entries 0 but th is 1, is the RTT with respect to , i.e., .2 Combining (1a), (5), (6), (7), (9), and (11), the TDTSGPC approach applied to a Hammerstein model can be fully described by the following equations ( is set to 0 without loss of generality): (12) (13) (14) (15) From the definition of obtain

,

, and

, we , thus

Combining with (15), we then obtain

(16) B. Closed-Loop Stability Here, the Popov criterion is applied to prove the stability of the TDTSGPC approach for constant delays. 2Note that the choice of the predictive control signal to be applied to the plant depends on the network-induced delay in the forward channel. This is different from conventional GPC applications where the first control signal is always used.

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Fig. 1. Popov theorem.

Fig. 2. Simplified block diagram of TDTSGPC.

Lemma 1 (Popov Criterion, See [8]): Suppose that in Fig. 1 is stable and . Then, the closed-loop . system is stable if In the case of constant delays, we have that , are all constant. Applying Lemma 1 to TDTSGPC and denote the characteristic polynomial of a by , we then obtain the transfer function following theorem. Theorem 1: Suppose that the linear part of the Hammerstein are located model is accurate and the roots of in the unit circle. Then, the closed-loop system of TDTSGPC is stable if there exists a positive constant such that the following is satisfied. 1) The input nonlinearity of the plant satisfies

Fig. 3. System response. 1) ( ; ) = (0; 0). 2) ( ; ) = (2; 3);.

(17) 2) The network induced delay satisfies (18) where

, and is the theoretical input value to the

CARIMA model. . Notice Proof: Without loss of generality, assume here that, for any column vector with comparable dimensions, by the definition of

.

Then, from (12)–(16), we obtain

(19) This is equivalent to the block diagram shown in Fig. 2. Thus, the theorem can be easily obtained by applying Lemma 1 to Fig. 2. IV. SIMULATION We give an example to illustrate the TDTSGPC approach in this section. The linear part of the system adopted is , and the input nonlinearity and the of the Hammerstein model is chosen as

Fig. 4. System response. 3) ( ; ) = (3; 7).

practical inverse of is , where is a random . This is introduced number with a uniform distribution in to represent the uncertainty in a practical implementation. From condition (1) of Theorem 1, we see that the parameter is 1 and the predictive horizon and control horizon are chosen as . It can be shown that the system is stable only for the first two cases according to Theorem 1 since for too large a time delay the system will not satisfy condition (2) in Theorem 1. The simulation results of three cases: 1) 2) and 3) are shown in Figs. 3 and 4 to illustrate the validity of the theoretical analysis. V. CONCLUSION In this paper, the two-step generalized predictive control approach, which is usually used in the controller design for the Hammerstein model, is integrated with a time-delay compensator to deal with networked control systems based on a Hammerstein model with random network-induced delays. This approach takes advantage of the characteristic of the network in an NCS such that a sequence of information can be packed

ZHAO et al.: NETWORKED PREDICTIVE CONTROL SYSTEMS BASED ON THE HAMMERSTEIN MODEL

to be transmitted simultaneously, so that the predictive control method can be easily implemented for NCSs. A theoretical result is presented for the stability of the system in the case of a constant time delay. Simulation work has also been done to illustrate the validity of the approach. Further research is still needed to analyze the stability conditions under random time delays, which is not addressed in this paper.

REFERENCES [1] J. Baillieul and P. J. Antsaklis, “Control and communication challenges in networked real-time systems,” Proc. IEEE, vol. 95, no. 1, pp. 9–27, Jan. 2007. [2] D. Yue, Q.-L. Han, and C. Peng, “State feedback controller design of networked control systems,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 11, pp. 640–644, Nov. 2004. [3] Y. Tipsuwan and M.-Y. Chow, “Control methodologies in networked control systems,” Control Eng. Practice, vol. 11, pp. 1099–1111, 2003. [4] J. P. Hespanha, P. Naghshtabrizi, and Y. G. Xu, “A survey of recent results in networked control systems,” Proc. IEEE, vol. 95, pp. 138–162, Jan. 2007.

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[5] J. Vörös, “Identification of hammerstein systems with time-varying piecewise-linear characteristics,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 52, no. 12, pp. 865–869, Dec. 2005. [6] K. J. Hunt, M. Munih, N. N. Donaldson, and F. M. D. Barr, “Investigation of the hammerstein hypothesis in the modeling of electrically stimulated muscle,” IEEE Trans. Biomed. Eng., vol. 45, no. 8, pp. 998–1009, Aug. 1998. [7] D. C. Evans, D. Rees, and D. L. Jones, “Identifying linear models of systems suffering nonlinear distortions, with a gas turbine application,” IET Control Theory Appl., vol. 142, no. 3, pp. 229–240, 1995. [8] B. C. Ding, S. Y. Li, and Y. Xi, “Stability analysis of generalized predictive control with input nonlinearity based on popov theorem,” ACTA Automatica SINICA, vol. 29, pp. 582–588, 2003. [9] J. P. Richard, “Time-delay systems: An overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003. [10] E. F. Camacho and C. Bordons, Model Predictive Control, 2nd ed. Berlin, Germany: Springer-Verlag, 2004. [11] G. Tao and P. V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York: Wiley, 1996. [12] G. P. Liu, Y. Xia, D. Rees, and W. Hu, “Networked predictive control of systems with random network delays in both forward and feedback channels,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1282–1297, Jun. 2007. -optimal [13] G. Daniel and D. M. Tilbury, “Packet-based control: The solution,” Automatica, vol. 42, no. 1, pp. 137–144, 2006.

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 3, JUNE 2008

A Predictive Control-Based Approach to Networked Hammerstein Systems: Design and Stability Analysis Yun-Bo Zhao, Guo-Ping Liu, Senior Member, IEEE, and David Rees

Abstract—In this paper, a predictive control-based approach is proposed for a Hammerstein-type system which is closed through some form of network. The approach uses a two-step predictive controller to deal with the static input nonlinearity of the Hammerstein system and a delay and dropout compensation scheme to compensate for the communication constraints in a networked control environment. Theoretical results are presented for the closed-loop stability of the system. Simulation examples illustrating the validity of the approach are also presented. Index Terms—Delay and dropout compensation scheme (DDCS), Hammerstein system, networked control systems (NCSs), predictive control, two-step approach.

I. I NTRODUCTION

A

CONTROL system is called a “networked control system” (NCS) when the direct connections used in conventional control systems between sensors, controllers, and actuators are replaced by some form of communication networks with limited resource [1]–[4]. This configuration, which is due to the network inserted, brings to the system lower cost, flexibility, the ability of remote control, etc., whereas the communication constraints of the network, e.g., the time delay of data exchange through the network (so-called “networkinduced delay”), data packet dropout, quantization, medium access constraint, etc., greatly degrade the performance of the control systems, even making the system unstable under certain conditions. Such a configuration presents a new challenge to conventional control theories [5]. The limits to the performance of control systems in a networked control environment are caused primarily by network-induced delay and data packet dropout [5]. These communication constraints can mean in NCSs that the control signal for the plant is delayed or even unavailable, which results in an open loop system. The desire to obtain a better performance than that resulting from holding the last available control signal or using zero control during open loop intervals in NCSs has led to a model-based control architecture [6]

Manuscript received May 9, 2007; revised October 9, 2007. This paper was recommended by Associate Editor W. Dixon. Y.-B. Zhao and D. Rees are with the Faculty of Advanced Technology, University of Glamorgan, CF37 1DL Pontypridd, U.K. (e-mail: [email protected]; [email protected]). G.-P. Liu is with the Faculty of Advanced Technology, University of Glamorgan, CF37 1DL Pontypridd, U.K., and also with the CSIS Laboratory, Chinese Academy of Sciences, Beijing 100080, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2008.918572

and to a predictive control-based control architecture [7], [9]–[12]. The key idea of the model-based approach is that the knowledge of the plant dynamics is used to reduce the usage of the network, whereas in the predictive control-based approach proposed in [9], the plant dynamics is further used to produce future control signals to actively compensate for the random network-induced delay in the forward channel with the use of a corresponding time-delay compensator at the actuator side. A better performance can be expected since the predictive control-based approach takes greater advantage of the knowledge available. In this paper, following the predictive control-based approach in [9], we extend its application to networked Hammerstein systems where a static nonlinear input process and random network-induced delays and data packet dropouts in both forward and backward channels exist. In order to deal with the static input nonlinearity of the Hammerstein system, a two-step design approach that is similar to that in [13] is applied, the key idea of which is to design for the linear part of the Hammerstein system first and then compensate for the input nonlinearity using an inverse process. The inaccuracy in compensation for the nonlinear input process is assumed to satisfy a sector constraint based on which the stability criteria of the closedloop system are obtained. Compared with previous results, the main advantage of the predictive controller designed in this paper is that only delayed sensing data are used, whereas in [9], the previous control signals up to the current step were all required, which data will be shown later to be hard to obtain in practice (Remark 2). To correspond to the new predictive controller, a novel compensation scheme for the communication constraints, which is called the delay and dropout compensation scheme (DDCS), is designed, which consists of three components: a matrix selector at the controller side to compensate for the network-induced delay in the backward channel, a delay compensator at the actuator side to compensate for the network-induced delay and data packet dropout in the forward channel, and a horizon adjustor for the controller to compensate for the network jitter by adjusting the control horizon according to current network condition (see Fig. 1 for the whole structure). The implementation of DDCS makes the predictive control-based approach work well in a network-based environment. The remainder of this paper is organized as follows. The design of the proposed approach is presented in Section II. Then, the theoretical results for the system stability and the simulation results are presented in Sections III and IV, respectively. The conclusions are given in Section V.

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ZHAO et al.: PREDICTIVE CONTROL-BASED APPROACH TO NETWORKED HAMMERSTEIN SYSTEMS

Fig. 1.

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Structure of networked predictive control system with input nonlinearity.

II. D ESIGN OF N ETWORKED P REDICTIVE C ONTROL S YSTEM W ITH I NPUT N ONLINEARITY The following single-input–single-output Hammerstein system S is considered in this paper:  (1a)  x(k + 1) = Ax(k) + bv(k) (1b) S : y(k) = cx(k)  v(k) = f (u(k)) (1c) where x ∈ Rn , u, v, y ∈ R, and f (·) : R → R is a memoryless static nonlinear function. In this section, we present first the design details of the twostep predictive control approach to system S and then the design of DDCS to compensate for the network-induced delays and data packet dropout when such a system is implemented in a networked control environment. A. Design of the Two-Step Predictive Control Approach The key idea of the typical two-step predictive control approach is to design an intermediate control signal v(k) of the linear part of system S [(1a) and (1b)] with a linear predictive control method (a linear generalized predictive control (LGPC) method is adopted in this paper) first and then obtain the real control signal u(k) for system S from the nonlinear relationship v(k) = f (u(k)) [10], [13]. In a networked control environment, the typical two-step predictive control approach is modified as follows with the consideration of the networkinduced delays. 1) Design of LGPC: In the presence of the network-induced delay, the following modified quadratic objective function is adopted: J(N1 , N2 , Nu ) =

N2

qj (ˆ y (k + j|k − τsc,k ) − ω(k + j))2

j=N1

+

Nu

rj ∆v 2 (k + j − 1) (2)

j=1

where N1 and N2 are the minimum and maximum prediction horizons, Nu is the control horizon, qj , N1 ≤ j ≤ N2 , and rj ,

1 ≤ j ≤ Nu , are weighting factors, ω(k + j), j = N1 , . . . , N2 , are the set points, ∆v(k) = v(k) − v(k − 1) is the control increment, and yˆ(k + j|k − τsc,k ), j = N1 , . . . , N2 , are the forward predictions of the system outputs, which are obtained on data up to time k − τsc,k and will be calculated in detail later, where τsc,k is the network-induced delay in the backward channel at time k. Let x ¯(k) = [ xT (k) v(k − 1) ]T , then system S can be represented by S as S :

¯x(k) + ¯b∆v(k) x ¯(k + 1) = A¯ y(k) = c¯x ¯(k)

(3a) (3b)

A b ¯ b ,b= , and c¯ = ( c 0 ). Thus, the 0 1 1 j step forward output prediction at time k is

where A¯ =

¯j

yˆ(k + j |k ) = c¯A x ¯(k ) +

j −1

c¯A¯j

−l −1¯

b∆v(k + l |k ).

l =0

Let j = j + τsc,k , k = k − τsc,k , and l = l + τsc,k . Then the forward output predictions at time k based on the information of the state on time k − τsc,k and control signals from time k − τsc,k − 1 are yˆ(k + j|k − τsc,k ) = c¯A¯j+τsc,k x ¯(k − τsc,k ) +

j−1

c¯A¯j−l−1¯b∆v(k + l|k − τsc,k ).

