Observer-based event-triggered control for continuous-time switched linear parameter-varying systems

Abstract

This paper researches the interval observer–based event-triggered control of switched linear parameter-varying (LPV) systems, which involve unknown external disturbance and measurement noise. First, we offer an event-triggering condition and a controller which are correlated with state estimations. Then, by analyzing the relationship between switching sequences and sampling sequences, we model the resulting systems as switched time-delay systems. It deserves to note that the mismatches of the switching signal for the controllers and the systems may occur due to the sampling of switching signal. Using the merging switching signal technique and the multiple Lyapunov–Krasovskii functional method, and thinking over both synchronous and asynchronous switching of the systems and the controllers, we acquire relevant conditions which are used to realize the stability of the whole switched LPV closed-loop systems under the mode-dependent average dwell time (MDADT) switching. Next, we give a specific algorithm to design the controller gains. Finally, a numerical example is provided to verify the superiority of the outcomes.

Keywords

Switched linear parameter-varying systems asynchronous switching event-triggered control stability

Introduction

Switched systems can not only solve the fundamental issues in practice, but also enhance the temporary features of systems, so many scholars are absorbed in studying the relevant problems of switched systems (Aravind and Balasubramaniam, 2022; Huang et al., 2020; Lin and Chen, 2021; Sun et al., 2004; Wang et al., 2016, 2021, 2022; Yang et al., 2022; Zhao and Hill, 2008). However, most papers are about linear time invariant (LTI) systems, such as Ma et al. (2017) and Yuan and Wu (2015). When the more complex nonlinear systems are studied, many scholars adopt the following method, that is, convert these systems into linear form based on the linear parameter-varying (LPV) transformation of nonlinear systems. Through such transformation, we can use the research method of LTI systems to study complex nonlinear systems, which also expands the application field of linear systems. In addition, in any case, we want to make the system models as close to the actual situation as possible. Therefore, in order to better deal with the changing parameters and external factors in the actual situation, these systems are modeled as LPV system models. From this point of view, LPV systems have certain advantages over LTI systems. Different from general linear systems, the state matrices of LPV systems involve parametric variables, which are related to time. Therefore, LPV systems are relatively complex. Recently, switched LPV systems have received attention and scholars have achieved some results (Lacerda and Agulhari, 2020; Li et al., 2019; Rios et al., 2015; Yang and Zhao, 2018; Zong et al., 2022).

In order to adapt to the rapid development of communication and other fields, network control has attracted extensive attention of scholars; it has the benefits of diverse functions and low costs. But at the same time, it also has defects, such as restricted communication bandwidth and invalid energy expenditure. For the sake of solving above problems, Dorf et al. (1962) proposed the event-triggered sampled-data control of adaptive systems, which can decrease the load of network transducers. Therefore, scholars have turned their attention to the study of event-triggering (Ai et al., 2022; Chen et al., 2022). In a broader sense, event-triggered control (ETC) schemes can be divided into continuous and periodic types (Liu, 2016). Nevertheless, the former have a shortcoming that the Zeno behavior is difficult to be eliminated, and from a practical point of view, it is not easy to be implemented. Periodic ETC schemes can make up for the deficiency of the former, so they have greater research significance (Xiang and Johnson, 2017). However, it is challenging to meet the condition that the switching between subsystems occurs simultaneously with sampling, so the conclusions in Xiang and Johnson (2017) are more conservative. Judging from this, the asynchronism between controllers and systems is worth considering for the related problems of ETC. Xiao et al. (2019) discussed the asynchronous problem for switched linear systems based on the ETC schemes. In Ren et al. (2018), the asynchronous case of controller and switching signal was studied through the average dwell time method. At present, the research on event-triggering of LPV systems is in its infancy, so there are few relevant results. De Souza et al. (2021) explored the ETC of discrete-time LPV systems with saturating actuators by utilizing two event-triggered schemes which are not related. In Zhu et al. (2018), two cases of ETC for switched LPV systems were considered with regard to whether to jointly design the event-triggered mechanism and controllers.

It should be noted that a better study of systems is inseparable from the acquisition of state information, which can often not be measured directly. To solve this problem, scholars put forward the concept of the state observer and studied it (De Oliveira and Pereira, 2021; Li and Xiang, 2014; Theis et al., 2020). But general state observers are unable to provide the estimation range of the states, so the interval observers came into being (Gouze et al., 2000; Ifqir et al., 2017a). To authors’ knowledge, there have not been researches on ETC based on interval observers for switched LPV systems.

Inspired by the above contents, this paper focuses on the interval observer–based ETC for switched LPV systems. First, the state estimation-based event-triggered scheme and the interval observer–based feedback controller are constructed. Second, we consider the connections between switching sequences and sampling sequences, and then model the resulting systems as switched time-delay systems. Next, by analyzing two different situations between the system switching signal and the controller switching signal (i.e. synchronization and asynchronization, which are caused by the sampling), in light of the merging switching signal and the multiple Lyapunov functional method, some sufficient conditions are given to make the switched LPV closed-loop systems stable under the mode-dependent average dwell time (MDADT) switching. Then, through the linearization procedure, we obtain the controller gains, and finally, a simulation example reflects the availability of the conclusions. The contributions of this paper can be generalized as follows. First, we design an event-triggered controller based on interval estimation, which makes full use of the state information. Then, contrasted with the continuous transmission of states to event trigger (Qi and Cao, 2018), the event-triggered scheme involved in this paper can avoid Zeno behavior as well as cutting the transmission cost. Finally, in Xiao et al. (2019), the stability conditions were reached in light of the linear properties of the function, and these conditions were subject to greater constraints. Whereas in this paper, the improved Lyapunov–Krasovskii functional is used, so the conditions obtained in this way are easier to compute.

