H∞ controller design of networked control systems with a new quantization structure

Applied Mathematics and Computation 376 (2020) 125070

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H∞ controller design of networked control systems with a new quantization structure Jia-Sheng Song a,b, Xiao-Heng Chang a,b,∗ a b

Institute of Automation, Bohai University, Jinzhou, Liaoning, 121013, China College of Engineering, Bohai University, Jinzhou, Liaoning, 121013, China

a r t i c l e

i n f o

Article history: Received 24 September 2019 Revised 21 December 2019 Accepted 19 January 2020

Keywords: Networked control systems Quantization Packet dropouts H∞ performance

a b s t r a c t This paper is concerned with the problem of H∞ static output-feedback control for networked control systems (NCSs) with quantization and Markov packet dropouts. A new quantization structure is proposed, and a mathematical treatment of this structure is given to illustrate the advantage for the quantization effects. Moreover, two Markov chains are adopted to describe the packet dropouts, not all the elements of the transition probability matrices are assumed to be known. Then, the closed-loop systems are modeled as the Markov jump linear systems (MJLSs) with partly unknown transition probabilities. An H∞ static output-feedback controller design method is given to ensure the stability of the corresponding closed-loop systems with proposed quantization structure and satisfy the H∞ performance for the external bounded disturbance in terms of linear matrix inequalities (LMIs). A simulation example is utilized to demonstrate the effectiveness and the applicability of developed approach. © 2020 Published by Elsevier Inc.

1. Introduction In recent years, the NCSs have attracted much attention due to the low costs, easy maintenance, convenient installation and increased flexibility. And the application of NCSs has widely existed in many areas, such as chemical plants, mechanical systems, hybrid systems and biological systems. Nevertheless, it should be noticed that, unlike the research on traditional control systems, some new thorny problems, such as packet dropouts and quantization, are raised in the research on NCSs owing to the fact that the insertion of communication networks in the feedback loop, which may lead to performance degradation or even destabilize the systems [1–8]. Due to the network congestion in data packet transmission, packet dropout becomes one of the practical issues during the study on NCSs. During the last few years, there has been considerable process in solving this problem. And various approaches have been reported in previous literature, such as the methods of using Bernoulli process [9–12] and Markov chain [13–15]. The Bernoulli process is usually adopted to describe the state of packet dropout and packet receipt with a given probability at current time. This method is widely used to cope with the problem of Kalman filtering [9] and outputfeedback control [12] due to its simpleness and convenience. However, it can be observed from the literature mentioned above that the process of packet transmission over communication networks is independent for each other. In other words, Bernoulli process are memoryless, which means that the state of the data packet transmitted in the channel at current time can’t affect the state at further time. And we all know that there exists the effect of the packet state from current time to ∗

Corresponding author at: Institute of Automation, Bohai University, Jinzhou, Liaoning, 121013, China. E-mail addresses: [email protected] (J.-S. Song), [email protected] (X.-H. Chang).

https://doi.org/10.1016/j.amc.2020.125070 0 096-30 03/© 2020 Published by Elsevier Inc.

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J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