(4)

l=−τsc,k

If the state vector x is not available, an observer must be included x ˆ(k + 1|k) = Aˆ x(k|k − 1) + bv(k) +L (ym (k) − cˆ x(k|k − 1))

(5)

where ym (k) is the measured output. If the plant is subject to white noise disturbances affecting the process and the output with known covariance matrices, the observer becomes a Kalman filter, and the gain L is calculated solving a Riccati equation.

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Let Yˆ (k|k−τsc,k ) = [ˆ y (k+N1 |k−τsc,k ) · · · yˆ(k + N2 |k − τsc,k )]T , ∆V (k|k − τsc,k ) = [∆v(k − τsc,k |k − τsc,k ) · · · ∆v(k + Nu − 1|k − τsc,k )]T . Then ¯(k − τsc,k ) +Fτsc,k ∆V (k|k − τsc,k ) Yˆ (k|k − τsc,k ) = Eτsc,k x (6) cA¯N1 +τsc,k )T · · · (¯ cA¯N2 +τsc,k )T ]T and where Eτsc,k = [ (¯ Fτsc,k is an (N2 − N1 + 1) × (Nu + τsc,k ) matrix with the non-null entries defined by (Fτsc,k )ij = c¯A¯N1 +τsc,k +i−j−1¯b, j − i ≤ N1 + τsc,k − 1. Note here that Eτsc,k and Fτsc,k vary with different τsc,k ’s. Let k = [ ω(k + N1 ) · · · ω(k + N2 ) ]T . Then the optimal predictive control increments from k to k + Nu − 1 can be calculated by letting ∂J(·)/∂∆V = 0 ¯(k − τsc,k ) (7) ∆V (k|k − τsc,k ) = Mτsc,k k − Eτsc,k x where ∆V (k|k−τsc,k ) = [∆v(k|k−τsc,k ) · · · ∆v(k+Nu − 1|k − τsc,k )]T , Mτsc,k = Hτsc,k (FτTsc,k QFτsc,k + R)−1 FτTsc,k Q, Q and R are diagonal matrices with Qi,i = qi and Ri,i = ri , respectively, and Hτsc,k = [0Nu ×τsc,k INu ×Nu ], with INu ×Nu being the identity matrix with rank Nu . Remark 1: Normally, the minimum prediction horizon can be set as one. Rewrite the maximum prediction horizon N2 as Np . The following constraint between Nu and Np needs to be always held in order to implement the LGPC method successfully: Nu ≤ N p .

(8)

Remark 2: In [9], the previous control signals v(k − 1), . . . , v(k − τsc,k ) are used to calculate the predictive control sequence at time k. However, this information is hard to obtain for the controller in practice due to the random networkinduced delays in both channels. As will be discussed further in Section II-B, in a networked predictive control environment, a sequence of future control signals is packed and sent to the actuator, and the actuator only selects one from the sequence according to the specific time delay in the forward channel. Therefore, the controller does not know the real control signal adopted by the actuator until it receives the information about the previous control signals applied to the actuator. Only in such a special case that, with no delay in the forward channel, the previous control signals are all known by the controller immediately. Therefore, in this paper, we develop a new method to deal with this problem, in which only the control and state (output) information at time k − τsc,k is used to generate the predictive control sequence, by including the control sequence from time k − τsc,k to k − 1 as part of the predictive control sequence. As a result, the forward predictive control sequence obtained depends only on delayed sensing data at time k − τsc,k (7), which is always available to the controller [see Assumption A3)], thus enabling the approach to be feasible in practice. 2) Nonlinear Input Process: Assuming that the nonlinear function f (·) is invertible and denoting its inverse by fˆ−1 (·), then ∆u(k) = fˆ−1 (∆v(k)) .

(9)

Thus, at every time instant k, the intermediate control increments ∆v(k), k = 1, 2, . . . , Nu , can be obtained from (7), and then, the real control increments ∆u(k), k = 1, 2, . . . , Nu , can be calculated from (9), thus enabling the control law to be derived for system S . If ∆u(k) can be calculated accurately using (9), thus enabling the function fˆ−1 (·) to be exactly known, then the system with compensation for the nonlinear input process is equivalent to LGPC, and the system is stable if and only if the linear part of system S with LGPC is stable. However, in practice, it is usually impossible to calculate u(k) that accurately, i.e., fˆ−1 (f (·)) = 1(·). This inaccuracy introduces to the LGPC a nonlinear disturbance, which makes the stability analysis difficult. ˆ −1 For simplicity of notation, let f (·) : RNu → RNu ˆ −1 with f (∆V (k|k − τsc,k )) = [fˆ−1 (∆v(k|k − τsc,k )) · · · fˆ−1 (∆v(k + Nu − 1|k − τsc,k ))]T . Then, from the earlier discussion, the real predictive control increment sequence for system S can be represented by ˆ −1 ∆U (k|k − τsc,k ) = f (∆V (k|k − τsc,k )

(10)

where ∆U (k|k − τsc,k ) = [∆u(k|k − τsc,k ) · · · ∆u(k + Nu − 1|k − τsc,k )]T . Remark 3: Note that the control increment, instead of the control signal itself, is used in the compensation for the nonlinear input process in (9). Although the use of control increments complicates the problem in that the past control increments are also needed to determine the current control increment, it is inevitable since the objective function to be optimized takes the form of control increments. In order to implement the predictive controller in this paper, the past control increments are sent to the controller as well as the state information [see Assumption A3)], which is different from conventional control systems. Note that for a system without a nonlinear input process (1c), it makes no difference whether the intermediate control increment or the intermediate control signal itself is used to calculate the real control signal, whereas for system S, generally, these two methods give different control input at time k, i.e., f (∆v(k)) = f (v(k)) − f (v(k − 1)). B. Design of DDCS To enable the predictive controller designed in this paper to work appropriately in a networked control environment, a DDCS is proposed to compensate for the network-induced delay and data packet dropout in NCSs. The following assumptions are first made for the DDCS design. A1) Each data packet containing the sensing data is sent with a time stamp to notify when it was sent from sensor to controller. This enables the network-induced delay in the backward channel for each data packet known to the controller. This information is then used to calculate the appropriate control predictions. A2) At every time instant k, the control predictions ∆U (k|k − τsc,k ) with time stamps k and τsc,k are


packed into one data packet and sent to the actuator. These time stamps are to notify the time when it was sent and also the network-induced delay in the backward channel which the calculation of the control predictions was based on. This enables the networkinduced delays in both channels for each control predictive sequence known to the actuator. A3) The information of the control increment signal actually applied to the plant is also sent to the controller. A4) The control horizon is chosen in such a way that the sum of the maximum network-induced delay in the forward channel (noted by τ¯ca ) and the maximum number of continuous data packet dropout (noted by χ) ¯ is bounded by Nu , i.e., τ¯ca + χ ¯ ≤ Nu − 1.

(11)

Remark 4: The data packet dropout is not treated as a long delay in this paper. They are simply ignored, and the measurement of the delay bound is only over those received successfully so that the delay bound can be assumed to be finite. In this way, the data packet dropout does not need to be specially treated. This can be compared with the approaches in [14] and [15], where the effect of the data packet dropout is explicitly considered. Based on the aforementioned assumptions, the three components of the DDCS, the matrix selector, the delay compensator, and the horizon adjustor, which are to deal with the networkinduced delay in the backward channel, network-induced delay and data packet dropout in the forward channel, and the network jitter, respectively, are presented in the following sections. 1) Compensation for the Random Network-Induced Delay in the Backward Channel—A Matrix Selector: Note that the matrices Eτsc,k , Fτsc,k , Mτsc,k , and Hτsc,k are all needed to implement the predictive controller in (7), which vary with τsc,k and, if computed online, will present a great computation burden for the controller and introduce additional computation delay to the system. Fortunately, although these matrices vary with the delay in the backward channel, they can be calculated offline since all the matrices are fixed for a given τsc . This advantage enables us to calculate offline all the matrices with respect to the specific τsc ’s, to store them in a device called the “matrix selector,” and to just choose the appropriate ones from the matrix selector when calculating online the predictive control increments according to the current value of the delay τsc,k , which is known to the controller from Assumption A1). In this way, the computation delay can be reduced to a certain extent. Let Esc = {E0 , E1 , . . . , Eτ¯sc }, Fsc = {F0 , F1 , . . . , Fτ¯sc }, Msc = {M0 , M1 , . . . , Mτ¯sc }, and Hsc = {H0 , H1 , . . . , Hτ¯sc }, where τ¯sc is the upper bound of the network-induced delay in the backward channel, then we have for any k (or τsc,k ), Eτsc,k ∈ Esc , Fτsc,k ∈ Fsc , Mτsc,k ∈ Msc , and Hτsc,k ∈ Hsc , respectively. For a practical implementation, these 4×(¯ τsc +1) matrices are calculated offline and stored in the matrix selector for online use.