This paper is arranged as follows. Section “Preliminaries” offers the problem formulation. Then, section “Main results” furnishes the major results. In section “Examples”, numerical examples are given and after that is section “Conclusion,” which sums up this paper.

Notations

Euclidean norm is represented as $| | \cdot | |$ . For a measurable and essentially bounded function $ℏ : R^{+} \to R^{q}$ , $∥ ℏ ∥_{[t_{1}, t_{2}]} = ess sup_{t_{1} \leq t \leq t_{2}} {| | ℏ (t) | |}$ , if $t_{2} = + \infty$ , we use $∥ ℏ ∥_{\infty}$ , and let $L_{\infty}^{q} = {ℏ | ∥ ℏ ∥_{\infty} < \infty}$ . Define $| x_{t} |_{cl} = sup_{- d \leq d (t) \leq 0} {| | x (t + d (t)) | |, | | \overset{\cdot}{x} (t + d (t)) | |}$ for known $d > 0$ . For any vectors $Y, Z \in R^{n}$ , $Y \leq Z$ means the size relationship of the corresponding position elements of the two vectors, so are the other inequality signs. $\partial_{ij}$ is the standard Kronecker function, that is, $\partial_{ij} = 0 (i \neq j)$ , $\partial_{ii} = 1$ . $E_{b}$ is a $(b \times 1)$ vector; its elements are ones.

Preliminaries

We consider the switched LPV system

\begin{matrix} \overset{\cdot}{x} (t) = A_{σ (t)} (ρ) x (t) + B_{σ (t)} (ρ) u_{σ (t)} (t) + D_{σ (t)} (ρ) ω_{σ (t)} (t) \\ y (t) = C_{σ (t)} x (t) + v_{σ (t)} (t) \end{matrix}

(1)

with

\begin{matrix} A_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) A_{ir} \\ B_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) B_{ir} \\ D_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) D_{ir} \end{matrix}

(2)

where $x (t) \in R^{n}$ is the state; $x (t_{0}) = x_{0}$ is the initial state; $y (t) \in R^{b}$ is the output; $u_{σ (t)} (t) \in R^{q}$ is the control; $ω_{σ (t)} (t) \in R^{l}$ and $v_{σ (t)} (t) \in R^{b}$ represent the disturbance and the measurement noise, respectively. $A_{σ (t)} (ρ)$ , $B_{σ (t)} (ρ)$ , and $D_{σ (t)} (ρ)$ are the matrices scheduled by measurable parameter vector $ρ (t) \in R^{n_{a}}$ ; s is the amount of vertices of polyhedron set $N$ that bounds $ρ (t)$ and the polyhedral coordinates $g_{r} (ρ)$ , $r \in J = {1, 2, \dots, s}$ conform to $\sum_{r = 1}^{s} g_{r} (ρ) = 1, g_{r} (ρ) \geq 0, r \in J$ ; $A_{ir}$ , $B_{ir}$ , $D_{ir} \geq 0$ , and $C_{i}$ are the constant matrices with known proper dimensions. In any limited interval, the system (1) has only a finite number of switches, and the state is continuous all along. $σ (t) : [0, \infty) \to O = {1, 2, \dots, o}$ is a piecewise constant function of time t and o is the quantity of subsystems. $σ (t)$ has the switching sequence ${(l_{0}, t_{0}), . . ., (l_{i}, t_{i}), . . ., | l_{i} \in O, i \in N}$ , which means running the $l_{i} th$ subsystem for $\forall t \in [t_{i}, t_{i + 1}), i \in N$ .

Here are some lemmas, definitions, and assumptions that will be used later.

Assumption 1. Some variable vectors about the system are bounded: $\underline{x} (0) \leq x (0) \leq \bar{x} (0)$ , $\underline{ω} \leq ω_{i} (t) \leq \bar{ω}$ , $| v_{i} (t) | \leq \bar{v} E_{b}$ , where $\underline{x} (0)$ , $\bar{x} (0)$ , $\underline{ω}$ , $\bar{ω}$ are the known vectors and $\bar{v} > 0$ is a scalar.

Remark 1. Assumption 1 is about initial state, external disturbance, and measurement noise, and their upper and lower bounds are known, which is reasonable in interval estimation theory. In specific cases, these values correspond to theoretical or actual boundary values (Gouze et al., 2000; Ifqir et al., 2017b).

Definition 1. For $0 \leq t \leq T$ , let $N_{σ i} (t, T)$ represent the number of switches of $σ (t)$ caused by activating the i th subsystem in interval $(t, T)$ So, $N_{σ} (t, T) = \sum_{i = 1}^{o} N_{σ i} (t, T)$ . For $τ_{ai} > 0$ and $N_{0 i} \geq 0$ , if expression $N_{σ i} (t, T) \leq N_{0 i} + T_{i} (t, T) / τ_{ai}$ is right, then $τ_{ai}$ is called MDADT and $N_{0 i}$ is known as mode-dependent chattering bound. These switching signals are marked as $S [τ_{ai}, N_{0 i}]$ (Liberzon, 2003).

Lemma 1. If there is a scalar $ς > 0$ and a positive definite matrix $A \in R^{a \times a}$ , then $2 x^{T} y \leq 1 / ς x^{T} Ax + ς y^{T} A^{- 1} y, x, y \in R^{a}$ (Ethabet et al., 2018).

Lemma 2. For system $\overset{\cdot}{x} = A (ρ) x + B (ρ) u + D (ρ) ω (t)$ , $x \in R^{n}$ , $ω (t) \in R^{l}$ , $u \in R^{q}$ with a Metzler matrix $A (ρ) \in R^{n \times n}$ . Any solution of it is element-wise nonnegative for all $t \geq t_{0}$ , if $x (t_{0}) \geq 0$ , $u \geq 0$ , $B (ρ) \geq 0$ , $D (ρ) \geq 0$ and $ω (t) \geq 0$ , $\forall ρ (t) \in R^{n_{a}}$ (Ifqir et al., 2017b).

