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

Abstract

This paper is concerned with the nonfragile predictive control problem for the networked control system (NCS) with input delay and network-induced time delays (NITDs). First, the transmission strategy governed by redundant channels is introduced to reduce the occurrence of data packet dropout between the sensor and the observer. Moreover, the event-triggered mechanism (ETM) is employed to save the network resources, where the maximum triggered interval is given. Subsequently, a nonfragile networked predictive control scheme (NPCS) is designed to address the input delay and the NITDs. Accordingly, the sufficient criterion ensuring mean-square exponential stability (MSES) is derived by employing the Lyapunov–Krasovskii stability theorem. Both the observer gain matrix (OGM) and the controller gain matrix (CGM) are gained through solving the nonlinear minimization problem with convex constraints. Finally, a simulation example is utilized to demonstrate the usefulness of the proposed main results.

Keywords

Event-triggered mechanism networked control system redundant channels predictive control mean-square exponential stability

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 $H_{\infty}$ controller has been proposed in (Arumugam et al., 2020) to achieve the desired $H_{\infty}$ 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 $n \times p$ matrices and the n-dimensional Euclidean space are represented by $R^{n \times p}$ 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 $a^{sc} (0 \leq a^{sc} \leq π^{sc})$ . When the signals from the predictive controller are transmitted to the actuator through the network, the NITD occurred is $a^{c a} (0 \leq a^{c a} \leq π^{c a})$ .

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

\begin{matrix} x (m + 1) = Ax (m) + Bu (m - d), \\ x (0) = \tilde{χ}, u (m) = 0, m \in (- d, 0) \end{matrix}

(1)

where $x (m) \in R^{n}$ and $u (m) \in R^{p}$ denote the state vector and control input, respectively. $A \in R^{n \times n}$ and $B \in R^{n \times p}$ are known real matrices. $d \in N^{+}$ represents the input delay. In fact, the system will not receive the control signal for $m \in [- d, 0)$ , that is, the control input is $u (m) = 0$ for $m \in [- d, 0)$ .

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 $y (m) \in R^{l}$ denotes measurement output and $Γ \overset{Δ}{=} α_{1} (m) C_{1} + \sum_{k = 2}^{M} {Π_{l = 1}^{k - 1} (1 - α_{l} (m)) α_{k} (m) C_{k}}$ . $C_{k} \in R^{l \times n}$ is known real matrix for $k \in {1, 2, \dots, M}$ . 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

\begin{matrix} Prob {α_{k} (m) = 1} = {\bar{α}}_{k}, \\ Prob {α_{k} (m) = 0} = 1 - {\bar{α}}_{k} \end{matrix}

(3)

where ${\bar{α}}_{k} \in [0, 1] (k = 1, 2, \dots, M)$ are known constant. Assume that $α_{p} (m)$ and $α_{q} (m)$ are mutually independent for $p \neq q (p, q \in {1, 2, \dots, M})$ .

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 ( $k \leq M$ ) if and only if the channel $\tilde{k}$ ( $\tilde{k} = 1, 2, \dots, k - 1$ ) 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

\begin{array}{l} \hat{x} (m + 1 | m) = A \hat{x} (m | m - 1) + B u (m - d) + L (y (m) \\ - \bar{Γ} \hat{x} (m | m - 1)) \end{array}

(4)

where $\hat{x} (m + 1 | m)$ denotes the one-step prediction for time $m + 1$ , and L is the OGM to be designed and $\bar{Γ} \overset{Δ}{=} {\bar{α}}_{1} C_{1} + \sum_{k = 2}^{M} {Π_{l = 1}^{k - 1} (1 - {\bar{α}}_{l}) {\bar{α}}_{k} C_{k}}$ .

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 $0 = m_{1} < m_{2} < \dots < m_{ρ} < \dots$ can be generated by

m_{ρ + 1} = {\begin{matrix} \min {m | m > m_{ρ}, f (m) > 0}, & m < \tilde{H} \\ m_{ρ} + H, & m \geq \tilde{H} \end{matrix}

(5)

where $\tilde{H} = m_{ρ} + H$ , $f (m) = f_{1} (m) - ε f_{2} (m)$ is the event generator function, $Φ > 0$ is a symmetric matrix, $ε \in (0, 1)$ is a constant, and H $(H > 1)$ stands for the maximum triggered interval, $f_{1} (m) = \hat{x} (m | m - 1) - \hat{x} (m_{ρ} | m_{ρ} - 1))^{T} Φ (\hat{x} (m | m - 1)$ $- \hat{x} (m_{ρ} | m_{ρ} - 1)$ , $f_{2} (m) = {\hat{x}}^{T} (m | m - 1) Φ \hat{x} (m | m - 1)$ . According to equation (5), for $m \in {m_{ρ}, m_{ρ} + 1, \dots, m_{ρ + 1} - 1}$ , we know that the released signal $\overset{⌣}{x} (m | m - 1)$ can be described by