703

2) Compensation for the Random Network-Induced Delay and Data Packet Dropout in the Forward Channel—A Delay Compensator: As presented in Assumption A2), the predictive control increment sequence ∆U (k|k − τsc,k ) is sent to the actuator all in one data packet. When a new sequence arrives at the actuator side, it is compared with the one already in the so-called “delay compensator” according to the time stamps (which notify the time when the sequences were sent from the controller), and only the one with the latest time stamp is stored. The delay compensator is specially designed for the actuator, and it can only store one control sequence (data packet) at any time. For example, denote the sequence that arrives at the actuator side as ∆U (k1 |k1 − τsc,k1 ) with a time stamp k1 and the one already in the delay compensator as ∆U (k2 |k2 − τsc,k2 ) with a time stamp k2 . Then, if k1 > k2 , ∆U (k2 |k2 − τsc,k2 ) will be replaced by ∆U (k1 |k1 − τsc,k1 ); otherwise, ∆U (k1 |k1 − τsc,k1 ) will be simply discarded, and the delay compensator remains unchanged. The comparison process is introduced at the actuator side due to the fact that different data packets may experience different delays in the forward channel, thereby producing a situation where, for example, a data packet sent earlier from the controller may arrive at the actuator later or may never arrive in the case of data packet dropout. As a result of the comparison process, the predictive control sequence stored in the delay compensator is always the latest one available at any specific time. As for the actuator, it can be either time driven or event driven. The difference between the two driven methods lies in the trigger method that initiates the actuator. For time-driven actuator, the actuator is trigged to work at regular intervals, no matter whether the delay compensator is updated or not, whereas for event-driven actuator, it is only trigged by the update of the delay compensator, i.e., a new predictive control sequence is stored in the delay compensator. Whatever method is used, the actuator selects the appropriate control increment signal which can compensate for current network-induced delay in the forward channel from the predictive control increment sequence in the delay compensator at every execution time instant and then applies it to the plant. The method to choose the appropriate control increment signal at a specific time will be explained in detail in the next section. It is necessary to point this out that the appropriate control increment is always available using the delay compensator provided that Assumption A4) holds. 3) Compensation for the Network Jitter—A Horizon Adjustor: A larger control horizon generally leads to a better performance for a typical GPC implementation, whereas in a networked control environment, a larger control horizon means a greater computation burden for the controller and, more severely, a greater communication burden for the network, since more control predictions are computed and transmitted through the network (note that the size of the control predictive sequence is proportional to the control horizon Nu ). This may result in network traffic congestion and makes the performance of the NCS worse on the contrary. Therefore, we argue that an appropriately chosen control horizon is important for the performance of the proposed approach, and hence, a

704


horizon adjustor is proposed in this paper, which adjusts the control horizon Nu by taking account of the current network performance. In the design of the horizon adjustor, the constraints for Nu [see (8) and (11)] should always be satisfied for the successful implementation of both the LGPC method and the delay compensator. Notice also that the period of updating the control horizon depends on the network conditions. A period of T can be used if the network condition does not change much during this period. The horizon adjustor using a period T can therefore be obtained as τca (t), χ(t)) ¯ Nu (kT ) = Nu ((k − 1)T ) + ψ (¯ Nu (t) = Nu (kT ),

t ∈ [kT (k + 1)T )

(12a) (12b)

with the constraints τ¯ca (t) + χ(t) ¯ + 1 ≤ Nu ≤ Np [constraints ¯ are the upper bounds of the (8) and (11)], where τ¯ca (t) and χ(t) network-induced delay and continuous data packet dropout in the forward channel during the next period of T , respectively, and ψ(·, ·) is an adjusting function to adjust the control horizon dynamically with the network conditions. Since the future network condition is unavailable in practice, previous information could be used instead. A simple form of ψ(·, ·) can then be ¯ = ρt ·(¯ τca (t) + χ(t) ¯ − τ¯ca (t − 1) − χ(t ¯ − 1)) ψ(¯ τca (t), χ(t)) (13) where ρt is an adjusting factor to reflect the extent of the network jitter. ρt will be set to be large if the network jitter is severe and vice versa. In the implementation of the horizon adjustor, Np remains to be a constant which results in Esc unchanged. What is required is to calculate different sets of Fτsc,k , Mτsc,k , and Hτsc,k with respect to different Nu ’s offline and store them in the matrix selector for online use. Np is chosen in such a way that the data packet containing the control predictions does not exceed the packet size limit of the network used even if Nu = Np , which enables the control predictions to be packed into one data packet. The two-step predictive control approach with DDCS can now be summarized as follows, within a specific period T of the horizon adjustor. S1) Calculation. The predictive controller calculates the intermediate predictive control increment sequence ∆V (k|k − τsc,k ) using (7) with the use of the proposed matrix selector and delayed information of states and control signals. The predictive control increment sequence ∆U (k|k − τsc,k ) is then obtained by compensating for the nonlinear input process using (10). S2) Forward transmission. ∆U (k|k − τsc,k ) is packed and sent to the actuator simultaneously with time stamps k and τsc,k . S3) Comparison. The delay compensator updates its information according to the time stamps once a data packet arrives.

Fig. 2.

Time delays of the control signal adopted by the actuator at time k.

S4) Execution. An appropriate control increment signal is picked out from the control sequence in the delay compensator and applied to the plant. S5) Backward transmission. The information of the applied control increment with the sensing state is sent to the controller. The structure of the predictive control-based approach with DDCS [the so-called “networked predictive control systems”(NPCSs)] is shown in Fig. 1. III. S TABILITY A NALYSIS In this section, the closed-loop formulation of such an NPCS with a nonlinear input process is derived, and then, the stability theorem is obtained by using a switched system theory under a sector constraint of the nonlinearity due to calculation inaccuracy. A. Closed-Loop System ∗ denote the network-induced delay in the forward Let τca,k channel of the predictive control increment sequence, from which the control signal is picked out by the actuator at time instant k. The time when the sequence was sent from the controller side can then be read from its time stamp as ∗ = max {j|∆U (j|j − τsc,j ) ∈ Γk } k ∗ = k − τca,k j

(14)

where Γk is the set of the predictive control increment sequences that is available during time interval (k − 1, k] at the actuator side, including not only the one in the delay compensator but also others that arrive at the actuator during this interval (see Fig. 2). From (10) and (14), the control signal adopted by the actuator at time k is obtained as ∆u(k) = ∆u (k|k − τk∗ ) ∗ ∗ = dT τ ∗ ∆U k − τca,k |k − τk ca,k

(15)

∗ where dτca,k is an Nu × 1 matrix with all entries being zero, ∗ except that (τca,k + 1)th is one, τk∗ is the round trip time with ∗ ∗ ∗ ∗ + τsc,k , and τsc,k = τsc,k∗ . respect to τca,k , i.e., τk∗ = τca,k From (7) and (10) and noticing for any vector V with an apˆ −1 propriate dimension, dT∗ f (V ) = fˆ−1 (dT∗ V ) recalling

τca,k

τca,k


ˆ −1 the definition of f (·); thus, we obtain (assume that the set point ω = 0 without loss of generality) ∗ ∗ ∆u(k) = dT τ ∗ ∆U k − τca,k |k − τk ca,k

ˆ −1 ∗ = dT ∆V (k − τca,k |k − τk∗ f ∗ τca,k

∗ = fˆ−1 dT ∆V (k − τca,k |k − τk∗ ∗ τca,k ∗ x ¯ (k − τk∗ ) = fˆ−1 −Kτ,k

(16)

∗ where Kτ,k = dT Mτsc,k Eτsc,k .1 The real control increment ∗ τca,k for linear system [(1a) and (1b)] at time k can then be obtained as ∗ ∆v(k) = f (∆u(k)) = f ◦ fˆ−1 −Kτ,k x ¯ (k − τk∗ ) (17)

where f ◦ fˆ−1 (·) = f (fˆ−1 (·)) is the composite function of f (·) and fˆ−1 (·). ¯T (k) ]T , w(k) = ∆v(k). Let X(k) = [ x ¯T (k − τ¯) · · · x Then the closed-loop system can be represented by   X(k + 1) = AX(k) + bw(k) (18a)

S∗ : −1 ∗ ˆ  w(k) = f ◦ f −K X(k) (18b) τ¯,k

T ∗ where b = [ 0n+1,1 · · · 0n+1,1 ¯bT n+1,1 ] , Kτ¯,k is a 1 × (¯ τ + 1) block matrix with a block size of 1 × (n + 1), and all ∗ (the set its blocks are zero, except that (¯ τ + 1 − τk∗ )th is Kτ,k ∗ of all the possible Kτ¯,k ’s will be denoted by K), and

0

n+1

  = A  

In+1 0n+1 0

 In+1 .. .

..

.

0n+1

As has been pointed out in Section II-A2, the compensation for the nonlinear input process using (9) is generally not accurate, and this inaccuracy introduces to the linear part of the system [see (1a) and (1b)] a nonlinear disturbance, which appears in the form of f ◦ fˆ−1 (·). Although, generally, f ◦ fˆ−1 (·) ≡ 1, it is reasonable to assume that the calculation error meets some accuracy requirement to a certain extent, which results in a sector constraint of the term f ◦ fˆ−1 (·), as described in Assumption A5) as follows.2 A5) The nonlinearity due to the calculation inaccuracy is supposed to satisfy a sector constraint, i.e., there exist 0 < ε ≤ ε¯ < ∞, s.t. ∀α ∈ R.

(19)

∗ that the value of Kτ,k varies with the delays in both channels, and thus, it has (¯ τca + 1)(¯ τsc + 1) different values in total. 2 Note that, although it is reasonable to place a sector constraint as in Assumption A5) to f ◦ fˆ−1 (·), it is somewhat conservative since the calculation of some strongly nonlinear function may not be that accurate and, thus, does not satisfy A5). 1 Note

This constraint can be denoted by f ◦ fˆ−1 (·) ∈ [ε, ε¯].

(20)

Notice here that, generally, 0 < ε ≤ 1 ≤ ε¯ < ∞. By using Assumption A4), we obtain that for any specific α ∈ R, there exists a real number εα , ε ≤ εα ≤ ε¯, such that f ◦ fˆ−1 (α) = εα α; (18b) can thereby be rewritten as w(k) = f ◦ fˆ−1 −Kτ¯∗,k X(k) = − εk Kτ¯∗,k X(k)

(21)

where εk ∈ [ε, ε¯] represents the compensation for the specific nonlinearity for the term Kτ¯∗,k X(k) at time k. Recalling (18a) and (21), the closed-loop system S ∗ can then be written as X(k + 1) = AX(k) + bw(k)

− εk bKτ¯∗,k X(k) = A = Λ εk , Kτ¯∗,k X(k)

(22)

− εk bK ∗ has where the closed-loop matrix Λ(εk , Kτ¯∗,k ) = A τ¯,k the form 0  I n+1

  ∗ Λ εk , Kτ¯,k =   

n+1

0n+1

···

0   .   In+1 ¯ A

B. Stability Analysis

εα ≤ f ◦ fˆ−1 (α) ≤ ε¯α,

705

0 ∗ −εk ¯bKτ,k

In+1 .. . ···

..

. 0n+1

0   .   In+1 ¯ A

∗ The position and value of the term −εk ¯bKτ,k depend on the specific delays in the both channels at time k, i.e., ∗ , j = τk∗ = 1, 2, . . . , τ¯, and (Λ(εk , Kτ¯∗,k ))τ¯+1,j = −εk ¯bKτ,k ∗ ∗ ¯ ¯ , if τk∗ = τ¯ + 1. (Λ(εk , Kτ¯,k ))τ¯+1,¯τ +1 = A − εk bKτ,k Theorem 1: The closed-loop system S ∗ is stable if A4) holds and there exists a positive definite solution P = P T > 0 for the τsc + 1) LMIs: following 2(¯ τca + 1)(¯ (23a) ΛT ε, Kτ¯∗,k P Λ ε, Kτ¯∗,k − P ≤ 0 ΛT ε¯, Kτ¯∗,k P Λ ε¯, Kτ¯∗,k − P ≤ 0 (23b)

where Kτ¯∗,k ∈ K. Proof: Let V (k) = X T (k)P X(k) be a Lyapunov function candidate, then the incremental V (k) for system S ∗ can be obtained using (22)

T ∆V (k) = X T (k) Λ εk , Kτ¯∗,k P Λ εk , Kτ¯∗,k − P X(k)

TP bKτ¯∗,k −εk Kτ¯∗T,k bTP A TP A−P = X T (k) A −εk A + ε2k Kτ¯∗T,k bT P bKτ¯∗,k X(k) X T (k)A εk , Kτ¯∗,k X(k) where εk ∈ [ε, ε¯] and Kτ¯∗,k ∈ K.