the next. Markov chain, a discrete-time stochastic process in which the current state can affect the next state, is able to solve this problem, for instance [16]. On the other hand, quantization is said to be another important issue on NCSs. In the early research of quantizer, Kalman investigated the quantization effect in sampled data control systems, and pointed out the fact using a finite-alphabet quantizer to quantized the stabilizing controller will result in that the feedback systems exhibit limit cycle and chaotic behavior. Since then the research on quantization mainly focuses on the understanding and mitigation of quantization effects. With the in-depth study of quantization issue, there have been two main approaches to solve this problem in recent years. The first approach considers the static quantizer which is memoryless with fixed quantization levels. Delchamps [17] employed the static uniform quantizer to study the time-invariant discrete linear unstable systems. Whereafter, Wong and Brockett [18] proposed the bounds on the number of quantization intervals needed to stabilize the linear systems. Elia and Mitter [19] showed that for quadratic stabilization of the discrete-time linear time-invariant systems, the coarsest, or least dense quantizer is logarithmic. The second approach is dynamic quantizer, which is considered to be advantageous as it scales the quantization levels dynamically, thereby increasing the attraction region and reduceing the steady limit cycle. In fact, the dynamic quantizer introduced in [20] can be treated as a device that combines a dynamic scaling factor with a static quantizer. Fu and Xie [21] had proposed a dynamic scaling method for quantized output feedback control to achieve stabilization using a finite-level quantizer. And Chang et al. [22] investigated the quantized feedback control problem for uncertain systems. It should be pointed out that, the dynamic quantizers have powerful functions, however the process is very complicated to handle in practice. It is worth mentioned that, the signal to be quantized is handled only once in the previous investigation. Thus, we are concerned about the other case that the signal is quantized multiple times and transmitted via multiple channels. Consider the above and the stability analyse [23–25] of the systems, this paper investigates the H∞ static outputfeedback control problem for discrete-time systems with output quantization and Markov packet dropouts. Considering a new quantization structure, we aim to design the H∞ static output-feedback controller to make the closed-loop systems stochastically stable and satisfy a prescribed H∞ performance. The contributions of this paper can be concluded as follows: 1. The problem of H∞ output-feedback control has been considered for NCSs with the effects of dynamic quantization and Markov packet dropouts. 2. A new quantization structure is proposed to reduce the quantization effects. The rest of the paper is organized as follows. Section 2 introduces the system to be studied, and some fundamentals about quantization and Markov chain are discussed, which are useful throughout this paper and give the basic idea of our setup. Further, the stability analysis with H∞ performance index γ is investigated, and the controller design is given in terms of LMIs in Section 3. Section 4 gives a simulation example to demonstrate the effectiveness and practicability of the approach. Section 5 draws some conclusions. Notations. Notations used in this paper are explained as follows. For real symmetric matrices V and Y, the notation V − Y ≤ 0 (or V − Y ≤ 0) means that the notation V − Y is negative definite (or negative semi-definite). The asterisk ∗ is referred to as a term that is induced by symmetry. Without special explanation, I and 0 respectively represent the identity matrix and zero matrix with appropriate dimensions. R⊥ stands for the null space of R. The superscript T denotes matrix transposition. 2. Problem formulation and preliminaries 2.1. Control system Consider the NCS shown in Fig. 1, the discrete-time linear system is expressed as:

x(k + 1 ) = Ax(k ) + Bu(k ) + B1 w(k ), z(k ) = Dx(k ) + Ew(k ), y(k ) = Cx(k ),

(1)

where x(k) ∈ Rn is the state, u(k) ∈ Rm is the control input, y(k) ∈ Rp is the measurement output, z(k) ∈ Rq is the regulated output, and w(k) ∈ Rr is the disturbance vector, respectively. A, B, B1 , C, D and E are known constant matrices with appropriate dimensions. Bounded random packet dropouts exist in the channels, as shown in Fig. 1. Assumptions : 1. 2. 3. 4.

The matrix B is of full column rank, i.e. rank(B ) = m. The maximum number of packet dropouts is bounded and known. The scaling factors of the quantization structure adopt the same value. Communication channels are free of delay.

The rest of Section 2 gives some descriptions of quantization and packet dropouts and presents a mathematical model of closed-loop system based on quantization and packet dropouts. 2.2. Quantization structure To demonstrate the main results, we adopt the dynamic quantizer to deal with the output signal of the system. The quantizer is defined as the form described in [22], which is composed of scaling factor and static quantizer. Defining that

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

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Fig. 1. A networked control system.

Fig. 2. The new quantization structure.

v ∈ Rl is signal variable to be quantized, we can obtain the description for quantizer by the mathematical model as follows:

q μ ( v ) = μq

v μ

,

μ>0

(2)

where μ is the parameter of the quantizer and can be computed independently. One such static quantizer is the uniform quantizer whose characteristic is that the quantization interval is equal, which is described by a function q: Rs → Rs with the following property:

z − q ( z )  ≤

 2

, if

z  ≤ M

(3)

where M and  stand for the range and the sensitivity of the static quantizer, respectively. Generally, in almost all of NCSs, the controller and the plant are separate, the signal from the sensor needs to be quantized before it is transmitted via the communication channel, therefore the effects of quantization are inevitable. If the effects of quantization is ignored, it would be difficult for us to avoid the degradation of system performance. However, in the previous literatures, the signal to be quantized is handled only once. Now, let us consider a new quantization structure as shown in Fig. 2. And the description of how the quantization structure deals with the input signal is shown as follows. The input signal of the proposed quantization structure is first quantized by qμb (v ). Then a bounded signal e(k) can be obtained by making a difference between the quantized signal qμb (y(k )) and y(k) before qμb (y(k )) is transmitted via Channel 1, i.e.,

e(k ) = y(k ) − qμb (y(k )).

