Robust finite-time contractive stability analysis of faulty linear network-based control systems with an aperiodic sampling and adaptive event-triggered scheme

Abstract

This study investigates the problem of finite-time contractive stability analysis and observer-based $H_{\infty}$ fault-tolerant control (FTC) for linear network-based control systems subject to network-induced time-varying delay. It is assumed that the faults occur in the both actuator and sensor components. For the sake of data transmission reduction, an aperiodic-sampling-based adaptive event-triggered scheme is used, in which the interval between two sampling instants varies within a certain known bound, and the event threshold is adjusted by using the adaptive rule. An unknown input observer (UIO) is used to estimate the system states and faults simultaneously. Then, using Lyapunov–Krasovskii stability theory, delay-dependent sufficient conditions for the observer-based FTC of the networked control system (NCS) are derived. These conditions are presented in the form of linear matrix inequalities (LMIs), ensuring that both the error system and the closed-loop NCS achieve finite-time contractive stability while simultaneously satisfying the $H_{\infty}$ performance index. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed design approach.

Keywords

Fault estimation fault-tolerant control finite-time contractive stability networked control system aperiodic sampling adaptive event-triggered scheme

Introduction

In the past few years, networked control systems (NCSs) have attracted significant theoretical and practical research interests. NCSs are feedback control systems wherein the control loops are closed through communication networks (Zhang et al., 2017). NCSs have many advantages when they are compared with traditional point-to-point systems, such as low cost, reduced system wiring, ease of maintenance, and high reliability (Zhang et al., 2019). The communication networks in control loops introduce some challenging problems, such as network-induced delays, data packet dropout, and quantization, which may degrade the system’s performance. In recent years, some researchers have studied the stability/stabilization problem and controller design for NCSs in the presence of network-induced imperfections (Deng et al., 2021; Halder et al., 2019; Su et al., 2021; Tan et al., 2015; Wang et al., 2023; Zhang et al., 2021; Zheng et al., 2022a, 2022b). It is noted that the communication bandwidth of NCSs is often limited. Consequently, it is important to find an effective method to reduce the amount of data transmission. In view of this point, event-trigger-based control methods have been developed to deal with this problem in the last few years. In event-triggered schemes, the data is decided to be transmitted only when the error between the current sampled instant and the last transmitted instant becomes larger than a certain threshold level. In most of the traditional event-triggered strategies, the threshold level is often assumed to be constant, which makes it hard to adapt the variation of system states (Li and Peng, 2022; Yan et al., 2019). Li et al. (2020d) presented an efficient adaptive event-triggered scheme and it has been shown that the network consumption was reduced efficiently. It is worth mentioning that most of the existing results are based on periodic sampling. While, due to some unpredictable environmental changes, the sampling period is not usually equidistant and deterministic. Stability analysis of NCSs with aperiodic sampling and networked-induced time-varying delay was addressed in Chen et al. (2016). Very recently, a stochastic-sampling-based adaptive event-triggered approach was proposed for NCSs in Xie et al. (2020), where the sampling period varies in an interval with probability and the event threshold is adjusted with the adaptive rule. It is shown that this strategy can decrease unnecessary data transmission more significantly.

On the other hand, actuator and sensor faults are inevitable in the realistic environment of NCSs. Hence, due to the increasing requirement for high safety and reliability, the fault-tolerant control (FTC) for NCSs has received increasing attention in recent years (Fang et al., 2021; He et al., 2016; Li et al., 2020b, 2020c). Generally, the FTC can be classified into two main categories: the passive FTC (PFTC) and the active FTC (AFTC) (Abbaspour et al., 2020). In the PFTC, the design is considered without fault information such as location, magnitude, and time occurrence and it treats faults in the way of an additional form of uncertainties (Lan and Patton, 2016). Therefore, this method is conservative and if the faults are outside the prescribed ranges, it cannot always guarantee the stability or desired performance of the system. Some studies have been done on PFTC in Liu et al. (2016a) and Zhang et al. (2016b). The AFTC is mainly based on fault detection and isolation (FDI) or fault estimation, and the controller is changed in an active way to compensate for the effect of faults. In recent years, several valuable works have studied the FDI problem of the NCSs (Jin et al., 2019; Kargar et al., 2019; Wang et al., 2016; Wu et al., 2016). In El Abbadi and Jamouli (2023), a fault detection filter was constructed for an NCS subject to packet dropout and time delay. The event-triggered fault detection problem for NCSs was studied in Atitallah et al. (2018), where the data packet dropout was taken into account in the designing procedure. Furthermore, a robust residual generation using a parity space framework and an adaptive threshold was proposed to overcome the limitations of static thresholds. In Huang and Pan (2020), the $H_{\infty}$ fault detection filter was designed for a class of nonlinear NCSs while considering the network imperfections such as random packet loss and delay.

Unlike FDI approaches, fault estimation can provide important details about a fault, including its location, magnitude, and duration. Thus, finding an effective method for fault estimation is of great importance. During the past few years, several effective fault estimation (FE) design methods have been proposed, for example, based on unknown input observer (UIO) (Lan and Patton, 2015, 2017), adaptive fault estimation (You et al., 2015), sliding mode observer (Chu and Li, 2018), and extended state observer (Shi et al., 2015). In Qian et al. (2016), the problem of observer-based robust FTC was studied for a class of uncertain nonlinear discrete-time systems subject to actuator fault. A study of self-triggered fault diagnosis and FTC was considered for NCSs with actuator fault in Qiu et al. (2018) to retain the input-to-state stability of the closed-loop system. In Zhang et al. (2016a), an adaptive fault observer was used to estimate the constant actuator fault in discrete-time linear NCS and then the predictive FTC was designed to compensate for the fault effect. In Lee et al. (2019), the fault estimation and decentralized control problem for an integrated NCS with faulty actuators was studied. In Wang et al. (2019), event-triggered fault estimation and FTC problem was considered for continuous-time dynamic NCSs with additive system fault. Wang et al. (2019) investigated the problem of fault estimation and FTC for linear NCSs experiencing additive actuator fault and the network-induced delay which was only taken into account in the sensor-to-controller communication path.

Note that most of the previous studies have concentrated on asymptotic stability where system state trajectories converge to an equilibrium point over an infinite time interval. Whereas, the main focus in practical applications is on the behavior of dynamic systems within a specified finite time frame. On the other hand, in the presence of state saturations, large values of the states are not acceptable. Therefore, we need to determine a condition under which the input/system states do not exceed a certain threshold within a fixed time interval. Considering this point, the finite-time contractive stability (FTCS) was proposed in Weiss and Infante (1967). Note that FTCS is a different concept from finite-time stability (FTS) presented in Li et al. (2020a); Ren et al. (2018). FTS only concerns with the boundedness of the system states, while in FTCS, the contraction characteristic is as important as the boundedness. FTCS ensures that the system’s trajectories will remain within a specified bound that is smaller than a bound on the initial state before reaching the terminal time. Recently, the analysis of FTCS and FTC was addressed for a nonlinear networked single-link flexible manipulator in Jani et al. (2023). This paper utilizes an event-triggered scheme to save network resources, relying on periodic sampling and the assumption of a constant threshold level, which poses difficulties in adapting to changes in system states. In Jani et al. (2022), the FTCS was extended to uncertain nonlinear NCSs subject to network-induced delay. It is notable that in the aforementioned work, the sampling period was assumed to be constant, whereby sampling was done at equidistant instants of time. Furthermore, the fault was considered just in the actuator component, while the possibility of the occurrence of a sensor fault is usually inevitable in practice. Also, the PFTC method was used to tolerate the effect of the actuator fault, which is valid only for the prescribed faults.

Motivated by the above discussion, this paper aims to compensate for the shortcomings of the previous works. To the best of the author’s knowledge, the problem of robust FTCS and observer-based active fault-tolerant controller design for linear NCSs with an aperiodic sampling and adaptive event-triggered scheme has not been investigated completely yet, especially for the case where the system model subjects to both sensor and actuator faults, external disturbances, and time-varying delay. Therefore, compared to the existing works, the main contributions of this study are listed as follows:

Due to some factors such as the influence of a noisy environment, the sampling period is usually aperiodic. Therefore, an aperiodic sampling is used in this paper, whereby the interval between two sampling instants varies within a certain known bound.

By aiming at transmission reduction and saving more network resources, a novel adaptive event-triggered communication scheme is proposed based on an aperiodic sampling, in which the event threshold is adjusted by using the adaptive rule.

Since the PFTC method is valid only for the prescribed faults, in this paper, the observer-based AFTC is studied to compensate the effects of faults immediately.

To reflect a more realistic environment of NCSs, in addition to the actuator fault, the sensor fault is considered in the system model, which makes the system model more general.

Therefore, to investigate this problem for linear NCSs, first, UIO is used, which can provide accurate estimations of system states and faults simultaneously. In the next step, by employing the proposed Lyapunov–Krasovskii functional, we derive delay-dependent sufficient conditions to ensure that both the error system and the closed-loop NCS achieve FTCS. Then an observer-based feedback controller is designed based on the derived results such that the fault tolerance is achieved and the $H_{\infty}$ performance index is satisfied in the closed system in the presence of external disturbance.

The content of the paper is structured as follows. Section “Problem formulation and preliminaries” provides the description of linear NCS with actuator and sensor faults and some preliminaries. The main results of the solution to the FTCS analysis and $H_{\infty}$ FTC problem are proposed in Section “Main results.” A simulation example is given in Section “Simulation results” that validates the proposed theoretical results and finally, Section “Conclusion” concludes the study and gives some future works.

Notations Throughout the paper, $P^{†}$ is pseudo inverse matrix satisfying $P P^{†} P = P$ . $λ_{\min} (P)$ and $λ_{\max} (P)$ are the minimum and maximum eigenvalues of $P$ , respectively. The space of all square-integrable vector functions over $[0, \infty)$ is denoted as $L_{2} [0, \infty)$ . In symmetric matrices, the symbol * represents the transpose of the symmetric part of a matrix.

Problem formulation and preliminaries

Plant description

Consider the following linear NCS with actuator and sensor faults:

\begin{matrix} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} u (t) + F_{0} f (t) + D_{0} w (t), \\ y (t) = C_{0} x (t) + G_{0} f (t), \end{matrix}

(1)

where $x (t) \in R^{n}$ represents the state vector of the NCS, $u (t) \in R^{m}$ and $y (t) \in R^{p}$ denote the control input and system output, respectively. $w (t) \in R^{n_{d}}$ is the unknown external disturbance and $f (t) \in R^{n_{h}}$ $(n_{h} \leq p)$ stands for the fault. Furthermore, $w (t)$ and $f (t)$ satisfy $∥ w (t) ∥ \leq \bar{w}$ and $∥ f (t) ∥ \leq \bar{f}$ with positive scalars $\bar{w}$ and $\bar{f}$ , respectively. $A_{0} \in R^{n \times n}$ , $B_{0} \in R^{n \times m}$ , $C_{0} \in R^{p \times n}$ , $D_{0} \in R^{n \times n_{d}}$ , $F_{0} \in R^{n \times n_{h}}$ , and $G_{0} \in R^{p \times n_{h}}$ are real known constant matrices.

Before we state the main results, the following assumptions should be noted.

Assumption 1. The pair $(A_{0}, C_{0})$ is detectable and the pair $(A_{0}, B_{0})$ is stabilizable.

Assumption 2. The matrix $B_{0}$ is of full-column rank, then there exists a singular value decomposition as follows:

B_{0} = U [\begin{matrix} ϒ \\ 0 \end{matrix}] V^{T},

where $U \in R^{n \times n}$ and $V \in R^{m \times m}$ are two orthogonal matrices, $ϒ = diag {υ_{1}, υ_{2}, \dots, υ_{m}}$ and $υ_{i}$ s $(i = 1, 2, \dots, m)$ are nonzero singular values of $B_{0}$ .