(24)

706


Notice that for any εk ∈ [ε, ε¯], there exists 0 ≤ λk ≤ 1 s.t. ε, and thus, we obtain by substituting this εk = λk ε + (1 − λk )¯ into (24) A εk , Kτ¯∗,k = λk A ε, Kτ¯∗,k + (1 − λk )A ε¯, Kτ¯∗,k − λk (1 − λk )(ε − ε¯)2 Kτ¯∗T,k bT P bKτ¯∗,k .

(25)

From (23a), (23b), and (24), A(ε, Kτ¯∗,k ) and A(¯ ε, Kτ¯∗,k ) are ∗ seminegative definite for all Kτ¯,k ∈ K. Notice that P is symmetric positive definite and that Kτ¯∗T,k bT P bKτ¯∗,k is semipositive definite as a symmetric matrix, thus enabling A(εk , Kτ¯∗,k ) to be seminegative definite for any εk ∈ [ε, ε¯] and Kτ¯∗,k ∈ K, which completes the proof. Remark 5: It is necessary to point this out that according to Assumption A5) and Theorem 1, what is required for the stability of the system is to satisfactorily meet the sector constraint in (20) no matter how the inverse function fˆ−1 (·) is calculated. It implies that the function f (·) does not need to be theoretically invertible as long as its inverse can be obtained by a numerical method and satisfies the sector constraint [one can refer to [16] and the references therein for more information of the calculation of fˆ−1 (·)]. Remark 6: When the horizon adjuster is also considered, the feedback gain Kτ¯∗,k in (16) will depend on a different control horizon Nu and can be rewritten as Kτ¯∗,Nu ,k . Thus, the set K now consists of all the possible Kτ¯∗,Nu ,k , mint (τca (t) + χ(t)) + 1 ≤ Nu ≤ Np . A similar stability criterion to Theorem 1 can then be obtained analogously. The following two special conditions are also considered for the stability of the closed-loop system. C1) The network-induced delays in both channels are constant 0 0 and τca , respectively). (noted by τsc C2) The calculation of the inverse of the nonlinear function is accurate. The following corollary can be easily obtained by using Theorem 1. Corollary 1: The closed-loop system S ∗ is stable if any one of the following three conditions holds. 1) A4) and C1) hold, and there exists a positive definite solution P = P T > 0 for the following two LMIs: (26a) ΛT ε, Kτ¯∗,k P Λ ε, Kτ¯∗,k − P ≤ 0 T ∗ ∗ Λ ε¯, Kτ¯,k P Λ ε¯, Kτ¯,k − P ≤ 0 (26b) 0 0 where τsc,k ≡ τsc , τca,k ≡ τca , and Kτ¯∗,k is therefore fixed. 2) C2) holds, and there exists a positive definite solution τca + 1)(¯ τsc + 1) LMIs: P = P T > 0 for the following (¯ ΛT 1, Kτ¯∗,k P Λ 1, Kτ¯∗ ,k − P ≤ 0 (27)

where Kτ¯∗,k ∈ K. 3) Both of C1) and C2) hold, and there exists a positive definite solution P = P T > 0 for the following LMI: ΛT 1, Kτ¯∗,k P Λ 1, Kτ¯∗ ,k − P ≤ 0 (28) 0 0 where τsc,k ≡ τsc , τca,k ≡ τca , and Kτ¯∗,k is therefore fixed.

Fig. 3.

State evolution using LQR method.

Fig. 4.

State evolution using the approach in this paper.

IV. S IMULATION In this section, a second-order Hammerstein model is adopted to illustrate the effectiveness of the proposed approach. The system matrices in (1a) and (1b) of system S are set as follows which is open-loop unstable: 0.98 0.1 0.04 A= b= c = ( 1 0 ). 0 1 0.1 We first use this linear system [see (1a) and (1b)] to illustrate the effectiveness of the proposed predictive controller and the compensation scheme DDCS for the communication constraints. In order to do this by comparison, the linear quadratic optimal (LQR) control method is used to design a state feedback law for this system without consideration of the communication constraints, which yields the feedback gain KLQR = [ 0.7044 1.3611 ]. The simulation result shows that it is unstable using this LQR control when there is random delays in both channels (the upper bounds of the delays are τ¯ = 3, τ¯ca = 2, and τ¯sc = 1, see Fig. 3), whereas it is stable


707

V. C ONCLUSION In this paper, a novel approach with the integration of the two-step predictive control method and a DDCS is proposed for a Hemmerstein system in a networked control environment. In the approach, the predictive controller for the linear part of the system is first designed by using delayed sensing data, and the nonlinear input can be viewed as a nonlinear disturbance after a compensation scheme. The communication constraints considered in this paper, i.e., random delays in both channels and data packet dropout in the forward channel, are dealt with by the DDCS, which consists of three components configured at both the controller and actuator sides. The stability theorem for the closed-loop system is obtained by using switched system theory. Simulation work has also been done to illustrate the effectiveness of the proposed approach. Fig. 5.

Random delays in the forward channel.

ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous referees for their insightful comments and suggestions which helped to improve this paper. R EFERENCES

Fig. 6.

Effectiveness of the compensation for the input nonlinear process.

using the proposed approach in this paper (Fig. 4). The random delays in the forward channel are shown in Fig. 5. Other parameters of the simulation are chosen as Nu = 8, Np = 10, and the initial state x0 = [ −1 −1 ]T . The delays in both channels are set to vary randomly within their upper bounds. Note the fact that with an inverse process to compensate for the static input nonlinearity in the Hammerstein system, from (18b), we know that the system performance only depends on the accuracy of this compensation process, i.e., the size of the sector constraint [ε, ε¯] for f ◦ fˆ−1 (·) [see (20)]. In this simulation, we set [ε, ε¯] = [0.5, 1.5], which means that there is approximately 50% error in the compensation for the input nonlinearity, whereas the input nonlinear function f (·) can be of any form provided that this compensation accuracy is satisfied. All the other parameters are set as the same as the aforementioned ones. Such a system with those parameters can be proved to be stable using Theorem 1. The effect of the compensation for the input nonlinearity is shown in Fig. 6, from which it is seen that the compensation accuracy for the input nonlinearity is effective.

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[15] P. Mhaskar, A. Gani, C. McFall, P. D. Christofides, and J. F. Davis, “Faulttolerant control of nonlinear process systems subject to sensor faults,” AIChE J., vol. 53, no. 3, pp. 654–668, Mar. 2007. [16] G. Tao and P. V. Kokotovic, Adaptive Control of Systems With Actuator and Sensor Nonlinearities. New York: Wiley, 1996.

Yun-Bo Zhao received the B.Sc. degree in mathematics from Shandong University, Shandong, China, in 2003, and the M.Sc. degree in systems theory from the Institute of Systems Science, Chinese Academy of Sciences, Beijing, in 2007. He is currently working toward the Ph.D. degree at the University of Glamorgan, Pontypridd, U.K. His research interests include networked control systems, model predictive control, Markov jump systems, and switched systems.

Guo-Ping Liu (M’97–SM’99) received the B.Eng. and M.Eng. degrees in electrical and electronic engineering from the Central South University of Technology (now the Central South University), Changsha, China, in 1982 and 1985, respectively, and the Ph.D. degree in control engineering from the University of Manchester, Manchester, U.K., in 1992. From 1992 to 1993, he did postdoctoral research with the University of York, York, U.K. In 1994, he was a Research Fellow with the University of Sheffield, Sheffield, U.K., and a Visiting Professor with the Central South University, Changsha, China. From 1996 to 2000, he was a Senior Engineer with GEC-Alsthom and ALSTOM and then a Principal Engineer and a Project Leader with ABB ALSTOM Power. From 2000 to 2003, he was a Senior Lecturer with the University of Nottingham, Nottingham, U.K. He has been a Professor with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, U.K., since 2004, and a Visiting Professor with the Chinese Academy of Sciences, Beijing, since 2000. He has worked in more than 50 academic research and industrial technology projects. He has more than 300 publications on control systems. He has authored or coauthored six books. He is a Chair of control engineering with the University of Glamorgan. He is the Editor-in-Chief of the International Journal of Automation and Computing. His main research areas include networked control systems, modeling and control of fuel cells, advanced control of industrial systems, nonlinear system identification and control, and multiobjective optimization and control. Prof. Liu is the General Chair of the 2007 IEEE International Conference on Networking, Sensing and Control. He was awarded the Alexander von Humboldt Research Fellowship in 1992. He received the best paper prize for applications at the UKACC International Conference on Control in 1998. His paper was shortlisted for the best application prize at the 14th IFAC World Congress in 1999.

David Rees received the B.Sc. (Hons.) degree in electrical engineering from the University of Wales, Swansea, U.K., in 1967, and the Ph.D. degree from the Council of National Academic Awards, U.K., in 1976. He is currently a part-time Reader with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, U.K., where he was previously the Director of Research and Associate Head in the School of Electronics. He has published over 170 journal and conference publications. He was the Joint Editor of Industrial Digital Control Systems (Peter Peregrinus, London: 1988) and has contributed to numerous monographs; the latest is three chapters in Dynamic Modelling of Gas Turbines—Identification, Simulation, Condition Monitoring and optimal Control (Springer, 2004). His current research interests include nonlinear modeling, system identification, and networked control systems. Dr. Rees is a Fellow of the Institution of Engineering and Technology and a past Chairman of the IEE Control Applications Professional Group. He was the recipient of an IEE Premium Award in 1996.

International Journal of Innovative Computing, Information and Control Volume x, Number 0x, x 2008

c ICIC International °2008 ISSN 1349-4198 pp. 0–0

A PREDICTIVE CONTROL BASED APPROACH TO NETWORKED WIENER SYSTEMS Yun-Bo Zhao, Guoping Liu and David Rees Faculty of Advanced Technology University of Glamorgan Pontypridd, CF37 1DL, United Kingdom {yzhao, gpliu,drees}@glam.ac.uk

Abstract. A predictive control based approach is proposed to deal with a Wiener type system which is closed through a network. In this approach, an output feedback predictive controller is designed using delayed sensing data with a specially designed state observer. The network constraints, i.e., the network-induced delay and data packet dropout, are compensated in both the forward and backward channels by taking advantage of the characteristics of both the predictive controller and the network transmission. Stability of the closed-loop system is derived by using the separation principle and switched system theory. Simulations illustrate the validity of the proposed approach. Keywords: Networked control systems, Predictive control, Wiener system, Network constraints

1. Introduction. Networked Control Systems (NCSs) is an emerging research area in recent years. Distinct from conventional control systems, where the links from sensor to controller (“backward channel”) and from controller to actuator (“forward channel”) are assumed to be connected directly with no data loss or delay through the links, in NCSs, instantaneous and perfect signals between these components are not achievable due to the inserted network [11, 13]. Despite the ability of remote and distribute control that such a configuration brings, the network constraints, i.e., the network-induced delays, data packet dropout, communication bandwidth limitation, data rate constraints, etc. in NCSs present a new challenge to conventional control theory [2, 5, 6, 10, 12]. A challenging aspect of the networked configuration is that we need to compensate for the negative effects of the network constraints to retain stability and performance of the system. For this purpose, a natural and necessary approach is to take advantage of all the information available on the network to design the controller rather than separate the design of the controller and network protocols. Preliminary work on this can be found in a number of publications under the name of “co-design” [3, 14, 15]. Following this idea, a model based control architecture was proposed in [9], where the knowledge of the plant dynamics was used to reduce the usage of the network. Furthermore, a predictive control based control architecture was also reported recently in [4, 8, 15]. In [8], knowledge of the plant dynamics was used to produce future control signals to actively compensate for the random network-induced delay in the forward channel with the use of a corresponding time delay compensator at the actuator side. A better performance can be expected since the predictive control based approach takes greater advantage of the knowledge available. 1

2

Y.-B.ZHAO, G.P.LIU AND D.REES

However, no results on nonlinear NCSs have been reported to date under such a predictive control based framework. In this paper, a modified predictive control based approach is applied to a Wiener type system with network constraints [1, 7]. For the output nonlinearity in the Wiener type system, an output feedback predictive controller is obtained using delayed sensing data with the help of a specially designed state observer. Unlike normal predictive control applications, where only the first predictive input of the predictive control sequence is applied to the plant, in this paper, the whole predictive control sequence is packed and sent to the actuator through the network and the appropriate predictive input is chosen by the actuator by a certain rule. With this modification, the conventional predictive control method can readily extend its application to the networked control environment, where the network-induced delay and data packet dropout are exactly compensated for. The stability of the closed-loop system is obtained by proving the stability of the proposed state observer under certain conditions and modeling the closed-loop system as a switched system. The remainder of the paper is organized as follows. Section 2 presents the design details of the proposed predictive based approach to Wiener systems with network constraints; Section 3 analyzes the stability of the closed-loop system; Section 4 gives a simple example to illustrate the validity of the proposed approach and Section 5 concludes the paper.

