Robust predictive control for networked control and ... - IEEE Xplore

www.ietdl.org Published in IET Control Theory and Applications Received on 8th October 2013 Revised on 5th February 2014 Accepted on 5th March 2014 doi: 10.1049/iet-cta.2013.0901

ISSN 1751-8644

Robust predictive control for networked control and application to DC-motor control Di Wu1 , Xi-Ming Sun1 , Wei Wang1 , Peng Shi2,3 1 School

of Control Science and Control Engineering, Dalian University ofTechnology, Dalian 116024, People’s Republic of China 2 School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia 3 College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia E-mail: [email protected]

Abstract: This study is concerned with the robust predictive control problem for a class of uncertain networked control systems (NCS) with time-varying delays and packet dropouts. A networked predictive control (NPC) scheme is proposed to compensate for the effect of time-varying delays and packet dropouts. The closed-loop NCS with uncertainties under the networked predictive controller are described as coupled switched control systems. Then by Lyapunov function approach, sufficient conditions are provided to guarantee the robust stability of the system in terms of linear matrix inequalities. As an application, the proposed NPC scheme is applied to DC-motor control to show the effectiveness of the proposed control scheme.

1

Introduction

The so-called networked control systems (NCS) [1, 2] mean that the inputs and outputs signals of control systems form a closed-loop via a real-time network, that is, the data between each nodes of control systems (sensors, actuators and controllers) are transmitted through the network. Compared with traditional control systems, NCS have several advantages, such as resource sharing, low cost, easily extensible, simple installation and maintenance and high reliability. However, because of the involvement of network, some imperfections in the transmission of instable network also arise, for example, quantisation noise, packet dropouts, network-induced delays and transmission channel constraints [3, 4]. These imperfections may degrade closed-loop performance, and even worse, may often harm the stability of the closed-loop control systems. It is always a key question to analyse stability of closedloop systems affected by time-varying transmission delays in NCS. These delays are usually produced by data exchanged through the network, and often exhibit random behaviour [5] including delays from sensors to controllers (the feedback channel) and delays from controllers to actuators (the forward channel). Also, it is noted that packet dropouts are closely related to the transmission delays. Thus, to overcome the bad network environments, numerous control schemes have been proposed (e.g. see [6–9]) where networked predictive control (NPC) technique is an important tool [10, 11], since it often effectively compensates the network-induced delays and consecutive packet dropouts and therefore has attracted much attentions [12–16]. Liu et al. [17] presented the sufficient conditions for the stability of the closed-loop 1312 © The Institution of Engineering and Technology 2014

system with only bounded time-varying delay in feedback channels, then in [18], the sufficient conditions are given to ensure the stability of the closed-loop system with timevarying delays in both feedback and forward channels. The key idea of NPC is as follows: based on received data, the observer and the network delay compensator calculate a series of predicted states and the predicted control signals, and then this series of control signals are packed and transmitted through a communication network to the plant side. Next, the networked predictive controller selects an appropriate input value from existing control prediction sequence on the actuator side. It is noted that almost all of practical control systems are non-linear with uncertainties which may come from random disturbances and modelling errors [3, 19, 20]. Therefore the robust stability for NCS with NPC seems more important. However, since in the design of NPC, the controller is heavily dependent on the accuracy of the considered NCS model, the robust stability problem for NPC is difficult and to now there are few papers available in the literature. In [21], NCS with uncertainties based on NPC has been considered, but the result is only limited to the constant delay case rather than the situation of time-varying delays. For time-varying delay case, the robust stability problem for NPC is more complicated since it is very difficult to make a prediction to the system behaviour, and therefore, to the best of our knowledge, there are few references on this respect available in the literature at present. This paper considers robust stability of NCS with NPC. First by introducing an error variable and by making a deep analysis of the relations between the size of two delays at two consecutive sampling points, a switched error model IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