Remark 1. Assumption 1 provides necessary conditions for fault estimation-based FTC systems. Furthermore, under Assumption 2, the singular value decomposition method can be used to deal with the nonlinear terms in the subsequent theorems.

Taking the fault vector as an auxiliary state vector, an augmented system is constructed as follows:

\begin{array}{l} E \dot{\bar{x}} (t) = Ā_{0} \bar{x} (t) + B_{0} u (t) + D_{0} w (t), \\ y (t) = {\bar{C}}_{0} \bar{x} (t), \end{array}

(2)

where $\bar{x} (t) \overset{Δ}{=} [x^{T} (t) f^{T} (t)]^{T} \in R^{n + n_{h}}$ , $E = [I_{n \times n} 0_{n \times n_{h}}]$ , $Ā_{0} = [A_{0} F_{0}],$ and ${\bar{C}}_{0} = [C_{0} G_{0}]$ .

The augmented state $\bar{x} (t)$ can be estimated by the following UIO:

\begin{matrix} \overset{\cdot}{z} (t) = Nz (t) + Tu (t) + Ly (t), \\ \hat{\bar{x}} (t) = z (t) + Hy (t), \end{matrix}

(3)

where $z (t) \in R^{n + n_{h}}$ and $\hat{\bar{x}} (t) \in R^{n + n_{h}}$ are the observer system state and the augmented state estimation, respectively. The designed matrices $N, T, L$ , and $H$ are of appropriate dimensions. Define the estimation error as $e (t) = \hat{\bar{x}} (t) - \bar{x} (t)$ and the auxiliary error as $ε (t) = z (t) - DE \bar{x} (t)$ . Differentiating $ε (t)$ with respect to time gives

\begin{matrix} \dot{ε} (t) = ż (t) - D E \dot{\bar{x}} (t) = N [z (t) - D E \bar{x} (t)] + [L {\bar{C}}_{0} + N D E \\ - D Ā_{0}] \bar{x} (t) + [T - D B_{0}] u (t) - D D_{0} w (t) . \end{matrix}

(4)

On the other hand, by adding and subtracting $DE \bar{x} (t)$ to the right side of $\hat{\bar{x}} (t)$ in (3), we can write

\hat{\bar{x}} (t) = ε (t) + [DE + H {\bar{C}}_{0}] \bar{x} (t) .

(5)

If there exists matrix $D$ such that

{\begin{matrix} L {\bar{C}}_{0} + N D E - D Ā_{0} = 0, \\ T - D B_{0} = 0, \\ D E + H {\bar{C}}_{0} = I, \end{matrix}

(6)

then equations (4) and (5) can be written as

\overset{\cdot}{ε} (t) = N [z (t) - DE \bar{x} (t)] - D D_{0} w (t) = N ε (t) - D D_{0} w (t),

(7)

e (t) = ε (t) .

(8)

If the error system (7) is finite-time contractively stable, then we can conclude that the original error system (8) is also finite-time contractively stable.

The conditions (6) can be written as

{\begin{matrix} [\begin{matrix} N & L \\ I & H \end{matrix}] [\begin{matrix} D E \\ {\bar{C}}_{0} \end{matrix}] = [\begin{matrix} D Ā_{0} \\ I \end{matrix}], \\ T = D B_{0} . \end{matrix}

(9)

Then by using the method presented in Darouach et al. (2011), the general solution to (9) is given by

\begin{array}{l} [\begin{matrix} N & L \\ I & H \end{matrix}] & = [\begin{matrix} D Ā_{0} \\ I \end{matrix}] {[\begin{matrix} D E \\ {\bar{C}}_{0} \end{matrix}]}^{†} + [\begin{matrix} Y_{0} \\ Y_{1} \end{matrix}] (I - [\begin{matrix} D E \\ {\bar{C}}_{0} \end{matrix}] {[\begin{matrix} D E \\ {\bar{C}}_{0} \end{matrix}]}^{†}), \end{array}

(10)

where $Y_{0}$ and $Y_{1}$ are arbitrary matrices of appropriate dimensions and matrix $D$ is calculated as follows:

D = G {[\begin{matrix} E \\ {\bar{C}}_{0} \end{matrix}]}^{†} [\begin{matrix} I \\ 0 \end{matrix}],

(11)

where $G$ is any full-row rank matrix such that

rank [\begin{matrix} G \\ {\bar{C}}_{0} \end{matrix}] = rank [\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}] = n + n_{h} .

(12)

Let us define the following matrices:

\begin{matrix} ν_{1} \overset{Δ}{=} {[\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}]}^{†} [\begin{matrix} I_{(n + n_{h}) \times (n + n_{h})} \\ 0_{p \times (n + n_{h})} \end{matrix}], \\ ν_{2} \overset{Δ}{=} {[\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}]}^{†} [\begin{matrix} 0_{(n + n_{h}) \times p} \\ I_{p \times p} \end{matrix}], \\ κ_{1} \overset{Δ}{=} (I - [\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}] {[\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}]}^{†}) [\begin{matrix} I_{(n + n_{h}) \times (n + n_{h})} \\ 0_{p \times (n + n_{h})} \end{matrix}], \\ κ_{2} \overset{Δ}{=} (I - [\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}] {[\begin{matrix} DE \\ {\bar{C}}_{0} \end{matrix}]}^{†}) [\begin{matrix} 0_{(n + n_{h}) \times p} \\ I_{p \times p} \end{matrix}] . \end{matrix}

(13)

Then from (9) and (10), we can construct the observer gains as follows:

\begin{array}{l} N & = D Ā_{0} ν_{1} + Y_{0} κ_{1}, \\ L & = D Ā_{0} ν_{2} + Y_{0} κ_{2}, \\ I & = ν_{1} + Y_{1} κ_{1}, \\ H & = ν_{2} + Y_{1} κ_{2}, \\ T & = D B_{0} . \end{array}

(14)

Event detector scheme

The communication bandwidth of network resources is often limited. Therefore, it is imperative to utilize the network resources in an efficient way. In a conventional time-triggered scheme, the sampling instants are distributed equidistantly in time, resulting in the transmission of all sampled data through the communication network. In general, a time-triggered mechanism leads to waste the network bandwidth consumption due to unnecessary data transmission. To mitigate this problem, we use the aperiodic-sampling-based adaptive event-triggered communication scheme which is adaptable to the variation of system states. The block diagram of overall system is shown in Figure 1, where the signal $\hat{\bar{x}} (t)$ has been sampled at instants $i_{j}, j = 0, 1, 2, \dots, \infty$ . It is assumed that the sampling interval $d_{l}$ is variable, which satisfies $0 < d_{m} \leq d_{l} \leq d_{M}$ , where $d_{m}$ and $d_{M}$ are two known positive constants.

Then the following adaptive event-triggered scheme is used to assess whether it is necessary to transmit the sampled data into the communication network.

t_{k + 1} = t_{k} + \sum_{s = 0}^{l^{*} - 1} d_{s} + \inf_{l} {d_{l} | e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}) \geq σ (t_{k}) {\hat{\bar{x}}}^{T} (t_{k}) Ω \hat{\bar{x}} (t_{k})},

(15)

where $Ω > 0$ is a weighting matrix and $e_{th} (i_{l}) = \hat{\bar{x}} (i_{l}) - \hat{\bar{x}} (t_{k})$ denotes the threshold error between the current sampled state $\hat{\bar{x}} (i_{l})$ and the last transmitted state $\hat{\bar{x}} (t_{k})$ . Note that $d_{0} = 0$ and $i_{l} = t_{k} + \sum_{s = 0}^{l} d_{s}$ , $l = 0, 1, \dots, l^{*} - 1$ represents the current sampling instant, and $l^{*}$ is the infimum $l$ that satisfies the condition $e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}) \geq σ (t_{k}) {\hat{\bar{x}}}^{T} (t_{k}) Ω \hat{\bar{x}} (t_{k})$ . Moreover, $σ (t_{k})$ is the adjustable threshold which can be determined by the following adaptive rule Jani et al. (2022):

\begin{matrix} σ (t_{k + 1}) = \max {σ (t_{k}) (1 - \frac{2 q_{1}}{π} \tan^{- 1} [q_{2} (∥ \hat{\bar{x}} (t_{k + 1}) ∥ \\ - ∥ \hat{\bar{x}} (t_{k}) ∥)]), σ_{m}}, \end{matrix}

(16)

where $q_{1}$ and $q_{2}$ are given positive constants, $σ_{m} \in [0, 1)$ is the lower bound of $σ (t_{k})$ and $σ (0) = σ_{m}$ .

Remark 2. The event-triggered condition (15) is influenced by the values of $σ (t_{k})$ , $\hat{\bar{x}} (i_{l})$ and $\hat{\bar{x}} (t_{k})$ . When $q_{1} = 0$ and $σ_{m} \neq 0$ , the communication mechanism (15) simplifies to the discrete event-triggered scheme described in Yan et al. (2019). Furthermore, in the special case where $σ_{m} = 0$ , this condition reverts to the conventional time-triggered scheme, resulting in the transmission of all sampled data to the communication network.

Remark 3. It is important to note that the minimum inter-event time in the presented event-triggered communication scheme (15) is $d_{m}$ . This ensures that an infinite number of events cannot occur within a finite time frame, thereby preventing the occurrence of the Zeno phenomenon.

Remark 4. From the condition (16), it can be seen that if $∥ \hat{\bar{x}} (t_{k + 1}) ∥ \leq ∥ \hat{\bar{x}} (t_{k}) ∥$ , then $σ (t_{k + 1}) > σ (t_{k})$ and the larger threshold is used to reduce the utilization of network resources and when $∥ \hat{\bar{x}} (t_{k + 1}) ∥ > ∥ \hat{\bar{x}} (t_{k}) ∥$ , one can obtain that $0 < 1 - (2 q_{1} / π) \tan^{- 1} [q_{2} (∥ \hat{\bar{x}} (t_{k + 1}) ∥ - ∥ \hat{\bar{x}} (t_{k}) ∥)] < 1$ and therefore, the smaller threshold is used to increase the transmission frequency.

Zero-order-holder (ZOH)

As shown in Figure 1, $τ_{t_{k}}^{sc}$ and $τ_{t_{k}}^{ca}$ denote the time-varying delay in the S–C and C–A links at the transmission instants $t_{k}$ , respectively. Note that since the feedback controller is static, $τ_{t_{k}}^{sc}$ and $τ_{t_{k}}^{ca}$ can be combined as $τ_{t_{k}} = τ_{t_{k}}^{sc} + τ_{t_{k}}^{ca}$ and it is assumed that $0 \leq τ_{t_{k}} \leq \bar{τ}$ , where $\bar{τ}$ is a positive known constant. Therefore, the data released at the transmission instant $t_{k}$ will arrive at the ZOH side at the time $t_{k} + τ_{t_{k}}$ . The ZOH operates in an event-trigger manner, meaning it uses the most recently transmitted signal and holds it until the next signal arrives.

The holding interval of ZOH $[t_{k} + τ_{t_{k}}, t_{k + 1} + τ_{t_{k + 1}})$ can be further divided into the following subintervals, as shown in Figure 2.

{\begin{matrix} if l^{*} = 1 \Rightarrow \\ I_{l} = [t_{k} + τ_{t_{k}}, t_{k + 1} + τ_{t_{k + 1}}), \\ if l^{*} \geq 2 \Rightarrow \\ I_{l} = {\begin{matrix} [t_{k} + \sum_{s = 0}^{l} d_{s} + τ_{t_{k}}, t_{k} + \sum_{s = 0}^{l} d_{s} + d_{l + 1} + τ_{t_{k}}), \\ l = 0, 1, 2, \dots, l^{*} - 2, \\ [t_{k} + \sum_{s = 0}^{l} d_{s} + τ_{t_{k}}, t_{k} + \sum_{s = 0}^{l} d_{s} + d_{l + 1} + τ_{t_{k + 1}}), \\ l = l^{*} - 1 . \end{matrix} \end{matrix}

Then, one can write

[t_{k} + τ_{t_{k}}, t_{k + 1} + τ_{t_{k + 1}}) = ⋃_{l = 0}^{l^{*} - 1} I_{l} .