(4)

Next, e(k) is quantized by qμs (v ) and transmitted to the controller via Channel 2. Finally, we can get the desired quantized signal Qμo (y(k )) by

Qμo (y(k )) = qμb (y(k )) + qμs (e(k )).

(5)

Fig. 2 shows the quantization structure, in which y(k) is input signal of quantization structure and Qμo (y(k )) is output, respectively. As described, the output signals is handled by the proposed quantization structure and transmitted over two communication channels, and then the signal is re-processed before entering the controller. In order to illustrate the mathematical characteristics of the proposed quantization structure by effective method, and considering the parameters of the quantizers, we define

M s = b ,

μb = μs = μo

(6)

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J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

where Ms and μs are the range and the scaling factor of quantizer qs (v), b and μb are the sensitivity and scaling factor of the quanizer qb (v). From (4), (5) and (6), the following equality can be obtained to express the quantized signal Qμo (y(k )),

Qμo (y(k )) = qμb (y(k )) + qμs (e(k ))

μb qb (μ−1 y(k )) + μs qs (μ−1 s e (k )) b −1 −1 −1 = μb qb (μ−1 y ( k )) + μ q ( μ s s s y (k ) − μs μb qb (μb y (k ))) b −1 −1 = μoqb (μo y(k )) + μoqs (μo e(k )) −1 −1 −1 = μo ((qb (μ−1 o y (k )) + μo y (k ) − μo y (k )) + qs (μo e (k ))) −1 −1 −1 = μo ((μo y(k ) − μo e(k )) + qs (μo e(k ))) −1 = y(k ) − μoμ−1 o e (k ) + μo qs (μo e (k )) −1 = y(k ) + μo (qs (μ−1 o e (k )) − μo e (k )) = y ( k ) + μo v 1 ( k ) =

qs (μ−1 o e (k ))

(7)

− μ−1 o e ( k ).

where v1 (k ) = Next, we define that the sensitivities of Q(v) and qs (v) are Q and δ , respectively. From the above, the following equation can be obtained,

Q = δ.

(8)

Remark 1. The new quantization structure is proposed via a extension of ordinary dynamic quantizer qμb (v ), i.e. adding a quantizer qμs (v ) to handle the quantization error e(k) of the quantizer qμb (v ), which can be expressed as

Q μo ( · ) = =

μo Q ( · ) μo (qb (· ) + qs (· ) ),

(9)

where · stands for the corresponding signal variable to be quantized. From (8), compared to ordinary quantizer qμb (· ), we can be effortless to find that, the proposed quantization structure has more satisfying quantization effects than ordinary quantizer qμb (· ). In this section, we have discussed the mathematical description of the proposed quantized structure without packet dropouts, which can reduce the effects of quantization to a certain extent. Next, we introduce the packet dropoouts and give a mathematical description of the proposed quantized structure under the case of existing packet dropouts in the communication channels. 2.3. Packet Dropouts Modeled by Markov Chains Packet dropout is one of the criterion for quality of service for the network, which is also an important issue on NCSs. In this paper, we mainly study the packet dropout caused by network congestion. In practice, the process of packet dropouts has the phenomenon that the states of contiguous data packets transmitted at each time are not independent for each other, that is to say, there exists effect of the state from current time to the next. To solve the effect of packet dropouts, the Markov chain is adopted to describe the process of packet dropouts. Due to the particularity of proposed quantization structure, it is difficult to simply describe the packet dropout process on the communication channels by the ordinary method mentioned in the previous literature. Based on this, we define the following independent cases for the output data packet of the proposed quantization structure, Case a: the packet transmitted via channel 1 is received, Case a¯ : the packet transmitted via channel 1 is dropped, Case b: the packet transmitted via channel 2 is received, Case b¯ : the packet transmitted via channel 2 is dropped. Since the packet receipts in case ab¯ carry a large information of the original signal, and in order to prevent system instability caused by excessive packet dropout probability, we model the states of packet receipts in cases ab and ab¯ as a Markov process and use the fact that the state of the packet at this moment can affect the state of the packet at the next. We assume there is a device ϒ which treats the case a¯ b as packet dropout. Base on packet dropouts and the quantization structure, the signals being transmitted are described as



ab : Qμ(1o) (· ) = μo (qb (· ) + qs (· ) ), ab¯ : Qμ(2o) (· ) = μoqb (· ).

