Event-triggered nonfragile predictive control for networked control system with redundant channels

Introduction

In recent years, accompanying with the rapid developments of communication and control techniques, the traditional point-to-point control system has exposed some problems such as complex wiring and maintenance difficulties, which is hard to adapt to the requirements of the complex engineering structure. As discussed in (Pang et al., 2022a; Zhang et al., 2020b), the networked control system (NCS) has the advantages such as low cost, maintenance convenience, and high flexibility, which is gradually replacing the point-to-point control system. At present, the NCS plays a vitally important role in many different fields such as aerospace (Pongsakornsathien et al., 2019), transportation (Lv et al., 2021), and military defense (Hu et al., 2021). In contrast to the traditional control system, it is more difficult to handle the control problem for NCS due to network-induced phenomena (NIP) occurring in the networks, which would weaken the whole performance of the control system and even lead to instability (Pang et al.,2021a, 2021b, 2022a, 2022b; Suo and Li, 2022). During the past decades, the analysis problem has received increasing attention for the NCS with NIP and a large of innovative results have been reported (Li et al., 2019a; Rouamel et al., 2020; Xu et al., 2018; Zhao et al., 2021a). In addition, due to the limitation of network resources, the network environment is usually difficult to realize the desired control performance of multiple control systems to access the network. Consequently, the researchers have made great efforts to discuss the transmission strategies (Wang et al., 2019; Zhao et al., 2021b) and communication protocols (Liu et al., 2021a; Shen et al., 2020a, 2020b, 2021).

Up to now, many kinds of strategies have been proposed to deal with the NIP. The transmission strategy governed by redundant channels is a highly effective method to reduce the occurrence of data packet dropout and increase the reliability of the information (Li et al., 2021; Liu et al., 2021b; Shen et al., 2020b; Zhang et al., 2017). Under this transmission scheme, the output feedback control problem has been investigated in (Shen et al., 2020a) for the networked nonlinear semi-Markov jump system, where the proposed method can reduce the adverse effects caused by data packet loss. On the contrary, the networked predictive control method (NPCM) has been utilized to compensate the network-induced time delays (NITDs) (see Liu, 2022a, 2022b; Pang et al., 2021a; Pang et al., 2022b; Zhang et al., 2020a). With the help of the NPCM, the closed-loop NCS has been modeled as a switch system in (Sun and Chen, 2014), where sufficient conditions ensuring stability have been derived and the adverse effects caused by NITDs have been overcome. Subsequently, a networked predictive control scheme (NPCS) has been presented in (Liu, 2016), where the necessary and sufficient condition has been established to guarantee both stability and consensus for the closed-loop multiagent system. Besides, the change in the system operating environment will lead to input time delay, which is also a problem to be addressed. Nevertheless, most aforementioned results concerning NPCM for NCS are only applicable to the case of unlimited network resources. It has not been paid enough attention to the study of NPCM under the limited communication ability.

Due to the implementation error or unknown noise (Tan et al., 2021), the controller gain may change resulting in lower control performance. To treat this issue, the nonfragile controller has been proposed to achieve the desired control effect. For instance, an observer-based nonfragile controller has been designed to stabilize the nonlinear system with external disturbances and actuator saturation in (Sakthivel et al., 2019), where additional parameter perturbations in the controller are characterized by norm-bounded uncertainties. Considering the occurrence of uncertainties in the controller, the observer-based nonfragile H ∞ controller has been proposed in (Arumugam et al., 2020) to achieve the desired H ∞ performance requirements. The nonfragile sampled-data controller has been designed in (Muthukumar et al., 2019), and the stabilization problem has been solved for uncertain NCS with additive time-varying delay. So far, the nonfragile controller has played a meaningful role in the NCS, so it is significant to study such controller design problem.

It is well recognized that the signals are transmitted under the time-driven mechanism in most cases, where this communication method does not take the limited communication resource into account. However, the communication bandwidth is a scarce resource which needs to be properly utilized (Yang et al., 2020). At present, the event-triggered mechanism (ETM) is popular as an effective communication strategy (Deng et al., 2021a; Gao et al., 2020; Han and Lian, 2021; Liang et al., 2021; Qu, et al., 2022; Sun et al., 2022; Tan et al., 2023). The results show that the network resources can be effectively saved and the network environment can be optimized by combining the ETM with the NPCM. For instance, by aid of the Lyapunov stability theorem, the predictive control problem has been solved in (Zou et al., 2017) for NCS based on the ETM, where the event-triggered generator (ETG) has been added to reduce unnecessary signal transmission between the sensor and the observer. Recently, the NPCS has been designed in (Deng et al., 2021b) based on ETM, which can not only effectively reduce the pressure of network bandwidth, but also deal with the negative impact due to the denial-of-service attack. Compared with the ETM mentioned above, the maximum trigger interval is constructed in (Yang et al., 2018). This is an effective way to avoid inaccurate prediction results caused by too-long trigger intervals. To sum up, it is significant to study the design problem of NPCS under the ETM. In addition, there are a few of researches on handling the design problem of nonfragile NPCM with better robustness, which constitutes another research motivation.