Figure 1: Wiener Systems Closed through Networks

2. Design of the Predictive Control Based Approach to Wiener Systems with network constraints. We consider the following Single-Input-Single-Output (SISO) Wiener type system S which is closed through some form of network (Fig.1) in this paper,  (1a)  x(k + 1) = Ax(k) + bu(k) (1b) S : y(k) = cx(k)  z(k) = f (y(k)) (1c)

3

where x ∈ Rn , u, y, z ∈ R, A ∈ Rn×n , b ∈ Rn×1 , c ∈ R1×n . The memoryless static nonlinear function f (·) is assumed to be invertible with its inverse denoted by fˆ−1 (·). Notice that fˆ−1 (·) can not be obtained accurately in practice which means ϕ(·) , fˆ−1 (f (·)) 6≡ 1(·). The approximate intermediate output y˜(k) (Fig.2) can thus be obtained as follows, y˜(k) = fˆ−1 (z(k)) = ϕ(y(k))

(2)

With this inverse process, the predictive controller for system S in a networked environment can then be obtained using a Linear Generalized Predictive Control (LGPC) method and a state observer as follows. 2.1. Design of the predictive controller using delayed data. Let the cost function be defined by J(N1 , N2 , Nu ) =

N2 X

2

qj (ˆ y (k + j|k − τsc,k ) − ω(y; k + j)) +

Nu X

rj ∆u2 (k + j − 1)

(3)

j=1

j=N1

where N1 and N2 are the minimum and maximum prediction horizons, Nu is the control horizon, qj , N1 ≤ j ≤ N2 and rj , 1 ≤ j ≤ Nu are weighting factors, ∆u(k) = u(k)−u(k−1) is the control increment, yˆ(k + j|k − τsc,k ), j = N1 , ..., N2 are the forward predictions of the system outputs, which are obtained on data up to time k − τsc,k , where τsc,k is the network-induced delay in the backward channel at time k; ω(y; k + j) is the set point with respect to y and can be obtained approximately by inverting corresponding set point ω(z; k + j) with respect to z, i.e., ω(y; k + j) = fˆ−1 (ω(z; k + j)), j = N1 , ..., N2

(4)

¯ Let x¯(k) = [xT (k) u(k − 1)]T , then system S can be rewritten as S, ½ ¯x(k) + ¯b∆u(k) x¯(k + 1) = A¯ (5a) S¯ : y(k) = c¯x¯(k) (5b) µ ¶ µ ¶ ¡ ¢ A b b where A¯ = , ¯b = , c¯ = c 0 . Thus the j 0 step forward output 0 1 1 0 prediction at time k is j 0 −1 0 yˆ(k + j |k ) = c¯A¯j x¯(k 0 ) +

0

0

0

X

c¯A¯j −l −1¯b∆u(k 0 + l0 ) 0

0

l0 =0

Let j 0 = j + τsc,k , k 0 = k − τsc,k , l0 = l + τsc,k , then the forward output predictions at time k based on the information of the state on time k − τsc,k and control signals from time k − τsc,k − 1 is ¯j+τsc,k

yˆ(k + j|k − τsc,k ) = c¯A

x¯(k − τsc,k ) +

j−1 X

c¯A¯j−l−1¯b∆u(k + l)

(6)

l=−τsc,k

Let Yˆ (k|k − τsc,k ) = [ˆ y (k + N1 |k − τsc,k ) · · · yˆ(k + N2 |k − τsc,k )]T , ∆U 0 (k|k − τsc,k ) = [∆u(k − τsc,k |k − τsc,k ) · · · ∆u(k + Nu − 1|k − τsc,k )]T , then Yˆ (k|k − τsc,k ) = Eτsc,k x¯(k − τsc,k ) + Fτsc,k ∆U 0 (k|k − τsc,k )

(7)

4


Figure 2: Predictive Based Approach to Wiener Systems

cA¯N1 +τsc,k )T · · · (¯ cA¯N2 +τsc,k )T ]T , Fτsc,k is a (N2 − N1 + 1) × (Nu + τsc,k ) where Eτsc,k = [(¯ matrix with the non-null entries defined by (Fτsc,k )ij = c¯A¯N1 +τsc,k +i−j−1¯b, j − i ≤ N1 + τsc,k − 1. Note here that Eτsc,k and Fτsc,k vary with different τsc,k s. Let $k (y; ·) = [ω(y; k+N1 ) · · · ω(y; k+N2 )]T , the optimal predictive control increments from k to k + Nu − 1 can then be calculated by letting ∂J(·)/∂∆U 0 = 0, ∆U (k|k − τsc,k ) = Mτsc,k ($k (y; ·) − Eτsc,k x¯(k − τsc,k ))

(8)

where ∆U (k|k − τsc,k ) = [∆u(k|k − τsc,k ) · · · ∆u(k + Nu − 1|k − τsc,k )]T , Mτsc,k = Hτsc,k (FτTsc,k QFτsc,k + R)−1 FτTsc,k Q, Q, R are diagonal matrices with Qi,i = qi , Ri,i = ri respectively and Hτsc,k = [0Nu ×τsc,k INu ×Nu ], INu ×Nu is the identity matrix with rank Nu . Since the system states are normally unavailable for the controller, we construct the ˆ following system S: ½ xˆ(k + 1) = Aˆ x(k) + bu(k) (9a) ˆ S: yˆ(k) = ϕ(cˆ x(k)) (9b) to observe the system states, xˆ(k + 1) = Aˆ x(k) + bu(k) + L(˜ y (k) − yˆ(k))

(10)

where xˆ(k) is the observed state at time k. Let xˆ¯(k) = [ˆ x(k) u(k − 1)], the real predictive control sequence can then be obtained as ∆U (k|k − τsc,k ) = Mτsc,k ($k (y; ·) − Eτsc,k xˆ¯(k − τsc,k ))

(11)

Remark 2.1. Notice that the calculation of the predictive control sequence ∆U (k|k −τsc,k ) in (11) is only based on the input and output data up to time k − τsc,k − 1. This can be compared with the one applied in [8] where the data from time k − τsc,k to k − 1 are used to determine the predictive controller, which certainly is hard to obtain by the controller in practice.

5

2.2. Design of the compensation scheme for network constraints. To take advantage of the characteristics of the network transmission and the predictive controller to compensate for the network constraints, i.e., network-induced delays and data packet dropout, the following assumptions are made: A1. A time stamp can be used for each data packet transmitted through the network to notify the time when it was sent; A2. The sum of the maximum network-induced delay in the forward channel (denoted by τ¯ca ) and the maximum number of continuous data packet dropout (denoted by χ) ¯ is bounded by the control horizon, i.e., τ¯ca + χ¯ ≤ Nu − 1

(12)

A3. Each control predictive sequence U (k|k − τsc,k ) is packed into one packet to be sent to the actuator. Remark 2.2. The network-induced delay in the backward channel for each data packet is known to the controller under assumption A1. Remark 2.3. The network-induced delays in both channels for each control predictive sequence are known to the actuator under assumptions A1 and A3. Note here that different from conventional predictive control implementations, where only the first predictive input is applied to the plant, in this paper, we generate a sequence of predictive inputs and send them in one data packet to the actuator. This is the key point of the proposed approach to compensate for the network constraints. With the assumptions above, we propose the following schemes to compensate for the network constraints in the backward and forward channels, respectively. 2.2.1. Compensation for the network constraints in the backward channel. From Remark 2.1 we know that the network-induced delay in the backward channel is known to the controller, which enables the predictive control sequence to be calculated (see equation (11)). However, as the matrices Eτsc,k , Fτsc,k , Mτsc,k , Hτsc,k in (11) vary with the networkinduced delay in the backward channel, it would be a great computation burden for the predictive controller if these matrices are calculated online. Fortunately, these matrices, actually, can be calculated off line since all the matrices are fixed for a given τsc . This advantage enables us to calculate off line all the matrices with respect to the specific τsc s, store them in the controller and just choose the appropriate ones when calculating online the predictive control increments, according to the current value of the delay τsc,k . 2.2.2. Compensation for the network constraints in the forward channel. In order to implement the compensation scheme to compensate for the network constraints in the forward channel, we introduce a cache for the actuator. When a new sequence arrives at the actuator side in one data packet as given in assumption A3, it is compared with the one already in the cache of the actuator according to the time stamps and only the latest one sent from the controller is stored. The cache is specially designed for the actuator and it can only store one control sequence (data packet) at any one time. The comparison process is introduced at the actuator side due to the fact that different data packets may experience different delays in the forward channel, thereby producing a situation where for example a data packet sent earlier from the controller may arrive at

6


the actuator later or may never arrive in the case of data packet dropout. As a result of the comparison process, the predictive control sequence stored in the cache of the actuator is always the latest one available at any specific time. At every execution time instant, the actuator picks out the appropriate control signal which can compensate for the current network-induced delay in the forward channel from the predictive control sequence and applies it to the plant. The method used to choose the appropriate control increment signal at a specific time will be further explained in the next section. It is necessary to point this out that the appropriate control increment is always available using the delay compensator if assumption A2 holds. The algorithm of the predictive control based approach to Wiener systems with compensation for network constraints can now be summarized as follows: S1. The predictive controller receives the delayed signals of output z(k −τsc, ) and control input ∆u(k − τsc,k ) and reads the current network-induced delay in the backward channel τsc,k ; S2. The predictive controller calculates the predictive control sequence ∆U (k|k − τsc,k ) through (11) using delayed data; S3. The predictive control sequence ∆U (k|k − τsc,k ) is packed and sent to the actuator simultaneously with time stamps k and τsc,k ; S4. The cache of the actuator updates its predictive control sequence according to the time stamps once a data packet arrives; S5. An appropriate control increment signal is picked out from the predictive control sequence and applied to the plant. The structure of the proposed approach is illustrated in Fig.3.

Figure 3: The structure of networked predictive control system

3. Stability of the proposed approach. In this section, we first prove that the state observer proposed in this paper is stable under certain conditions. This fact enables us to construct the stability theorem for the closed loop system.