\overset{⌣}{x} (m | m - 1) = \hat{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

\begin{matrix} \hat{x} (m + 1 - π^{sc} | m - π^{sc}) = A \hat{x} (m - π^{sc} | m - 1 - π^{sc}) \\ + Bu (m - d - π^{sc}) + L (y (m - π^{sc}) \\ - \bar{Γ} \hat{x} (m - π^{sc} | m - 1 - π^{sc})) \end{matrix}

(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 $\tilde{x} (m - π^{s c} | m - 1 - π^{s c})$ received by the predictive controller are expressed by

\tilde{x} (m - π^{sc} | m - 1 - π^{sc}) = \hat{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

\tilde{x} (m - π^{sc} | m - 1 - π^{sc}) = \tilde{x} (m - 1 - π^{sc} | m - 2 - π^{sc})

(9)

When $\tilde{x} (m - π^{sc} | m - 1 - π^{sc})$ is determined by equation (8), the state $\tilde{x} (m | m - π^{sc})$ can be predicted by

\begin{array}{l} \tilde{x} (m + n - π^{s c} | m - π^{s c}) = A \tilde{x} (m + n - π^{s c} - 1 | m - π^{s c}) \\ + B u (m + n - π^{s c} - 1 - d) \end{array}

(10)

for $n = 2, \dots, π$ , where $π = π^{sc} + π^{ca} + d$ . Furthermore, when $\tilde{x} (m - π^{sc} | m - 1 - π^{sc})$ is determined by equation (9), the state $\tilde{x} (m | m - π^{sc})$ can be predicted by

\tilde{x} (m | m - π^{sc}) = A \tilde{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) = \tilde{K} \tilde{x} (m | m_{ρ} - π)

(12)

where $m_{ρ}$ ( $m_{ρ} \leq t$ ) is the latest triggered moment, $\tilde{K} = K + Δ K (m)$ , 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 $F^{T} (m) F (m) \leq I$ .

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

\begin{matrix} x (m) = A^{s (m) - 1} x (m + 1 - s (m)) + \sum_{n = 1}^{s (m) - 1} A^{s (m) - n} \\ Bu (m + n - s (m) - d) \end{matrix}

(13)

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

\begin{matrix} \tilde{x} (m | m - s (m)) = A^{s (m) - 1} \tilde{x} (m - s (m) + 1 | m - s (m)) \\ + \sum_{n = 1}^{s (m) - 1} A^{s (m) - n} Bu (m + n - s (m) - d) \end{matrix}

(14)

Combining equation (13) with (14) yields

\begin{matrix} \tilde{x} (m | m - s (m)) = A^{s (m) - 1} \tilde{x} (m - s (m) + 1 | m - s (m)) \\ + x (m) - A^{s (m) - 1} x (m + 1 - s (m)) \end{matrix}

(15)

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

\begin{matrix} u (m - d) = \tilde{K} A^{s (m) - 1} \tilde{x} (m - s (m) + 1 | m - s (m)) \\ + \tilde{K} x (m) - \tilde{K} A^{s (m) - 1} x (m + 1 - s (m)) \end{matrix}

(16)

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

\begin{matrix} x (m + 1) = (A + B \tilde{K}) x (m) - B \tilde{K} A^{σ (m) + π - 2} (x (m - ϱ (m)) \\ - \tilde{x} (m - ϱ (m) | m - ϱ (m) - 1)) \end{matrix}

(17)

where $ϱ (m) = s (m) - 1$ , $σ (m) = m - m_{ρ} + 1$ . Based on the previous analysis, it is not hard to get $π - 1 \leq ϱ (m) \leq π + H - 1$ and $σ (m) \in I \overset{Δ}{=} {1, 2, \dots, H + 1}$ .

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

\begin{matrix} \hat{x} (m + 1 | m) = - B \tilde{K} A^{σ (m) + π - 2} (x (m - ϱ (m)) \\ - \tilde{x} (m - ϱ (m) | m - ϱ (m) - 1)) \\ + (A - L \bar{Γ}) \hat{x} (m | m - 1) + (B \tilde{K} + L Γ) x (m) \end{matrix}

(18)

Letting $ζ (m) = [x^{T} (m) {\hat{x}}^{T} (m | m - 1)]^{T}$ , $ϱ_{1} = π - 1$ , $ϱ_{2} = π + H - 1$ , the following augmented system with delay can be obtained according to equations (17) and (18) as

\begin{matrix} ζ (m + 1) = {\tilde{A}}_{m} ζ (m) - {\tilde{B}}_{σ (m)} ζ (m - ϱ (m)) + {\tilde{C}}_{σ (m)} \vec{x} (m - ϱ (m)) \\ ζ (m) = ϕ (m), m \in [- ϱ_{2}, 0] \end{matrix}