Inspired by the aforementioned discussions, we aim to design the event-triggered nonfragile NPCS under the redundant channels such that the mean-square exponential stability (MSES) can be guaranteed for the conducted NCS. The major contributions are summarized as follows. (1) A nonfragile NPCS is proposed to compensate input delay and NITDs, the redundant channel transmission strategy is used to deal with data packet dropout, and the ETM is used to reduce resource consumption. (2) By taking an important property of the ETM proposed in this paper, the closed-loop system is transformed into a switched system with time delay. In virtue of the Lyapunov–Krasovskii stability theorem, the desired control performance analysis is conducted, and new sufficient condition ensuring MSES is acquired for the closed-loop system in form of matrix inequalities. (3) The observer gain matrix (OGM) and controller gain matrix (CGM) are acquired by solving nonlinear minimization problem with convex constraints. The rest of this paper is described as follows. In “Problem formulation and preliminaries” section, the control system model and transmission strategy governed by redundant channels are introduced, respectively. Moreover, the event-triggered NPCS is designed. The main theorem is presented in “Main results” section to address the problem of MSES for NCS with input delay and NITDs. A simulation example is shown in “An illustrative example” section to illustrate the main results. The conclusion of this paper is given in “Conclusions” section.

Notations: The notations used are awfully standard in this paper. The set of all n × p matrices and the n-dimensional Euclidean space are represented by R n × p and R n , respectively. The inverse and transpose of matrix Q are denoted by Q − 1 and Q T , respectively. If Q is a positive-definite symmetric matrix, then we write as Q > 0 . Let 0 and I be the zero matrix and the identity matrix with appropriate dimensions, respectively. In a symmetric matrix, “*” is employed to describe the symmetric term. If x is a random variable, E { x } stands for the mathematical expectation of x. The maximum and minimum eigenvalues of the matrix Q are denoted by λ max ( Q ) and λ min ( Q ) . Besides, ⊗ represents the Kronecker product of matrix.

Problem formulation and preliminaries

The block diagram is shown in Figure 1 for the NCS under the redundant channel transmission scheme and the ETM discussed in this paper. In Figure 1, the signals in the sensor will be transmitted to the observer through the network, and it is assumed that only the data packet dropout phenomenon may occur. When the released signals by the ETG are transmitted to the predictive controller through the network, the NITD occurred is a sc ( 0 ≤ a sc ≤ π sc ) . When the signals from the predictive controller are transmitted to the actuator through the network, the NITD occurred is a c a ( 0 ≤ a c a ≤ π c a ) .

OPEN IN VIEWER

Figure 1.

Diagram of the NCS under the redundant channel transmission scheme and the ETM.

The plant in Figure 1 is described by a linear system with input delay

x ( m + 1 ) = Ax ( m ) + Bu ( m − d ) , x ( 0 ) = χ ~ , u ( m ) = 0 , m ∈ ( − d , 0 )

(1)

where x ( m ) ∈ R n and u ( m ) ∈ R p denote the state vector and control input, respectively. A ∈ R n × n and B ∈ R n × p are known real matrices. d ∈ N + represents the input delay. In fact, the system will not receive the control signal for m ∈ [ − d , 0 ) , that is, the control input is u ( m ) = 0 for m ∈ [ − d , 0 ) .

To reduce the negative impact of data packet dropout between the sensor and the observer, the redundant channel transmission scheme is introduced to obtain more effective signals for the observer. To be more specific, under the M redundant channels, the measurement output received by the observer is described in the following form

y ( m ) = Γ x ( m )

(2)

where y ( m ) ∈ R l denotes measurement output and Γ = Δ α 1 ( m ) C 1 + ∑ k = 2 M { Π l = 1 k − 1 ( 1 − α l ( m ) ) α k ( m ) C k } . C k ∈ R l × n is known real matrix for k ∈ { 1 , 2 , ⋯ , M } . The random variables α k ( m ) are used to describe the data packet dropout phenomenon in the k th channel and satisfy the Bernoulli distribution with

Prob { α k ( m ) = 1 } = α ¯ k , Prob { α k ( m ) = 0 } = 1 − α ¯ k

(3)

where α ¯ k ∈ [ 0 , 1 ] ( k = 1 , 2 , ⋯ , M ) are known constant. Assume that α p ( m ) and α q ( m ) are mutually independent for p ≠ q ( p , q ∈ { 1 , 2 , ⋯ , M } ) .

Remark 1. On the basis of equation (2), we know that the transmission channel will be determined based on the value of α k ( m ) . If α 1 ( m ) = 1 , it indicates that the channel 1 is capable of transmitting signals at time m, so the signals will be transmitted through the channel 1. If α 1 ( m ) = 0 , it means that signals can not be via the channel 1 at time m, then we will consider whether the signals can be transmitted via the channel 2. Therefore, the signals will be transmitted through the channel k ( k ≤ M ) if and only if the channel k ~ ( k ~ = 1 , 2 , ⋯ , k − 1 ) cannot transmit signals and the channel k can transmit signals. Obviously, this transmission strategy can reduce the occurrence of data packet dropout.