7

3.1. Observer error. Let the observer error e(k) = x(k) − xˆ(k). From equations (1a), (9a) we obtain e(k + 1) = x(k + 1) − xˆ(k + 1) = Ae(k) − L(˜ y (k) − yˆ(k))

(13)

Assume ϕ(·) ∈ C 1 , then by mean value theorem, y˜(k) − yˆ(k) = ϕ(cx(k)) − ϕ(cˆ x(k)) = cϕ0 (ξk ))e(k)

(14)

where ξk ∈ [min{cx(k), cˆ x(k)} max{cx(k), cˆ x(k)}]. Combining equations (13) and (14) yields e(k + 1) = (A − Lcϕ0 (ξk ))e(k) (15) Notice that though ϕ(·) 6≡ 1(·), it is reasonable to assume that the compensation for the nonlinear function f (·) is smooth, which means there exists ε > 0 s.t. |ϕ0 (α) − 1| ≤ ε, ∀α ∈ R. Thus the dynamics of the observer error can be obtained as e(k + 1) =(A − Lc − ζk Lc)e(k) =Aζk e(k)

(16)

where Aζk = A − Lc − ζk Lc, |ζk | ≤ ε. Theorem 3.1 (Observer Error). The observer error converges to 0 if there exists a positive definite solution Pe = PeT > 0 to the following two LMIs ATε Pe Aε − Pe ≤ 0 AT−ε Pe A−ε − Pe ≤ 0

(17)

where Aε = A − Lc − εLc and A−ε = A − Lc + εLc. Proof Let V (k) = eT (k)Pe e(k) be a Lyapunov function candidate. Notice the fact that for any ζk , there exists 0 ≤ λk ≤ 1 such that ζk = λk ε + (1 − λk )(−ε). Thus by simple calculation, the incremental V for system (16) can be obtained as ∆V (k + 1) =eT (k)Γζk e(k) =eT (k)(λk Γε + (1 − λk )Γ−ε − 4λk (1 − λk )(Lc)T Pe Lc)e(k)

(18)

where Γζk = ATζk Pe Aζk − Pe . Noticing that λk (1 − λk ) ≥ 0 and (Lc)T Pe Lc is semi positive definite, it yields that ∆V (k) is decreasing which completes the proof. ¥ ∗ 3.2. Closed-loop stability. Let τca,k denote the network-induced delay in the forward channel of the predictive control sequence, from which the control signal is picked out by the actuator at time instant k. The time when the sequence was sent from the controller can then be read from its time stamp as ∗ k ∗ = k − τca,k = max{j|∆U (j|j − τsc,j ) ∈ Γk } j

(19)

where Γk is the set of the predictive control increment sequences that are available during time interval (k − 1, k] at the actuator side, including not only the one in the cache of the actuator but others that arrive at the actuator during this interval.

8


From equations (11), (19), the control signal adopted by the actuator at time k is obtained as ∗ ∆u(k) = dTτca,k ∆U (k − τca,k |k − τk∗ ) ∗

= −dTτca,k Mτk∗ Eτk∗ xˆ¯(k − τk∗ ) ∗ = −Στk xˆ¯(k − τk∗ )

(20)

∗ ∗ ∗ ∗ where dτca,k is a Nu ×1 matrix with all entries 0 except the (τca,k +1)th is 1, τk∗ = τca,k +τsc,k , ∗ T τsc,k = τsc,k∗ , Στk = dτca,k Mτk∗ Eτk∗ and the set point is assumed to be 0 without loss of gen∗ erality.

Let e¯(k) = x¯(k) − xˆ¯(k) = [e(k) 0]T , then µ

¶

e¯(k + 1) = A¯ξk e¯(k)

(21)

A − Lcϕ0 (ξk ) 0 . 0 0 Let Z(k) = [¯ xT (k − τ¯) · · · x¯T (k) e¯(k − τ¯) · · · e¯(k)]T , then the closed loop system can be represented by where A¯ξk =

Ã where Λξk ,τk =



Λ22 ξk

    =   

Λ11 τk 0

Λ12 τk Λ22 ξk

Z(k + 1) = Λξk ,τk Z(k)  0n+1 In+1   ! In+1   ... , Λ11 τk =    In+1  · · · −Στk · · · A¯ 

0n+1 In+1 0n+1 In+1 ... ... ...



(22)

    ,   

    , and Λ12 τk is a block matrix with all its entries   

In+1 A¯ξk (blocks) 0 except (Λ12 τ −1)×(¯ τ −τk∗ +1) = −Στk . τk )(¯

Theorem 3.2 (Closed-loop stability). The closed loop system is stable if (17) holds and τca + 1)(¯ τsc + 1) there exists a positive definite solution Pc = PcT > 0 for the following (¯ LMIs T 11 (Λ11 τk ) Pc Λτk − Pc ≤ 0

(23)

Proof: Noticing the block-triangular structure of the system matrix Λξk ,τk for the closedloop system, we see that the state observer can be designed separately without influencing the stability of the system and the closed-loop system is stable if we can guarantee the

9

stability of the state observer (Theorem 3.1) and the following system, X(k + 1) = Λ11 τk X(k)

(24)

where X(k) = [¯ xT (k − τ¯) · · · x¯T (k)]. Let V (k) = X T (k)Pc X(k) be a Lyapunov function candidate, then the incremental V (k) for system (24) is T 11 ∆V (k) = X T (k)((Λ11 τk ) Pc Λτk − Pc )X(k)

which completes the proof using equation (23).

¥

Remark 3.1. Notice that the two conditions ((17) and (23)) that guarantee the stability of the closed-loop system are with respect to the compensation accuracy for the nonlinearity and the influence of the network constraints respectively.

0.15

0.3 State evolution without compensation for network constraints State evolution with compensation for network constraints

0.1

State evolution without compensation for the nonlinearity State evolution with compensation for the nonlinearity 0.2

0.05

0.1

0 x1(k)

x1(k)

0 −0.05

−0.1 −0.1 −0.2

−0.15

−0.3

−0.2 −0.25

0

20

40

60

80

100

120

140

−0.4

0

20

40

k

(a) A comparison between with/without compensation for network constraints

60 k

80

100

120

(b) A comparison between with/without compensation for output nonlinearity

Figure 4: The validity of the proposed approach 4. Simulation. An example is given in this section to illustrate the validity of the proposed approach. For this purpose, a second order plant model in discrete time with a static nonlinear output process and random delays in both channels and data packet dropout in the forward channel, is adopted, µ ¶ µ ¶ ¡ ¢ 0.8 0.1 0.05 A= ,b = ,c = 1 0 . 0 1 0.2 Other parameters of the simulation are chosen as τ¯ = 8, τ¯ca = 4, τ¯sc + χ¯ = 4, Nu = 8, Np = 10, ε = 0.5 and the initial state x(0) = x0 = [−0.1 0.2]T . The delays in both channels are set to vary randomly within their upper bounds. Such a system using the proposed approach in this paper can be proven to be stable under Theorem 3.2. Two cases which illustrate the validity of the compensation for the network constraints (Fig.4a) and the compensation for the output nonlinearity (Fig.4b) respectively, are shown in Fig.4. In both cases, all the other parameters remain the same and only the evolution of the first state of the system is illustrated. The simulation results show that the system

10


is stable with the compensation scheme while unstable without it, which illustrate the validity of the proposed approach in this paper. 5. Conclusion. In this paper, we propose a predictive control based approach to deal with a Wiener type system which is closed through a network. In this approach, a state observer is designed to derive the predictive controller using delayed sensing data, and with the use of time stamps for each data packet, the negative effects of the networkinduced delay and data packet dropout in both channels are also compensated for. The deriving closed-loop system is proved to be stable under certain conditions related to the compensation for the nonlinear process and the network constraints. The effects of the compensation for the nonlinearity and network constraints are also illustrated by simulations. REFERENCES [1] Aberkane S., J. C. Ponsart and D. Sauter, Output feedback H∞ control of a class of stochastic hybrid systems with Wiener process via convex analysis, Int. J. Innov. Comp. Inf. Control, vol.2, no.6, pp.1179-1196, 2006. [2] Baillieul, J. and P. J. Antsaklis, Control and communication challenges in networked real-time systems, IEEE Proc., vol.95, no.1, pp.9-27, 2007. [3] Branicky, M. S., V. Liberatore and S. M. Phillips, Networked control system co-simulation for codesign, Proc. 2003 American control conference, Denver, USA, pp.3341-3346, June 2003. [4] Goodwin, G. C., H. Haimovich, D. E. Quevedo and J. S. Welsh, A moving horizon approach to networked control system design, IEEE Trans. Autom. Control, vol.49, no.9, pp.1427-1445, 2004. [5] Hespanha,J. P., P. Naghshtabrizi, and Y. Xu, A survey of recent results in networked control systems, IEEE Proc., vol.95, no.1, pp.138-162, 2007. [6] Hu, S. and W. Y. Yan, Stability robustness of networked control systems with respect to packet loss, Automatica, vol.43, no.7, pp.1243-1248, 2007. [7] Kemih K. and O. Tekkouk, Constrained generealized predictive control with estimation by genetic algorithm for a magnetic levitation systems, Int. J. Innov. Comp. Inf. Control, vol.2, no.3, pp.543552, 2006. [8] Liu, G. P., J. X. Mu, D. Rees, and S. C. Chai, Design and stability analysis of networked control systems with random communication time delay using the modified MPC, Int. J. Control, vol.79, no.4, pp.288-297, 2006. [9] Montestruque, L. A. and P. J. Antsaklis, On the model based control of networked systems, Automatica, vol.39, no.10, pp.837-1843, 2003. [10] Nair, G. N., F. Fagnani, S. Zampieri, and R.J. Evans, Feedback control under data rate constraints: An overview, IEEE Proc., vol.95, no.1, pp.108–137, 2007. [11] Tipsuwan, Y. and M. Y. Chow, Control methodologies in networked control systems, Control Eng. Practice, vol.11, no.10, pp.1099-1111, 2003. [12] Wang, Y., and Z. Sun, H∞ control of networked control system via LMI approach, Int. J. Innov. Comp. Inf. Control, vol.3, no.2, pp.343–352, 2007. [13] Yang, T. C., Networked control systems: A brief survey, IET Control Theory Appl., vol.153, no.4, pp.403-412, 2006. [14] Zhang, L. and D. H. Varsakelis, Communication and control co-design for networked control systems, Automatica, vol.42, no.6, pp.953-958, 2006. [15] Zhao, Y.-B., G. P. Liu, and D. Rees, Integrated predictive control and scheduling co-design for networked control systems, IET Control Theory Appl., In press.

Techset Composition Ltd, Salisbury Doc: {IEE}CTA/Articles/Pagination/CTA55037.3d

www.ietdl.org Published in IET Control Theory and Applications Received on 29th September 2007 Revised on 27th February 2008 doi: 10.1049/iet-cta:20070363

ISSN 1751-8644

Improved predictive control approach to networked control systems Y.-B. Zhao G.P. Liu D. Rees Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, UK E-mail: [email protected]

Abstract: A predictive control-based approach is proposed to networked control systems. In this approach, an improved predictive controller is designed using delayed sensing data and a compensation scheme is proposed to overcome the negative effects of the network-induced delays and data packet dropouts in both the forward and backward channels. The proposed approach is easy to be implemented in practice compared with previous results in that only delayed data of the control inputs are used to derive the forward control predictions. The stability of the closed-loop system is obtained by modelling the system as a time delay system with structural uncertainties. Simulations show that the proposed approach is superior to the previous results in the situation where only delayed data are used.