Lemma 3. Assume K is a positive definite matrix, U is a symmetric matrix, and $λ > 0$ is a scalar, then $- U^{T} KU \leq λ^{2} K^{- 1} - 2 λ U$ (Qi et al., 2020).

Main results

Interval observers

Assumption 2. There are gain matrices $L_{ir}$ such that the matrices $A_{ir} - L_{ir} C_{i} (\forall i \in O, r \in J)$ are Metzler. In addition, interval observers follow the same switching rules as the system (1).

Theorem 1. Under the premise of Assumptions 1 and 2, if

Θ_{i} A_{ir} - W_{ir} C_{i} + A_{ir}^{T} Θ_{i} - C_{i}^{T} W_{ir}^{T} + \frac{1}{ζ_{i}} Θ_{i} < 0

(3)

with matrices $Θ_{i} > 0$ , $W_{ir}$ , and scalars $ζ_{i} > 0, i \in O, r \in J$ , then in term of the subsystem i, the interval observer is given below

\begin{matrix} \overset{\cdot}{\bar{x}} = [A_{i} (ρ) - L_{i} (ρ) C_{i}] \bar{x} + B_{i} (ρ) u_{i} + L_{i} (ρ) y \\ + | L_{i} (ρ) | \bar{v} E_{b} + D_{i} (ρ) \bar{ω} \\ \overset{\cdot}{\underline{x}} = [A_{i} (ρ) - L_{i} (ρ) C_{i}] \underline{x} + B_{i} (ρ) u_{i} + L_{i} (ρ) y \\ - | L_{i} (ρ) | \bar{v} E_{b} + D_{i} (ρ) \underline{ω} \end{matrix}

(4)

with

L_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) L_{ir}

(5)

In addition, the observer gains $L_{ir}$ are presented as $L_{ir} = Θ_{i}^{- 1} W_{ir}$ .

Proof. By analogy with the evidence of Theorem 1 and Theorem 2 in He and Xie (2015), we can obtain the outcome here.□

Controllers

In subsequent process, the algorithmic rule as described below was used to decide whether the latest sampling messages can be transmitted

\begin{matrix} ℸ_{r_{d + 1}} = min {{\hat{ℸ}}_{r_{d + 1}}, ℸ_{r_{d}} + [\frac{τ_{d}}{d}] d}, k \geq 0 \\ {\hat{ℸ}}_{r_{d + 1}} = min_{ℸ_{j} > ℸ_{r_{d}}} {ℸ_{j} | \tilde{φ} (\tilde{e} (ℸ_{j}), \tilde{x} (ℸ_{r_{d}})) \geq 0} \end{matrix}

(6)

where $\tilde{φ} (\tilde{e} (ℸ_{j}), \tilde{x} (ℸ_{r_{d}})) = {\tilde{e}}^{T} (ℸ_{j}) {\tilde{℧}}_{σ (ℸ_{r_{d}})} \tilde{e} (ℸ_{j}) - {\tilde{ξ}}_{σ (ℸ_{r_{d}})} {\tilde{x}}^{T} (ℸ_{r_{d}}) {\tilde{℧}}_{σ (ℸ_{r_{d}})} \tilde{x} (ℸ_{r_{d}})$ , $0 < {\tilde{ξ}}_{σ (ℸ_{r_{d}})} < 1$ , ${\tilde{℧}}_{σ (ℸ_{r_{d}})} = diag {{\bar{℧}}_{σ (ℸ_{r_{d}})}, {\underline{℧}}_{σ (ℸ_{r_{d}})}}$

\tilde{x} (ℸ_{r_{d}}) = [\begin{matrix} \bar{x} (ℸ_{r_{d}}) \\ x (ℸ_{r_{d}}) \end{matrix}], \tilde{e} (ℸ_{j}) = [\begin{matrix} \tilde{\bar{e}} (ℸ_{j}) \\ \underline{\tilde{e}} (ℸ_{j}) \end{matrix}] = [\begin{matrix} \bar{x} (ℸ_{j}) - \bar{x} (ℸ_{r_{d}}) \\ \underline{x} (ℸ_{j}) - \underline{x} (ℸ_{r_{d}}) \end{matrix}]

$τ_{d}$ is the dwell time, $ℸ_{r_{d}}$ is the event-triggered transmission moment, and $0 < d \leq τ_{d}$ is the sampling period.

Remark 2. It is easy to know from equation (6) that there is at least one sample in the successive event-triggered interval, which shows that the next event-triggered time has a lower bound from the previous event-triggered time, so the Zeno behavior is avoided.

Next, based on equations (4) and (6), we establish the controller below for follow-up research

u_{σ (ℸ_{r_{d}})} = {\bar{K}}_{σ (ℸ_{r_{d}})} \bar{x} (ℸ_{r_{d}}) + {\underline{K}}_{σ (ℸ_{r_{d}})} \underline{x} (ℸ_{r_{d}}), t \in [ℸ_{r_{d}}, ℸ_{r_{d + 1}})

(7)

where ${\bar{K}}_{σ (ℸ_{r_{d}})}$ and ${\underline{K}}_{σ (ℸ_{r_{d}})}$ are the feedback control gains.

