Design of Networked Control Systems With Packet Dropouts

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 7, JULY 2007

Design of Networked Control Systems With Packet Dropouts Jing Wu and Tongwen Chen

Abstract—This note is concerned with stability and controller design of networked control systems (NCSs) with packet dropouts. New NCS models are provided considering both single- and multiple-packet transmissions. Both sensor-to-controller (S/C) and controller-to-actuator (C/A) packet dropouts are modeled and their history behavior is described by different independent Markov chains. In term of the given models, sufficient conditions for stochastic stability are derived in the form of linear matrix inequalities (LMIs) and corresponding control laws are given. Numerical examples illustrate the effectiveness of the results. Index Terms—Markov chains, networked control systems (NCSs), packet dropout, stochastic stability.

I. INTRODUCTION In networked control systems (NCSs), control loops are closed through real-time networks. Such networked systems bring new functionalities that were not available in the past, such as low cost, reduced system wiring, simple system diagnosis and maintenance, and increased system agility. However, the insertion of communication networks in feedback control loops makes the NCS analysis and synthesis complex; see [1]–[3] and the references therein, where much attention has been paid to the delayed data packets of an NCS due to network transmissions. In fact, data packets through networks suffer not only transmission delays, but also, possibly, transmission loss/packet dropout [4], [5]; the latter is a potential source of instability and poor performance in NCSs because of the critical real-time requirement in control systems. How such packet dropout affects stability and performance of NCSs is an issue focused in this note. Prior work, examining the effect of dropouts on system stability and performance, can be roughly categorized into three types based on the resulting closed-loop systems: switching systems [6], asynchronous dynamical systems (ADSs) [7], and jump linear systems with Markov chains [8]–[12]. It is noticed that all the stability conditions and controller designs given in the aforementioned references are derived based on the assumption that packet dropout exists only in the sensor-to-controller (S/C) side. The effect of controller-to-actuator (C/A) packet dropouts is neglected due to the complicated NCS modeling. Recently, some results were obtained in [13]–[16], where ADSs were introduced to model NCSs with packet dropouts on both S/C and C/A sides [13], [14], a switching system was used to model NCSs [15], and a linear system with stochastic variables was discussed in [16] to describe NCSs with both-side packet dropouts. Moreover, the controller gain in [13] is obtained by solving bilinear matrix inequalities (BMIs) with rate constraints on the occurrence of events, while in [14] the controller gain is chosen in advance by a pole placement method considering rate constraints on the occurrence of discrete states of a dynamical system. In [16], linear/nonlinear LQG optimal Manuscript received February 23, 2006; revised August 1, 2006 and January 12, 2007. Recommended by Associate Editor M. Fujita. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: tchen@ece. ualberta.ca). Color versions of one or more of the figures in this note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2007.900839

Fig. 1. NCS with data packet dropout via state feedback.

controllers were designed to minimize a cost function according to the transmission control protocol (TCP) and user datagram protocol (UDP). Those references discuss packet dropouts in the single-packet transmission. As to the jump linear system approach, to our best knowledge, no work has been done at present for modeling NCSs with both S/C and C/A packet dropouts histories simultaneously. Note that packet dropouts defined in the aforementioned references have two cases, dropped or sent successfully, which are modeled as a Bernoulli or a two-state Markov chain process. In this note, Markov chains are introduced to describe S/C and C/A packet dropouts. The Markov chains in our note describe the quantity of packet dropouts between current time k and its latest successful transmission instead of only the information on if a packet is dropped or not, which is different from the aforementioned references. By this definition, the number of states of Markov chains is larger than two and the history of packet dropouts can be seen clearly. Under consideration of network packet size constraints, new models of NCSs with packet dropouts are presented according to the single- and multiple-packet transmissions. By augmenting the state vector, the resulting closedloop system can be transformed to a standard jump linear system with time delays, which enables us to apply the results of jump linear systems to the analysis and synthesis of such NCSs. Sufficient conditions for stochastic stability are given and corresponding controller design steps are provided. Examples are finally given to show the effectiveness of our method. This note is organized as follows. Section II introduces the basic preliminary of our setup in Fig. 1. Sections III and IV consider the modeling of NCSs with packet dropouts in single- and multiple-packet transmissions, respectively. According to the resulting NCSs, the stochastic stabilities and controller designs are discussed. Section V provides two numerical examples to illustrate the effectiveness of our results. Finally, Section VI gives some concluding remarks. II. PROBLEM FORMULATION Consider the NCS setup with data packet dropouts in Fig. 1, where sensors, controllers, and actuators are clock-driven. The linear timeinvariant (LTI) plant we consider here is

x(k + 1) = 8x(k) + 0u(k)

(1)

 is the input. 8 and 0 are where x(k) 2 0), instead of how many designed control signals are dropped (the value of dkca ). This classification can simplify the modeling of the closed-loop system since the control input u(k) will not be updated no matter what value dkca > 0 will be. That is, the control signal u(k) will be the same when dkca = 1; 2; 3; . . . ; d2 . Another advantage of this classification is to avoid introducing this unknown dkca in the augmented state vectors and controller design. Thus, we replace u(k) with (k 0 dkca ), the deriving method being the same (2) instead of u(k) = u as the iteration method for x (k). Concatenating plant and controller state vectors to obtain a global vector z(k) = [xT (k)uT (k 0 1)]T by (7)–(8), we can obtain the closed-loop system for the NCS with single-packet transmissions in Fig. 1 as

8 0 (dkca ) x(k) 0  (dkca ) u(k 0 1) (1 0  (dkca ))0F (dksc ) 0 x (k 0 dksc ) + ca sc (1 0  (dk ))F (dk ) 0 u (k 0 dksc 0 1) ca = A (dk ) z(k) + B (dkca ; dksc ) z (k 0 dksc ) :

z(k + 1) =

where d1 and d2 are nonnegative integers. We model dksc and dkca as two homogeneous independent Markov chains, which take values in S1 = f0; 1; . . . ; d1 g and S2 = f0; 1; . . . ; d2 g with the generators 1 = (ij ) and 2 = (mn ), respectively. The transition probabilities of dksc (jumping from mode i to j ) and dkca (jumping from mode m to n) are defined by

if j

>0

where

III. MODELING AND CONTROLLER DESIGN OF NCSS WITH SINGLE-PACKET TRANSMISSIONS

ij = 0; mn = 0;

if dkca = 0 otherwise dkca

Note that x (k) = x(k 0 dksc ), which can be easily derived by iterations based on (3). To simplify the expression of the closed-loop system, we introduce a function ( 1 ) to combine the previous closed-loop system as

Moreover, due to the bandwidth and packet size constraints of the network, the packet transmission can be classified into two types, singleand multiple-packet transmissions. By this classification, we will have two new NCS models for the setup in Fig. 1, which are given in the following sections.

ij = Pr (dksc+1 = j j dksc = i) mn = Pr (dkca+1 = n j dkca = m )

8x(k) + 0F (dksc )x(k); 8x(k) + 0u(k 0 1);

(5)

(9)

Remark 2: The resulting closed-loop system in (9) is a jump linear system with two modes (dksc and dkca ) and one mode-dependent time-varying delay dksc , where their transitions are described by two Markov chains, which give the history behavior of S/C and C/A packet dropouts, respectively. This also enables us to apply the results of jumping linear systems with time-delays to the analysis and synthesis of such NCSs. Before proceeding, we need the following definition. Definition 1 [17]: The free nominal jump discrete-time system in  (9) is said to be stochastically stable, if for all finite zk = ' 2