Remark 2. Compared with the single-channel transmission strategy, the redundant channel transmission strategy can effectively reduce the probability of packet loss and increase the availability of data. Note that this scheme may increase the communication burden and the computational complexity.

By employing the information of measurement output, the following state observer is designed

x ^ ( m + 1 | m ) = A x ^ ( m | m − 1 ) + B u ( m − d ) + L ( y ( m ) − Γ ¯ x ^ ( m | m − 1 ) )

(4)

where x ^ ( m + 1 | m ) denotes the one-step prediction for time m + 1 , and L is the OGM to be designed and Γ ¯ = Δ α ¯ 1 C 1 + ∑ k = 2 M { Π l = 1 k − 1 ( 1 − α ¯ l ) α ¯ k C k } .

To effectively reduce the communication burden and save network resources, an observer-based ETG is employed to reduce the transmissions of signals in the feedback channel from the sensor to the controller. The triggered instants 0 = m 1 < m 2 < ⋯ < m ρ < ⋯ can be generated by

m ρ + 1 = { min { m | m > m ρ , f ( m ) > 0 } , m < H ~ m ρ + H , m ≥ H ~

(5)

where H ~ = m ρ + H , f ( m ) = f 1 ( m ) − ε f 2 ( m ) is the event generator function, Φ > 0 is a symmetric matrix, ε ∈ ( 0 , 1 ) is a constant, and H ( H > 1 ) stands for the maximum triggered interval, f 1 ( m ) = x ^ ( m | m − 1 ) − x ^ ( m ρ | m ρ − 1 ) ) T Φ ( x ^ ( m | m − 1 ) − x ^ ( m ρ | m ρ − 1 ) , f 2 ( m ) = x ^ T ( m | m − 1 ) Φ x ^ ( m | m − 1 ) . According to equation (5), for m ∈ { m ρ , m ρ + 1 , ⋯ , m ρ + 1 − 1 } , we know that the released signal x ⌣ ( m | m − 1 ) can be described by

x ⌣ ( m | m − 1 ) = x ^ ( m ρ | m ρ − 1 )

(6)

Remark 3. For a given Φ in equation (5), the adjustable parameters H and ε determine the number of times triggered. The ETM becomes the time-triggered mechanism when H = 1 or ε = 0 . Meanwhile, the network burden may aggravate, which is not applicable to a weak network environment and degrades the system performance. If H and ε are too large, a few of data packets are sent through the network and signals in the controller are not updated in time, which might degrade the system performance. Therefore, we need to set the appropriate H and ε in the ETG to balance the system performance and the network resource consumption.

Remark 4. There have been many reports showing that event-triggered control for continuous time is more widely used than discrete time (Tan et al., 2022, 2023). It is worth mentioning that the study of the event triggering for discrete systems is still meaningful. With the expansion of the research field of system theory and the wide application of computer technology, the theory of discrete system has been developed into a branch of control theory parallel to the continuous system, which has become an important part of control theory. But the study of the discrete system is still not perfect, especially in the aspect of networked predictive control. Besides, the control of a continuous system is also achieved by discretizing the control of the system in practice. Therefore, the ETM is introduced into the study of networked predictive control for discrete system, which not only enriches the NPCM for the such system but also promotes its practical application.

Remark 5. The Zeno behavior is not expected to exist in the triggered-control scheme. As the research object of this paper is discrete system, the lower bound of inter-execution times by the proposed ETM is larger than 1. This means that Zeno behavior can be avoided.

If there is no ETG in the feedback channel or all signals in the observer can reach the controller through the ETG, then the data received by the controller from the observer are

x ^ ( m + 1 − π sc | m − π sc ) = A x ^ ( m − π sc | m − 1 − π sc ) + Bu ( m − d − π sc ) + L ( y ( m − π sc ) − Γ ¯ x ^ ( m − π sc | m − 1 − π sc ) )

(7)

More generally, due to the existence of an event-triggered generator in the feedback channel, the following two cases need to be considered.

Case 1: The data from the observer are received in the predictive controller at the time m. The latest data x ~ ( m − π s c | m − 1 − π s c ) received by the predictive controller are expressed by

x ~ ( m − π sc | m − 1 − π sc ) = x ^ ( m ρ − π sc | m ρ − 1 − π sc )

(8)

Case 2: No data reach the controller through the ETG at the time m. The data stored in the predictive controller at the previous moment will be used, that is

x ~ ( m − π sc | m − 1 − π sc ) = x ~ ( m − 1 − π sc | m − 2 − π sc )

(9)

When x ~ ( m − π sc | m − 1 − π sc ) is determined by equation (8), the state x ~ ( m | m − π sc ) can be predicted by

x ~ ( m + n − π s c | m − π s c ) = A x ~ ( m + n − π s c − 1 | m − π s c ) + B u ( m + n − π s c − 1 − d )