For the convenience of analysis, we separate the interval into smaller intervals for discussion, specifically, let $[ℸ_{r_{d}}, ℸ_{r_{d + 1}}) = ⋃_{l_{d} = 0}^{g_{d} - 1} [ℸ_{p_{l}}, ℸ_{p_{l} + 1})$ with $p_{l} = r_{d} + l_{d}$ , $l_{d} = 0, 1, \dots, g_{d} - 1$ , in which $g_{d} = r_{d + 1} - r_{d}$ . We define the delay $d (t) = t - ℸ_{p_{l}}$ , $t \in [ℸ_{p_{l}}, ℸ_{p_{l} + 1})$ , which is caused by sampling and implies $d (t) \in [0, d)$ . Besides, let $ℓ (t) = t - ℸ_{r_{d}}$ , $t \in [ℸ_{r_{d}}, ℸ_{r_{d + 1}})$ , then it infers $ℓ (t) \in [0, τ_{d})$ . Set $\tilde{e} (ℸ_{p_{l}}) = \tilde{x} (ℸ_{p_{l}}) - \tilde{x} (ℸ_{r_{d}}) = \tilde{x} (t - d (t)) - \tilde{x} (ℸ_{r_{d}})$ , $σ' (t) = σ (ℸ_{r_{d}})$ , and $\tilde{σ} (t) = (σ (t), σ' (t))$ .

Stability synthesis

Lemma 4. Let $σ (t) \in S [τ_{ai}, N_{0 i}]$ and $σ' (t) = σ (t - ℓ (t))$ with $ℓ (t) \in [0, τ_{d})$ . Then, the relational expressions listed below hold, $σ' (t) \in S [τ_{ai}, N_{0 i} + τ_{d} / τ_{ai}]$ , $\tilde{σ} (t) = (σ (t), σ' (t)) \in S [τ_{ai} / 2, 2 N_{0 i} + τ_{d} / τ_{ai}]$ (Vu and Morgansen, 2010; Wang et al., 2020).

Lemma 5. If $σ (t) \in S [τ_{ai}, N_{0 i}]$ and $σ' (t) = σ (t - ℓ (t))$ , $0 \leq ℓ (t) \leq τ_{d}, \forall t \geq 0$ . Let $T_{si} (t_{0}, t), i \in O$ express the operation time of the i th subsystem with the i th controller over $(t_{0}, t)$ , $T_{i} (t_{0}, t)$ is the operation time of the i th subsystem in the interval $(t_{0}, t)$ , and then mark $T_{ui} (t_{0}, t) = T_{i} (t_{0}, t) - T_{si} (t_{0}, t)$ . For $α > 0, β > 0$ and $\bar{α} \in [0, α]$ , if

τ_{d} (α + β) \leq (α - \bar{α}) τ_{ai}

(8)

then

- α T_{si} (t_{0}, t) + β T_{ui} (t_{0}, t) \leq c_{i} - \bar{α} T_{i} (t_{0}, t), \forall t \geq t_{0} \geq 0

(9)

where $c_{i} = (α + β) N_{0 i} τ_{d}$ (Wang et al., 2020).

We will prove the conclusion below.

Theorem 2. For given matrices $L_{ir}$ , $i \in O$ , $r \in J$ , and known constants $d > 0$ , $ℶ_{s} > 0$ , $ℶ_{a} > 0$ and $μ_{i} \geq 1$ , $0 < {\tilde{ξ}}_{j} < 1 (i, j \in O)$ , if the existence of matrices $P_{ij} > 0$ , $Q_{ij} > 0$ , $R_{ij} > 0$ , and ${\tilde{℧}}_{j} > 0$ , ${\bar{K}}_{j}$ and ${\underline{K}}_{j}$ make the following conditions true for $\forall i, j \in O$ , $r \in J$

P_{ij} \leq μ_{i} P_{jj}, e^{(ℶ_{s} + ℶ_{a}) d} Q_{ij} \leq μ_{i} Q_{jj}, e^{(ℶ_{s} + ℶ_{a}) d} R_{ij} \leq μ_{i} R_{jj}

(10)

P_{ii} \leq μ_{i} P_{ij}, R_{ii} \leq μ_{i} R_{ij}

(11)

[\begin{matrix} Ψ_{ijr} & P_{ij} B_{ir} K_{j} F + Λ_{ij} R_{ij} & - P_{ij} B_{ir} K_{j} & {\overset{⌣}{A}}_{ir}^{T} \\ * & - Λ_{ij} R_{ij} + {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} F & - {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} & F^{T} K_{j}^{T} B_{ir}^{T} \\ * & * & ({\tilde{ξ}}_{j} - 1) {\tilde{℧}}_{j} & - K_{j}^{T} B_{ir}^{T} \\ * & * & * & - \frac{1}{2 d} R_{ij}^{- 1} \end{matrix}] < 0

(12)

where

\begin{matrix} Ψ_{ijr} = {\overset{⌣}{A}}_{ir}^{T} P_{ij} + P_{ij} {\overset{⌣}{A}}_{ir} + Q_{ij} + ℶ_{ij} P_{ij} - Λ_{ij} R_{ij} + P_{ij} \\ P_{ij} = diag {P_{ij}, P_{ij}, P_{ij}, P_{ij}}, Q_{ij} = diag {Q_{ij}, Q_{ij}, Q_{ij}, Q_{ij}} \\ R_{ij} = diag {R_{ij}, R_{ij}, R_{ij}, R_{ij}}, ℶ_{ij} = \partial_{ij} ℶ_{s} + (\partial_{ij} - 1) ℶ_{a} \\ B_{ir} = diag {B_{ir}, B_{ir}, 0, 0}, Λ_{ij} = {\begin{matrix} \frac{1}{d}, & i \neq j \\ \frac{e^{- ℶ_{s} d}}{d}, & i = j \end{matrix} \\ F = [\begin{matrix} I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \end{matrix}], K_{j}^{T} = [\begin{matrix} {\hat{K}}_{j}^{T} & 0 \end{matrix}], {\hat{K}}_{j} = [\begin{matrix} {\bar{K}}_{j} & {\underline{K}}_{j} \\ {\bar{K}}_{j} & {\underline{K}}_{j} \end{matrix}] \\ {\overset{⌣}{A}}_{ir} = [\begin{matrix} {\tilde{A}}_{ir} & {\tilde{C}}_{ir} \\ 0 & {\bar{A}}_{ir} \end{matrix}], {\tilde{A}}_{ir} = [\begin{matrix} A_{ir} & 0 \\ 0 & A_{ir} \end{matrix}] \\ {\bar{A}}_{ir} = [\begin{matrix} A_{ir} - L_{ir} C_{i} & 0 \\ 0 & A_{ir} - L_{ir} C_{i} \end{matrix}], {\tilde{C}}_{ir} = [\begin{matrix} - L_{ir} C_{i} & 0 \\ 0 & L_{ir} C_{i} \end{matrix}] \end{matrix}