(10)

Then, the sensitivity of Q(m) ( · ), m ∈ {1, 2}, is



ab : 1 = δ, ab¯ : 2 = .

(11)

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Remark 2. If there is not the assumption that treat the case a¯ b as packet dropout, for the signals being transmitted in case a¯ b, based on packet dropouts and quantization structure, it is described as

a¯ b : Qμ(3o) (· ) = μoqs (· ). However, we all know that, when the case a¯ b occurs, the signal transmitted to the controller is a quantization signal of e( · ), which not only has no control ability itself, but also may affect the performance of the control system. Remark 3. It can be noticed from (10) that the sensitivity of the proposed quantization structure depends on whether communication channels are free of packet dropout or not. When case ab¯ occurs, the proposed quantization structure will degenerate into a ordinary quantizer qμb (v ). Further, some descriptions of packet dropouts are given as follows. 1 ≤ r(k) ≤ d stands for the number of packet dropouts over the feedback loop and s(k ) = m represents the sensitivity of the quantization structure at current instant k. Further, r(k) and s(k) are described as two homogeneous Markov chains, independent of each other. And  and  shown in the following are two transition probability matrices of r(k) and s(k) respectively, where πi j = prob{r (k + 1 ) = j | r (k ) = i} and λmn = prob{s(k + 1 ) = n | s(k ) = m }(m, n = {1, 2} ) which mean that the probabilities of r(k) and s(k) jumping from state i to j and from state m to n are π ij and λmn , respectively.

⎡

π11

⎢ ? =⎢ ⎣ . ..

πd1

?

π22 .. .

πd2

··· ··· .. . ···

⎤ π1 d π2 d ⎥ λ11 , = .. ⎥ ⎦ λ21 .

λ12 , λ22

(12)

?

(i ) where ? represents each unknown element. For convenience, ∀i ∈ S, we denote S = SK(i ) + SUK with (i ) SUK  { j : πi j is unknown},

SK(i )  { j : πi j is known}.

(13)

If SK(i ) = ∅, it can be obtained that

SK(i ) = {K1 , K2 , . . . , Kl }, l ∈ {1, 2, . . . , d − 2},

(14)

where Kt ∈ N + , t ∈ {1, 2, . . . , l }, stands for the index of the t − th known transition probability in the i − th row of matrix . And we should be note that d 

πi j = 1 ,



πi j = πK(i) ,

j∈SK(i )

j=1

2 

λmn = 1.

(15)

n=1

Remark 4. Comparing (7) and (10), the method describing the state of the packet receipts with the Markov chain is more complicated than the method only treating the case ab as the packet receipts. However, in view of the particularity of the proposed quantization structure, (10) can more accurately describe the changes of signals. Remark 5. Moreover, the case that unknown elements exist in  is excluded, since if we have one known element, the other can be naturally calculated from the known element in each row. 2.4. Controller In order to compensate for packet dropouts, it is necessary that the feedback control law at the time k should depend on r(k) and s(k). Thus, the static output-feedback controller to be designed can be expressed as

u(k ) = Ki,m Qμ(mo ) (y(k )) = Ki,m (y(k ) + Qμ(mo ) (y(k )) − y(k )) = Ki,mCx(k ) + Ki,m v(k ),

(16)

(m ) where v(k ) = Qμ (y(k )) − y(k ). o

From the view point of the zero-order hold (ZOH), the closed-loop system can be expressed as an MJLS by considering the utilized control signal,

x(k + 1 ) = (Ai + Bi Ki,mC )x(k ) + Bi Ki,m v(k ) + B1 w(k ),

(17)

z(k ) = Dx(k ) + Ew(k ),

(18)

where Ai =

Aj

and Bi =

 j−1 r=0

Ar B.

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J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

Next, we will propose some lemmas which are essential to the proofs of theorems. Lemma 1 [26]. For the discrete-time system with w(k ) = 0, if there exist a set of matrices Pi > 0, i = 1, 2, . . . , d such that

ATi

d 

πi j Pj Ai − Pi < 0.