(10)

for n = 2 , ⋯ , π , where π = π sc + π ca + d . Furthermore, when x ~ ( m − π sc | m − 1 − π sc ) is determined by equation (9), the state x ~ ( m | m − π sc ) can be predicted by

x ~ ( m | m − π sc ) = A x ~ ( m − 1 | m − π sc ) + Bu ( m − 1 − d )

(11)

To compensate the NITDs and input delay, the following nonfragile controller is designed based on the signals governed by the ETG

u ( m − d ) = u ( m | m ρ − π ) = K ~ x ~ ( m | m ρ − π )

(12)

where m ρ ( m ρ ≤ t ) is the latest triggered moment, K ~ = K + Δ K ( m ) , K is the CGM to be designed, Δ K ( m ) satisfies the structure: Δ K ( m ) = DF ( m ) E , D and E are constant matrices, F ( m ) is unknown time-varying matrix satisfying F T ( m ) F ( m ) ≤ I .

Remark 6. It is noted that the controller gain contains the perturbation term Δ K . The purpose of introducing Δ K is mainly to increase the insensitivity of the controller in practical application and cope with the possible influences from data inaccuracy, measurement error, and noise on the control performance.

For the sake of illustration, set s ( m ) = m − ( m ρ − π ) . According to equation (1), it is clear that

x ( m ) = A s ( m ) − 1 x ( m + 1 − s ( m ) ) + ∑ n = 1 s ( m ) − 1 A s ( m ) − n Bu ( m + n − s ( m ) − d )

(13)

From equations (10) and (11), we have

x ~ ( m | m − s ( m ) ) = A s ( m ) − 1 x ~ ( m − s ( m ) + 1 | m − s ( m ) ) + ∑ n = 1 s ( m ) − 1 A s ( m ) − n Bu ( m + n − s ( m ) − d )

(14)

Combining equation (13) with (14) yields

x ~ ( m | m − s ( m ) ) = A s ( m ) − 1 x ~ ( m − s ( m ) + 1 | m − s ( m ) ) + x ( m ) − A s ( m ) − 1 x ( m + 1 − s ( m ) )

(15)

Therefore, the control input equation (12) can be rewritten as follows

u ( m − d ) = K ~ A s ( m ) − 1 x ~ ( m − s ( m ) + 1 | m − s ( m ) ) + K ~ x ( m ) − K ~ A s ( m ) − 1 x ( m + 1 − s ( m ) )

(16)

Based on equations (1) and (16), under the event-triggered NPCS and redundant channels, the closed-loop NCS is written as follows

x ( m + 1 ) = ( A + B K ~ ) x ( m ) − B K ~ A σ ( m ) + π − 2 ( x ( m − ϱ ( m ) ) − x ~ ( m − ϱ ( m ) | m − ϱ ( m ) − 1 ) )

(17)

where ϱ ( m ) = s ( m ) − 1 , σ ( m ) = m − m ρ + 1 . Based on the previous analysis, it is not hard to get π − 1 ≤ ϱ ( m ) ≤ π + H − 1 and σ ( m ) ∈ I = Δ { 1 , 2 , ⋯ , H + 1 } .

Putting equation (16) into (4), we can achieve

x ^ ( m + 1 | m ) = − B K ~ A σ ( m ) + π − 2 ( x ( m − ϱ ( m ) ) − x ~ ( m − ϱ ( m ) | m − ϱ ( m ) − 1 ) ) + ( A − L Γ ¯ ) x ^ ( m | m − 1 ) + ( B K ~ + L Γ ) x ( m )

(18)

Letting ζ ( m ) = [ x T ( m ) x ^ T ( m | m − 1 ) ] T , ϱ 1 = π − 1 , ϱ 2 = π + H − 1 , the following augmented system with delay can be obtained according to equations (17) and (18) as

ζ ( m + 1 ) = A ~ m ζ ( m ) − B ~ σ ( m ) ζ ( m − ϱ ( m ) ) + C ~ σ ( m ) x → ( m − ϱ ( m ) ) ζ ( m ) = ϕ ( m ) , m ∈ [ − ϱ 2 , 0 ]

(19)

where

x → ( m − ϱ ( m ) ) = [ x ¯ T ( m − ϱ ( m ) ) x ¯ T ( m − ϱ ( m ) ] T , x ¯ ( m − ϱ ( m ) ) = x ^ ( m − ϱ ( m ) | m − ϱ ( m ) − 1 ) − x ~ ( m − ϱ ( m ) | m − ϱ ( m ) − 1 ) , A ~ m = [ A + B K ~ 0 B K ~ + L Γ A − L Γ ¯ ] , B ~ σ ( m ) = [ B K ~ A σ ( m ) + π − 2 − B K ~ A σ ( m ) + π − 2 B K ~ A σ ( m ) + π − 2 − B K ~ A σ ( m ) + π − 2 ] , C ~ σ ( m ) = [ B K ~ A σ ( m ) + π − 2 0 0 B K ~ A σ ( m ) + π − 2 ]

Now, it can be seen that the augmented system (equation (19)) is a switched system with time delay. σ ( m ) : [ 0 , + ∞ ) → I is the switching signal and ϱ ( m ) denotes time-varying delay. Let σ ( m ) = i , i ∈ I . For the sake of simplicity, set

β k ( m ) = { α 1 ( m ) , k = 1 ∑ s = 2 k − 1 ( 1 − α s ( m ) ) α k ( m ) , k = 2 , ⋯ M β ( m ) = col { β 1 ( m ) , β 2 ( m ) , ⋯ , β M ( m ) } , Σ = E { β ( m ) β T ( m ) } .