(19)

where

\begin{matrix} \vec{x} (m - ϱ (m)) = [{\bar{x}}^{T} (m - ϱ (m)) {\bar{x}}^{T} (m - ϱ (m)]^{T}, \\ \bar{x} (m - ϱ (m)) = \hat{x} (m - ϱ (m) | m - ϱ (m) - 1) \\ - \tilde{x} (m - ϱ (m) | m - ϱ (m) - 1), \\ {\tilde{A}}_{m} = [\begin{matrix} A + B \tilde{K} & 0 \\ B \tilde{K} + L Γ & A - L \bar{Γ} \end{matrix}], \\ {\tilde{B}}_{σ (m)} = [\begin{matrix} B \tilde{K} A^{σ (m) + π - 2} & - B \tilde{K} A^{σ (m) + π - 2} \\ B \tilde{K} A^{σ (m) + π - 2} & - B \tilde{K} A^{σ (m) + π - 2} \end{matrix}], \\ {\tilde{C}}_{σ (m)} = [\begin{matrix} B \tilde{K} A^{σ (m) + π - 2} & 0 \\ 0 & B \tilde{K} A^{σ (m) + π - 2} \end{matrix}] \end{matrix}

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

\begin{matrix} β_{k} (m) = {\begin{matrix} α_{1} (m), k = 1 \\ \sum_{s = 2}^{k - 1} (1 - α_{s} (m)) α_{k} (m), k = 2, \dots M \end{matrix} \\ β (m) = col {β_{1} (m), β_{2} (m), \dots, β_{M} (m)}, \\ Σ = E {β (m) β^{T} (m)} . \end{matrix}

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

Definition 1. For $m \geq m_{0}$ , if the solution $x (m)$ of augmented system (equation (19)) satisfies (Liu et al., 2016)

E {∥ x (m) ∥^{2}} \leq κ λ^{- (m - m_{0})} sup_{m_{0} - ϱ_{2} \leq θ \leq 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 $F (m)$ ( $F^{T} (m) F (m) \leq I$ ), 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.

Propose sufficient condition ensuring the MSES for the augmented system (equation (19)).

Design gain matrices L and K such that the input delay and NITDs can be compensated by the NPCS proposed based on the ETM.

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 $X = [\begin{matrix} X_{11} & X_{12} \\ * & X_{22} \end{matrix}] \geq 0$ , real matrices $N_{i}$ ( $i = 1, 2$ ) and scalar $ε > 0$ such that

\tilde{Ψ} = [\begin{matrix} Ψ & D \\ D^{T} & - κ I \end{matrix}] < 0

(20)

Θ = [\begin{matrix} X_{11} & X_{12} & N_{1} \\ X_{12}^{T} & X_{22} & N_{2} \\ N_{1}^{T} & N_{2}^{T} & λ^{- ϱ_{2}} Z_{i} \end{matrix}] \geq 0

(21)

where

Ψ = [\begin{matrix} Ψ_{11} & Ψ_{21}^{T} \\ Ψ_{21} & Ψ_{22} \end{matrix}],

D = col {0, 0, 0, \tilde{D}, \sqrt{ϱ_{2}} \tilde{D}, \sqrt{2} \tilde{D}, \sqrt{2 ϱ_{2}} \tilde{D}, 0, 0, 0, 0},

\tilde{D} = col {BD, BD}, Ψ_{11} = \vec{Ψ} + κ H^{T} H,

H = [H_{1} 0 \dots 0], H_{1} = [E 0], Ψ_{21} = [Ψ_{211} 0 0 0 0],

Ψ_{22} = diag {- P^{- 1}, - Z^{- 1}, - {\vec{Σ}}_{1}, - {\vec{Σ}}_{2}, - P^{- 1}, - Z^{- 1}},

{\vec{Σ}}_{1} = Σ^{- 1} \otimes P_{2}^{- 1}, {\vec{Σ}}_{2} = Σ^{- 1} \otimes Z_{2}^{- 1},

\vec{Ψ} = [\begin{matrix} Ψ_{111}^{1} & Ψ_{111}^{2} & 0 & {\bar{A}}^{T} & \sqrt{ϱ_{2}} {(\bar{A} - I)}^{T} \\ * & Ψ_{111}^{3} & 0 & {\tilde{B}}_{i}^{T} & \sqrt{ϱ_{2}} {\tilde{B}}_{i}^{T} \\ * & * & - Φ_{2} & {\tilde{C}}_{i}^{T} & \sqrt{ϱ_{2}} {\tilde{C}}_{i}^{T} \\ * & * & * & - P^{- 1} & 0 \\ * & * & * & * & - Z^{- 1} \end{matrix}],

Ψ_{211} = col {\sqrt{2} {\tilde{A}}_{1}, \sqrt{2 ϱ_{2}} {\tilde{A}}_{1}, C, \sqrt{ϱ_{2}} C, {\bar{A}}_{2}, \sqrt{ϱ_{2}} {\bar{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},

\bar{A} = [\begin{matrix} A + BK & 0 \\ BK + L \bar{Γ} & A - L \bar{Γ} \end{matrix}],

{\tilde{A}}_{1} = [\begin{matrix} A + BK & 0 \\ BK & A - L \bar{Γ} \end{matrix}],

C = [\begin{matrix} {\bar{C}}^{T} {(I_{M} \otimes L)}^{T} \\ 0 \end{matrix}], {\bar{A}}_{2} = [\begin{matrix} 0 & 0 \\ L \bar{Γ} & 0 \end{matrix}],

Φ_{1} = [\begin{matrix} 2 ε Φ & 0 \\ 0 & 0 \end{matrix}], Φ_{2} = [\begin{matrix} Φ & 0 \\ 0 & Φ \end{matrix}],