www.ietdl.org is proposed. Next for the closed-loop system, a coupled switched system model is given. Then, based on the common Lyapunov function technique, sufficient conditions for stability of such a system are developed. Finally, the effectness of the proposed method is shown by both a numerical academic example and an experimental example on a practical DC-motor-based test-rig. The remainder of this paper is organised as follows. The model of NCS with networked predictive control is presented in Section 2. A novel coupled switched system model is then presented to describe the considered NCS under NPC in Section 3, followed by the stability analysis for this model in Section 4. A numerical example and a DC-motor-based experimental plant are provided to illustrate the effectiveness of the proposed approach in Section 5. The conclusions are drawn in the last section. Notations: R and N denote the sets of real number and non-negative integer, respectively. Rn is defined as the ndimensional Euclidean space. The Schur stable matrix means that the norm of all eigenvalues of this matrix are less than one. In and On denote the identity matrix and zero matrix with dimensions n × n, respectively. in a matrix is used as an ellipsis for terms induced by symmetry. The notation X < Y , where X and Y are symmetric matrices, means that Y − X is positive definite. The matrix diag{a, . . . , a} means the diagonal matrix. Matrix P > 0(< 0) denotes a positive (negative) definite matrix P.

2

System description and NPC

Consider the following linear discrete-time plant with uncertainties xt+1 = (A + At )xt + But yt = Cxt

(1)

2.1

NCS existing in feedback channels

Suppose that systems (1) and (2) are controlled through the network. The diagram of NCS can be seen in Fig. 1. In this subsection, suppose that network only exists in feedback channels while the network in forward channels is assumed to be ideal, and the following assumptions are adopted. Assumption 1: The pair (A, B) is completely controllable and the pair (A, C) is completely observable. Assumption 2: Denote by kt the time-varying delay in feedback channels. Assume kt ≤ kt−1 + 1 with kt ∈ {0, 1, . . . , N1 } for a positive integer N1 . Assumption 3: The number of consecutive packet dropouts is bounded by N2 , where N2 ≤ N1 is a positive integer. Assumption 4: All packets in the NCS are transmitted with time stamps. Remark 1: Assumptions 1–3 are standard assumptions in the literature of NCS, (e.g. see, [12, 17, 18, 21, 22]). Assumption 4 has been used in several papers, such as [22] and so on. By using this assumption, the network-induced delays in feedback channels for each packets will be known to the controller, and then based on the size of these delays, one can calculate appropriate predictive state values and control values. Assumption 2 is the same as the one in [17, 18], and it is clear that the packet disorder cannot happen in such NCS. Under Assumption 1, the state-feedback controller is designed by a model control method, such as pole assignment, linear quadratic Gaussian (LQG) and H∞ control method and so on. The control law is constructed as

(2)

ut+1 = Kxt+1

(3)

where xt ∈ Rn , ut ∈ Rm and yt ∈ Rl are the system state, input and output, respectively, and A, B and C are constant matrices with appropriate dimensions. At is a real matrix function representing the uncertainties, and is assumed to be norm-bounded with the form

where K ∈ Rm×n . Revisit the NPC scheme proposed in [17]. Firstly, the onestep ahead observer is designed. When the measured output yt arrives at the controller, the following observer is adopted

At = DFt E

xˆ t+1|t = Aˆxt + But + L( yt − C xˆ t )

where D, E are known constant real matrices of appropriate dimensions and Ft is an unknown matrix function satisfying FtT Ft ≤ I .

where xˆ t+1|t ∈ R is the observer state based on the system output yt and L ∈ Rn×l is the observer gain which can be designed by observer design approaches.

Fig. 1

(4)

n

Diagram of NCS

IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

1313 © The Institution of Engineering and Technology 2014

www.ietdl.org Owing to the existence of time-varying delays in feedback channels, the measured output yt often cannot be used at t instant, while yt−kt may reach the controller. By using the data stored in the controller and the measured value yt−kt , rewrite the observer (4) by the following form xˆ t−kt +1|t−kt = Aˆxt−kt + But−kt + L(yt−kt − C xˆ t−kt )

(5)

where xˆ t−kt is defined as xˆ t−kt |t−1−kt−1 to denote the latest predicted value at the time t − 1, with kt−1 standing for the delay at time t − 1. And if t − kt = t − 1 − kt−1 , let xˆ t−kt = xˆ t−1−kt−1 . Then based on the nominal system of uncertain system (1), that is xt+1 = Axt + But

(6)

We construct the estimate state with the following form xˆ t−kt +1+i|t−kt = Aˆxt−kt +i|t−kt + But−kt +i ,

i ∈ {1, . . . , kt } (7)