Before proceeding, we introduce some definitions and lemmas that will be very useful for subsequent analysis of MSES.

Definition 1. For m ≥ m 0 , if the solution x ( m ) of augmented system (equation (19)) satisfies (Liu et al., 2016)

E { ∥ x ( m ) ∥ 2 } ≤ κ λ − ( m − m 0 ) sup m 0 − ϱ 2 ≤ θ ≤ m 0 E { ∥ x ( θ ) ∥ 2 }

then, equation (19) is said to be mean-square exponential stable, where decay coefficient κ > 0 is a constant and λ > 1 represents the decay rate.

Lemma 1. For known matrices R = R T , M , N and unknown matrix F ( m ) ( F T ( m ) F ( m ) ≤ I ), the inequality R + M F ( m ) N + N T F T ( m ) M T < 0 holds, if and only if there exists scalar ε > 0 , such that R + ε − 1 M M T + ε N T N < 0 (Feng and Hao, 2020).

The objective of this paper is to deal with the issue of predictive control for NCS (equation (1)) with input delay and NITDs under the redundant channel transmission scheme (equation (2)) and the ETM (equation (5)). More precisely, the following two issues need to be handled.

Main results

In this part, based on the Lyapunov–Krasovskii stability theorem and the matrix analysis technique, sufficient condition is attained to ensure the MSES of the augmented system (equation (19)). Moreover, the OGM and CGM are obtained by solving the nonlinear minimization problem with convex constraints.

Analysis of MSES

Theorem 1. For known scalar λ > 1 , the augmented system (equation (19)) is mean-square exponential stable, if there exist matrices P 1 > 0 , P 2 > 0 , Z 1 > 0 , Z 2 > 0 , Q > 0 , and X = [ X 11 X 12 * X 22 ] ≥ 0 , real matrices N i ( i = 1 , 2 ) and scalar ε > 0 such that

Ψ ~ = [ Ψ D D T − κ I ] < 0

(20)

Θ = [ X 11 X 12 N 1 X 12 T X 22 N 2 N 1 T N 2 T λ − ϱ 2 Z i ] ≥ 0

(21)

where

Ψ = [ Ψ 11 Ψ 21 T Ψ 21 Ψ 22 ] ,

D = col { 0 , 0 , 0 , D ~ , ϱ 2 D ~ , 2 D ~ , 2 ϱ 2 D ~ , 0 , 0 , 0 , 0 } ,

D ~ = col { BD , BD } , Ψ 11 = Ψ → + κ H T H ,

H = [ H 1 0 ⋯ 0 ] , H 1 = [ E 0 ] , Ψ 21 = [ Ψ 211 0 0 0 0 ] ,

Ψ 22 = diag { − P − 1 , − Z − 1 , − Σ → 1 , − Σ → 2 , − P − 1 , − Z − 1 } ,

Σ → 1 = Σ − 1 ⊗ P 2 − 1 , Σ → 2 = Σ − 1 ⊗ Z 2 − 1 ,

Ψ → = [ Ψ 111 1 Ψ 111 2 0 A ¯ T ϱ 2 ( A ¯ − I ) T * Ψ 111 3 0 B ~ i T ϱ 2 B ~ i T * * − Φ 2 C ~ i T ϱ 2 C ~ i T * * * − P − 1 0 * * * * − Z − 1 ] ,

Ψ 211 = col { 2 A ~ 1 , 2 ϱ 2 A ~ 1 , C , ϱ 2 C , A ¯ 2 , ϱ 2 A ¯ 2 } ,

Ψ 111 1 = − λ − 1 P + ( ϱ 2 − ϱ 1 + 1 ) Q + ϱ 2 Z + ϱ 2 X 11 + N 1 + N 1 T ,

Ψ 111 2 = ϱ 2 X 12 − N 1 + N 2 T , P = diag { P 1 , P 2 } , Z = diag { Z 1 , Z 2 } ,

Ψ 111 3 = − λ ϱ 2 Q + Φ + ϱ 2 X 22 − N 2 − N 2 T ,

A ¯ = [ A + BK 0 BK + L Γ ¯ A − L Γ ¯ ] ,

A ~ 1 = [ A + BK 0 BK A − L Γ ¯ ] ,

C = [ C ¯ T ( I M ⊗ L ) T 0 ] , A ¯ 2 = [ 0 0 L Γ ¯ 0 ] ,

Φ 1 = [ 2 ε Φ 0 0 0 ] , Φ 2 = [ Φ 0 0 Φ ] ,

C ¯ = [ C 1 C 2 ⋯ C M ]

Proof. Letting η ( m ) = ζ ( m + 1 ) − ζ ( m ) , one has

ζ ( m ) − ζ ( m − ϱ ( m ) ) − ∑ m − ϱ ( m ) m − 1 η ( m ) = 0

(22)