\bar{C} = [C_{1} C_{2} \dots C_{M}]

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

ζ (m) - ζ (m - ϱ (m)) - \sum_{m - ϱ (m)}^{m - 1} η (m) = 0

(22)

Choose the Lyapunov functional

V (m) = \sum_{i = 1}^{4} V_{i} (m)

(23)

where

\begin{matrix} V_{1} (m) = ζ^{T} (m) P ζ (m), \\ V_{2} (m) = \sum_{l = m - ϱ (m)}^{m - 1} λ^{l - m + 1} ζ^{T} (l) Q ζ (l), \\ V_{3} (m) = \sum_{θ = - ϱ_{2} + 1}^{- ϱ_{1}} \sum_{s = m - 1 + θ}^{m - 1} λ^{s - m + 1} ζ^{T} (s) Q ζ (s), \\ V_{4} (m) = \sum_{θ = - ϱ_{2} + 1}^{0} \sum_{s = m - 1 + θ}^{m - 1} λ^{s - m + 1} η^{T} (s) Z η (s) . \end{matrix}

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

\begin{matrix} E {Δ V_{1} (m)} \\ = ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} P {\tilde{B}}_{i} ζ (m - ϱ (m)) + E {ζ^{T} (m) {\tilde{A}}_{m}^{T} P {\tilde{A}}_{m} ζ (m)} \\ + {\vec{x}}^{T} (m - ϱ (m)) {\tilde{C}}_{i}^{T} P {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + 2 ζ^{T} (m) {\bar{A}}^{T} \\ P {\tilde{B}}_{i} ζ (m - ϱ (m)) \\ + 2 ζ^{T} (m) {\bar{A}}^{T} P {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) - λ^{- 1} ζ^{T} (m) P ζ (m) \\ + 2 ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} P {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) \end{matrix}

(24)

\begin{matrix} E {Δ V_{2} (m)} \\ = E {\sum_{l = m - ϱ (m) + 1}^{m} λ^{m - l} ζ^{T} (l) Q ζ (l) - \sum_{l = m - ϱ (m)}^{m - 1} λ^{m - l} ζ^{T} (l) Q ζ (l)} \\ \leq ζ^{T} (m) Q ζ (m) - λ^{ϱ_{2}} ζ^{T} (m - ϱ (m)) Q ζ (m - ϱ (m)) \end{matrix}

(25)

\begin{matrix} E {Δ V_{3} (m)} \\ = E {\sum_{θ = - ϱ_{2} + 1}^{- ϱ_{1}} \sum_{s = m + θ}^{m} λ^{s - m} ζ^{T} (s) Q ζ (s) - \sum_{θ = - ϱ_{2} + 1}^{- ϱ_{1}} \sum_{s = m - 1 + θ}^{m - 1} λ^{s - m} ζ^{T} (s) Q ζ (s)} \\ = E {\sum_{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))} \\ \leq (ϱ_{2} - ϱ_{1}) ζ^{T} (m) Q ζ (m) \end{matrix}

(26)

\begin{matrix} E {Δ V_{4} (m)} \\ = E {\sum_{θ = - ϱ_{2} + 1}^{0} \sum_{s = m + θ}^{m} η^{T} (s) Z η (s) - \sum_{θ = - ϱ_{2} + 1}^{0} \sum_{s = m - 1 + θ}^{m - 1} λ^{m - s - 1} η^{T} (s) Z η (s)} \\ \leq ϱ_{2} E {η^{T} (m) Z η (m)} - \sum_{θ = m - ϱ (m)}^{m - 1} λ^{- ϱ_{2}} η^{T} (s) Z η (s) \\ = ϱ_{2} E {ζ^{T} (m) {\tilde{A}}_{m}^{T} Z {\tilde{A}}_{m} ζ (m)} + ϱ_{2} {ζ^{T} (m) Z ζ (m) - 2 ζ^{T} (m) {\bar{A}}^{T} Z ζ (m) \\ + {\vec{x}}^{T} (m - ϱ (m)) {\tilde{C}}_{i}^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} Z {\tilde{B}}_{i} ζ (m - ϱ (m)) \\ + 2 ζ^{T} (m) (\bar{A} - I)^{T} Z {\tilde{B}}_{i} ζ (m - ϱ (m)) + 2 ζ^{T} (m) (\bar{A} - I)^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) \\ + 2 ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m))} - \sum_{θ = m - ϱ (m)}^{m - 1} λ^{- ϱ_{2}} η^{T} (s) Z η (s) \end{matrix}

(27)

For brevity, we set

{\tilde{A}}_{1} = [\begin{matrix} A + B \tilde{K} & 0 \\ B \tilde{K} & A - L \bar{Γ} \end{matrix}], {\tilde{A}}_{2} = [\begin{matrix} 0 & 0 \\ L Γ & 0 \end{matrix}]