(19)

j=0

Then the system is stochastically stable. Lemma 2. (S-procedure): Given quadratic functions for ∈ Rn , S0 ( ) = T Q0 , S1 ( ) = T Q1 , . . . , Sn ( ) = T Qn , Qm = QTm , m = 1, 2, . . . , n. Then, we have S0 ( ) < 0 with S1 ( ) ≥ 0, S2 ( ) ≥ 0, . . . , Sn ( ) ≥ 0, if there exist scalars ρ 1 > 0, ρ2 > 0, . . . , ρ n > 0 satisfying

Q 0 + ρ1 Q 1 + ρ2 Q 2 + . . . + ρ n Q n < 0 .

(20)

3. Main results In this section, to discuss the effects of new quantization structure on the performance of the control system in the case of packet dropouts, we first consider the stability of system (17) without disturbance w(k), and give the design method of corresponding controller in Theorem 1 according to LMIs. Further, Theorem 2 is presented to ensure the system (17) and (18) with disturbance w(k) is stable and meets an H∞ performance. The main results and corresponding proofs are shown as follows. Theorem 1. Consider the discrete-time system (17) with disturbance w(k ) = 0. For known quantization range M, sensitivities m and partly unknown transition probabilities, the corresponding system is stochastically stable if there exist scalar θ > 1, matrices T > 0, R , N , and M Pi,m = Pi,m i,m , i ∈ S, m = 1, 2, such that i,m i,m

⎡

⎤

−Pi,m + θ C T C

∗

0

− 4M2

∗

Gi,m Ai + Bi Ni,mC

Bi Ni,m

i,m

⎢ ⎣

∗ 2

m

⎥ ⎦ < 0,

(i ) where Gi,m = Bi Mi,m (BTi Bi )−1 BTi + Ri,m B⊥ , i,m = −Gi,m − GTi,m + PK(i,m ) + (1 − πK(i ) )Pj,n , ∀ j ∈ SUK and PK(i,m ) = i

πi j λmn Pj,n . The controller gain matrices are Ki,m = M−1 N . i,m i,m

(21) 2

n=1



(i ) j∈SK

(22)

Proof. Consider the property of quantizer expressed by (2),

   y (k )     μo  ≤ M.

(23)

We select a scalar variable θ > 1 to ensure the following equality holds,

√

μo =

θ

M

|y(k )|.

(24)

Base on the proposed quantization structure and packet dropouts, it is effortless to obtain,

|v(k )| = |Qμ(mo ) (y(k )) − y(k )| m ≤ μo = =

m 2

2 √

θ

M √

m θ 2M

|y(k )|

|Cx(k )|.

(25)

That is,

vT ( k )v ( k ) ≤

2m θ 4M 2

xT (k )C T Cx(k ).

(26)

It can be obtained that

ξ T (k )m ξ (k ) ≥ 0, where



x (k ) ξ (k ) = , v (k )

(27)



CT C 1 = 0

∗

2

4M − 2 θ m

.

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

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In order to get the stability condition of system (17), we choose the form of Lyapunov function as follows

V (k, i, m ) = xT (k )Pi,m x(k )

(28) T Pi,m

with positive definite matrices Pi,m = > 0. Assume w(k ) = 0, the following can be obtained



V (k, i, m ) = ξ (k ) T

ATi,m

2  d 



πi j λmn Pj,n Ai,m ξ (k ) − xT (k )Pi,m x(k ) < 0,

(29)

n=1 j=1

where Ai,m = [Aˆ i,m Bˆi,m ], Aˆ i,m = Ai + Bi Ki,mC and Bˆi,m = Bi Ki,m .   πi j And from (15), we know that 2n=1 λmn = 1 and (i ) (i ) = 1. Via Lemma 1, the stability condition can be rewritten j∈SUK 1−π K

as

ATi,m

2  d 

πi j λmn Pj,n Ai,m − diag{Pi,m , 0}

n=1 j=1

⎛

= ATi,m ⎝PK(i,m ) +

2 

λmn

(i ) j∈SUK

n=1

=

2 

λmn

 (i ) j∈SUK

n=1



⎞ πi j (1 − πK(i) )Pj,n ⎠Ai,m − diag{Pi,m , 0} 1 − πK(i )

πi j (ATi,m (PK(i,m) + (1 − πK(i) )Pj,n )Ai,m − diag{Pi,m , 0} ) < 0, 1 − πK(i )

(30)

which is equivalent to (i ) i,m  ATi,m (PK(i,m) + (1 − πK(i) )Pj,n )Ai,m − diag{Pi,m , 0} < 0, ∀ j ∈ SUK .