Choose the Lyapunov functional

V ( m ) = ∑ i = 1 4 V i ( m )

(23)

where

V 1 ( m ) = ζ T ( m ) P ζ ( m ) , V 2 ( m ) = ∑ l = m − ϱ ( m ) m − 1 λ l − m + 1 ζ T ( l ) Q ζ ( l ) , V 3 ( m ) = ∑ θ = − ϱ 2 + 1 − ϱ 1 ∑ s = m − 1 + θ m − 1 λ s − m + 1 ζ T ( s ) Q ζ ( s ) , V 4 ( m ) = ∑ θ = − ϱ 2 + 1 0 ∑ s = m − 1 + θ m − 1 λ s − m + 1 η T ( s ) Z η ( s ) .

Letting E { Δ V ( m ) } = E { V ( m + 1 ) − λ − 1 V ( m ) } and along with the solution of equation (19), we have

E { Δ V 1 ( m ) } = ζ T ( m − ϱ ( m ) ) B ~ i T P B ~ i ζ ( m − ϱ ( m ) ) + E { ζ T ( m ) A ~ m T P A ~ m ζ ( m ) } + x → T ( m − ϱ ( m ) ) C ~ i T P C ~ i x → ( m − ϱ ( m ) ) + 2 ζ T ( m ) A ¯ T P B ~ i ζ ( m − ϱ ( m ) ) + 2 ζ T ( m ) A ¯ T P C ~ i x → ( m − ϱ ( m ) ) − λ − 1 ζ T ( m ) P ζ ( m ) + 2 ζ T ( m − ϱ ( m ) ) B ~ i T P C ~ i x → ( m − ϱ ( m ) )

(24)

E { Δ V 2 ( m ) } = E { ∑ l = m − ϱ ( m ) + 1 m λ m − l ζ T ( l ) Q ζ ( l ) − ∑ l = m − ϱ ( m ) m − 1 λ m − l ζ T ( l ) Q ζ ( l ) } ≤ ζ T ( m ) Q ζ ( m ) − λ ϱ 2 ζ T ( m − ϱ ( m ) ) Q ζ ( m − ϱ ( m ) )

(25)

E { Δ V 3 ( m ) } = E { ∑ θ = − ϱ 2 + 1 − ϱ 1 ∑ s = m + θ m λ s − m ζ T ( s ) Q ζ ( s ) − ∑ θ = − ϱ 2 + 1 − ϱ 1 ∑ s = m − 1 + θ m − 1 λ s − m ζ T ( s ) Q ζ ( s ) } = E { ∑ θ = − ϱ 2 + 1 − ϱ 1 ( ζ T ( m ) Q ζ ( m ) − λ θ − 1 ζ T ( m + θ − 1 ) Q ζ ( m + θ − 1 ) ) } ≤ ( ϱ 2 − ϱ 1 ) ζ T ( m ) Q ζ ( m )

(26)

E { Δ V 4 ( m ) } = E { ∑ θ = − ϱ 2 + 1 0 ∑ s = m + θ m η T ( s ) Z η ( s ) − ∑ θ = − ϱ 2 + 1 0 ∑ s = m − 1 + θ m − 1 λ m − s − 1 η T ( s ) Z η ( s ) } ≤ ϱ 2 E { η T ( m ) Z η ( m ) } − ∑ θ = m − ϱ ( m ) m − 1 λ − ϱ 2 η T ( s ) Z η ( s ) = ϱ 2 E { ζ T ( m ) A ~ m T Z A ~ m ζ ( m ) } + ϱ 2 { ζ T ( m ) Z ζ ( m ) − 2 ζ T ( m ) A ¯ T Z ζ ( m ) + x → T ( m − ϱ ( m ) ) C ~ i T Z C ~ i x → ( m − ϱ ( m ) ) + ζ T ( m − ϱ ( m ) ) B ~ i T Z B ~ i ζ ( m − ϱ ( m ) ) + 2 ζ T ( m ) ( A ¯ − I ) T Z B ~ i ζ ( m − ϱ ( m ) ) + 2 ζ T ( m ) ( A ¯ − I ) T Z C ~ i x → ( m − ϱ ( m ) ) + 2 ζ T ( m − ϱ ( m ) ) B ~ i T Z C ~ i x → ( m − ϱ ( m ) ) } − ∑ θ = m − ϱ ( m ) m − 1 λ − ϱ 2 η T ( s ) Z η ( s )

(27)

For brevity, we set

A ~ 1 = [ A + B K ~ 0 B K ~ A − L Γ ¯ ] , A ~ 2 = [ 0 0 L Γ 0 ]

Subsequently, it is not hard to get

E { ζ T ( m ) A ~ m T P A ~ m ζ ( m ) } = E { ζ T ( m ) ( A ~ 1 + A ~ 2 ) T P ( A ~ 1 + A ~ 2 ) ζ ( m ) } = ζ T ( m ) A ~ 1 T P A ~ 1 ζ ( m ) + E { ζ T ( m ) A ~ 2 T P A ~ 2 ζ ( m ) } + 2 ζ T ( m ) A ~ 1 T P A ¯ 2 ζ ( m )