Subsequently, it is not hard to get

\begin{matrix} E {ζ^{T} (m) {\tilde{A}}_{m}^{T} P {\tilde{A}}_{m} ζ (m)} = E {ζ^{T} (m) ({\tilde{A}}_{1} + {\tilde{A}}_{2})^{T} P ({\tilde{A}}_{1} + {\tilde{A}}_{2}) ζ (m)} \\ = ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\tilde{A}}_{1} ζ (m) + E {ζ^{T} (m) {\tilde{A}}_{2}^{T} P {\tilde{A}}_{2} ζ (m)} \\ + 2 ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\bar{A}}_{2} ζ (m) \end{matrix}

(28)

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

\begin{matrix} E {ζ^{T} (m) {\tilde{A}}_{2}^{T} P {\tilde{A}}_{2} ζ (m)} \\ = ζ^{T} (m) E {[\begin{matrix} Γ^{T} L^{T} P_{2} L Γ & 0 \\ 0 & 0 \end{matrix}]} ζ (m) \\ = ζ^{T} (m) [\begin{matrix} {\bar{C}}^{T} {(I \otimes L)}^{T} (Σ \otimes P_{2}) (I \otimes L) \bar{C} & 0 \\ 0 & 0 \end{matrix}] ζ (m) \\ = ζ^{T} (m) C^{T} (Σ \otimes P_{2}) C ζ (m) \end{matrix}

(29)

Combining equation (28) with (29) yields

\begin{matrix} E {ζ^{T} (m) {\tilde{A}}_{m}^{T} P {\tilde{A}}_{m} ζ (m)} = ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\tilde{A}}_{1} ζ (m) + ζ^{T} (m) C^{T} Σ \otimes P_{2} C ζ (m) \\ + 2 ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\bar{A}}_{2} ζ (m) \end{matrix}

(30)

In the same way, one has

\begin{matrix} E {ζ^{T} (m) {\tilde{A}}_{m}^{T} Z {\tilde{A}}_{m} ζ (m)} = ζ^{T} (m) {\tilde{A}}_{1}^{T} Z {\tilde{A}}_{1} ζ (m) + ζ^{T} (m) \\ C^{T} Σ \otimes Z_{2} C ζ (m) + 2 ζ^{T} (m) {\tilde{A}}_{1}^{T} Z {\bar{A}}_{2} ζ (m) \end{matrix}

(31)

It should be noticed that

\begin{matrix} 2 [ζ^{T} (m) N_{1} + ζ^{T} (m - ϱ (m)) N_{2}] \times \\ [ζ (m) - ζ (m - ϱ (m)) - \sum_{l = m - ϱ (m)}^{m - 1} η (l)] = 0 \end{matrix}

(32)

\begin{matrix} \sum_{l = m - ϱ_{2}}^{m - 1} ζ_{1}^{T} (m) X ζ_{1} (m) - \sum_{l = m - ϱ (m)}^{m - 1} ζ_{1}^{T} (m) X ζ_{1} (m) \\ = ϱ_{2} ζ_{1}^{T} (m) X ζ_{1} (m) - \sum_{l = m - ϱ (m)}^{m - 1} ζ_{1}^{T} (m) X ζ_{1} (m) \geq 0 \end{matrix}

(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

\begin{matrix} E {Δ V (m)} \\ \leq ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\tilde{A}}_{1} ζ (m) + ζ^{T} (m) {\tilde{A}}_{1}^{T} P {\tilde{A}}_{1} ζ (m) + ζ^{T} (m) {\bar{A}}_{2}^{T} P {\bar{A}}_{2} ζ (m) \\ + ζ^{T} (m) C^{T} Σ \otimes P_{2} C ζ (m) + ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} P {\tilde{B}}_{i} ζ (m - ϱ (m)) \\ + {\vec{x}}^{T} (m - ϱ (m)) {\tilde{C}}_{i}^{T} P {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + 2 ζ^{T} (m) {\bar{A}}^{T} P {\tilde{B}}_{i} ζ (m - ϱ (m)) \\ + 2 ζ^{T} (m) {\bar{A}}^{T} P {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + 2 ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} P {\tilde{C}}_{i} \vec{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) {\tilde{A}}_{1}^{T} Z {\tilde{A}}_{1} ζ (m) + ϱ_{2} ζ^{T} (m) {\tilde{A}}_{1}^{T} Z {\tilde{A}}_{1} ζ (m) \\ + ϱ_{2} ζ^{T} (m) {\bar{A}}_{2}^{T} Z {\bar{A}}_{2} ζ (m) + ϱ_{2} ζ^{T} (m) C^{T} Σ \otimes Z_{2} C ζ (m) + ϱ_{2} ζ^{T} (m) Z ζ (m) \end{matrix}