(17)

Define

τ (t) = t - i_{l}, t \in [t_{k} + τ_{t_{k}}, t_{k + 1} + τ_{t_{k + 1}}),

(18)

where $τ (t)$ is a piecewise-linear function and by considering $i_{l} = t_{k} + \sum_{s = 0}^{l} d_{s}$ , $l = 0, 1, \dots, l^{*} - 1$ , it can be observed that for $t \in I_{l}$ , $τ (t)$ satisfies

{\begin{matrix} if l^{*} = 1 \Rightarrow {\begin{matrix} 0 \leq τ_{t_{k}} \leq τ (t) \leq d_{1} + τ_{t_{k + 1}} \leq d_{M} + \bar{τ}, \end{matrix} \\ if l^{*} \geq 2 \Rightarrow {\begin{matrix} 0 \leq τ_{t_{k}} \leq τ (t) \leq d_{l + 1} + τ_{t_{k}} \leq d_{M} + \bar{τ}, \\ l = 0, 1, 2, \dots, l^{*} - 2, \\ 0 \leq τ_{t_{k}} \leq τ (t) \leq d_{l + 1} + τ_{t_{k + 1}} \leq d_{M} + \bar{τ}, \\ l = l^{*} - 1 . \end{matrix} \end{matrix}

Obviously,

0 \leq τ (t) \leq d_{M} + \bar{τ} : = τ_{m}, st \in [t_{k} + τ_{t_{k}}, t_{k + 1} + τ_{t_{k + 1}}) .

(19)

Therefore, by considering the above discussion, the general form of the control law can be described as

u (t) = K \hat{\bar{x}} (t_{k}) = K \hat{\bar{x}} (t - τ (t)) - K e_{th} (i_{l}),

(20)

where $K = [k_{x}, k_{f}]$ represents the controller gain, consisting of the nominal controller gain $k_{x}$ and the fault compensation gain $k_{f}$ .

Substituting (20) into (1) yields the closed-loop system as follows:

\begin{matrix} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} K \hat{\bar{x}} (t - τ (t)) - B_{0} K e_{th} (i_{l}) + F_{0} f (t) + D_{0} w (t), \\ y (t) = C_{0} x (t) + G_{0} f (t) . \end{matrix}

(21)

The resulting system (21) can be written as:

\begin{matrix} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} k_{x} x (t - τ (t)) + B_{0} Ke (t - τ (t)) - B_{0} K e_{th} (i_{l}) \\ + F_{0} f (t) + B_{0} k_{f} f (t - τ (t)) + D_{0} w (t), \\ y (t) = C_{0} x (t) + G_{0} f (t) . \end{matrix}

(22)

Using (8), the augmented system composed of (22) and (7) is

\begin{matrix} \overset{\cdot}{η} (t) = {\tilde{A}}_{0} η (t) + {\tilde{A}}_{1} η (t - τ (t)) + {\tilde{A}}_{2} e_{th} (i_{l}) + {\tilde{F}}_{1} f (t) \\ + {\tilde{F}}_{2} f (t - τ (t)) + \tilde{D} w (t), \\ y_{c} (t) = \tilde{C} η (t) . \end{matrix}

(23)

where $y_{c} (t)$ is the regulated output, $η (t) = {[\begin{matrix} x^{T} (t) & ε^{T} (t) \end{matrix}]}^{T}$ and

\begin{matrix} {\tilde{A}}_{0} = [\begin{matrix} A_{0} & 0 \\ 0 & N \end{matrix}], {\tilde{A}}_{1} = [\begin{matrix} B_{0} k_{x} & B_{0} K \\ 0 & 0 \end{matrix}], {\tilde{A}}_{2} = [\begin{matrix} - B_{0} K \\ 0 \end{matrix}], \\ {\tilde{F}}_{1} = [\begin{matrix} F_{0} \\ 0 \end{matrix}], {\tilde{F}}_{2} = [\begin{matrix} B_{0} k_{f} \\ 0 \end{matrix}], \tilde{D} = [\begin{matrix} D_{0} \\ - D D_{0} \end{matrix}], \tilde{C} = [\begin{matrix} C_{x} & 0 \\ 0 & C_{ε} \end{matrix}] \end{matrix}

and also $C_{x}$ and $C_{ε}$ are regulated output coefficient matrices.

The following preliminaries are presented for subsequent analysis.

Lemma 1. (Liu et al., 2016b). Let $S \in R^{n \times n}$ be a symmetric matrix, and let $x \in R^{n}$ , then the following inequality holds:

λ_{\min} (S) x^{T} x \leq x^{T} Sx \leq λ_{\max} (S) x^{T} x .

(24)

Definition 1. (Jani et al., 2022). For given positive constants $c_{0}, r_{1}, r_{2}$ , $t_{s}$ , and $T_{f}$ with $r_{1} < c_{0} < r_{2}$ and $t_{s} \in (0, T_{f})$ , system (23) is said to be finite-time contractively stable with respect to $(c_{0}, r_{1}, r_{2}, t_{s}, T_{f})$ if for every $η (s)$ , $\sup_{- τ_{m} \leq s \leq 0} {η^{T} (s) η (s),$ ${\overset{\cdot}{η}}^{T} (s) \overset{\cdot}{η} (s)} < c_{0}^{2}$ implies that $η^{T} (t) η (t) < r_{2}^{2}$ , $\forall t \in [0, T_{f}]$ and moreover $η^{T} (t) η (t) < r_{1}^{2}$ , $\forall t \in [t_{s}, T_{f}]$ .

Main results

The purpose of this study is to design the feedback controller gain matrix $K$ and the observer gains to ensure that the augmented system (23) is contractively stable in finite-time when $w (t) = 0$ and meantime, for any nonzero $w (t) \in L_{2} [0, T_{f}]$ , the augmented system (23) satisfies the following $H_{\infty}$ performance index under zero state response.

\int_{0}^{T_{f}} y_{c}^{T} (s) y_{c} (s) ds \leq γ^{2} \int_{0}^{T_{f}} w^{T} (s) w (s) ds .

(25)

In the following, we present two theorems for solving this problem.

Theorem 2. For given positive scalars $γ$ , $ρ_{1}$ , $ρ_{2}$ , $α$ , $σ_{m}$ , and $τ_{m}$ , the augmented system (23) achieves FTCS with respect to $(c_{0}, r_{1}, r_{2}, t_{s}, T_{f})$ and the $H_{\infty}$ index (25) is satisfied, if there exist scalars $λ_{i} > 0$ ( $i = 1, \dots, 4$ ) and symmetric positive definite matrices $P = diag {P_{1}, P_{2}}$ , $Q = diag {Q_{1}, Q_{2}}$ , $R = diag {R_{1}, R_{2}}$ , and $Ω = diag {Ω_{1}, Ω_{2}}$ and matrices $M$ and $W$ such that the following matrix inequalities hold:

Π_{j} : = [\begin{matrix} Π_{j, 11} & Π_{j, 12} & Π_{j, 13} \\ * & Π_{j, 22} & Π_{j, 23} \\ * & * & Π_{j, 33} \end{matrix}] < 0,

(26)

c_{0}^{2} (λ_{2} + τ_{m} λ_{3} + \frac{τ_{m}^{2}}{2} λ_{4}) e^{- α T_{f}} - r_{2}^{2} λ_{1} < 0,

(27)

c_{0}^{2} (λ_{2} + τ_{m} λ_{3} + \frac{τ_{m}^{2}}{2} λ_{4}) e^{- α t_{s}} - r_{1}^{2} λ_{1} < 0,

(28)

λ_{1} I < P < λ_{2} I, Q < λ_{3} I, R < λ_{4} I,

(29)

where

\begin{matrix} Π_{j, 11} \\ = [\begin{matrix} Π_{11} & Π_{12} & M_{3}^{T} & Π_{14} & M_{5}^{T} + W_{1} & Π_{16} & M_{7}^{T} & Π_{18} & Π_{19} \\ * & Π_{22} & - M_{3}^{T} & Π_{24} & - M_{5}^{T} + W_{2} & Π_{26} & - M_{7}^{T} & - M_{8}^{T} & - M_{9}^{T} \\ * & * & Π_{33} & 0 & W_{3} & - W_{3} & 0 & 0 & 0 \\ * & * & * & Π_{44} & W_{4} & - W_{4} - σ_{m} Ω & 0 & 0 & Π_{49} \\ * & * & * & * & Π_{55} & - W_{5} + W_{6}^{T} & W_{7}^{T} & W_{8}^{T} & W_{9}^{T} \\ * & * & * & * & * & Π_{66} & - W_{7}^{T} & - W_{8}^{T} & Π_{69} \\ * & * & * & * & * & * & Π_{77} & 0 & 0 \\ * & * & * & * & * & * & * & - ρ_{1} \bar{f} I & 0 \\ * & * & * & * & * & * & * & * & Π_{99} \end{matrix}], \end{matrix}

\begin{array}{l} Π_{11} = P_{1} A_{0} + A_{0}^{T} P_{1} + (Q_{1} + α P_{1}) + M_{1} + M_{1}^{T}, \\ Π_{12} = P_{1} B_{0} k_{x} - M_{1} + M_{2}^{T}, \\ Π_{14} = - P_{1} B_{0} K + M_{4}^{T}, \\ Π_{16} = P_{1} B_{0} K + M_{6}^{T} - W_{1}, \\ Π_{18} = P_{1} F_{0} + M_{8}^{T}, \\ Π_{19} = P_{1} B_{0} k_{f} + M_{9}^{T}, \\ Π_{22} = σ_{m} Ω_{1} - M_{2} - M_{2}^{T}, \\ Π_{24} = - σ_{m} [\begin{matrix} Ω_{1} & 0 \end{matrix}] - M_{4}^{T}, \\ Π_{26} = σ_{m} [\begin{matrix} Ω_{1} & 0 \end{matrix}] - M_{6}^{T} - W_{2}, \\ Π_{33} = - e^{- α τ_{m}} Q_{1}, \\ Π_{44} = (σ_{m} - 1) Ω, \\ Π_{49} = - σ_{m} {[\begin{matrix} 0 & Ω_{2} \end{matrix}]}^{T}, \\ Π_{55} = P_{2} (D Ā_{0} ν_{1} + Y_{0} κ_{1}) + {(D Ā_{0} ν_{1} + Y_{0} κ_{1})}^{T} P_{2} \\ + (Q_{2} + α P_{2}) + W_{5} + W_{5}^{T}, \\ Π_{66} = σ_{m} Ω - W_{6} - W_{6}^{T}, \\ Π_{69} = σ_{m} {[\begin{matrix} 0 & Ω_{2} \end{matrix}]}^{T} - W_{9}^{T}, \\ Π_{77} = - e^{- α τ_{m}} Q_{2}, \\ Π_{99} = - ρ_{2} \bar{f} I + σ_{m} Ω_{2}, \\ Π_{j, 12} = [\begin{matrix} τ_{m} \hat{M} & τ_{m} \hat{W} & τ_{m} Â_{1 P}^{T} & τ_{m} Â_{2 P}^{T} \end{matrix}], \\ Â_{1 P} = [\begin{matrix} P_{1} A_{0} & P_{1} B_{0} k_{x} & 0 & - P_{1} B_{0} K & 0 & P_{1} B_{0} K & 0 & P_{1} F_{0} & P_{1} B_{0} k_{f} \end{matrix}], \\ Â_{2 P} = [\begin{matrix} 0 & 0 & 0 & 0 & P_{2} (D Ā_{0} ν_{1} + Y_{0} κ_{1}) & 0 & 0 & 0 & 0 \end{matrix}], \\ Π_{j, 13} = [\begin{matrix} C_{x}^{T} & 0 & P_{1} D_{0} + M_{10}^{T} \\ 0 & 0 & - M_{10}^{T} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & C_{ε}^{T} & W_{10}^{T} - P_{2} D D_{0} \\ 0 & 0 & - W_{10}^{T} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], \\ Π_{j, 22} = d i a g {- τ_{m} e^{- α τ_{m}} R_{1}, - τ_{m} e^{- α τ_{m}} R_{2}, - τ_{m} P_{1} R_{1}^{- 1} P_{1}, - τ_{m} P_{2} R_{2}^{- 1} P_{2}}, \\ Π_{j, 23} = [\begin{matrix} 0 & 0 & τ_{m} M_{10}^{T} \\ 0 & 0 & τ_{m} W_{10}^{T} \\ 0 & 0 & τ_{m} P_{1} D_{0} \\ 0 & 0 & - τ_{m} P_{2} D D_{0} \end{matrix}], \\ Π_{j, 33} = d i a g {- I, - I, - γ^{2} I} . \end{array}