(13)

Then, for arbitrary switching signal whose MDADT meets the following inequality

τ_{ai} > τ_{ai}^{★} = \frac{4 \ln μ_{i} + τ_{d} (ℶ_{s} + ℶ_{a})}{ℶ_{s}}

(14)

the system (1) with the controller (7) is input-to-state stable (ISS).

Proof. We adopt the following three parts to certify this result.

Part 1. Let $\bar{e} (t) = \bar{x} (t) - x (t)$ , $\underline{e} (t) = x (t) - \underline{x} (t)$ and consider equation (4) with controller (7), it holds

\overset{\cdot}{\tilde{x}} = {\tilde{A}}_{σ} (ρ) \tilde{x} + {\tilde{C}}_{σ} (ρ) e + {\tilde{B}}_{\tilde{σ}} (ρ) \tilde{x} (ℸ_{r_{d}}) + {\tilde{ν}}_{σ}, t \in [ℸ_{p_{l}}, ℸ_{p_{l} + 1})

(15)

where

\begin{matrix} {\tilde{A}}_{σ} (ρ) = [\begin{matrix} A_{σ} (ρ) & 0 \\ 0 & A_{σ} (ρ) \end{matrix}], e = [\begin{matrix} \bar{e} \\ \underline{e} \end{matrix}] \\ {\tilde{B}}_{\tilde{σ}} (ρ) = [\begin{matrix} B_{σ} (ρ) {\bar{K}}_{σ'} & B_{σ} (ρ) {\underline{K}}_{σ'} \\ B_{σ} (ρ) {\bar{K}}_{σ'} & B_{σ} (ρ) {\underline{K}}_{σ'} \end{matrix}] \\ {\tilde{ν}}_{σ} = [\begin{matrix} L_{σ} (ρ) v_{σ} + | L_{σ} (ρ) | \bar{v} E_{b} + D_{σ} (ρ) \bar{ω} \\ L_{σ} (ρ) v_{σ} - | L_{σ} (ρ) | \bar{v} E_{b} + D_{σ} (ρ) \underline{ω} \end{matrix}] \\ {\tilde{C}}_{σ} (ρ) = [\begin{matrix} - L_{σ} (ρ) C_{σ} & 0 \\ 0 & L_{σ} (ρ) C_{σ} \end{matrix}] \end{matrix}

Based on the aforementioned contents, we can also get error systems

\overset{\cdot}{e} = {\bar{A}}_{σ} (ρ) e + {\bar{v}}_{σ}

(16)

where

\begin{matrix} {\bar{A}}_{σ} (ρ) = [\begin{matrix} A_{σ} (ρ) - L_{σ} (ρ) C_{σ} & 0 \\ 0 & A_{σ} (ρ) - L_{σ} (ρ) C_{σ} \end{matrix}] \\ {\bar{v}}_{σ} = [\begin{matrix} L_{σ} (ρ) v_{σ} + | L_{σ} (ρ) | \bar{v} E_{b} + D_{σ} (ρ) \bar{ω} - D_{σ} (ρ) ω_{σ} \\ - L_{σ} (ρ) v_{σ} + | L_{σ} (ρ) | \bar{v} E_{b} + D_{σ} (ρ) ω_{σ} - D_{σ} (ρ) \underline{ω} \end{matrix}] \end{matrix}

Part 2. Utilize the Lyapunov–Krasovskii functional $V (t) = V_{\tilde{σ}} (t)$ with

\begin{matrix} V_{\tilde{σ}} = η^{T} P_{\tilde{σ}} η + \int_{t - d (t)}^{t} e^{ℶ_{\tilde{σ} (t)} (s - t)} η^{T} (s) Q_{\tilde{σ}} η (s) ds \\ + \int_{- d}^{0} \int_{t + θ}^{t} e^{ℶ_{\tilde{σ} (t)} (s - t)} {\overset{\cdot}{η}}^{T} (s) R_{\tilde{σ}} \overset{\cdot}{η} (s) dsd θ \end{matrix}

(17)

where $η^{T} = [\begin{matrix} {\tilde{x}}^{T} & e^{T} \end{matrix}]$ , $P_{\tilde{σ}}, Q_{\tilde{σ}}$ , and $R_{\tilde{σ}}$ are $4 n \times 4 n$ block diagonal matrices composed of $2 n$ dimensional matrix blocks $P_{\tilde{σ}}, Q_{\tilde{σ}}$ , and $R_{\tilde{σ}}$ , respectively.