Next, the predictive control value will be given by replacing xt+1 in (3) with xˆ t+1|t−kt at any time t, that is ut+1 = K xˆ t+1|t−kt

(8)

Remark 2: When data dropouts happen at time t, that is, yt is lost at time t, the predictive states related to yt can not be constructed. Therefore the introduction of signal xt+i|t−kt , i ∈ {1, . . . , N1 } is necessary and the NPC scheme proposed in the paper is more feasible than those in [12, 17, 21]. 2.2

NCS existing in both two channels

In [21], time-varying delays and packet dropouts in both forward and feedback channels are regarded as the round trip time delay which can be seen as a kind of constant delay. In order to compensate delays in forward channels, we adopt buffer and selector between the forward network and the plant, see Fig. 1 and make the following assumption. Assumption 5: Denote by it the time-varying delay including packet dropouts in forward channels. Assume it ∈ {0, . . . , N } for a positive integer N .

output of selector can be shown as ut = S(it )ut−it |t−it +1−kt−it +1

(10)

where ¯ m×it Im O ¯ m×(N −it ) ] S(it ) = [O ¯ m×it denotes the matrix and defined as [Om Om . . . Om ] . O it

3

Model of NCS under NPC

In this section, a novel coupled switched system model will be presented to describe the above-mentioned NCS model with NPC. Firstly, define a new auxiliary variable et−kt = xt−kt − xˆ t−kt , which means the observer error between xt−kt and xˆ t−kt . Because of the existence of many packet dropouts, the delay in feedback channels is still time-varying, which makes the auxiliary variable time-varying. Another auxiliary variable d is needed, and is defined as d = kt−1 − kt ∈ {−1, . . . , N1 }. 3.1

Model for NCS existing in feedback channels

Following the description in Section 2.1, we will show the switched model for NCS under its NPC scheme counteracting time-varying delays and packet dropouts in feedback channels. This time-varying situation can be summarised into two cases with variable d.

• Case I. Without new packet used In this case, there are no new packets arriving at the controller at t instant. Here, kt = kt−1 + 1, and d = −1. It is easy to see that the output data applied in the observer at time t is the same to that at t − 1. The closed-loop system can be rewritten into the following form xt+1 = (A + A)xt + But = (A + BK)xt + Axt − BK(xt − xˆ t|t−1−kt−1 ) = (A + BK)xt − BKAkt−1 (xt−kt−1 − xˆ t−kt−1 |t−1−kt−1 ) kt−1 −1

− BK

A j Axt−j−1 + Axt

j=0

= (A + BK)xt − BKAkt−1 (A − LC)et−1−kt−1 Under Assumption 5, from the observer (5), the predictive length in (7) will be changed from kt to kt + N , and the predictive control value ut+1+N will be calculated at time instant t as (9) ut+1+N = K xˆ t+1+N |t−kt Then predictor will pack ut+1 , ut+2 , . . . , ut+N +1 as a control packet ut+1|t−kt and then this packet will be transmitted through the network. Elements ut+1 , ut+2 , . . . , ut+N in ut+1|t−kt are calculated before the time t and they are same as elements with the same subscript in packet ut|t−1−kt−1 . Owing to the network-induced delay and packet dropouts in forward channels, the control packet arrives at the plant side can be expressed by ut−it |t−it +1−kt−it +1 . Next, we adopt the method in [23, 24], and utilise buffer and selector. As depicted in Fig. 1, when the control packet arrives at time t, this unit will first store this packet in the buffer, and then the control packet ut will be selected and transmitted to the plant. The 1314 © The Institution of Engineering and Technology 2014

− BK

kt−1


(11)

j=0

And for kt = kt−1 + 1, the observer error system in this case is (12) et−kt = et−kt−1 −1 = In et−kt−1 −1 With the time-varying delay in time t − 1, the cascade system (11) and (12) is a switched system with N1 subsystems.