\begin{matrix} + ϱ_{2} {\vec{x}}^{T} (m - ϱ (m)) {\tilde{C}}_{i}^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + ζ^{T} (m) {\bar{A}}^{T} P \bar{A} ζ (m) \\ + ϱ_{2} ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} Z {\tilde{B}}_{i} ζ (m - ϱ (m)) + ζ (m - ϱ (m)) Φ_{1} ζ (m - ϱ (m)) \\ + 2 ϱ_{2} ζ^{T} (m) (\bar{A} - I)^{T} Z {\tilde{B}}_{i} ζ (m - ϱ (m)) + ϱ_{2} ζ^{T} (m) (\bar{A} - I)^{T} Z (\bar{A} - I) ζ (m) \\ + 2 ϱ_{2} ζ^{T} (m) (\bar{A} - I)^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) - {\vec{x}}^{T} (m - ϱ (m)) Φ_{2} \vec{x} (m - ϱ (m)) \\ + 2 ϱ_{2} ζ^{T} (m - ϱ (m)) {\tilde{B}}_{i}^{T} Z {\tilde{C}}_{i} \vec{x} (m - ϱ (m)) + ϱ_{2} ζ_{1}^{T} (m) X ζ_{1} (m) \\ - \sum_{θ = m - ϱ (m)}^{m - 1} λ^{- ϱ_{2}} η^{T} (s) Z η (s) - \sum_{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}] \\ \times [ζ (m) - ζ (m - ϱ (m)) - \sum_{l = m - ϱ (m)}^{m - 1} η (l)] \\ = ξ_{1}^{T} (m) Ξ ξ_{1} (m) - \sum_{l = m - ϱ (m)}^{m - 1} ξ_{2}^{T} (m, l) Θ ξ_{2} (m, l) \end{matrix}

(34)

where

\begin{matrix} ξ_{1} (m) = [ζ^{T} (m) ζ^{T} (m - ϱ (m)) {\vec{x}}^{T} (m - ϱ (m))]^{T}, \\ ξ_{2} (m, l) = [ζ_{1}^{T} (m) η^{T} (l)]^{T}, \\ Ξ = [\begin{matrix} Ξ_{11} & Ξ_{12} & Ξ_{13} \\ * & Ξ_{22} & Ξ_{23} \\ * & * & Ξ_{33} \end{matrix}], \\ Ξ_{11} = Ψ_{111}^{1} + 2 {\tilde{A}}_{1}^{T} P {\tilde{A}}_{1} + {\bar{A}}_{2}^{T} P {\bar{A}}_{2} + C^{T} (Σ \otimes P_{2}) C^{T} \\ + 2 ϱ_{2} {\tilde{A}}_{1}^{T} Z {\tilde{A}}_{1} + ϱ_{2} {\bar{A}}_{2}^{T} Z {\bar{A}}_{2} + ϱ_{2} C^{T} (Σ \otimes Z_{2}) C \\ + {\bar{A}}^{T} P \bar{A} + ϱ_{2} (\bar{A} - I)^{T} Z (\bar{A} - I), \\ Ξ_{12} = Ψ_{111}^{2} + {\bar{A}}^{T} P {\tilde{B}}_{i} + ϱ_{2} (\bar{A} - I)^{T} Z {\tilde{B}}_{i}, \\ Ξ_{13} = {\bar{A}}^{T} P {\tilde{C}}_{i} + ϱ_{2} (\bar{A} - I)^{T} Z {\tilde{C}}_{i}, \\ Ξ_{22} = Ψ_{111}^{3} + {\tilde{B}}_{i}^{T} P {\tilde{B}}_{i} + ϱ_{2} {\tilde{B}}_{i}^{T} Z {\tilde{B}}_{i}, \\ Ξ_{23} = {\tilde{B}}_{i}^{T} P {\tilde{C}}_{i} + ϱ_{2} {\tilde{B}}_{i}^{T} Z {\tilde{C}}_{i}, \\ Ξ_{33} = {\tilde{C}}_{i}^{T} P {\tilde{C}}_{i} + ϱ_{2} {\tilde{C}}_{i}^{T} Z {\tilde{C}}_{i} - Φ_{2} \end{matrix}

With the help of Schur complement lemma, we know that $\tilde{Ψ} < 0$ 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 $\tilde{Ξ} = [\begin{matrix} {\tilde{Ξ}}_{11} & * \\ {\tilde{Ξ}}_{21} & Ψ_{22} \end{matrix}] < 0$ , where

\begin{matrix} {\tilde{Ξ}}_{11} = [\begin{matrix} Ψ_{111}^{1} & Ψ_{111}^{2} & 0 & {\bar{A}}^{T} & \sqrt{ϱ_{2}} {(\bar{A} - I)}^{T} \\ * & Ψ_{111}^{3} & 0 & {\tilde{B}}_{i}^{T} & \sqrt{ϱ_{2}} {\tilde{B}}_{i}^{T} \\ * & * & - Φ_{2} & {\tilde{C}}_{i}^{T} & \sqrt{ϱ_{2}} {\tilde{C}}_{i}^{T} \\ * & * & * & - P_{j}^{- 1} & 0 \\ * & * & * & * & - Z^{- 1} \end{matrix}] \\ {\tilde{Ξ}}_{21} = [{\tilde{Ξ}}_{21}^{1} 0000], \\ {\tilde{Ξ}}_{21}^{1} = col {\sqrt{2} {\tilde{A}}_{1}, \sqrt{2 ϱ_{2}} {\tilde{A}}_{1}, C, \sqrt{ϱ_{2}} C, {\bar{A}}_{2}, \sqrt{ϱ_{2}} {\bar{A}}_{2}} \end{matrix}

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

\begin{matrix} a E {∥ ζ (m) ∥^{2}} \leq E {V (m)} \leq \dots \leq λ^{- m} E {V (0)} \\ \leq b λ^{- m} E {∥ ζ (0) ∥^{2}} \end{matrix}

(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}} \leq \frac{b}{a} λ^{- m} sup_{- ϱ_{2} \leq θ \leq 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.