Then, for $t \in [ℸ_{p_{l}}, ℸ_{p_{l} + 1})$ , it infers

\overset{\cdot}{η} = {\overset{⌣}{A}}_{σ} (ρ) η + {\hat{B}}_{\tilde{σ}} (ρ) η (t - d (t)) - {\overset{⌣}{B}}_{\tilde{σ}} (ρ) \tilde{e} (t - d (t)) + {\overset{⌣}{ν}}_{σ}

(18)

where

\begin{matrix} {\overset{⌣}{A}}_{σ} (ρ) = [\begin{matrix} {\tilde{A}}_{σ} (ρ) & {\tilde{B}}_{σ} (ρ) \\ 0 & {\bar{A}}_{σ} (ρ) \end{matrix}], {\hat{B}}_{\tilde{σ}} (ρ) = [\begin{matrix} {\tilde{B}}_{\tilde{σ}} (ρ) & 0 \\ 0 & 0 \end{matrix}] \\ {\overset{⌣}{B}}_{\tilde{σ}}^{T} (ρ) = [{\tilde{B}}_{\tilde{σ}}^{T} (ρ) 0], {\overset{⌣}{ν}}_{σ}^{T} = [{\tilde{ν}}_{σ}^{T} {\bar{ν}}_{σ}^{T}] \end{matrix}

For any $t > 0$ , let $t_{0} = £_{0} < £_{1} < \dots < £_{N_{\tilde{σ} (0, t)}} < t$ be the switching instants of $\tilde{σ} (t)$ in the interval $(t_{0}, t)$ , then there are three cases of $£_{q}$ , which are discussed below. Note $σ (ℸ_{r_{d}}) = j$ , $σ (t_{i}) = i$ , then one has $\tilde{σ} (t) = (j, j)$ , $\forall t \in [ℸ_{r_{d}}, t_{i})$ and $\tilde{σ} (t) = (i, j)$ , $\forall t \in [t_{i}, ℸ_{r_{d + 1}})$ .

The relationship between $σ$ , $σ'$ , and $\tilde{σ}$ is shown in Figure 1.

Subsequently, let us testify that for any instant $£_{q}$

V_{\tilde{σ} (£_{q})} (£_{q}) \leq μ_{\tilde{σ} (£_{q})} V_{\tilde{σ} (£_{q}^{-})} (£_{q}^{-}), μ_{\tilde{σ} (£_{q})} = μ_{i}^{2}

(19)

(1) The system (1) switches at time $£_{q}$ , that is, $£_{q} = t_{i}$

\begin{matrix} V_{\tilde{σ} (£_{q})} (£_{q}) = η^{T} (£_{q}) P_{ij} η (£_{q}) \\ + \int_{£_{q} - d (£_{q})}^{£_{q}} e^{- ℶ_{a} (s - £_{q})} η^{T} (s) Q_{ij} η (s) ds \\ + \int_{- d}^{0} \int_{£_{q} + θ}^{£_{q}} e^{- ℶ_{a} (s - £_{q})} {\overset{\cdot}{η}}^{T} (s) R_{ij} \overset{\cdot}{η} (s) dsd θ \\ \leq μ_{i} η^{T} (£_{q}^{-}) P_{jj} η (£_{q}^{-}) \\ + μ_{i} \int_{£_{q} - d (£_{q})}^{£_{q}} e^{ℶ_{s} (s - £_{q})} η^{T} (s) Q_{jj} η (s) ds \\ + μ_{i} \int_{- d}^{0} \int_{£_{q} + θ}^{£_{q}} e^{ℶ_{s} (s - £_{q})} {\overset{\cdot}{η}}^{T} (s) R_{jj} \overset{\cdot}{η} (s) dsd θ \\ = μ_{i} V_{\tilde{σ} (£_{q}^{-})} (£_{q}^{-}) \end{matrix}

(20)

(2) $£_{q}$ is sampling transmission instant of $σ'$ , that is, $£_{q} = ℸ_{r_{d + 1}}$ :

From the expression of $d (t)$ , it stresses $d (ℸ_{r_{d + 1}}) = 0$ , $d (ℸ_{r_{d + 1}}^{-}) = d$ , then

\begin{matrix} V_{\tilde{σ} (£_{q})} (£_{q}) = V_{ii} (ℸ_{r_{d + 1}}) \\ = η^{T} (ℸ_{r_{d + 1}}) P_{ii} η (ℸ_{r_{d + 1}}) \\ + \int_{ℸ_{r_{d + 1}} - d (ℸ_{r_{d + 1}})}^{ℸ_{r_{d + 1}}} e^{ℶ_{s} (s - ℸ_{r_{d + 1}})} η^{T} (s) Q_{ii} η (s) ds \\ + \int_{- d}^{0} \int_{ℸ_{r_{d + 1}} + θ}^{ℸ_{r_{d + 1}}} e^{ℶ_{s} (s - ℸ_{r_{d + 1}})} {\overset{\cdot}{η}}^{T} (s) R_{ii} \overset{\cdot}{η} (s) dsd θ \\ = \int_{- d}^{0} \int_{ℸ_{r_{d + 1}} + θ}^{ℸ_{r_{d + 1}}} e^{ℶ_{s} (s - ℸ_{r_{d + 1}})} {\overset{\cdot}{η}}^{T} (s) R_{ii} \overset{\cdot}{η} (s) dsd θ \\ + η^{T} (ℸ_{r_{d + 1}}) P_{ii} η (ℸ_{r_{d + 1}}) \\ \leq μ_{i} η^{T} (ℸ_{r_{d + 1}}^{-}) P_{ij} η (ℸ_{r_{d + 1}}^{-}) \\ + μ_{i} \int_{ℸ_{r_{d + 1}}^{-} - d (ℸ_{r_{d + 1}}^{-})}^{ℸ_{r_{d + 1}}^{-}} e^{- ℶ_{a} (s - ℸ_{r_{d + 1}}^{-})} η^{T} (s) Q_{ij} η (s) d \\ + μ_{i} \int_{- d}^{0} \int_{ℸ_{r_{d + 1}}^{-} + θ}^{ℸ_{r_{d + 1}}^{-}} e^{- ℶ_{a} (s - ℸ_{r_{d + 1}}^{-})} {\overset{\cdot}{η}}^{T} (s) R_{ij} \overset{\cdot}{η} (s) dsd θ \\ = μ_{i} V_{\tilde{σ} (ℸ_{r_{d + 1}}^{-})} (ℸ_{r_{d + 1}}^{-}) \end{matrix}