• Case II. New packet used In this situation, a new output data will be used, which means kt ≤ kt−1 , in other words, d ∈ {0, . . . , kt−1 }, where d = 0 implies the case of constant delay, while d ∈ {1, . . . , kt−1 } means the occurrences of several consecutive IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

www.ietdl.org packet dropouts. The relation between the observer errors et−kt and et−1−kt−1 will be described as

with kt−1 ∈ {0, . . . , N1 − 1} and d = −1, it is given as ⎡ A + BK On On ⎢ In Ax (kt−1 , d) = ⎢ .. ⎣ ... . On ··· BxT (kt−1 , d) = −BKAkt−1 (A − LC) Ae (kt−1 , d) = In , Be (kt−1 , d) = On

et−kt = xt−kt − xˆ t−kt = xt−kt−1 +d − xˆ t−kt−1 +d = (A + A)xt−kt−1 +d−1 + But−kt−1 +d−1 − (Aˆxt−kt−1 +d−1|t−1−kt−1 + But−kt−1 −1+d ) = A(xt−kt−1 +d−1 − xˆ t−kt−1 +d−1|t−kt−1 −1 ) + Axt−kt−1 +d−1

On

· · · On

· · · On

On

and

··· = Ad (A − LC)et−1−kt−1 +

⎤ · · · On · · · On ⎥ .. ⎥ .. . . ⎦ In O n

d

j

A Axt−kt−1 +d−j−1

(13)

j=0

Combined (11)–(13), the closed-loop system can be completely formed into a coupled system consisting of an extended state subsystem and an observer error auxiliary subsystem. This system is also in fact a switched system as follows X¯ t+1 X¯ t = (A¯ σ (kt−1 ,d) + A¯ σ (kt−1 ,d) ) et−kt et−1−kt−1

(14)

−BKA · · · −BKAkt−1 A · · · ··· ··· ··· ··· .. .. .. .. . . . . On ··· ··· ··· ··· BxT (kt−1 , d) = On · · · · · · On Ae (kt−1 , d) = On , Be (kt−1 , d) = On · · · A ⎢ On ⎢ . ⎣ . .

⎡ A + BK On On ⎢ In Ax (kt−1 , d) = ⎢ .. ⎣ ... . On ··· BxT (kt−1 , d) = −BKAkt−1 (A − LC)

σ : {0, . . . , N1 } × {−1, . . . , kt−1 } → Nσ with a finite integer set Nσ , and

A¯ σ (kt−1 ,d) =

Ax (kt−1 , d) Be (kt−1 , d)

Ax (kt−1 , d) Be (kt−1 , d)

Bx (kt−1 , d) Ae (kt−1 , d)

Bx (kt−1 , d) Ae (kt−1 , d)

T T X¯ Tt+1 = [xt+1 , . . . , xt−N ] 1

⎤ On On ⎥ .. ⎥ ⎦ . On On

On

Then for the second case of σ (kt−1 , d) with kt−1 ∈ {0, . . . , N1 } and d ∈ {0, . . . , kt−1 }, it is given as (see the following two matrices and equations at the bottom of the page)

where the witched signal σ is related to the time-varying delay kt−1 and the auxiliary variable d, and is defined as

A¯ σ (kt−1 ,d) =

Ax (kt−1 , d) =

⎡

⎤ · · · On · · · On ⎥ .. ⎥ .. . . ⎦ In O n On

· · · On

Also, it is noted that with the form of uncertainties in (1) and (2), the uncertain term A¯ σ (kt−1 ,d) in (14) can be rewritten as A¯ σ (kt−1 ,d) = Dσ (kt−1 ,d) Ftσ (kt−1 ,d) Eσ (kt−1 ,d) where Ftσ (kt−1 ,d) = diag{Ft , . . . , Ft },

Eσ (kt−1 ,d) = diag{E, . . . , E}

and The detailed definition on the sub-matrices of A¯ x (kt−1 , d) and A¯ x (kt−1 , d) are as follows. Noting that based on Assumption 2, we in fact have d ≤ kt−1 . Therefore for the switching signal σ (kt−1 , d), we only need to consider the case of kt−1 ∈ {0, . . . , N1 }, and d ∈ {−1, 0, . . . , kt−1 }. Next, consider σ (kt−1 , d) with two cases. For the first case of σ (kt−1 , d)

⎡

Dσ (kt−1 ,−1)

D −BKD · · · −BKAkt−1 D ··· ··· ··· ⎢On ⎢ . .. .. .. . =⎢ . . . ⎢ . ⎣O ··· ··· ··· n On ··· ··· ···