(31)

Performing a congruence transformation obtains that

ξ T (k )i,m ξ (k ) < 0.

(32)

Further, from (27), a sufficient condition for i,m < 0 is obtained via S-procedure,

i,m + θ m < 0.

(33)

Using Schur’s complement, we have

⎡

−Pi,m + θ C T C

∗

0

− 4M2

⎢ ⎣

2

Bˆi,m

⎥ ⎦ < 0.

∗

m

Aˆ i,m

⎤

∗ −(

PKi,m

(i )

+ (1 − πK )Pj,n )

(34)

−1

We find that there exist nonlinear term in the inequality matrix (34), and cannot solve that by LMIs directly. Further, from the fact that (X − V )X −1 (X − V )T ≥ 0, we have −V X −1V T ≤ −V − V T + X when X = X T > 0. According to this, pre- and post-multiplying by diag{I, I, Gi,m } and diag{I, I, GTi,m }, one can be obtained that

⎡

−Pi,m + θ C T C

∗

0

− 4M2

∗

Gi,m Ai + Bi Mi,m Ki,mC

Bi Mi,m Ki,m

i,m

⎢ ⎣

∗ 2

m

⎤

⎥ ⎦ < 0.

(35)

For the nonlinear terms in (35), by defining Ni,m = Mi,m Ki,m , the inequality in (21) can be obtained. Then a modedependent stabilizing controller can be obtained, such that system (17) without w(k) is stochastically stable. Further, desired controller gain matrices are computed as

Ki,m = M−1 N . i,m i,m

(36) 

Remark 6. From (7), we know that the quantization effect caused by proposed quantization structure is better than that of the ordinary quantizer. However, it should be pointed out that, the difference between proposed quantization structure and ordinary quantizer in the form of signal transmission leads to the fact that the quantized signals is transmitted via two communication channels. Compared with the previous literature, due to the packet dropouts in both channels, it is necessary to add a homogeneous Markov chain s(k) to describe the states of the late-arrival packets. In view of this problem, we model the states of the packet receipts as a Markov process. Therefore, based on packet dropouts, the closed-loop system with proposed quantization structure can be viewed as an MJLS.

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J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

Theorem 1 gives a sufficient condition for the stochastic stability of NCSs with new quantization structure and Markov packet dropouts, and not all the elements of the transition probability matrices are assumed to be known. Next, the following corollary for the transition probability matrices without unknown elements could be reduced from Theorem 1 directly. Corollary 1. Consider the discrete-time system (17) with disturbance w(k ) = 0. For the known quantization range M, sensitivities m and known transition probabilities, the corresponding system is stochastically stable if there exist scalar θ > 1, matrices T > 0, R¯ ¯ i,m , N¯ i,m , i ∈ S, m = 1, 2, such that Pi,m = Pi,m i,m , M

⎡

⎤

−Pi,m + θ C T C

∗

0

− 4M2

∗

G¯ i,m Ai + Bi N¯ i,mC

Bi N¯ i,m

¯ i,m 

⎢ ⎣

∗ 2

m

⎥ ⎦ < 0,

(37)

¯ i,m (BT Bi )−1 BT + R¯ i,m B⊥ ,  ¯ i,m = −G¯ i,m − G¯ Ti,m + where Gi,m = Bi M i i i

Ki,m =

2

n=1

d j=1

πi j λmn Pj,n . And the controller gain matrices are

¯ −1 N¯ i,m . M i,m

(38)

In Theorem 1 and Corollary 1, the stability conditions have been provided for system (17) without disturbance w(k). Next, considering the external disturbance, we will propose the effective methods to solve H∞ control problem for discrete-time system (17) and (18) in the rest of this section. Theorem 2. Consider the discrete-time system (17) and (18). For known quantization range M, sensitivities m and partly unT > 0, Y , R ˆi,m and Ui,m , i ∈ S, known transition probabilities, given a scalar γ > 0, if there exist scalar θ > 1, matrices Pi,m = Pi,m i,m m = 1, 2, such that

⎡

−Pi,m + θ C T C + DT D

∗

0

− 4M2

ET D

0

−γ 2 I + E T E

Gˆ i,m Ai + BiYi,mC

BiYi,m

Gˆ i,m B1

⎢ ⎢ ⎢ ⎣

where Gˆ i,m =

∗ 2

m

BiUi,m (BTi Bi )−1 BTi

+

∗

Rˆi,m B⊥ , i

∗

⎤

⎥ ⎥ ⎥ < 0, ∗ ⎦ ∗

(39)

i,m

(i ) i,m = −Gˆ i,m − Gˆ Ti,m + PK(i,m) + (1 − πK(i) )Pj,n , ∀ j ∈ SUK and PK(i,m ) =

2

n=1



(i ) j∈SK

πi j λmn Pj,n , the corresponding system is stochastically stable with the prescribed H∞ performance index γ , and the controller gain matrices are obtained by −1 Ki,m = Ui,m Yi,m .