(21)

(3) The system (1) switches at $£_{q}$ when the event-triggering conditions (6) are met, that is, $£_{q} = ℸ_{r_{d + 1}} = t_{i}$

Imitate the process (2) above, we achieve

V_{\tilde{σ} (£_{q})} (£_{q}) = V_{ii} (£_{q}) \leq μ_{i}^{2} V_{\tilde{σ} (£_{q}^{-})} (£_{q}^{-})

(22)

Taking what has been discussed into consideration, equation (19) is valid for any $£_{q}$ .

Part 3. The inference of the stability of the closed-loop system (18) is given below. According to equation (13), we know that for any $i, j \in O$ , $ℶ_{ij} = - ℶ_{a} (i \neq j)$ , $ℶ_{ii} = ℶ_{s}$ .

(1) Think over the case that a switching occurs in the interval $[ℸ_{r_{d}}, ℸ_{r_{d + 1}})$ .

(a) When $t \in [ℸ_{p_{l}}, ℸ_{p_{l} + 1}) \subset [t_{i}, ℸ_{r_{d + 1}})$ , we know $\tilde{σ} (t) = (i, j)$ , the derivative of equation (17) with respect to time t is

\begin{matrix} {\overset{\cdot}{V}}_{ij} + ℶ_{ij} V_{ij} = 2 η^{T} P_{ij} \overset{\cdot}{η} + η^{T} Q_{ij} η - ℶ_{a} η^{T} P_{ij} η \\ - \int_{t - d}^{t} e^{- ℶ_{a} (s - t)} {\overset{\cdot}{η}}^{T} (s) R_{ij} \overset{\cdot}{η} (s) ds \\ + d {\overset{\cdot}{η}}^{T} R_{ij} \overset{\cdot}{η} \end{matrix}

(23)

From the Jensen inequalities, it follows that

- \int_{t - d}^{t} e^{- ℶ_{a} (s - t)} {\overset{\cdot}{η}}^{T} (s) R_{ij} \overset{\cdot}{η} (s) ds \leq - \frac{1}{d} {\hat{η}}^{T} [\begin{matrix} R_{ij} & - R_{ij} \\ - R_{ij} & R_{ij} \end{matrix}] \hat{η}

(24)

where ${\hat{η}}^{T} = [\begin{matrix} η^{T} (t) & η^{T} (t - d (t)) \end{matrix}]$ .

Combine equations (18), (23), and (24) with Lemma 1, one gets

\begin{matrix} {\overset{\cdot}{V}}_{ij} - ℶ_{a} V_{ij} \leq 2 η^{T} P_{ij} U_{ij} (ρ) Γ + 2 η^{T} P_{ij} {\overset{⌣}{ν}}_{i} - ℶ_{a} η^{T} P_{ij} η \\ + η^{T} Q_{ij} η - \frac{1}{d} {\hat{η}}^{T} [\begin{matrix} R_{ij} & - R_{ij} \\ - R_{ij} & R_{ij} \end{matrix}] \hat{η} \\ + d {[U_{ij} (ρ) Γ + {\overset{⌣}{ν}}_{i}]}^{T} R_{ij} [U_{ij} (ρ) Γ + {\overset{⌣}{ν}}_{i}] \\ + {\tilde{ξ}}_{j} {\tilde{x}}^{T} (ℸ_{r_{d}}) {\tilde{℧}}_{j} \tilde{x} (ℸ_{r_{d}}) - {\tilde{e}}^{T} (ℸ_{p_{l}}) {\tilde{℧}}_{j} \tilde{e} (ℸ_{p_{l}}) \\ = Γ^{T} Ξ_{ij} (ρ) Γ + 2 η^{T} P_{ij} {\overset{⌣}{ν}}_{i} \\ + d {[U_{ij} (ρ) Γ + {\overset{⌣}{ν}}_{i}]}^{T} R_{ij} [U_{ij} (ρ) Γ + {\overset{⌣}{ν}}_{i}] \\ \leq Γ^{T} [Π_{ij} (ρ) + 2 {dU}_{ij}^{T} (ρ) R_{ij} U_{ij} (ρ)] Γ + γ (| | \overset{⌣}{ν} | |) \end{matrix}

(25)

where

\begin{matrix} Ξ_{ij} (ρ) = [\begin{matrix} Ψ_{ij} (ρ) - P_{ij} & P_{ij} B_{i} (ρ) K_{j} F + \frac{1}{d} R_{ij} & - P_{ij} B_{i} (ρ) K_{j} \\ * & - \frac{1}{d} R_{ij} + {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} F & - {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} \\ * & * & ({\tilde{ξ}}_{j} - 1) {\tilde{℧}}_{j} \end{matrix}] \\ Π_{ij} (ρ) = Ξ_{ij} (ρ) + [\begin{matrix} P_{ij} & 0 & 0 \\ * & 0 & 0 \\ * & * & 0 \end{matrix}], F = [\begin{matrix} I & 0 \end{matrix}] \\ γ (| | \overset{⌣}{ν} | |) = max_{i, j \in M} {{\overset{⌣}{ν}}_{i}^{T} (t) P_{ij} {\overset{⌣}{ν}}_{i} (t) + 2 d {\overset{⌣}{ν}}_{i}^{T} (t) R_{ij} {\overset{⌣}{ν}}_{i} (t)} \\ B_{i} (ρ) = diag {B_{i} (ρ), B_{i} (ρ), 0, 0}, Γ = {[\begin{matrix} {\hat{η}}^{T} & {\tilde{e}}^{T} (ℸ_{p_{l}}) \end{matrix}]}^{T} \\ Ψ_{ij} (ρ) = {\overset{⌣}{A}}_{i}^{T} (ρ) P_{ij} + P_{ij} {\overset{⌣}{A}}_{i} (ρ) + Q_{ij} - ℶ_{a} P_{ij} - \frac{1}{d} R_{ij} + P_{ij} \\ U_{ij} (ρ) = [\begin{matrix} {\overset{⌣}{A}}_{i} (ρ) & B_{i} (ρ) K_{j} F & - B_{i} (ρ) K_{j} \end{matrix}] \end{matrix}