· · · On ··· ··· . .. . .. ··· ··· ··· ···

⎤ On On ⎥ .. ⎥ ⎥ . ⎥ O⎦ n

On

Ae (kt−1 , d) = Ad (A − LC), Be (kt−1 , d) = On · · · On On ⎤ ⎡ A −BKA · · · −BKAkt−1 A · · · On ··· ··· ··· · · · On ⎥ ⎢ On Ax (kt−1 , d) = ⎢ . .. ⎥ . . . .. ⎦ ⎣ . .. .. .. . . . On ··· ··· ··· · · · On T Bx (kt−1 , d) = On · · · · · · On , Ae (kt−1 , d) = On Be (kt−1 , d) = On · · · A · · · Ad A · · · On



www.ietdl.org 3.2

and ⎡

Dσ (kt−1 ,d)

D −BKD · · · −BKAkt−1 D · · · O ··· ··· ··· ··· ⎢ n ⎢ . . . . .. .. .. .. =⎢ . ⎢ .. ⎣O ··· ··· ··· ··· n On ··· D ··· Ad D

⎤

On On · · · On ⎥ .. .. ⎥ ⎥ . . ⎥ ··· O ⎦ n

We now give the model for the case of time-varying delays and packet dropouts existing in both forward and feedback channels. The control value used at time t is constructed at time t − N − 1 − kt−N −1 and (11) can be rewritten as follows

· · · On

xt+1 = (A + BK)xt − BKAN +kt−N −1 (A − LC)et−N −1−kt−N −1

with d ∈ {0, . . . , N1 }.

N +kt−N −1

As a special case, we omit packet dropout phenomenon and only consider the constant delay case, which means d = 0. Suppose that the constant delay size is k, where k ∈ {0, . . . , N1 }. Then the switched system (14) will be reduced to the system under the switching signal σ (k, 0) and it can be written as the following form Xˆ t+1 Xˆ t ¯ ¯ = (Ak + Ak ) (15) et−k et−1−k where

A¯ k

− BK

Remark 3: From the above description, it is clear that the novel system model (14) just reflects the relation between the system state and the observer auxiliary variable. It is indeed a coupled switched system, in which the switching signals are closely related to the networked delays and the difference of delay size between two adjacent times, that is the auxiliary variable d. The model is different from the one constructed in [21] where only constant delay are considered.

A¯ k

Extended model for two channels

⎡ A + BK On On ⎢ In ⎢ . .. ⎢ = ⎢ .. . ⎣ O · · · n On ··· ⎡ A −BKA ··· ⎢ On ⎢ . .. ⎢ . =⎢ . . ⎣O · · · n On ···

· · · On · · · On .. .. . . In O n On O n

⎤ −BKAk (A − LC) On ⎥ ⎥ .. ⎥ ⎥ . ⎦ O n

A − LC

⎤ · · · −BKAk A On ··· ··· On ⎥ .. ⎥ .. .. ⎥ . . . ⎥ ··· ··· On ⎦ ··· ··· On

T T Xˆ Tt = [xt+1 , . . . , xt−k−1 ]

A¯ k can be further rewritten as the form of the normbounded uncertainties ¯ k F¯ tk E¯ k A¯ k = D where ⎤ D −BKD · · · −BKAk D On ··· ··· ··· On ⎥ ⎢On ⎢ . .. ⎥ . . . ⎥ ⎢ ¯ .. .. .. Dk = ⎢ .. . ⎥ ⎣O ··· ··· ··· On ⎦ n On ··· ··· ··· On F¯ tk = diag{Ft , . . . , Ft }, E¯ k = diag{E, . . . , E} ⎡

Remark 4: It is noted that in [21], the same model as (15) was proposed. Therefore the NPC model (14) with timevarying delays includes the existing ones with constant delay as a special case. 1316 © The Institution of Engineering and Technology 2014


(16)

j=0

Similar to Case I in Section 3.1, we will obtain the observer error subsystem as follows et−N −kt−N = et−N −1−kt−N −1

(17)

Similarly, as Case II in Section 3.1, the relationship between the observer error et−N −kt−N and et−N −1−kt−N −1 is expressed in the following form et−N −kt−N = Ad (A − LC)et−N −1−kt−N −1 +

d

Aj Axt−N −kt−N −1 +d−j−1

(18)

j=0

where d ∈ {0, . . . , kt−N −1 }. Then, by replacing the system state and the system matrix, we can obtain the same switched system as the form (14). Dimensions of the switched system matrix in (14) will be enlarged in this situation. They are related to the total length of the bound range of delays in both forward and feedback channels. Remark 5: As it has been shown above, the switched system (14) is adopted to describe the closed-loop NCS under time-varying delays that may be multiples of the transmission interval. In [25], the switched method is also used to study the stability problem for NCS with communication constraints, delays and transmission intervals. However, network-induced delays in [25] are restricted to be smaller than transmission intervals. Based on the system model proposed in this section, we will give stability analysis in the following.