1

Introduction

Networked control systems (NCSs) are control systems where the links from sensor to controller (‘backward channel’) and from controller to actuator (‘forward channel’) are not connected directly but closed through some form of network. In NCSs, the network introduces communication constraints to the control system such as the network-induced delay, data packet dropout, data packet disorder, data rate constraint and so on, which requires the designer to develop novel approaches to overcome these negative effects to retain stability and meet other performance requirements[1 –8]. One technique recently proposed for NCSs is the predictive control approach, as, for example, in [9 – 11]. It is noticed that most of the existing results using predictive control consider it as merely a control approach, whereas in [11], the dynamics of the plant is explicitly used to derive a sequence of forward control predictions, which are sent to the actuator simultaneously and the actuator chooses the appropriate one to compensate for the delays. In this way, the delays in both channels can be theoretically exactly compensated for and, thus, better performance can be expected. In [11], the design of the predictive controller is based on the previous control inputs up to the last step. However, this information is not easy to obtain for certain IET Control Theory Appl., pp. 1 – 7 doi: 10.1049/iet-cta:20070363

conditions because of the delays in both channels, which restricts the application of the approach proposed in practice. In this paper, an improved predictive control-based approach is proposed to NCSs. In this approach, the deriving predictive controller is only based on delayed sensing data which is always available to the controller, thus enabling the approach to be feasible in practice. The compensation scheme proposed in [11] is redesigned to handle delays (data packet dropout as well) in both channels at the same time, which cannot be obtained using the previous approach if only the delayed data are used. The corresponding closed-loop system is modelled as a time delay system with structural uncertainties, for which fruitful results of delay-dependent stability analysis can be applied to analyse the stability [12, 13]. This can be compared with the previous results in [11], where the stability of the derived closed-loop system is obtained using switched system theory and the number of the LMIs required to guarantee stability is proportional to the size of the delays. The remainder of the paper is organised as follows. Section 2 presents the design details of the proposed approach; Section 3 analyses the stability of the corresponding closed-loop system; Section 4 gives examples

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1 The Institution of Engineering and Technology 2008

www.ietdl.org to illustrate the effectiveness of the proposed approach and Section 5 concludes the paper.

2 Design of the proposed approach The following multi-input multi-output (MIMO) linear time-invariant system S is considered in this paper S:

x(k þ 1) ¼ Ax(k) þ Bu(k) y(k) ¼ Cx(k)

where x(k) [ Rn , u(k) [ Rm , B [ Rnm and C [ Rrn .

y(k) [ Rr ,

(1a) (1b) A [ Rnn ,

For the system considered above, the design details of the improved predictive controller are presented first in this section, following which the compensation scheme in [11] is redesigned to deal with communication constraints in both channels. The proposed approach relaxes the requirements compared with previous approaches and thus is easier to be implemented in practice.

(2)

where Jk,tsc,k is the objective function at time k, tsc,k the corresponding delay in the backward channel, DU 0 (kjk tsc,k ) ¼ [Du(k tsc,k jk tsc,k ) Du(k þ Nu 1jk tsc,k )]T the forward control increment sequence, Y^ (kjk tsc,k ) ¼ [^y(k þ 1jk tsc,k ) y^ (k þ Np jk tsc,k )]T the predictive output trajectory which will be shown to be based only on the state at time k tsc,k and DU 0 (kjk tsc,k ); Q and R are constant weighting matrices and Np and Nu are the prediction and control horizons respectively. The so-called ‘forward’ predictive control increment sequence here actually includes as part of it the ‘backward’ control increment signals from time k tsc,k to k 1. The objective function is chosen in this way because

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(3a) (3b) 0

and

The predictive outputs at time k based on the state at time k tsc,k and the control increment sequences from k tsc,k can then be obtained by iteration y^ (k þ jjk tsc,k ) ¼ C A jþtsc,k x (k tsc,k ) j1 X

þ l jk tsc,k ), j ¼ 1, 2, . . . , Np C A jl 1 BDu(k

Thus Y^ (kjk tsc,k ) ¼ Etsc,k x (k tsc,k ) þ Ftsc,k DU 0 (kjk tsc,k ) (4) where Ftsc,k is a block lower triangular matrix with its nont þij ji B, null elements defined by (Ftsc,k )ij ¼ C A sc,k tsc,k þ1 T tsc,k þNp T T tsc,k and Etsc,k ¼ [(C A ) (C A ) ] . Let DU (kjk tsc,k ) ¼ [Du (kjk tsc,k ) Du (k þ Nu 1jk tsc,k )]T denote the optimal control increment sequence from k to k þ Nu 1. It can be calculated by substituting (4) – (2) and optimising Jk,tsc,k , which turns out to be state feedback control DU (kjk tsc,k ) ¼ Ktsc,k x (k tsc,k )

Jk,tsc,k ¼ Y^ T (kjk tsc,k )QY^ (kjk tsc,k ) þ DU 0T (kjk tsc,k )

The Institution of Engineering and Technology 2008

x (k þ 1) ¼ A x (k) þ BDu(k) S0: y(k) ¼ C x (k) A B cB where A ¼ , B¼ , C ¼ C 0 I I Du(k) ¼ u(k) u(k 1).

l ¼tsc,k

In a typical model predictive control implementation, the predictive controller determines a sequence of forward control signals at each control interval that optimise future open-loop plant behaviour and only the first control input is actually applied to the plant. The entire optimisation is repeated at every subsequent control interval, which enables the controller to deal with uncertainties. In such a typical implementation, the forward predictive outputs are based on previous outputs up to the last step. However, in NCSs, the previous outputs to a certain time are unavailable to the controller because of the network-induced delay and data packet dropout in the backward channel. The objective function for open-loop optimisation in this paper is therefore defined as follows

2

The objective here is to minimise the objective function (2) to derive the optimal control predictions. For this purpose, we rewrite system S as S 0 by letting x (k) ¼ [x(k) u(k 1)]T ,

þ

2.1 Design of the predictive controller

RDU 0 (kjk tsc,k )

of the fact that those ‘backward’ control increment signals are unavailable for the controller at time k under the proposed predictive-based approach in this paper and in [11], even though they have already been applied to the plant. See Remark 2 and Section 2.2 for more details. Q1

where Ktsc,k ¼ Mtsc,k (FtTsc,k QFtsc,k þ R)1 FtTsc,k QEtsc,k Mtsc,k ¼ (0mNu mtsc ImNu mNu ).

(5) and

Remark 1 (state observer): If the state vector x is not available, an observer must be included x^ (k þ 1) ¼ A x^ (k) þ Bu(k) þ L(Cx(k) C x^ (k))

(6)

where x^ (k) is the observed state at time k.

Remark 2: In [11], state feedback uk ¼ K x^ kjktsc,k is also used, where K is artificially chosen without consideration of the communication constraints and x^ kjktsc,k depends on ‘the state estimation x^ ktsc,k jktsc,k 1 , the past control input up to uk1 and the past output up to yktsc,k of the system’. However, as we will specify later in the following IET Control Theory Appl., pp. 1 – 8 doi: 10.1049/iet-cta:20070363

www.ietdl.org subsection, under the compensation scheme in the forward channel in [11], the whole sequence of the optimal forward control increments DU (kjk tsc,k ) is sent to the actuator and only one of them is chosen to be applied to the plant. Thus, unless information from the actuator is received, we have no idea which control prediction was really used if the data packets in the forward channel were randomly delayed. Hence, the use of the previous control inputs implies an additional communication channel which can send the applied control inputs to the controller efficiently. Without such a channel, the approaches proposed in these publications are only applicable to such a situation where there is no delay or data packet dropout in the forward channel. To relax this requirement, we redesigned the predictive controller in this paper by optimising an objective function which includes as part of it the previous control increment sequence from k tsc,k to k 1 [i.e. DU 0 (kjk tsc,k )]. As a result, the forward control predictions at time k are only based on data up to time k tsc,k [see (5)), which is always feasible in practice. In the next section, it is shown that such an improved predictive controller with the modified compensation scheme gives state feedback control where the feedback gain varies with the delays in both channels. This varying feedback gain scheme is shown to be superior to the fixed feedback gain approach in [11] in such a situation where only the delayed data are used.

2.2 Design of the compensation scheme for communication constraints To take advantage of the characteristics of the packet-based transmission in a networked environment and the proposed predictive controller to compensate for the network constraints considered in this paper, that is, the networkinduced delay and data packet dropout, we make the following two assumptions similar to those in [11], (A1) All the components in the system including the controller, the actuator and the sensor are time-synchronised and each data packet transmitted through the network uses a time stamp to notify the time when it was sent, (A2) Each optimal control increment sequence DU (kjk tsc,k ) is packed into one data packet to be sent to the actuator; and a further assumption that is not explicitly included in [11], and yet very important to implement the proposed approach, with which the compensation for the data packet dropout can also be realised. (A3) The sum of the maximum network-induced delays in the forward channel (backward channel) and the maximum number of continuous data packet dropouts is upper bounded by tca (tsc accordingly) and

tca Nu 1 IET Control Theory Appl., pp. 1 – 7 doi: 10.1049/iet-cta:20070363

(7)

Remark 3: The network-induced delay in the backward channel for each data packet is known to the controller under assumption (A1) and the network-induced delays in both channels for each control predictive sequence are known to the actuator under assumptions (A1) and (A2). Remark 4: In this paper, the dropped data packets are simply ignored which is common in a real-time application but not regarded as an infinite delay. The so-called ‘maximum network-induced delay’ in assumption (A3) is only measured over those data packets that are received successfully. Hereafter, the term ‘network-induced delay’ instead of ‘network-induced delay and data packet dropout’ is used for simplicity since the effect of the data packet dropout does not need any special treatment using the compensation scheme described below under assumption (A3). With the above assumptions, we propose a new compensation scheme for the network-induced delay in the backward channel, and redesign the compensation scheme at the actuator side compared with [11] as follows. (1) Compensation for the communication constraints in the backward channel: The network-induced delay in the backward channel is known to the controller (Remark 3), which enables the predictive control sequence to be calculated equation (5). However, as the matrices Etsc,k , Ftsc,k and Mtsc,k in (5) vary with tsc,k , it would be a great computation burden for the predictive controller if these matrices are calculated online. Fortunately, these matrices, actually, can be calculated offline since all the matrices are fixed for a given delay. This advantage enables us to calculate offline all the matrices with respect to the specific tsc s, store them in the controller and just choose the appropriate ones when calculating online the predictive control increments, according to the current value of the delay tsc,k . Let Esc ¼ {E0 , E1 , . . . , Etsc }, Fsc ¼ {F0 , F1 , . . . , Ftsc } and Msc ¼ {M0 , M1 , . . . , Mtsc }, then we have for any k (or tsc,k ), Etsc,k [ Esc , Ftsc,k [ Fsc and Mtsc,k [ Msc , respectively. For a practical implementation, these matrices are calculated offline and stored in the matrix selector for online use. (2) Compensation for the communication constraints in the forward channel: As in [11], a cache is used at the actuator side which can only store one predictive control increment sequence [or one data packet, see assumption (A2)] at any one time. When a new sequence arrives at the actuator side, it is compared with the one already in the cache according to the time stamps and only the latest one sent from the controller is stored. This comparison process is introduced since different data packets may experience different delays in the forward channel, thereby producing a situation where, for example, a data packet sent earlier from

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www.ietdl.org the controller may arrive at the actuator later or may never arrive in the case of a data packet dropout. As a result of the comparison process, the predictive control sequence stored in the cache of the actuator is always the latest one available at any specific time. At every execution time instant, the actuator selects the appropriate control prediction which can compensate for the current network-induced delay in the forward channel from the predictive control increment sequence in the cache and applies it to the plant. It is necessary to point out that the appropriate control increment is always available provided assumption (A3) holds. The algorithm of the predictive control-based approach can now be summarized as follows.