Proof. The proof includes two parts. In the first part, we need to prove that the augmented system (23) is contractively stable in finite-time when $w (t) = 0$ . To do this, consider the following multiple Lyapunov–Krasovskii functional:

\begin{matrix} V (t) = η^{T} (t) P η (t) + \int_{t - τ_{m}}^{t} e^{- α (t - s)} η^{T} (s) Q η (s) d s \\ + \int_{- τ_{m}}^{0} \int_{t + θ}^{t} e^{- α (t - s)} {\overset{\cdot}{η}}^{T} (s) R \overset{\cdot}{η} (s) d s d θ, \end{matrix}

(30)

where

P = [\begin{matrix} P_{1} & 0 \\ * & P_{2} \end{matrix}], Q = [\begin{matrix} Q_{1} & 0 \\ * & Q_{2} \end{matrix}], and R = [\begin{matrix} R_{1} & 0 \\ * & R_{2} \end{matrix}]

are symmetric positive definite matrices.

In the following, for the sake of convenience, we consider $V (t)$ as

V (t) = V_{x} (t) + V_{ε} (t),

(31)

where

\begin{matrix} V_{x} (t) = x^{T} (t) P_{1} x (t) + \int_{t - τ_{m}}^{t} e^{- α (t - s)} x^{T} (s) Q_{1} x (s) ds \\ + \int_{- τ_{m}}^{0} \int_{t + θ}^{t} e^{- α (t - s)} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) dsd θ, \\ V_{ε} (t) = ε^{T} (t) P_{2} ε (t) + \int_{t - τ_{m}}^{t} e^{- α (t - s)} ε^{T} (s) Q_{2} ε (s) ds \\ + \int_{- τ_{m}}^{0} \int_{t + θ}^{t} e^{- α (t - s)} {\overset{\cdot}{ε}}^{T} (s) R_{2} \overset{\cdot}{ε} (s) dsd θ . \end{matrix}

One can get the derivative of (31) as follows:

\begin{matrix} \overset{\cdot}{V} (t) = {\overset{\cdot}{V}}_{x} (t) + {\overset{\cdot}{V}}_{ε} (t) \\ = {\overset{\cdot}{x}}^{T} (t) P_{1} x (t) + x^{T} (t) P_{1} \overset{\cdot}{x} (t) + x^{T} (t) (Q_{1} + α P_{1}) x (t) \\ - e^{- α τ_{m}} x^{T} (t - τ_{m}) Q_{1} x (t - τ_{m}) \\ + τ_{m} {\overset{\cdot}{x}}^{T} (t) R_{1} \overset{\cdot}{x} (t) - \int_{t - τ_{m}}^{t} e^{- α (t - s)} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds \\ + {\overset{\cdot}{ε}}^{T} (t) P_{2} ε (t) + ε {(t)}^{T} P_{2} \overset{\cdot}{ε} (t) + ε^{T} (t) (Q_{2} + α P_{2}) ε (t) \\ - e^{- α τ_{m}} ε^{T} (t - τ_{m}) Q_{2} ε (t - τ_{m}) + τ_{m} {\overset{\cdot}{ε}}^{T} (t) R_{2} \overset{\cdot}{ε} (t) \\ - \int_{t - τ_{m}}^{t} e^{- α (t - s)} {\overset{\cdot}{ε}}^{T} (s) R_{2} \overset{\cdot}{ε} (s) ds - α V (t) \\ \leq 2 x^{T} (t) P_{1} [A_{0} x (t) + B_{0} k_{x} x (t - τ (t)) + B_{0} K ε (t - τ (t)) \\ - B_{0} K e_{th} (i_{l}) + F_{0} f (t) \\ + B_{0} k_{f} f (t - τ (t))] + x^{T} (t) (Q_{1} + α P_{1}) x (t) \\ - e^{- α τ_{m}} x^{T} (t - τ_{m}) Q_{1} x (t - τ_{m}) \\ + τ_{m} {\overset{\cdot}{x}}^{T} (t) R_{1} \overset{\cdot}{x} (t) - \int_{t - τ (t)}^{t} e^{- α (t - s)} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds \\ + 2 ε^{T} (t) P_{2} N ε (t) + ε^{T} (t) (Q_{2} + α P_{2}) ε (t) \\ - e^{- α τ_{m}} ε^{T} (t - τ_{m}) Q_{2} ε (t - τ_{m}) + τ_{m} {\overset{\cdot}{ε}}^{T} (t) R_{2} \overset{\cdot}{ε} (t) \\ - \int_{t - τ (t)}^{t} e^{- α (t - s)} {\overset{\cdot}{ε}}^{T} (s) R_{2} \overset{\cdot}{ε} (s) ds - α V (t) . \end{matrix}

(32)

It is easy to see that

\int_{t - τ (t)}^{t} e^{- α (t - s)} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds \geq \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds,

(33)

\int_{t - τ (t)}^{t} e^{- α (t - s)} {\overset{\cdot}{ε}}^{T} (s) R_{2} \overset{\cdot}{ε} (s) ds \geq \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\overset{\cdot}{ε}}^{T} (s) R_{2} \overset{\cdot}{ε} (s) ds,

(34)

On the other hand, based on the Leibniz–Newton formula, we have the following equations for matrices of appropriate dimensions:

2 ξ^{T} (t) \hat{M} [x (t) - x (t - τ (t)) - \int_{t - τ (t)}^{t} \overset{\cdot}{x} (s) ds] = 0,

(35)

2 ξ^{T} (t) \hat{W} [ε (t) - ε (t - τ (t)) - \int_{t - τ (t)}^{t} \overset{\cdot}{ε} (s) ds] = 0,

(36)

where $ξ (t) = [x^{T} (t), x^{T} (t - τ (t)), x^{T} (t - τ_{m}), e_{th}^{T} (i_{l}), ε^{T} (t),$ $ε^{T} (t - τ (t)), ε^{T} (t - τ_{m}), f^{T} (t), f^{T} (t - τ (t))]^{T}$ , $\hat{M} = [M_{1}^{T}, M_{2}^{T}, M_{3}^{T},$ $M_{4}^{T}, M_{5}^{T}, M_{6}^{T}, M_{7}^{T}, M_{8}^{T}, M_{9}^{T}]^{T}$ and $\hat{W} = [W_{1}^{T}, W_{2}^{T}, W_{3}^{T}, W_{4}^{T}, W_{5}^{T}, W_{6}^{T},$ $W_{7}^{T}, W_{8}^{T}, W_{9}^{T}]^{T}$ .

Furthermore, from the event-triggered condition (15), one can get that

e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}) \geq σ_{m} {\hat{\bar{x}}}^{T} (t_{k}) Ω \hat{\bar{x}} (t_{k}),

(37)

Therefore, when the current sampled data is not transmitted, we have

\begin{matrix} e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}) < σ_{m} {\hat{\bar{x}}}^{T} (t_{k}) Ω \hat{\bar{x}} (t_{k}) \\ = σ_{m} {[\hat{\bar{x}} (t - τ (t)) - e_{th} (i_{l})]}^{T} Ω [\hat{\bar{x}} (t - τ (t)) - e_{th} (i_{l})] \\ = σ_{m} {\hat{\bar{x}}}^{T} (t - τ (t)) Ω \hat{\bar{x}} (t - τ (t)) \\ - 2 σ_{m} {\hat{\bar{x}}}^{T} (t - τ (t)) Ω e_{th} (i_{l}) + σ_{m} e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}) \\ = σ_{m} {[ε (t - τ (t)) + \bar{x} (t - τ (t))]}^{T} Ω [ε (t - τ (t)) + \bar{x} (t - τ (t))] \\ - 2 σ_{m} {[ε (t - τ (t)) + \bar{x} (t - τ (t))]}^{T} Ω e_{th} (i_{l}) \\ + σ_{m} e_{th}^{T} (i_{l}) Ω e_{th} (i_{l}), \end{matrix}

(38)

where $Ω = diag {Ω_{1}, Ω_{2}}$ .

Thus, by incorporating (33) to (38) in (32), it follows that

\begin{matrix} \dot{V} (t) \leq σ_{m} {[ε (t - τ (t)) + \bar{x} (t - τ (t))]}^{T} Ω [ε (t - τ (t)) + \bar{x} (t - τ (t))] \\ - 2 σ_{m} {[ε (t - τ (t)) + \bar{x} (t - τ (t))]}^{T} Ω e_{t h} (i_{l}) \\ + (σ_{m} - 1) e_{t h}^{T} (i_{l}) Ω e_{t h} (i_{l}) \\ + 2 x^{T} (t) P_{1} [A_{0} x (t) + B_{0} k_{x} x (t - τ (t)) + B_{0} K ε (t - τ (t)) \\ - B_{0} K e_{t h} (i_{l}) + F_{0} f (t) \\ + B_{0} k_{f} f (t - τ (t))] + x^{T} (t) (Q_{1} + α P_{1}) x (t) \\ - e^{- α τ_{m}} x^{T} (t - τ_{m}) Q_{1} x (t - τ_{m}) \\ + τ_{m} ξ^{T} (t) Â_{1}^{T} R_{1} Â_{1} ξ (t) - \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\dot{x}}^{T} (s) R_{1} \dot{x} (s) d s \\ + 2 ε {(t)}^{T} P_{2} N ε (t) + ε^{T} (t) (Q_{2} + α P_{2}) ε (t) \\ - e^{- α τ_{m}} ε^{T} (t - τ_{m}) Q_{2} ε (t - τ_{m}) \\ + τ_{m} ξ^{T} (t) Â_{2}^{T} R_{2} Â_{2} ξ (t) - \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\dot{ε}}^{T} (s) R_{2} \dot{ε} (s) d s \\ + 2 ξ^{T} (t) \hat{M} [x (t) - x (t - τ (t)) - \int_{t - τ (t)}^{t} \dot{x} (s) d s] \\ + 2 ξ^{T} (t) \hat{W} [ε (t) - ε (t - τ (t)) - \int_{t - τ (t)}^{t} \dot{ε} (s) d s] - α V (t) \\ \leq ξ^{T} (t) [Π_{j, 11} + τ_{m} \hat{M} {(e^{- α τ_{m}} R_{1})}^{- 1} {\hat{M}}^{T} \\ + τ_{m} \hat{W} {(e^{- α τ_{m}} R_{2})}^{- 1} {\hat{W}}^{T} + τ_{m} Â_{1}^{T} R_{1} Â_{1} \\ + τ_{m} Â_{2}^{T} R_{2} Â_{2}] ξ (t) - α V (t) + ρ_{1} f^{T} (t) f (t) \\ + ρ_{2} f^{T} (t - τ (t)) f (t - τ (t)) \\ - \int_{t - τ (t)}^{t} [ξ^{T} (t) \hat{M} + {\dot{x}}^{T} (s) {(e^{- α τ_{m}} R_{1})}^{T}] \\ \times {(e^{- α τ_{m}} R_{1})}^{- 1} [{\hat{M}}^{T} ξ (t) + (e^{- α τ_{m}} R_{1}) \dot{x} (s)] d s \\ - \int_{t - τ (t)}^{t} [ξ^{T} (t) \hat{W} + {\dot{ε}}^{T} (s) {(e^{- α τ_{m}} R_{2})}^{T}] {(e^{- α τ_{m}} R_{2})}^{- 1} \\ \times \hat{W} [^{T} ξ (t) + (e^{- α τ_{m}} R_{2}) \dot{ε} (s)] d s \\ \leq ξ^{T} (t) [Π_{j, 11} + τ_{m} \hat{M} {(e^{- α τ_{m}} R_{1})}^{- 1} {\hat{M}}^{T} \\ + τ_{m} \hat{W} {(e^{- α τ_{m}} R_{2})}^{- 1} {\hat{W}}^{T} \\ + τ_{m} Â_{1}^{T} R_{1} Â_{1} + τ_{m} Â_{2}^{T} R_{2} Â_{2}] ξ (t) + β - α V (t), \end{matrix}

(39)

where ${\overset{⌢}{A}}_{1} = [A_{0} B_{0} k_{x} 0 - B_{0} K 0 B_{0} K 0 F_{0} B_{0} k_{f}]$ , ${\overset{⌢}{A}}_{2} = [0 0 0 0 N 0 0 0 0]$ and $β = (ρ_{1} + ρ_{2}) \bar{f}$ .