4

Stability analysis

Under the above analysis, we are in the position to give the main result. Theorem 1: The uncertain discrete-time systems (1) and (2) with time-varying delays in feedback channels is stable under the proposed predictive controller (8) if there exist a matrix P ∈ Rn(N1 +3)×n(N1 +3) and a scalar > 0, such that the following LMIs ⎡ ⎢ ⎣

−P + Dσ (kt−1 ,d) DσT(kt−1 ,d)

A¯ σ (kt−1 ,d) P

−P

On(N1 +3)

⎤

⎥ PEσT(kt−1 ,d) ⎦ < 0 −In(N1 +3) (19)

hold for kt−1 ∈ {0, . . . , N1 } and d ∈ {0, . . . , kt−1 }, where A¯ σ (kt−1 ,d) , Dσ (kt−1 ,d) and Eσ (kt−1 ,d) are defined above. IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

www.ietdl.org In the following, for convenience of analysis, we use σ instead of σ (kt−1 , d). Proof: For the inequality

5

(A¯ σ + A¯ σ ) P (A¯ σ + A¯ σ ) − P T

−1

−1

0 and a scalar > 0, such that ⎡ ⎤ ¯ kD ¯ kT A¯ k P On(N1 +3) −P + D ⎢ ⎥ (25) −P P E¯ kT ⎦ < 0 ⎣ −I ¯ k and E¯ k are the matrix defined in (15). holds, where A¯ k , D IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

Simulation and experiment on DC-motor

In this section, a numerical example and a practical example on DC-motor will be given, respectively, to show the effectiveness of the proposed NPC scheme. Example 1: The numerical example. Consider (1) and (2) with the following system matrices ⎡ ⎤ −0.85 0.2710 −0.4880 ⎢ ⎥ 0.1 0.24 ⎦ A = ⎣0.4820 0.0020 0.3681 0.7070 ⎡ ⎤ 0.5 0.1

0.1 0.2 1 ⎢ ⎥ B = ⎣0.3 −0.4⎦ , C = 0.4 0.3 0.1 0.2 0.5 The parameters of uncertain matrices ⎡ 0.0390 −0.0560 ⎢ 0.0444 D = ⎣ 0.0088 −0.0636 −0.0950 ⎡ −0.2656 −09863 ⎢ E = ⎣−1.1878 −0.5186 −2.2023

0.3273

in (1) are ⎤ 0.0781 ⎥ 0.0569 ⎦ , −0.0822 ⎤ 0.2341 ⎥ 0.0215 ⎦ −1.0039

The eigenvalues of A are −1.0187, 0.2098 and 0.7659, and it is obvious that the system is unstable. The system is controllable and observable since the controllability matrix and the observability matrix are full-rank, and then design the matrices K and L by pole assignment to ensure the stability of the closed-loop system free of network delay. The matrices are given as

0.1060 −0.567 0.0117 K= 0.0837 −0.0471 0.0074 ⎡ ⎤ −2.2223 −0.6502 0.5179 ⎦ L = ⎣ 0.0930 0.309

0.1495

Choosing = 1, and then solving the LMIs in Theorem 1 by using the Matlab-LMI control toolbox lead to a feasible solution with N1 = 6 and N = 2. The control trajectories and measurable output trajectories with the initial conditions x(0) = [3 1 2]T , xˆ (0) = [0 0 0]T are presented in Figs. 2 and 3, respectively. The simulation results reveal that the closed-loop is stable. Example 2: The experiment on DC-motor. Next, we will consider a DC-motor speed control system to illustrate the stability and robustness of NCS. The DCmotor system consists of a DC-motor and a motor driver (see Fig. 4). The sensor and the controller are connected through a network, and the communication protocol between them is UDP. The objective of this control system is to make the DC-motor rotate speed follow the reference input. In this paper, the reference input voltage for the motor driver is 3.5 V and the desired output of the DC-motor is 770 rpm. 1317 © The Institution of Engineering and Technology 2014

www.ietdl.org 900

2 u1 u2

1.5

800 700 Reference & Outputs (rpm)