Design of observer and controller gains

It should be noted that it is pretty difficult to get the OGM and CGM from the matrix inequalities (equations (20) and (21)) because the formula (20) is not a linear matrix inequality due to the existence of P, $P^{- 1}$ , Z, and $Z^{- 1}$ . Then, we will handle this problem by employing the cone complementary linearization algorithm in (Liu et al., 2017).

For convenience, we set $Y = diag {Y_{1}, Y_{2}} = Z^{- 1}$ and $W = diag {W_{1}, W_{2}} = P^{- 1}$ . It is not difficult to get that $W_{2} = P_{2}^{- 1}$ , $Y_{2} = Z_{2}^{- 1}$ . By replacing the term $Ψ_{22}$ in equation (20) by ${\tilde{Ψ}}_{22} = diag {- W, - Y, - Σ^{- 1} \otimes W_{2}, - Σ^{- 1} \otimes Y_{2}, - W, - Y}$ , it converts $\tilde{Ψ} < 0$ to ${\tilde{Ψ}}^{1} < 0$ , where ${\tilde{Ψ}}^{1} = [\begin{matrix} Ψ^{1} & D \\ D^{T} & - κ I \end{matrix}]$ and $Ψ^{1} = [\begin{matrix} Ψ_{11} & Ψ_{21}^{T} \\ Ψ_{21} & {\tilde{Ψ}}_{22} \end{matrix}]$ . The OGM and CGM can be obtained by solving the following nonlinear minimization problem

\begin{matrix} minimize trace (PW + ZY) \\ subject to {\tilde{Ψ}}^{1} < 0, \\ [\begin{matrix} P & I \\ I & W \end{matrix}] \geq 0, \\ [\begin{matrix} Z & I \\ I & Y \end{matrix}] \geq 0 \end{matrix}

(39)

Remark 7. So far, we have developed a new nonfragile NPCS under the transmission of the redundant channel and event-triggered communication criteria. The condition in Theorem 1 guarantees the MSES of the addressed NCS. Besides, we can obtain the OGM and CGM by solving equation (39). In Theorem 1, we can observe the effect of redundant channel transmission strategy and ETM on system stability. To be specific, $Σ$ reflects the influence of redundant channel transmission strategy on system stability and the stability criterion involves parameters $Φ$ and $ε$ in the ETM.

Remark 8. In this paper, we mainly answered the following two questions: (1) how to propose an effective NPCS to deal with the delay influence caused by the network? (2) how to ensure stability with theoretical analysis by proposing a new control method? Compared with the existing NPCS in (Sun and Chen, 2014; Liu, 2016; Yang et al., 2018; Pang et al., 2021b), the major effort has been devoted to designing the nonfragile NPCS based on ETM in the case of communication limitation. The differences compared with existing literature are as follows. (1) The effects of gain perturbation, communication limitation, and NIP are considered comprehensively when analyzing the system performance. (2) The closed-loop system is modeled as a switching system and its MSES problem is analyzed. (3) New effective nonfragile NPCS is proposed to compensate the effects of the input time delay and NITDs.

An illustrative example

In this section, a simulation example is used to illustrate the effectiveness of the designed nonfragile NPCS based on the ETM. To be more specific, for the NCS described by equations (1) and (2), let $M = 3$ and select the following parameter matrices:

\begin{matrix} A = [\begin{matrix} - 0.71 & 0.21 & 0.38 \\ 0.78 & - 0.58 & - 0.36 \\ 0.36 & 0.26 & 0.68 \end{matrix}], B = [\begin{matrix} 0.73 & - 0.46 \\ - 0.65 & 0.25 \\ 0.26 & - 0.71 \end{matrix}], \\ C_{1} = C_{2} = C_{3} = [0.43 - 0.12 0.08] \end{matrix}

Assume input delay $d = 2$ , the bounds of NITDs $π^{sc} = π^{ca} = 1$ , then we have $π = 4$ . The parameters are selected as $Φ = I$ , $H = 5$ , $ε = 0.1$ in the ETM. The initial values of NCS are set to $\tilde{χ} = {[2.14.41.7]}^{T}$ , $u (0) = [0.30.2]^{T}$ , and $\hat{x} (0) = [0 0 0]^{T}$ . The expectations of $α_{1} (m)$ , $α_{2} (m)$ , and $α_{3} (m)$ are ${\bar{α}}_{1} = 0.9$ , ${\bar{α}}_{2} = 0.7$ , and ${\bar{α}}_{3} = 0.6$ , respectively. The other parameters are selected as $λ = 1.3$ , $D = [- 0.010]^{T}$ , $F = [0.7 \sin (t) 00.8 \cos (t)]$ , $E = diag {0.0100}$ . The state trajectories of the open-loop system are depicted in Figure 2, where $x_{r} (m)$ ( $r = 1, 2, 3$ ) is the r th element of state $x (m)$ . Obviously, the open-loop system is unstable.