Since $R_{i} = R_{i}^{T} > 0 (i = 1, 2)$ and $e^{- α τ_{m}} > 0$ then $[ξ^{T} (t) \hat{M} + {\overset{\cdot}{x}}^{T} (s) {(e^{- α τ_{m}} R_{1})}^{T}] {(e^{- α τ_{m}} R_{1})}^{- 1} [{\hat{M}}^{T} ξ (t) + (e^{- α τ_{m}} R_{1}) \overset{\cdot}{x} (s)] \geq 0$ and $[ξ^{T} (t) \hat{W} + {\overset{\cdot}{ε}}^{T} (s) {(e^{- α τ_{m}} R_{2})}^{T}] {(e^{- α τ_{m}} R_{2})}^{- 1} [{\hat{W}}^{T} ξ (t) +$ $(e^{- α τ_{m}} R_{2}) \overset{\cdot}{ε} (s)] \geq 0$ hold true. Therefore, it is easy to see that when $λ_{\min} (- {\overset{\lor}{Π}}_{j, 1}) ∥ ξ (t) ∥^{2} > β$ ( $λ_{\min} (- {\overset{\lor}{Π}}_{j, 1})$ is the minimum eigenvalue of $- [Π_{j, 11} + τ_{m} \hat{M} {(e^{- α τ_{m}} R_{1})}^{- 1} {\hat{M}}^{T} + τ_{m} \hat{W} {(e^{- α τ_{m}} R_{2})}^{- 1} {\hat{W}}^{T} + τ_{m} Â_{1}^{T} R_{1} Â_{1} + τ_{m} Â_{2}^{T} R_{2} Â_{2}]$ ), then we have $\overset{\cdot}{V} (t) + α V (t) < 0$ . Thus, if a feasible solution exists for $[Π_{j, 11} + τ_{m} \hat{M} {(e^{- α τ_{m}} R_{1})}^{- 1} {\hat{M}}^{T} + τ_{m} \hat{W} {(e^{- α τ_{m}} R_{2})}^{- 1} {\hat{W}}^{T} + τ_{m} Â_{1}^{T}$ $R_{1} Â_{1} + τ_{m} Â_{2}^{T} R_{2} Â_{2}] < 0$ , one can get that

\overset{\cdot}{V} (t) + α V (t) < 0 .

(40)

The inequality $[Π_{j, 11} + τ_{m} \hat{M} {(e^{- α τ_{m}} R_{1})}^{- 1} {\hat{M}}^{T} + τ_{m} \hat{W}$ ${(e^{- α τ_{m}} R_{2})}^{- 1} {\hat{W}}^{T} + τ_{m} Â_{1}^{T} R_{1} Â_{1} + τ_{m} Â_{2}^{T} R_{2} Â_{2}] < 0$ is equivalent to (41) due to the Schur complement and congruence transformation using block-diagonal matrix $diag (I, I, I, I, I, I, I, I, I, I, I,$ $P_{1}, P_{2})$ . It is important to note that ${\bar{Π}}_{j, 1}$ is a sub-block of matrix $Π_{j}$ . Therefore, the inequality (26) implies (41).

{\bar{Π}}_{j, 1} : = [\begin{matrix} Π_{j, 11} & Π_{j, 12} \\ * & Π_{j, 22} \end{matrix}] < 0 .

(41)

By performing the integration of both sides of (40) from $0$ to $t$ , we can derive

V (t) < e^{- α t} V (0),

(42)

By using Lemma 1 and (30), one can get that the following relations hold:

\begin{matrix} V (0) \leq η^{T} (0) P η (0) + \int_{- τ_{m}}^{0} {su p_{- τ_{m} \leq s \leq 0} {e^{α s} η^{T} (s) Q η (s)}} ds \\ + \int_{- τ_{m}}^{0} \int_{θ}^{0} {su p_{- τ_{m} \leq s \leq 0} {e^{α s} {\overset{\cdot}{η}}^{T} (s) R \overset{\cdot}{η} (s)}} dsd θ \\ \leq η^{T} (0) P η (0) + \int_{- τ_{m}}^{0} η^{T} (0) Q η (0) ds \\ + \int_{- τ_{m}}^{0} \int_{θ}^{0} {\overset{\cdot}{η}}^{T} (0) R \overset{\cdot}{η} (0) dsd θ \\ \leq λ_{\max} (P) {η^{T} (0) η (0)} + λ_{\max} (Q) η^{T} (0) η (0) \int_{- τ_{m}}^{0} ds \\ + λ_{\max} (R) {\overset{\cdot}{η}}^{T} (0) \overset{\cdot}{η} (0) \int_{- τ_{m}}^{0} \int_{θ}^{0} dsd θ \\ = λ_{\max} (P) {η^{T} (0) η (0)} + τ_{m} λ_{\max} (Q) {η^{T} (0) η (0)} \\ + \frac{τ_{m}^{2}}{2} λ_{\max} (R) {{\overset{\cdot}{η}}^{T} (0) \overset{\cdot}{η} (0)} . \end{matrix}

(43)

On the other hand, it can be derived from (30) that

V (t) \geq η^{T} (t) P η (t) \geq λ_{\min} (P) {η^{T} (t) η (t)} .

(44)

From (42) to (44), it is apparent to see that $V (t)$ satisfies

η^{T} (t) η (t) \leq \frac{V (t)}{λ_{1}} < \frac{e^{- α t} V (0)}{λ_{1}} \leq \frac{λ_{2} e^{- α t} c_{0}^{2}}{λ_{1}},

(45)

where

\begin{matrix} λ_{1} = λ_{\min} (P), \\ λ_{2} = λ_{\max} (P) + τ_{m} λ_{\max} (Q) + \frac{τ_{m}^{2}}{2} λ_{\max} (R), \end{matrix}

(46)

Therefore, it can be easily seen that if the inequalities (27) and (28) hold, then the augmented system (23) is contractively stable in finite-time when $w (t) = 0$ . □

In the next part, we establish the $H_{\infty}$ performance (25) for nonzero $w (t) \in L_{2} [0, \infty)$ . To do this, let define

J (t) \overset{Δ}{=} \int_{0}^{t} (y_{c}^{T} (s) y_{c} (s) - γ^{2} w^{T} (s) w (s)) ds .

(47)

Given that $V (t) \geq 0$ and assuming zero initial conditions, it follows that

\begin{matrix} J (t) = \int_{0}^{t} (y_{c}^{T} (s) y_{c} (s) - γ^{2} w^{T} (s) w (s) + \overset{\cdot}{V} (s) + α V (s)) ds \\ - \int_{0}^{t} (\overset{\cdot}{V} (s) + α V (s)) ds \\ \leq \int_{0}^{t} (y_{c}^{T} (s) y_{c} (s) - γ^{2} w^{T} (s) w (s) + \overset{\cdot}{V} (s) + α V (s)) ds . \end{matrix}

Thus, a sufficient condition for meeting the $H_{\infty}$ performance index (25) is to establish

V_{o} (t) = y_{c}^{T} (t) y_{c} (t) - γ^{2} w^{T} (t) w (t) + \overset{\cdot}{V} (t) + α V (t) < 0 .

(48)

Taking the time derivative of (31) and using (33), (34), and the Leibniz–Newton formula with properly dimensioned matrices, one can obtain:

\begin{array}{l} V_{o} (t) \leq y_{c}^{T} (t) y_{c} (t) - γ^{2} w^{T} (t) w (t) + σ_{m} [ε (t - τ (t)) \\ + \bar{x} (t - τ (t))]^{T} Ω [ε (t - τ (t)) + \bar{x} (t - τ (t))] \\ - 2 σ_{m} {[ε (t - τ (t)) + \bar{x} (t - τ (t))]}^{T} Ω e_{t h} (i_{l}) \\ + (σ_{m} - 1) e_{t h}^{T} (i_{l}) Ω e_{t h} (i_{l}) + 2 x^{T} (t) P_{1} [A_{0} x (t) \\ + B_{0} k_{x} x (t - τ (t)) + B_{0} K ε (t - τ (t)) - B_{0} K e_{t h} (i_{l}) \\ + F_{0} f (t) + B_{0} k_{f} f (t - τ (t)) + D_{0} w (t)] \\ + x^{T} (t) (Q_{1} + α P_{1}) x (t) - e^{- α τ_{m}} x^{T} (t - τ_{m}) \\ \times Q_{1} x (t - τ_{m}) + τ_{m} ζ^{T} (t) A_{1}^{T} R_{1} A_{1} ζ (t) \\ - \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\dot{x}}^{T} (s) R_{1} \dot{x} (s) d s + 2 ε {(t)}^{T} P_{2} [N ε (t) \\ - D D_{0} w (t)] ε (t) + ε^{T} (t) (Q_{2} + α P_{2}) ε (t) - e^{- α τ_{m}} \\ \times ε^{T} (t - τ_{m}) Q_{2} ε (t - τ_{m}) + τ_{m} ζ^{T} (t) A_{2}^{T} R_{2} A_{2} ζ (t) \\ - \int_{t - τ (t)}^{t} e^{- α τ_{m}} {\dot{ε}}^{T} (s) R_{2} \dot{ε} (s) d s + 2 ζ^{T} (t) M [x (t) \\ - x (t - τ (t)) - \int_{t - τ (t)}^{t} \dot{x} (s) d s] \\ + 2 ζ^{T} (t) W [ε (t) - ε (t - τ (t)) - \int_{t - τ (t)}^{t} \dot{ε} (s) d s] \\ \leq ζ^{T} (t) [{\tilde{Π}}_{j, 11} + C_{1}^{T} C_{1} + τ_{m} M {(e^{- α τ_{m}} R_{1})}^{- 1} M^{T} \\ + τ_{m} W {(e^{- α τ_{m}} R_{2})}^{- 1} W^{T} + τ_{m} A_{1}^{T} R_{1} A_{1} \\ + τ_{m} A_{2}^{T} R_{2} A_{2}] ζ (t) + ρ_{1} f^{T} (t) f (t) + ρ_{2} f^{T} (t - τ (t)) \\ \times f (t - τ (t)) - \int_{t - τ (t)}^{t} [ζ^{T} (t) M + {\dot{x}}^{T} (s) (e^{- α τ_{m}} R_{1})^{T}] \\ \times {(e^{- α τ_{m}} R_{1})}^{- 1} [M^{T} ζ (t) + (e^{- α τ_{m}} R_{1}) \dot{x} (s)] d s \\ - \int_{t - τ (t)}^{t} [ζ^{T} (t) W + {\dot{ε}}^{T} (s) (e^{- α τ_{m}} R_{2})^{T}] (e^{- α τ_{m}} R_{2})^{- 1} \\ \times [W^{T} ζ (t) + (e^{- α τ_{m}} R_{2}) \dot{ε} (s)] d s \\ \leq ζ^{T} (t) [{\tilde{Π}}_{j, 11} + C_{1}^{T} C_{1} + τ_{m} M {(e^{- α τ_{m}} R_{1})}^{- 1} M^{T} \\ + τ_{m} W {(e^{- α τ_{m}} R_{2})}^{- 1} W^{T} + τ_{m} A_{1}^{T} R_{1} A_{1} \\ + τ_{m} A_{2}^{T} R_{2} A_{2}] ζ (t) + β . \end{array}