1

Inputs

0.5 0 −0.5 −1

600 500 400

200

−2

100 0

20

40

60

80

0

100

time (sec)

Fig. 2

Output without uncertainty

300

−1.5

−2.5

Reference input Output with uncertainty

0

2

4

6

8

10

Time (sec)

Control inputs u1 and u2

Fig. 5 Outputs and reference input of DC-motor plant with local control

3

900

y1 y2

2.5

800 2

700 Reference & Output (rpm)

Outputs

1.5 1 0.5 0 −0.5

400

Reference input Output without uncertainty

300

Output with uncertainty

100 0

20

40

60

80

100

time (sec)

Fig. 3

500

200

−1 −1.5

600

0

0

2

4

6

8

10

Time (sec)

Measurable outputs y1 and y2 Fig. 6

Outputs and reference input of DC-motor plant with NPC

The linear dynamic system can be re-written in terms of the state-space and output equation as follows: (1) and (2) without uncertainties effects ⎡ ⎤ 0.2842 −0.0719 0 0 0⎦ , A=⎣ 1 0 1 0 ⎡ ⎤ ⎡ ⎤T 1 13.3500 B = ⎣0⎦ , C = ⎣178.5000⎦ 0 35.7800 Fig. 4

Then the feedback gain K and the observer gain L are designed by using pole assignment method and are given as

DC-motor plant

According to the experimental measurements, which were carried out with the sampling period 0.25 s, under certain conditions, the identified model of the plant was found as follows 13.35z −2 + 178.5z −3 + 35.78z −4 G(z −1 ) = 1 − 0.2842z −1 + 0.0719−2 1318 © The Institution of Engineering and Technology 2014

(26)

K = −0.3158 0.0181 0.0500 T L = 0.0005 −0.0024 −0.0132 which ensure that the closed-loop system free of time delays and uncertainties is Schur stable. In this paper, we consider the norm-bounded uncertainties which is different from that IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

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Fig. 7

Simulation diagram

in [21]. Set the matrices in A are ⎡

⎤ −0.2734 0.0664 0.9596 D = ⎣−0.9619 −0.0189 −0.2727⎦ 0 −0.9976 0.0690 ⎡ ⎤ −0.0520 0.0010 0 0 −0.0501 0⎦ E=⎣ 0 0 0

7

This work was supported by the National Natural Science Foundation of China under Grants 61174058 and 61325014.

8 1 2

Given = 1, it is found that the LMI conditions in Theorem 1 are satisfied with the time-varying delay bound 1.5 s. Then the step response simulations of DC-motor system with its predictive controller are considered below. The initial speed of the plant is set to be 0 rpm, the initial states of the observer are set to be [0, 0, 0], and the reference input of the motor driver is set as 3.5 V (see the reference curves in Figs. 5 and 6). The local control and the intranet control of DC-motor plant are considered, respectively. The local control system is composed of the plant, one-step-ahead observer and controller, and the outputs of the DC motor system with and without uncertainty are shown in Fig. 5, respectively. The diagram of NPC is shown in Fig. 7. It is tested from Theorem 1 that the stability of DC-motor plant via network control can be guaranteed for the maximum delay bound 1.5 s in feedback channels under the proposed NPC, and the outputs of the NCS with and without uncertainty are given in Fig. 6, respectively.

6

Conclusions

This paper has considered the robustness of linear uncertain NCS under the proposed NPC scheme. The NCS is under the influences of time-varying delays and packet dropouts in both forward and feedback channels. By introducing an NPC scheme, a coupled switched system is proposed to describe the NCS with time-varying delays, packet dropouts and uncertainties. Then a sufficient stability criterion for the considered NCS has been proposed, which covers the constant delay as a special case. Results for two illustrative examples have been given, the second one being a laboratory-scale, DC-motor based test-rig. Both simulation and physical plant experimental results have demonstrated considerably improved effectiveness of the proposed novel design techniques. IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1312–1320 doi: 10.1049/iet-cta.2013.0901

Acknowledgments

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10

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