(40)

Proof. It can be obtained from Theorem 1 that system (17) without w(k) is stochastically stable via utilizing Schur complement and S-procedure, if LMIs (21) hold. In the presence of bounded disturbance w(k), the system (17) and (18) is required to guarantee the given H∞ performance. Accordingly, we adopt the Lyapunov function formed as (28). And then the difference of V(k, i, m) is deduced as follows:

V (k, i, m ) = ηT (k )AˆTi,m

2  d 

πi j λmn Pj,n Aˆi,m η (k ) − xT (k )Pi,m x(k ),

(41)

n=1 j=1

where η (k ) = [ xT (k ) vT (k ) wT (k ) ]T and Aˆi,m = [ Aˆ i,m Bˆi,m B1 ]. To establish the H∞ performance of system (17) and (18), consider the following H∞ performance index:

J 

∞ 

[ z T ( k ) z ( k ) − γ 2 w T ( k ) w ( k )] .

(42)

k=0

Due to V (0, i, m ) = 0,

J = ≤

∞  k=0 ∞ 

[zT (k )z(k ) − γ 2 wT (k )w(k ) + V (k, i, m )] − V (∞, i, m ) + V (0, i, m ) [zT (k )z(k ) − γ 2 wT (k )w(k ) + V (k, i, m )]

k=0

=

∞ 

ηT (k )¯ i,m η (k ) < 0,

k=0

2

d

(43)

πi j λmn Pj,n Aˆi,m + diag{−Pi,m , 0 − γ 2 I} + [ D 0 E ]T [ D 0 E ]. From Theorem 1, we know that (43) equivalent to i,m  AˆTi,m (PK(i,m ) + (1 − πK(i ) )Pj,n )Aˆi,m + diag{−Pi,m , 0 − γ 2 I} + (i ) [ D 0 E ]T [ D 0 E ] < 0, ∀ j ∈ SUK . ¯ i,m = AˆT where  i,m

n=1

j=1

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

9

Considering S-procedure, for (17), we can rewrite inequality (26) as

ηT (k )m η (k ) ≥ 0, where m = diag{C T C, −

(44) 4M 2

2m θ

, 0}.

Then, the H∞ performance can be satisfied via S-procedure,

i,m + θ m < 0.

(45)

By employing Schur complement, one can be obtained

⎡

−Pi,m + θ C T C + DT D

∗

0

− 4M2

ET D Aˆ i,m

⎢ ⎢ ⎢ ⎣

⎤

∗

∗

∗

∗

0

−γ 2 I + E T E

∗

Bˆi,m

B1

−(PK(i,m ) + (1 − πK(i ) )Pj,n )−1

2

m

⎥ ⎥ ⎥ < 0. ⎦

(46)

Performing a congruence transformation to (46) via diag{I, I, I, Gˆ i,m } and diag{I, I, I, Gˆ Ti,m } gives rise to

⎡

−Pi,m + θ C T C + DT D

∗

0

− 4M2

ET D Gˆ i,m Aˆ i,m

⎢ ⎢ ⎢ ⎣

⎤

∗

∗

∗

∗

0

−γ 2 I + E T E

∗

Gˆ i,m Bˆi,m

Gˆ i,m B1

−Gˆ i,m (PK(i,m ) + (1 − πK(i ) )Pj,n )−1 Gˆ Ti,m

2

m

⎥ ⎥ ⎥ < 0. ⎦

(47)