(49)

where

\begin{matrix} {\tilde{Π}}_{j, 11} = [\begin{matrix} {\tilde{Π}}_{11} & {\tilde{Π}}_{12} \\ * & {\tilde{Π}}_{22} \end{matrix}], \\ {\tilde{Π}}_{11} = Π_{11}, \\ {\tilde{Π}}_{12} = \\ {[\begin{matrix} {(P_{1} D_{0})}^{T} + M_{10} & - M_{10} & 0 & 0 & W_{10} - {(P_{2} D D_{0})}^{T} & - W_{10} & 0 & 0 & 0 \end{matrix}]}^{T}, \end{matrix}

\begin{matrix} {\tilde{Π}}_{22} = - γ^{2} I, \\ C_{1} = [\begin{matrix} C_{x} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{ε} & 0 & 0 & 0 & 0 & 0 \end{matrix}], \end{matrix}

and $ζ (t) = [x^{T} (t) x^{T} (t - τ (t)) x^{T} (t - τ_{m}) e_{th}^{T} (i_{k}) ε^{T} (t) ε^{T} (t - τ (t))$ $ε^{T} (t - τ_{m}) f^{T} (t) f^{T} (t - τ (t)) w^{T} (t)]^{T}$ , $M = [M_{1}^{T} M_{2}^{T} M_{3}^{T} M_{4}^{T} M_{5}^{T}$ $M_{6}^{T}$ $M_{7}^{T} M_{8}^{T} M_{9}^{T} M_{10}^{T}]^{T}$ , $W = [W_{1}^{T} W_{2}^{T} W_{3}^{T} W_{4}^{T} W_{5}^{T} W_{6}^{T} W_{7}^{T} W_{8}^{T} W_{9}^{T} W_{10}^{T}]^{T}$ , $A_{1} = [A_{0} B_{0} k_{x} 0 - B_{0} K 0 B_{0} K 0 F_{0} B_{0} k_{f} D_{0}]$ , $A_{2} = [0 0 0 0 N 0 0 0$ $0 - D D_{0}]$ and $β = (ρ_{1} + ρ_{2}) \bar{f}$ .

It is easy to see that $V_{o} (t) < 0$ holds when $λ_{\min} (- {\tilde{Π}}_{j}) ∥ ζ (t) ∥^{2}$ $> β$ , where $λ_{\min} (- \tilde{Π_{j}})$ is the minimum eigenvalue of $- [{\tilde{Π}}_{j, 11} + C_{1}^{T} C_{1} + τ_{m} M {(e^{- α τ_{m}} R_{1})}^{- 1} M^{T} + τ_{m} W {(e^{- α τ_{m}} R_{2})}^{- 1}$ $W^{T} + τ_{m} A_{1}^{T} R_{1} A_{1} + τ_{m} A_{2}^{T} R_{2} A_{2}]$ . By emplying Schur complement and congruence transformation using block-diagonal matrix $diag (I, I, I, I, I, I, I, I, I, I, I, I, P_{1}, P_{2}, I)$ , the inequality $[{\tilde{Π}}_{j, 11} + C_{1}^{T} C_{1} + τ_{m} M {(e^{- α τ_{m}} R_{1})}^{- 1} M^{T} + τ_{m} W {(e^{- α τ_{m}} R_{2})}^{- 1} W^{T}$ $+ τ_{m} A_{1}^{T} R_{1} A_{1} + τ_{m} A_{2}^{T} R_{2} A_{2}] < 0$ is equivalent to ${\tilde{Π}}_{j} < 0$ , where

\begin{array}{l} {\tilde{Π}}_{j} : = [\begin{matrix} {\tilde{Π}}_{j, 11} & {\tilde{Π}}_{j, 12} \\ * & {\tilde{Π}}_{j, 22} \end{matrix}] < 0, \\ {\tilde{Π}}_{j, 12} = [\begin{matrix} τ_{m} M & τ_{m} W & τ_{m} A_{1 P}^{T} & τ_{m} A_{2 P}^{T} & C_{1}^{T} \end{matrix}], \\ A_{1 P} = \\ [\begin{matrix} P_{1} A_{0} & P_{1} B_{0} k_{x} & 0 & - P_{1} B_{0} K & 0 & P_{1} B_{0} K & 0 & P_{1} F_{0} & P_{1} B_{0} k_{f} & P_{1} D_{0} \end{matrix}], \\ A_{2 P} = [\begin{matrix} 0 & 0 & 0 & 0 & P_{2} (D Ā_{0} ν_{1} + Y_{0} κ_{1}) & 0 & 0 & 0 & 0 & - P_{2} D D_{0} \end{matrix}], \\ {\tilde{Π}}_{j, 22} = d i a g {- τ_{m} e^{- α τ_{m}} R_{1}, - τ_{m} e^{- α τ_{m}} R_{2}, - τ_{m} P_{1} R_{1}^{- 1} P_{1}, \\ - τ_{m} P_{2} R_{2}^{- 1} P_{2}, - I} . \end{array}

(50)

Then the inequality (26) is obtained by pre- and post-multiplication of $[I_{1}^{T}, I_{2}^{T}, I_{3}^{T}, I_{4}^{T}, I_{5}^{T}, I_{6}^{T}, I_{7}^{T}, I_{8}^{T}, I_{9}^{T}, I_{11}^{T}, I_{12}^{T}, I_{13}^{T}, I_{14}^{T},$ $I_{15}^{T}, I_{10}^{T}]^{T}$ and its tranpose to $\tilde{Π_{j}} < 0$ , where $I_{i}$ denotes a matrix of order $1 \times 15$ (15 blocks with appropriate dimensions). The $i$ th block of $I_{i}$ is the identity matrix and all the other blocks are zero. Therefore, if (26) holds, $J (t) < 0$ is established and the proof is completed.

Theorem 2 gives a sufficient condition for the FTCS of the closed-loop system (23) with $H_{\infty}$ performance (25). However, it should be noted that since the terms $P_{1} B_{0} k_{x}, P_{1} B_{0} K, P_{1} B_{0} k_{f}$ , $P_{2} Y_{0}$ and $P_{i} R_{i}^{- 1} P_{i} (i = 1, 2)$ in Theorem 2 are nonlinear, so the inequality (26) cannot be directly solved with LMI toolboxes. Therefore, Theorem 3 is provided to solve the FTCS and $H_{\infty}$ fault-tolerant controller design problem based on an LMI.

Theorem 3. The augmented system (23) achieves FTCS with respect to $(c_{0}, r_{1}, r_{2}, t_{s}, T_{f})$ and the $H_{\infty}$ index (25) is satisfied, if for given positive scalars $γ$ , $ρ_{1}$ , $ρ_{2}$ , $α$ , $σ_{m}$ , and $τ_{m}$ , there exist scalars $λ_{i} > 0$ ( $i = 1, \dots, 4$ ) and symmetric positive definite matrices $P = diag {P_{1}, P_{2}}$ , $Q = diag {Q_{1}, Q_{2}}$ , $R = diag {R_{1}, R_{2}}$ , $Ω = diag {Ω_{1}, Ω_{2}}$ and matrices $M$ , $W$ , ${\hat{P}}_{1}, Z_{1} = [Z_{1, 11}, Z_{1, 12}]$ , and $Z_{2}$ such that

P_{1} B_{0} = B_{0} {\hat{P}}_{1},

(51)

Φ_{j} : = [\begin{matrix} Φ_{j, 11} & Φ_{j, 12} & Φ_{j, 13} \\ * & Φ_{j, 22} & Φ_{j, 23} \\ * & * & Φ_{j, 33} \end{matrix}] < 0,

(52)

c_{0}^{2} (λ_{2} + τ_{m} λ_{3} + \frac{τ_{m}^{2}}{2} λ_{4}) e^{- α T_{f}} - r_{2}^{2} λ_{1} < 0,

(53)

c_{0}^{2} (λ_{2} + τ_{m} λ_{3} + \frac{τ_{m}^{2}}{2} λ_{4}) e^{- α t_{s}} - r_{1}^{2} λ_{1} < 0,

(54)

λ_{1} I < P < λ_{2} I, Q < λ_{3} I, R < λ_{4} I,

(55)

where

\begin{matrix} Φ_{j, 11} = [\begin{matrix} Φ_{11} & Φ_{12} & M_{3}^{T} & Φ_{14} & M_{5}^{T} + W_{1} & Φ_{16} & M_{7}^{T} & Φ_{18} & Φ_{19} \\ * & Φ_{22} & - M_{3}^{T} & Φ_{24} & - M_{5}^{T} + W_{2} & Φ_{26} & - M_{7}^{T} & - M_{8}^{T} & - M_{9}^{T} \\ * & * & Φ_{33} & 0 & W_{3} & - W_{3} & 0 & 0 & 0 \\ * & * & * & Φ_{44} & W_{4} & - W_{4} - σ_{m} Ω & 0 & 0 & Φ_{49} \\ * & * & * & * & Φ_{55} & - W_{5} + W_{6}^{T} & W_{7}^{T} & W_{8}^{T} & W_{9}^{T} \\ * & * & * & * & * & Φ_{66} & - W_{7}^{T} & - W_{8}^{T} & Φ_{69} \\ * & * & * & * & * & * & Φ_{77} & 0 & 0 \\ * & * & * & * & * & * & * & - ρ_{1} \bar{f} I & 0 \\ * & * & * & * & * & * & * & * & Φ_{99} \end{matrix}], \\ Φ_{11} = P_{1} A_{0} + A_{0}^{T} P_{1} + (Q_{1} + α P_{1}) + M_{1} + M_{1}^{T}, \\ Φ_{12} = B_{0} Z_{1, 11} - M_{1} + M_{2}^{T}, \\ Φ_{14} = - B_{0} Z_{1} + M_{4}^{T}, \\ Φ_{16} = B_{0} Z_{1} + M_{6}^{T} - W_{1}, \\ Φ_{18} = P_{1} F_{0} + M_{8}^{T}, \\ Φ_{19} = B_{0} Z_{1, 12} + M_{9}^{T}, \\ Φ_{22} = σ_{m} Ω_{1} - M_{2} - M_{2}^{T}, \\ Φ_{24} = - σ_{m} [\begin{matrix} Ω_{1} & 0 \end{matrix}] - M_{4}^{T}, \\ Φ_{26} = σ_{m} [\begin{matrix} Ω_{1} & 0 \end{matrix}] - M_{6}^{T} - W_{2}, \\ Φ_{33} = - e^{- α τ_{m}} Q_{1}, \\ Φ_{44} = (σ_{m} - 1) Ω, \\ Φ_{49} = - σ_{m} {[\begin{matrix} 0 & Ω_{2} \end{matrix}]}^{T}, \\ Φ_{55} = P_{2} D Ā_{0} ν_{1} + Z_{2} κ_{1} + {(P_{2} D Ā_{0} ν_{1} + Z_{2} κ_{1})}^{T} \\ + (Q_{2} + α P_{2}) + W_{5} + W_{5}^{T}, \\ Φ_{66} = σ_{m} Ω - W_{6} - W_{6}^{T}, \\ Φ_{69} = σ_{m} {[\begin{matrix} 0 & Ω_{2} \end{matrix}]}^{T} - W_{9}^{T}, \\ Φ_{77} = - e^{- α τ_{m}} Q_{2}, \\ Φ_{99} = - ρ_{2} \bar{f} I + σ_{m} Ω_{2}, \\ Φ_{j, 12} = [\begin{matrix} τ_{m} \hat{M} & τ_{m} \hat{W} & τ_{m} Â_{1 P}^{T} & τ_{m} Â_{2 P}^{T} \end{matrix}], \\ Â_{1 P} = [\begin{matrix} P_{1} A_{0} & B_{0} Z_{1, 11} & 0 & - B_{0} Z_{1} & 0 & B_{0} Z_{1} & 0 & P_{1} F_{0} & B_{0} Z_{1, 12} \end{matrix}], \\ Â_{2 P} = [\begin{matrix} 0 & 0 & 0 & 0 & P_{2} D Ā_{0} ν_{1} + Z_{2} κ_{1} & 0 & 0 & 0 & 0 \end{matrix}], \\ Φ_{j, 13} = [\begin{matrix} C_{x}^{T} & 0 & P_{1} D_{0} + M_{10}^{T} \\ 0 & 0 & - M_{10}^{T} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & C_{ε}^{T} & W_{10}^{T} - P_{2} D D_{0} \\ 0 & 0 & - W_{10}^{T} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], \\ Φ_{j, 22} = d i a g {- τ_{m} e^{- α τ_{m}} R_{1}, - τ_{m} e^{- α τ_{m}} R_{2}, τ_{m} (R_{1} - P_{1}^{T} - P_{1}), τ_{m} (R_{2} - P_{2}^{T} - P_{2})} \\ Φ_{j, 23} = [\begin{matrix} 0 & 0 & τ_{m} M_{10}^{T} \\ 0 & 0 & τ_{m} W_{10}^{T} \\ 0 & 0 & τ_{m} P_{1} D_{0} \\ 0 & 0 & - τ_{m} P_{2} D D_{0} \end{matrix}], \\ Φ_{j, 33} = d i a g {- I, - I, - γ^{2} I} . \end{matrix}