(28)

Then, by virtue of the properties of Kronecker product, we deal with the term E { ζ T ( m ) A ~ 2 T P A ~ 2 ζ ( m ) } as follows

E { ζ T ( m ) A ~ 2 T P A ~ 2 ζ ( m ) } = ζ T ( m ) E { [ Γ T L T P 2 L Γ 0 0 0 ] } ζ ( m ) = ζ T ( m ) [ C ¯ T ( I ⊗ L ) T ( Σ ⊗ P 2 ) ( I ⊗ L ) C ¯ 0 0 0 ] ζ ( m ) = ζ T ( m ) C T ( Σ ⊗ P 2 ) C ζ ( m )

(29)

Combining equation (28) with (29) yields

E { ζ T ( m ) A ~ m T P A ~ m ζ ( m ) } = ζ T ( m ) A ~ 1 T P A ~ 1 ζ ( m ) + ζ T ( m ) C T Σ ⊗ P 2 C ζ ( m ) + 2 ζ T ( m ) A ~ 1 T P A ¯ 2 ζ ( m )

(30)

In the same way, one has

E { ζ T ( m ) A ~ m T Z A ~ m ζ ( m ) } = ζ T ( m ) A ~ 1 T Z A ~ 1 ζ ( m ) + ζ T ( m ) C T Σ ⊗ Z 2 C ζ ( m ) + 2 ζ T ( m ) A ~ 1 T Z A ¯ 2 ζ ( m )

(31)

It should be noticed that

2 [ ζ T ( m ) N 1 + ζ T ( m − ϱ ( m ) ) N 2 ] × [ ζ ( m ) − ζ ( m − ϱ ( m ) ) − ∑ l = m − ϱ ( m ) m − 1 η ( l ) ] = 0

(32)

∑ l = m − ϱ 2 m − 1 ζ 1 T ( m ) X ζ 1 ( m ) − ∑ l = m − ϱ ( m ) m − 1 ζ 1 T ( m ) X ζ 1 ( m ) = ϱ 2 ζ 1 T ( m ) X ζ 1 ( m ) − ∑ l = m − ϱ ( m ) m − 1 ζ 1 T ( m ) X ζ 1 ( m ) ≥ 0

(33)

where ζ 1 ( m ) = [ ζ T ( m ) ζ T ( m − ϱ ( m ) ] T

Based on the previous analysis, it follows from equations (24)–(27) and equation (30)–(33) that

E { Δ V ( m ) } ≤ ζ T ( m ) A ~ 1 T P A ~ 1 ζ ( m ) + ζ T ( m ) A ~ 1 T P A ~ 1 ζ ( m ) + ζ T ( m ) A ¯ 2 T P A ¯ 2 ζ ( m ) + ζ T ( m ) C T Σ ⊗ P 2 C ζ ( m ) + ζ T ( m − ϱ ( m ) ) B ~ i T P B ~ i ζ ( m − ϱ ( m ) ) + x → T ( m − ϱ ( m ) ) C ~ i T P C ~ i x → ( m − ϱ ( m ) ) + 2 ζ T ( m ) A ¯ T P B ~ i ζ ( m − ϱ ( m ) ) + 2 ζ T ( m ) A ¯ T P C ~ i x → ( m − ϱ ( m ) ) + 2 ζ T ( m − ϱ ( m ) ) B ~ i T P C ~ i x → ( m − ϱ ( m ) ) − λ − 1 ζ T ( m ) P ζ ( m ) + ζ T ( m ) Q ζ ( m ) − λ ϱ 2 ζ T ( m − ϱ ( m ) ) Q ζ ( m − ϱ ( m ) ) + ( ϱ 2 − ϱ 1 ) ζ T ( m ) Q ζ ( m ) + ϱ 2 ζ T ( m ) A ~ 1 T Z A ~ 1 ζ ( m ) + ϱ 2 ζ T ( m ) A ~ 1 T Z A ~ 1 ζ ( m ) + ϱ 2 ζ T ( m ) A ¯ 2 T Z A ¯ 2 ζ ( m ) + ϱ 2 ζ T ( m ) C T Σ ⊗ Z 2 C ζ ( m ) + ϱ 2 ζ T ( m ) Z ζ ( m )