(S4) The current sensing data with the control input are sent to the controller. The structure of the proposed approach is illustrated in Fig. 1.

3 Stability of the proposed approach In this section, we first give the explicit expression of the closed-loop system under the proposed predictive controlbased approach in this paper, and then analyse the stability of the closed-loop system using the results of the delay- Q2 dependent stability analysis [12].

3.1 Closed-loop system (S1) At time k, if the predictive controller receives the delayed data of state x(k tsc,k ) [or y(k tsc,k ) if the state is unavailable) and the control input Du(k tsc,k ), then the following are done. (S1a) The current network-induced delay in the backward channel tsc,k is read (S1b) The predictive control increment DU (kjk tsc,k ) using (5) is calculated.

sequence

(S1c) DU (kjk tsc,k ) is packed and sent to the actuator simultaneously with the time stamps k and tsc,k .

It has already been stated that there may be more than one predictive control increment sequence available for the actuator at time k because of different delays those sequences experienced, but only the latest one is stored in the cache after the comparison process. If we denote the delay in the forward channel of this predictive control increment sequence by tca,k and the corresponding delay in the backward channel by tsc,k , then the predictive control increment sequence applied at time k is calculated at time k tca,k based on data up to time k tk , that is, DU (k tca,k jk tk ), where tk ¼ tca,k þ tsc,k . Thus, the predictive control increment really applied at time k is Du(k) ¼ Du (kjk tk ) ¼ dtT DU (k tca,k jk tk )

If no data packet is received at time k, then let k ¼ k þ 1, and wait for the next time instant.

ca,k

¼ dtT Kt x (k tk ) ca,k

sc,k

(8)

(S2) The cache of the actuator updates its predictive control increment sequence according to the time stamps once a data packet arrives.

where dtT is a 1 Nu block matrix with all entries 0 ca,k except the tca,k th block, the identity matrix with rank m.

(S3) An appropriate control increment prediction is selected from the predictive control increment sequence and applied to the plant.

Notice that in practice there is at least a one-step delay in both the forward and backward channels, and the possible control increment input at time k should be one entry out

Figure 1 Structure of the improved predictive control-based approach 4

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IET Control Theory Appl., pp. 1 – 8 doi: 10.1049/iet-cta:20070363

www.ietdl.org where B m ¼ k B is a constant matrix and Dt ,t ¼ kt =k. It ca,k sc,k k is easy to conclude that kDt ,t k 1, 8 1 tca,k ca,k sc,k t¯ ca , 1 tsc,k t¯ sc .

of the following matrix 0

k11 x (k 2)

k21 x (k 3)

B B U k ¼ B B @

k12 x (k 3) k22 x (k 4) .. .. . . k1tca x (k t ca 1) k2tca x (k t ca 2) 1 kt sc 1 x (k t sc 1) C kt sc 2 x (k t sc 2) C C C .. C . A kt sc t¯ ca x (k t ca t¯ sc )

.. .

With the modelling of the closed-loop system in (10) and (11), the delay-dependent analysis which has been explored a lot recently (see, e.g., in [14 – 16]) can now be applied to derive a stability criterion.

Theorem 1: If there exists Pi ¼ PiT . 0, i ¼ 1, 2, 3, X12

X11

0, Ni , i ¼ 1, 2 with appropriate X22 dimensions and l . 0 satisfying the following two LMIs

X ¼

T X12

where the gain matrix of Uk is 0 B B B B B @

0

1

B T @ X12 N1T

k21 kt sc 1 C k22 kt sc 2 C C .. .. .. C . . . C A k2t¯ ca kt¯ sc t¯ ca 1 01 .. B. C ¼ @ .. 0t¯ ca (Nu t¯ ca ) A(K1 Kt¯ sc ) . 1 0

k11 k12 .. . k1t¯ ca 0

0

F11

B B FT12 B B H T (A I ) @ T B m P1

The control increment applied at time k can then be represented by ,t ca,k sc,k

x (k

P3

F12

(A I )T H

F22 þ lI

0

0

H T B m H

0

(12)

P1 B m

1

C 0 C C . 0 (13) H B m C A lI

F22 ¼ P2 N2 N2T þ t X22 H ¼ P1 þ t P3

Let Gt ¼ {ktsc ,tca jtsc þ tca ¼ 2 t¯ sc þ t¯ ca , then k the closed-loop system under the proposed approach can now be written as ,t ca,k sc,k

C N2 A 0

F12 ¼ N2T N1 þ t X12

tk

tk ),

N2T

1

þ (A I )T P1 þ N1 þ N1T þ t X11

(9)

t x (k þ 1) ¼ A x (k) Bk

X22

N1

F11 ¼ (t 1)P2 þ P1 (A I )

x (k tk ), 1 tsc,k t sc , 1 tca,k t¯ ca

tk },

X12

then the closed-loop system (10) is stable. Here

where Ki , 1 i t¯ sc are obtained using (5).

Du(k) ¼ kt

X11

¯ Let d1 ¼ 2, d2 ¼ t¯ , A ¼ A, and DAd (k) ¼ t ,t in Theorem 7.3 in [12], then the above Bk ca,k sc,k theorem can be obtained using the same techniques as in [12]. A

Proof:

kt

,t ca,k sc,k

[ Gt k

(10)

Remark 5: In [11], the feedback gain is designed without consideration of the network-induced delays and, thus, fixed for different delays. This can be compared with the approach in this paper where for different delays, different feedback gains apply (9). Simulations illustrate the superiority of this varying gain scheme compared with the fixed gain scheme in [11].

3.2 Stability analysis Let k ¼ max2t t¯ ca þt¯ sc kkt ,t k where k k denotes the k tca,k,tsc,k can be represented by Euclidean norm. Then Bk

Remark 6: The key idea to find the stability criteria in [11] is to model the closed-loop system as an augmented switched system X (k þ 1) ¼ L(t)X (k) where the system matrix L(t) varies with different delays. As a result, the number of the LMIs that guarantee the stability of the closed-loop system is proportional to the size of the delays (Theorem 2 in [11]). In this paper, by designing a different predictive controller (5), the closed-loop system can then be modelled as a time delay system with structural uncertainties and the LMIs required to ensure the stability are reduced to two.

ca,k sc,k

t Bk

,t ca,k sc,k

¼ B m Dt


,t ca,k sc,k

(11)

Remark 7: If a state observer as in Remark 1 is also involved, let z(k) ¼ [xT (k) x^ T (k)]T ¼ [xT (k) uT (k 1) x^ T (k)]T , then

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www.ietdl.org the closed-loop system can be written as ~ z(k þ 1) ¼ Az(k) B~ t

,t ca,k sc,k

z(k tk )

(14)

where 0

A

B

0

1

B C A~ ¼ @ 0 I 0 A, LC B A LC 0 Bkmn Bkm1 tca,k ,tsc,k tca,k ,tsc,k B mn m1 B kt ,t B~ t ,t ¼ B ktca,k ,tsc,k ca,k sc,k ca,k sc,k @ mn Bkt ,t Bkm1 t ,t ca,k sc,k

ca,k sc,k

0

1

C 0C C, ktca,k ,tsc,k [ Gtk A 0

m1 and kt ,t ¼ [kmn tca,k ,tsc,k ktca,k ,tsc,k ]. Thus, a similar stability ca,k sc,k criterion to Theorem 1 can be obtained analogously.

4

Figure 2 Example 1: The system is unstable using the approach in this paper when t¯sc ¼ 2 and t¯ca ¼ 1

Simulation

Two examples are presented in this section to illustrate the effectiveness of the proposed approach in this paper.

Example (Example 1 in [11]): The system matrices adopted are as follows 0

1:0100 0:2710

B A ¼ @ 0:4820 0:1000 0:0020 0:3681 1 2 3 C¼ 4 3 1

0:4880

1

C 0:2400 A, 0:7070

0

5

B B ¼ @3 5

5

1

C 2 A, 4

In [11], the above system is illustrated to be stable with t sc ¼ 2, t ca ¼ 1 0

1

0:3614 0:3326 L ¼ @ 0:0332 0:0576 A 0:2481 0:0750

Figure 3 Example 1: The system is stable using the varying feedback gain scheme when t¯sc ¼ 1 and t¯ca ¼ 1

and a fixed feedback gain K ¼

0:5858 0:5550

0:1347 0:0461

0:4543 0:4721

However, using the approach proposed in this paper, this system is unstable with the same t¯ sc , t¯ ca and L. (see Fig. 2. Other parameters: Nu ¼ 8 and Np ¼ 10.) This fact seems to mean the approach in [11] is better than the approach in this paper, but we need to remember that the approach in [11] takes advantage of more information to design the predictive controller and some of the information used is not easy to obtain in practice (Remark 2). On the other hand, the simulation results do illustrate that the varying feedback gain scheme in this paper is superior to the previous fixed feedback gain scheme (Remark 5), where the same system is stable using the approach in this paper 6

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Figure 4 Example 1: The system is unstable using the fixed feedback gain scheme when t¯sc ¼ 1 and t¯ca ¼ 1 IET Control Theory Appl., pp. 1 – 8 doi: 10.1049/iet-cta:20070363

www.ietdl.org [3] VARSAKELIS D.H. : ‘Stabilization of networked control systems with access constraints and delays’. Proc. 45th IEEE Conf. Decision and Control, December 2006, pp. 1123 – 1128 [4] YANG T.C.: ‘Networked control systems: a brief survey’, IEE Proc., Control Theory Appl, 2006, 153, (4), pp. 403– 412 [5] WANG Y.-L., YANG G.-H.: ‘H1 control of networked control systems with time delay and packet disordering’, IET Control Theory Appl., 2007, 1, (5), pp. 1344 – 1354

Figure 5 Example 2: The system is stable using the approach in this paper when tsc ¼ 3 and t¯ca ¼ 2 when t¯ sc ¼ t¯ sc ¼ 1 (Fig. 3) and yet is unstable using the same state feedback (10) with the fixed K above (Fig. 4).

Example 2: The system matrices are set as A¼

0:7 0:3

0:2 , 0:5

B¼

0:05 , 0:2

C¼ 1

0

This system can be shown using Theorem 1 to be stable under t sc ¼ 3, t ca ¼ 2, Nu ¼ 8, and Np ¼ 10. The simulation result is illustrated in Fig. 5.

5

Conclusion

A predictive control-based approach to NCSs was recently reported in [11], the implementation of which needs an additional communication channel and, thus, is not easy to be applied in practice. To deal with this problem, an improved predictive controller is designed with a modified compensation scheme, which is feasible in practice and can handle communication constraints in both the backward and forward channels. The stability criteria of the corresponding closed-loop system consists of only two LMIs which is easy to check compared with the previous results where the number of the LMIs required was proportional to the size of the delays. Simulations show that the proposed approach is superior to the previous results in the situation where only the delayed data are used.

6

References

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