Furthermore, the controller gain matrix is calculated by

K = {\hat{P}}_{1}^{- 1} Z_{1},

(56)

and the observer gains can be calculated from (14).

Proof. First, to linearize the nonlinear terms $P_{1} B_{0} K$ , $P_{1} B_{0} k_{x}$ , and $P_{1} B_{0} k_{f}$ in Theorem 2, we introduce the equality constraint $P_{1} B_{0} = B_{0} {\hat{P}}_{1}$ . Therfore, the terms $P_{1} B_{0} K$ , $P_{1} B_{0} k_{x}$ , and $P_{1} B_{0} k_{f}$ are transformed to $B_{0} {\hat{P}}_{1} K$ , $B_{0} {\hat{P}}_{1} k_{x}$ , and $B_{0} {\hat{P}}_{1} k_{f}$ , respectively. Then, by defining matrices $Z_{1} = {\hat{P}}_{1} K, Z_{1, 11} = {\hat{P}}_{1} k_{x}, Z_{1, 12} = {\hat{P}}_{1} k_{f}, Z_{2} = P_{2} Y_{0}$ and by noting that the inequality $- P_{i} R_{i}^{T} P_{i} \leq R_{i} - P_{i}^{T} - P_{i} (i = 1, 2)$ holds, it is evident that $Π_{j} < 0$ is equivalent to $Φ_{j} < 0$ . Consequently, the proof is completed. □

Remark 5. The necessary conditions for the feasibility of LMIs in Theorem 3 are that the system (1) has to be detectable and stabilizable. Therefore, Assumption 1 provides some standard requirements for fault estimation-based FTC systems. Furthermore, to deal with the nonlinear terms in Theorem 2, we use the equality constraint (51). To solve (51), we employ a singular value decomposition method. This method proposes that the matrix $B$ has to be full-column rank, which is presented in Assumption 2. If Assumption 2 is not satisfied, the other methods such as the optimization problem presented in Corless and Tu (1998) must be used to solve the the equality constraint (51). However, it should be noted that this method has one more design parameter for solving the LMIs.

Remark 6. Note that (51) is difficult to solve with LMI toolboxes. We employ the method presented in Jani et al. (2023). Under the condition of Assumption 2, if matrix $P_{1}$ satisfies

P_{1} = U [\begin{matrix} P_{11} & 0 \\ 0 & P_{22} \end{matrix}] U^{T},

where $P_{11} > 0$ and $P_{22} > 0$ , then there exists matrix $\hat{P_{1}}$ satisfying $P_{1} B_{0} = B_{0} {\hat{P}}_{1}$ . It is equivalent to:

P_{1} U [\begin{matrix} ϒ \\ 0 \end{matrix}] V^{T} = U [\begin{matrix} ϒ \\ 0 \end{matrix}] V^{T} {\hat{P}}_{1} .

Thus,

{\hat{P}}_{1}^{- 1} = V ϒ^{- 1} P_{11}^{- 1} ϒ V^{T} .

Therefore, we have the controller gain matrix $K$ as:

K = V ϒ^{- 1} P_{11}^{- 1} ϒ V^{T} Z_{1} .

(57)

Remark 7. It should be pointed out that all of the LMIs in Theorem 3 are computed offline and once the LMIs are solved, the gain of the controller and observer matrices are obtained from (57) and (14), respectively. Therefore, for online computational, we only need to check the adaptive event-trigger condition and since we use the discrete one, only in discrete times, the event-trigger condition is checked and if the condition is satisfied, the data is transmitted to the network. Therefore, it does not need to monitor all of the states continuously. In this way, the computation and communication resources are only used when needed.

Simulation results

In this section, we illustrate the effectiveness of the proposed approach using a direct current (DC) motor example taken from Lan and Patton (2016). Consider the state space model of a DC motor as:

\begin{matrix} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} u (t), \\ y (t) = C_{0} x (t), \end{matrix}

(58)

with states $x = [i_{a} ω]^{T}$ , control input $u = v_{a}$ , output $y (t)$ , and

A_{0} = [\begin{matrix} - \frac{R_{a}}{L_{a}} - \frac{K_{v}}{L_{a}} \\ \frac{K_{m}}{J_{i}} - \frac{β_{0}}{J_{i}} \end{matrix}], B_{0} = [\begin{matrix} \frac{1}{L_{a}} \\ 0 \end{matrix}], C_{0} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],

where $i_{a}$ , $ω$ , and $υ_{a}$ denote the armature current, the angular velocity, and the armature voltage, respectively. $R_{a}$ is the armature resistance and $L_{a}$ is the inductance. $K_{v}$ and $K_{m}$ are the voltage and motor constants, respectively.

Taken from Lan and Patton (2016), the parameters of the DC motor are assumed to be: $R_{a} = 1.2$ , $L_{a} = 0.05$ , $K_{v} = 0.6$ , $K_{m} = 0.6$ , $J_{i} = 0.1352$ , and $β_{0} = 0.3$ . Consider the offset fault of the armature current sensor as the sensor fault, and the voltage sensor gain fault of $v_{a}$ as the actuator fault. Therefore by considering the external disturbance $w (t)$ , the model (58) now becomes

\begin{matrix} \overset{\cdot}{x} (t) = A_{0} x (t) + B_{0} u (t) + F_{0} f (t) + D_{0} w (t), \\ y (t) = C_{0} x (t) + G_{0} f (t), \end{matrix}

(59)

where

F_{0} = [\begin{matrix} \frac{1}{10 L_{a}} \\ 0 \end{matrix}], G_{0} = [\begin{matrix} - 1 \\ 2 \end{matrix}], D_{0} = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}] .

(60)

The external disturbance vector $w (t)$ is chosen as band-limited white noise with power 0.1 and the fault $f (t)$ is given by

f (t) = {\begin{matrix} 0, & 0 \leq t \leq 7, \\ 2, & 10 < t \leq 15, \\ 0, & 15 < t \leq 20 . \end{matrix}

The parameters $d_{m}, d_{M}, τ_{m}$ , and $σ_{m}$ are selected as $d_{m} = 0.01 second$ , $d_{M} = 0.03 second$ , $τ_{m} = 0.05 second$ , and $σ_{m} = 0.15$ . For given $α = 0.3$ , $c_{0} = 2$ , $r_{1} = 1$ , $r_{2} = 4$ , $t_{s} = 2$ , and $T_{f} = 20$ , we solve (51) to (55) in Theorem 3 and using equations (6), the following solutions are obtained:

\begin{matrix} D = [\begin{matrix} 0.7200 & - 1.0900 \\ - 0.0200 & 1.5900 \\ 1.0800 & 0.3400 \end{matrix}], \\ N = [\begin{matrix} - 2.0257 & 0.6469 & - 0.7243 \\ 1.2679 & - 3.1251 & 0.9875 \\ 0.9965 & - 0.9491 & - 1.8728 \end{matrix}], \\ L = [\begin{matrix} - 19.8636 & - 9.2118 \\ 5.4943 & 2.7271 \\ - 23.1250 & - 10.4825 \end{matrix}], \\ T = [\begin{matrix} - 14.4 \\ - 0.4 \\ 21.6 \end{matrix}], H = [\begin{matrix} 0.5410 & 0.2705 \\ - 0.1448 & - 0.0724 \\ - 1.1624 & - 0.0812 \end{matrix}], \\ K = [\begin{matrix} - 0.0174 & 0.1382 & - 0.1000 \end{matrix}] . \end{matrix}

In addition, the minimum allowable value of $γ$ is determined to be as $γ_{\min} = 0.3$ .

In simulations, the initial conditions are set as $x (0) = [0.5 0.5]^{T}$ and $z (0) = [0.5 0.01 - 0.5]^{T}$ . The value of $\hat{\bar{x}} (0)$ is calculated using the equation $\hat{\bar{x}} (0) = z (0) + H \bar{C} \bar{x} (0)$ .

Figure 3 illustrates the comparison of the state trajectories of the closed-loop system using the proposed method with those obtained from the asymptotic stability method and the FTS method presented in Li et al. (2020a). The results clearly demonstrate that our proposed approach achieves FTCS for the closed-loop system trajectories during the specified time when $w (t) = f (t) = 0$ , while with the other two methods, it may take longer time to achieve the equilibrium point. It is important to note that the FTS method addresses only the “boundedness” characteristic of a dynamical system, which means that the system trajectories will remain within a specified threshold, which is greater than the initial state bound for a defined time interval and there is no guarantee for the contraction of the system trajectories. Suppose that $w (t)$ and $f (t)$ occur. Figure 4 depicts the estimation performance. It can be seen that the proposed method can estimate the actual system states and faults successfully. Furthermore, the trajectories of the closed-loop system states are illustrated in Figure 5 and it is obvious that under event-triggered condition (15) and in the presence of network-induced time-varying delay, the system states are controlled to achieve FTCS in the fault-tolerant framework and in the presence of the external disturbance, the proposed fault-tolerant $H_{\infty}$ approach can recover disturbance effect and $γ_{\min} = 0.3$ is obtained. The ratio $\int_{0}^{T_{f}} y_{c}^{T} (s) y_{c} (s) ds / \int_{0}^{T_{f}} w^{T} (s) w (s) ds$ is depicted in Figure 6. From this figure, one can see that $\int_{0}^{T_{f}} y_{c}^{T} (s) y_{c} (s) ds / \int_{0}^{T_{f}} w^{T} (s) w (s) ds < 0.08$ and therefore, the $H_{\infty}$ disturbance attenuation level ( $\sqrt{0.08} = 0.2828$ ) is less than the required $γ = 0.3$ . Figure 7 plots the released instants throughout the simulation duration. It is noteworthy that only $476$ instants $(26.14 %)$ of the total $1821$ sampled instants are transmitted to the controller. This reduction in data transmission highlights the efficiency of the proposed method compared to the traditional time-triggered method. Furthermore, Figure 8 shows the histogram of the released intervals. This demonstrates that the significant amount of released intervals in Figure 8 is more than $0.01 second$ , while in the periodic time-triggered scheme, all of the sampled instants are transmitted with $0.01 second$ released interval. This means that the proposed method in this paper can considerably reduce network consumption. Also, the variation of threshold $σ (t_{k})$ is indicated in Figure 9. It is clear that as $σ$ increases, the amount of released instants decreases, which helps to optimize the use of communication resources. For the specified values of $α = 0.3$ , $t_{s} = 2$ and $T_{f} = 20$ , the number of released instants and $γ_{\min}$ are compared for different values of $σ$ in Table 1. It reveals that as $σ$ increases, the number of released instants decreases, and $γ_{\min}$ increases. Furthermore, Table 2 lists the minimum allowable of $t_{s}$ for various values of $α$ . From this table, one can get that as $α$ increases, the value of $t_{s_{\min}}$ decreases. Therefore, it is concluded that the proposed method is capable of estimating and tolerating the faults in the finite-time interval and also, the $H_{\infty}$ performance is satisfied, which in turn, validates the theoretical results.