Utilizing Yi,m = Ui,m Ki,m and the method in Theorem 1 yields one matrix inequality (39) which demonstrates the closedloop system derived from (17) and (18) is stochastically stable with desired H∞ performance. And the controller gain matrices can be given by (40).  The proposed method in Theorem 2 reduces to the following corollary, subject to all the transition probabilities are assumed to be known. Corollary 2. Consider the discrete-time system (17) and (18). For known quantization range M, sensitivities m and the known T > 0, R ˜i,m , Y˜i,m , and U˜i,m , i ∈ S, transition probabilities, given a scalar γ > 0, if there exist scalar θ i,m > 1, matrices Pi,m = Pi,m m = 1, 2, such that

⎡

−Pi,m + θ C T C + DT D

∗

0

− 4M2

ET D

0

−γ 2 I + E T E

G˜ i,m Ai + BiY˜i,mC

BiY˜i,m

G˜ i,m B1

⎢ ⎢ ⎢ ⎣

∗ 2

∗

∗

m

⎤

⎥ ⎥ ⎥ < 0, ∗ ⎦ ∗

(48)

˜ i,m

 d ˜ i,m = −G˜ i,m − G˜ T + 2 where G˜ i,m = BiU˜i,m (BTi Bi )−1 BTi + R˜i,m B⊥ , n=1 j=1 πi j λmn Pj,n , the corresponding system is stochastii i,m cally stable with the prescribed H∞ performance index γ , and the controller gain matrices can be obtained by −1 ˜ Ki,m = U˜i,m Yi,m .

(49)

4. Numerical example In this section, in order to illustrate the effectiveness and practicability of the method proposed in this paper, we consider F-404 aircraft engine system as the object of the simulation example. For the F-404 engine system, follow [27], we select the sampling period T = 0.5s and the uncertain parameter δ (t ) = 0.5, thus the following discete-time linear system can be obtained:



x (k + 1 ) =

y (k ) =



0.5227 0.1463 0.0641

−0.16 0.2

z(k ) = −0.7



0.8 0.4 0.9



0.5099 0.2406 −0.0484 x(k ) + 0.2062 0.3638 0.1695

0 1.0513 0







1.8944 0.9842 0.1480 u(k ) + 0.2245 w(k ), 0.1211 0.4145

0.4 x ( k ), −0.48



0.7 x ( k ) + 0.5w ( k ).

(50)

In the system (50), x1 (k) and x2 (k) are used to denote the horizontal position, and x3 (k) represents the altitude of the aircraft. The control inputs u1 (k) and u2 (k) stand for the engine thrust and the flight path angle, respectively. We suppose that the maximum number of packet dropouts is d = 2, the sensitivity of the quantization structure is 1 = 0.01 or 2 = 1, and the quantization range M is 100.

10

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

Fig. 3. Number of the packet dropouts.

Fig. 4. The modes of quantization sensitivities.

From the quantization structure, the transmission probability matrices are given by



=

0.9 ?



0.1 0.6 , = ? 0.9

0.4 . 0.1

We employ the Matlab LMI toolbox to cope with the LMIs (39) with γ = 2 in Theorem 2. Then the desired controller gain matrices can be given via (40),



K1,1 =

K2,1 =





−6.6834 −1.4179

−1.3898 −4.0526 , K1,2 = 0.8970 −0.4440

−1.6421 , 0.4804

−6.6188 −1.3835

−1.5218 −4.0717 , K2,2 = 0.8338 −0.4683

−1.6844 . 0.4712

and the designed scalar θ = 1.8437.

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

11

Fig. 5. The responses of the states x(k).

Fig. 6. The responses of controller output u(k). √

For discrete system corresponding to (50), we adopt x(0 ) = [1 0 0], v(k ) = 2mM θ Cx(k ), and w(k ) = sin(5k )e−k . Fig. 3 and 4 give the changes in packet dropout numbers and quantization sensitivities. Figs. 5 and 6 show the simulation responses of the resulting system. By showing the calculation of the controller gain matrices and the response diagrams of system states and controller outputs, this simulation example proves the design method developed can meet the H∞ performance for F-404 engine system with proposed quantization structure. 5. Conclusions The problem of static output-feedback control is addressed for linear discrete-time system with quantization and packet dropouts in this paper. A new quantization structure has been presented and the corresponding mathematical treatment for the proposed quantization structure has been offered to demonstrate the advantage for the quantization effects. Next, the numbers of packet dropouts and states of late-arrival packets are modeled as two Markov chains. Further, the closedloop system is described as an MJLS, and the sufficient conditions of stochastic stabilization for the system are proposed

12

J.-S. Song and X.-H. Chang / Applied Mathematics and Computation 376 (2020) 125070

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