Figure 2.

State trajectories of open-loop system.

For comparison with existing control methods, three cases are discussed in this example.

Case 1: For this case, the effects from both input delay and NITDs are tackled simultaneously, the nonfragile NPCS is employed. The controller is given by equation (12), that is

u (m - d) = (K + Δ K (m)) \tilde{x} (m | m_{ρ} - π)

By solving the nonlinear minimization problem (equation (39)) with convex constraints, the OGM and CGM are obtained as follows

L = [\begin{matrix} - 0.7379 \\ - 0.1997 \\ - 0.1352 \end{matrix}], K = [\begin{matrix} 0.2141 & 0.0547 & - 0.3178 \\ 0.2179 & 0.0456 & - 0.2735 \end{matrix}]

Figure 3 describes the trajectories of the control input under the designed nonfragile NPCS, the corresponding state trajectories of the closed-loop system are displayed in Figure 4, where $x_{r} (m)$ ( $r = 1, 2, 3$ ) and $u_{r} (m)$ ( $r = 1, 2$ ) are the r th element of state $x (m)$ and control input $u (m)$ , respectively. Figure 5 shows the event-triggered times. At this time, the state trajectories are convergent.

Figure 3.

Trajectories of control input in Case 1.

Figure 4.

State trajectories of the closed-loop system in Case 1.

Figure 5.

Event-triggered times.

Case 2: Only the input delay is compensated. The following control method proposed in (Tan et al., 2019) is adopted

u (m - d) = K \hat{x} (m | m - d - 1)

By solving the linear matrix inequality (equation (21)) of Theorem 2 in (Tan et al. (2019)), we obtain

K = [\begin{matrix} 1.6479 & - 0.5284 & - 0.0759 \\ 1.1043 & 0.0840 & 0.8941 \end{matrix}]

It can be seen from Figure 6 that the state trajectories of the closed-loop system are convergent when there are no NITDs. If there are NITDs, the control effect is shown in Figure 7. Clearly, the desired stability has not been achieved.

Figure 6.

State trajectories of the closed-loop system without NITDs in Case 2.

Figure 7.

State trajectories of the closed-loop system without compensating NITDs in Case 2.

Case 3: The input delay and NITDs are compensated. In this case, the following NPCS mentioned in (Li et al., 2019b) is used

u (m - d) = K \hat{x} (m | m - d - 1)

The OGM and CGM are obtained by the eigenstructure assignment method as follows

K = [\begin{matrix} 1.0318 & 0.1830 & 0.6096 \\ 1.0365 & 0.4834 & 2.2255 \end{matrix}], L = [\begin{matrix} - 1.2081 \\ 1.6947 \\ 0.8459 \end{matrix}]

Similar to Case 1, the trajectories of the closed-loop system shown in Figure 8 are convergent.

Figure 8.

State trajectories of the closed-loop system in Case 3.

The results under the different cases show that the NPCM can effectively compensate the input delay and NITDs. The ETM can effectively reduce the frequency of signal transmission in the network while ensuring the control performance of the system.

The statistical properties of random variables $α_{1} (m)$ , $α_{2} (m)$ , and $α_{3} (m)$ are changed for the comparison purpose. To increase the occurrence of data packet dropout, let ${\bar{α}}_{1} = 0.6$ , ${\bar{α}}_{2} = 0.4$ , and ${\bar{α}}_{3} = 0.3$ , all other parameters are the same as mentioned above. Figures 9 and 10 depict the control input trajectories and state trajectories in this case, respectively. The convergence speed of the state trajectories becomes slow, which also means that the occurrence of data packet dropout will affect the performance of the system.

Figure 9.

Trajectories of control input with compensating under ${\bar{α}}_{1} = 0.6$ , ${\bar{α}}_{2} = 0.4$ , and ${\bar{α}}_{3} = 0.3$ .

Figure 10.

State trajectories of the closed-loop system with compensating under ${\bar{α}}_{1} = 0.6$ , ${\bar{α}}_{2} = 0.4$ , and ${\bar{α}}_{3} = 0.3$ .

Conclusions

The problem of predictive control has been investigated for the NCS with input delay and NITDs in this paper. The transmission scheme regulated by redundant channels has been adopted, which has enhanced the reliability of signal transmission. The nonfragile NPCS based on the ETM has been proposed to compensate input delay and NITDs and reduce the frequency of signal transmission. By using augmented technology, the closed-loop system has been converted to a switched system. By the aid of the Lyapunov stability theorem, sufficient condition ensuring MSES has been given for the closed-loop system. Moreover, the OGM and CGM have been obtained by solving the nonlinear minimization problem. Finally, the effectiveness of the nonfragile NPCS proposed has been verified by providing an illustrative example. Further research includes extending the event-triggered nonfragile predictive control method to nonlinear systems (Tan, 2017; Tan et al., 2021).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China (grant nos. 12071102 and 61673141) and the Natural Science Foundation of Heilongjiang Province of China (grant no. ZD2022F003).

ORCID iDs

Hairui Zhao

Dongyan Chen

Data accessibility statement

All data generated or analyzed during this study are included in this published article.

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