Table 1.

The number of released instants and $γ_{\min}$ for different values of $σ$

$σ$	0	0.15	0.3
The number of released instants	1821	476	241
$γ_{\min}$	0.25	0.3	0.98

Table 2.

The minimum value of $t_{s}$ for different values of $α$ .

$α$	0.1	0.3	0.5
${t_{s}}_{\min} (seconds)$	5	2	1.3

Conclusion

The problem of FTCS analysis and $H_{\infty}$ FTC of linear NCSs subject to time-varying delay was studied in this paper. For the sake of data transmission reduction, an aperiodic event-triggered scheme is used, in which the interval between two sampling instants varies within a certain known bound, and the event threshold is adjusted by using the adaptive rule. The states of the system and faults were estimated simultaneously by using a UIO. Using the proposed Lyapunov–Krasovskii functional, sufficient delay-dependent conditions were derived in the form of LMIs. Since the obtained conditions were difficult to solve with LMI toolboxes, the singular value decomposition method was applied to convert the conditions to LMIs. Then, the observer-based $H_{\infty}$ fault-tolerant controller was designed to ensure the FTCS of the faulty linear NCS while achieving the $H_{\infty}$ performance. Simulation results illustrated the effectiveness of the proposed method. Finally, it should be pointed out that in this paper, FTCS can be guaranteed for NCSs subject to time-varying delay. While, from a practical viewpoint and considering the unreliable network channel, the influence of other networked-induced imperfections such as data packet dropout and cyber attacks shall be taken into consideration to reduce the conservatism. Furthermore, the efficacy of the proposed method in the presence of input saturation shall be carefully studied. Consequently, the mentioned points may evoke interesting topics for further works.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Farzaneh Jani

Hamed Kharrati

References

Abbaspour

Mokhtari

Sargolzaei

, et al. (2020) A survey on active fault-tolerant control systems. Electronics 9: 1513.

Atitallah

Davoodi

Meskin

(2018) Event-triggered fault detection for networked control systems subject to packet dropout. Asian Journal of Control 20: 2195–2206.

Chen

Meng

Sun

(2016) Stability analysis of networked control systems with aperiodic sampling and time-varying delay. IEEE Transactions on Cybernetics 47: 2312–2320.

Chu

(2018) Event-triggered fault estimation and sliding mode fault-tolerant control for a class of nonlinear networked control systems. Journal of the Franklin Institute 355: 5475–5502.

Corless

(1998) State and input estimation for a class of uncertain systems. Automatica 34: 757–764.

Darouach

Boutat-Baddas

Zerrougui

(2011)

H_{\infty}

observers design for a class of nonlinear singular systems. Automatica 47: 2517–2525.

Deng

Yin

(2021) Event-triggered predictive control for networked control systems with DoS attacks. Information Sciences 542: 71–91.

El Abbadi

Jamouli

(2023) Fault detection of a networked control system and its application to a DC motor. International Journal of Control, Automation and Systems 21: 1769–1779.

Fang

Ding

Liu

, et al. (2021) Fault tolerant sampled-data h? Control for networked control systems with probabilistic time-varying delay. Information Sciences 544: 395–414.

10.

Halder

Bose

Gupta

(2019) Stability and performance analysis of networked control systems: A lifted sample-time approach with

L_{2}

induced norm. ISA Transactions 86: 62–72.

11.

Wang

Qin

, et al. (2016) Active fault-tolerant control for an internet-based networked three-tank system. IEEE Transactions on Control Systems Technology 24: 2150–2157.

12.

Huang

Pan

(2020) Fault detection for nonlinear networked control systems with sensor saturation and random faults. IEEE Access 8: 92541–92551.

13.

Jani

Hashemzadeh

Baradarannia

, et al. (2022) Robust finite-time contractive fault tolerant control of uncertain nonlinear network-based systems with adaptive event-triggered communication scheme. Iranian Journal of Science and Technology, Transactions of Electrical Engineering 46: 141–155.

14.

Jani

Hashemzadeh

Baradarannia

, et al. (2023) Robust event-triggered finite-time control of faulty networked flexible manipulator under external disturbance. Journal of Vibration and Control 29: 317–333.

15.

Jin

, et al. (2019) Event-triggered fault detection and diagnosis for networked systems with sensor and actuator faults. IEEE Access 7: 95857–95866.

16.

Kargar

Zarei

Razavi-Far

(2019) Robust fault detection filter design for nonlinear networked control systems with time-varying delays and packet dropout. Circuits, Systems, and Signal Processing 38: 63–84.

17.

Lan

Patton

(2015) Integrated design of robust fault estimation and fault-tolerant control for linear systems. In: 2015 54th IEEE conference on decision and control (CDC), 15 December 2015, pp. 5105–5110. Osaka, Japan: IEEE.

18.

Lan

Patton

(2016) A new strategy for integration of fault estimation within fault-tolerant control. Automatica 69:48–59.

19.

Lan

Patton

(2017) Integrated fault estimation and fault-tolerant control for uncertain Lipschitz nonlinear systems. International Journal of Robust and Nonlinear Control 27: 761–780.

20.

Lee

Huang

Kung

(2019) Faults estimation and decentralized control for an integrated networked control system with faulty actuators. In: 2019 IEEE 6th international conference on industrial engineering and applications (ICIEA), 12 April 2019, pp. 12–16. Tokyo, Japan: IEEE.

21.

Peng

(2022) Event-triggered-based adaptive sliding mode control for networked linear control systems. Transactions of the Institute of Measurement and Control 44: 2024–2036.

22.

(2020a) Event-triggered finite-time bounded and finite-time stability for networked control systems under DoS attacks. International Journal of Systems Science 51: 2820–2836.

23.

Yang

, et al. (2020b) Passivity-based event-triggered fault tolerant control for nonlinear networked control system with actuator failures and DoS jamming attacks. Journal of the Franklin Institute 357: 9288–9307.

24.

Tang

, et al. (2020c) Event-based fault-tolerant control for networked control systems applied to aircraft engine system. Information Sciences 512: 1063–1077.

25.

Wang

Zhai

, et al. (2020d) Event-triggered observer-based robust

H_{\infty}

control for networked control systems with unknown disturbance. International Journal of Robust and Nonlinear Control 30: 2671–2688.

26.

Liu

Qiu

Cui

, et al. (2016a) Fault–tolerant

H_{\infty}

control for networked control systems with randomly occurring missing measurements. Neurocomputing 175: 459–465.

27.

Liu

, et al. (2016b) On sliding mode control for networked control systems with semi-Markovian switching and random sensor delays. Information Sciences 337: 44–58.

28.

Qian

Peng

Yang

(2016) Reduced-order observer-based fault estimation and fault-tolerant control for a class of discrete Lipschitz nonlinear systems. Optimal Control Applications and Methods 37: 1236–1262.

29.

Qiu

Wen

, et al. (2018) Self-triggered fault estimation and fault tolerant control for networked control systems. Neurocomputing 272: 629–637.

30.

Ren

Zong

(2018) Event-triggered finite-time control for networked switched linear systems with asynchronous switching. IEEE Transactions on Systems, Man, and Cybernetics: Systems 48: 1874–1884.

31.

Shi

Liu

Zhang

(2015) Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements. IEEE Transactions on Industrial Electronics 62: 5910–5918.

32.

Wang

Chang

, et al. (2021) Event-triggered sliding mode control of networked control systems with Markovian jump parameters. Automatica 125: 109405.

33.

Tan

Zhang

(2015) Stabilization of networked control systems with both network-induced delay and packet dropout. Automatica 59: 194–199.

34.

Wang

Fei

Wang

, et al. (2019) Event-triggered fault estimation and fault-tolerant control for networked control systems. Journal of the Franklin Institute 356: 4420–4441.

35.

Wang

Hua

Qiu

(2023) Robust stability and

H_{\infty}

control for networked control systems with transmission delay and its application to 2 dof laboratory helicopter. Journal of the Franklin Institute 360: 2827–2847.

36.

Wang

Han

(2016) Fault detection filter design for data reconstruction-based continuous-time networked control systems. Information Sciences 328: 577–594.

37.

Weiss

Infante

(1967) Finite time stability under perturbing forces and on product spaces. IEEE Transactions on Automatic Control 12: 54–59.

38.

Lam

, et al. (2016) Fault detection for nonlinear networked systems based on quantization and dropout compensation: An interval type-2 fuzzy-model method. Neurocomputing 191: 409–420.

39.

Xie

(2020) Stabilisation of networked control systems under a novel stochastic-sampling-based adaptive event-triggered scheme. IET Control Theory & Applications 14: 1158–1169.

40.

Yan

Shen

Nguang

, et al. (2019) Event-triggered

H_{\infty}

control of networked control systems with distributed transmission delay. IEEE Transactions on Automatic Control 65: 4295–4301.

41.

You

Wang

, et al. (2015) Robust fast adaptive fault estimation for systems with time-varying interval delay. Journal of the Franklin Institute 352: 5486–5513.

42.

Zhang

Shi

Wang

, et al. (2017) Analysis and synthesis of networked control systems: A survey of recent advances and challenges. ISA Transactions 66: 376–392.

43.

Zhang

Pang

Zhou

, et al. (2016a) Active fault tolerant control for networked control systems with actuator fault. In: 2016 35th Chinese control conference (CCC), 27 July 2016, pp. 7521–7525. Chengdu, China: IEEE.

44.

Zhang

Chen

, et al. (2016b) Fault tolerant control for uncertain networked control systems with induced delays and actuator saturation. IEEE Access 4: 6574–6584.

45.

Zhang

Nguang

Yan

(2021) Event-triggered

H_{\infty}

control for networked control systems under denial-of-service attacks. Transactions of the Institute of Measurement and Control 43: 1077–1087.

46.

Zhang

Han

, et al. (2019) Networked control systems: a survey of trends and techniques. IEEE/CAA Journal of Automatica Sinica 7: 1–17.

47.

Zheng

Zhang

Lam

, et al. (2022a) LMIs-based stability analysis and fuzzy-logic controller design for networked systems with sector nonlinearities: Application in tunnel diode circuit. Expert Systems with Applications 198: 116627.

48.

Zheng

Zhang

Sun

, et al. (2022b) Robust stability analysis and feedback control for networked control systems with additive uncertainties and signal communication delay via matrices transformation information method. Information Sciences 582: 258–286.