+ ϱ 2 x → T ( m − ϱ ( m ) ) C ~ i T Z C ~ i x → ( m − ϱ ( m ) ) + ζ T ( m ) A ¯ T P A ¯ ζ ( m ) + ϱ 2 ζ T ( m − ϱ ( m ) ) B ~ i T Z B ~ i ζ ( m − ϱ ( m ) ) + ζ ( m − ϱ ( m ) ) Φ 1 ζ ( m − ϱ ( m ) ) + 2 ϱ 2 ζ T ( m ) ( A ¯ − I ) T Z B ~ i ζ ( m − ϱ ( m ) ) + ϱ 2 ζ T ( m ) ( A ¯ − I ) T Z ( A ¯ − I ) ζ ( m ) + 2 ϱ 2 ζ T ( m ) ( A ¯ − I ) T Z C ~ i x → ( m − ϱ ( m ) ) − x → T ( m − ϱ ( m ) ) Φ 2 x → ( m − ϱ ( m ) ) + 2 ϱ 2 ζ T ( m − ϱ ( m ) ) B ~ i T Z C ~ i x → ( m − ϱ ( m ) ) + ϱ 2 ζ 1 T ( m ) X ζ 1 ( m ) − ∑ θ = m − ϱ ( m ) m − 1 λ − ϱ 2 η T ( s ) Z η ( s ) − ∑ l = m − ϱ ( m ) m − 1 ζ 1 T ( m ) X ζ 1 ( m ) + 2 [ ζ T ( m ) N 1 + ζ T ( m − ϱ ( m ) ) N 2 ] × [ ζ ( m ) − ζ ( m − ϱ ( m ) ) − ∑ l = m − ϱ ( m ) m − 1 η ( l ) ] = ξ 1 T ( m ) Ξ ξ 1 ( m ) − ∑ l = m − ϱ ( m ) m − 1 ξ 2 T ( m , l ) Θ ξ 2 ( m , l )

(34)

where

ξ 1 ( m ) = [ ζ T ( m ) ζ T ( m − ϱ ( m ) ) x → T ( m − ϱ ( m ) ) ] T , ξ 2 ( m , l ) = [ ζ 1 T ( m ) η T ( l ) ] T , Ξ = [ Ξ 11 Ξ 12 Ξ 13 * Ξ 22 Ξ 23 * * Ξ 33 ] , Ξ 11 = Ψ 111 1 + 2 A ~ 1 T P A ~ 1 + A ¯ 2 T P A ¯ 2 + C T ( Σ ⊗ P 2 ) C T + 2 ϱ 2 A ~ 1 T Z A ~ 1 + ϱ 2 A ¯ 2 T Z A ¯ 2 + ϱ 2 C T ( Σ ⊗ Z 2 ) C + A ¯ T P A ¯ + ϱ 2 ( A ¯ − I ) T Z ( A ¯ − I ) , Ξ 12 = Ψ 111 2 + A ¯ T P B ~ i + ϱ 2 ( A ¯ − I ) T Z B ~ i , Ξ 13 = A ¯ T P C ~ i + ϱ 2 ( A ¯ − I ) T Z C ~ i , Ξ 22 = Ψ 111 3 + B ~ i T P B ~ i + ϱ 2 B ~ i T Z B ~ i , Ξ 23 = B ~ i T P C ~ i + ϱ 2 B ~ i T Z C ~ i , Ξ 33 = C ~ i T P C ~ i + ϱ 2 C ~ i T Z C ~ i − Φ 2

With the help of Schur complement lemma, we know that Ψ ~ < 0 means

Ψ + κ − 1 D D T + κ H T H < 0

(35)

According to Lemma 1, it is not hard to get that equation (35) is equivalent to the following inequality

Ψ + D F ( m ) H + H T F T ( m ) D T < 0

(36)

Furthermore, the above inequality can be rewritten as Ξ ~ = [ Ξ ~ 11 * Ξ ~ 21 Ψ 22 ] < 0 , where

Ξ ~ 11 = [ Ψ 111 1 Ψ 111 2 0 A ¯ T ϱ 2 ( A ¯ − I ) T * Ψ 111 3 0 B ~ i T ϱ 2 B ~ i T * * − Φ 2 C ~ i T ϱ 2 C ~ i T * * * − P j − 1 0 * * * * − Z − 1 ] Ξ ~ 21 = [ Ξ ~ 21 1 0000 ] , Ξ ~ 21 1 = col { 2 A ~ 1 , 2 ϱ 2 A ~ 1 , C , ϱ 2 C , A ¯ 2 , ϱ 2 A ¯ 2 }

By applying the Schur complement lemma again, one has Ξ < 0 from Ξ ~ < 0 . Evidently, it is clear from Ξ < 0 and Θ ≥ 0 that E { Δ V ( m ) } < 0 . After some straightforward algebraic manipulations, one further has

a E { ∥ ζ ( m ) ∥ 2 } ≤ E { V ( m ) } ≤ ⋯ ≤ λ − m E { V ( 0 ) } ≤ b λ − m E { ∥ ζ ( 0 ) ∥ 2 }

(37)

where a = λ min ( P ) and b = λ max ( P ) + ( ϱ 2 + ( ϱ 2 − ϱ 1 ) ( 1 + ϱ 2 ) ) λ max ( Q ) + ϱ 2 ( 1 + ϱ 2 ) λ max ( Z ) . It follows immediately from equation (37) that

E { ∥ ζ ( m ) ∥ 2 } ≤ b a λ − m sup − ϱ 2 ≤ θ ≤ 0 E { ∥ ζ ( θ ) ∥ 2 }

(38)

Therefore, we know that the augmented system (equation (19)) is mean-square exponential stable according to Definition 1. The proof of this theorem is now complete.