By equations (2) and (5), it can be judged that

\begin{matrix} {\overset{⌣}{A}}_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) {\overset{⌣}{A}}_{ir}, B_{i} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) B_{ir} \\ U_{ij} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) U_{ijr}, Ψ_{ij} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) Ψ_{ijr} \end{matrix}

(26)

Then utilizing the Schur complement and considering known conditions, we deduce that $Π_{ij} (ρ) + 2 {dU}_{ij}^{T} (ρ) R_{ij} U_{ij} (ρ) < 0$ , which means ${\overset{\cdot}{V}}_{ij} \leq ℶ_{a} V_{ij} + γ (| | \overset{⌣}{ν} | |)$ .

(b) When $t \in [ℸ_{p_{l}}, ℸ_{p_{l} + 1}) \subset [ℸ_{r_{d}}, t_{i})$ , we have $\tilde{σ} (t) = (j, j)$ . Perform operations similar to those described above, we reach

{\overset{\cdot}{V}}_{jj} + ℶ_{s} V_{jj} \leq Γ^{T} [Π_{jj} (ρ) + 2 {dU}_{jj}^{T} (ρ) R_{jj} U_{jj} (ρ)] Γ + γ (| | \overset{⌣}{ν} | |)

(27)

where

\begin{matrix} Π_{jj} (ρ) = [\begin{matrix} Ψ_{jj} (ρ) & P_{jj} B_{j} (ρ) K_{j} F + \frac{e^{- ℶ_{s} d}}{d} R_{jj} & - P_{jj} B_{j} (ρ) K_{j} \\ * & - \frac{e^{- ℶ_{s} d}}{d} R_{jj} + {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} F & - {\tilde{ξ}}_{j} F^{T} {\tilde{℧}}_{j} \\ * & * & ({\tilde{ξ}}_{j} - 1) {\tilde{℧}}_{j} \end{matrix}] \\ Ψ_{ij} (ρ) = {\overset{⌣}{A}}_{i}^{T} (ρ) P_{ij} + P_{ij} {\overset{⌣}{A}}_{i} (ρ) + Q_{ij} - ℶ_{a} P_{ij} - \frac{1}{d} R_{ij} + P_{ij}, \\ γ (| | \overset{⌣}{ν} | |) = max_{j \in M} {{\overset{⌣}{ν}}_{j}^{T} (t) P_{jj} {\overset{⌣}{ν}}_{j} (t) + 2 d {\overset{⌣}{ν}}_{j}^{T} (t) R_{jj} {\overset{⌣}{ν}}_{j} (t)} \\ B_{j} (ρ) = diag {B_{j} (ρ), B_{j} (ρ), 0, 0} \\ U_{jj} (ρ) = [\begin{matrix} {\overset{⌣}{A}}_{j} (ρ) & B_{j} (ρ) K_{j} F & - B_{j} (ρ) K_{j} \end{matrix}] \end{matrix}

Analogically, we gain that

\begin{matrix} {\overset{⌣}{A}}_{j} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) {\overset{⌣}{A}}_{jr}, B_{j} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) B_{jr} \\ U_{jj} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) U_{jjr}, Ψ_{jj} (ρ) = \sum_{r = 1}^{s} g_{r} (ρ) Ψ_{jjr} \end{matrix}

(28)

Then, with the aid of equations (12), (13), and the Schur complement, we draw that $Π_{jj} (ρ) + 2 {dU}_{jj}^{T} (ρ) R_{jj} U_{jj} (ρ) < 0$ , which further points out ${\overset{\cdot}{V}}_{jj} \leq - ℶ_{s} V_{jj} + γ (| | \overset{⌣}{ν} | |)$ .

(2) When no switching occurs in the interval $[ℸ_{r_{d}}, ℸ_{r_{d + 1}})$ , the certification resembles the above process of $t \in [ℸ_{r_{d}}, t_{i})$ . Consequently, for any $t \in [ℸ_{r_{d}}, ℸ_{r_{d + 1}})$ , equation (27) can be derived.

In brief, the following inequalities are valid in appropriate intervals

{\overset{\cdot}{V}}_{\tilde{σ}} \leq - ℶ_{\tilde{σ}} V_{\tilde{σ}} + γ (| | \overset{⌣}{ν} | |), \forall t \in [ℸ_{r_{d}}, ℸ_{r_{d + 1}})

(29)

Through integrating equation (29), we acquire

V_{\tilde{σ} (t)} \leq e^{- ℶ_{\tilde{σ} (£_{q})} (t - £_{q})} V_{\tilde{σ} (£_{q})} (£_{q}) + \int_{£_{q}}^{t} e^{- ℶ_{\tilde{σ} (£_{q})} (t - s)} γ (| | \overset{⌣}{ν} | |) ds

(30)

Then, take a series of iterative operations on equations (19) and (30), we get

\begin{matrix} V_{\tilde{σ} (t)} \leq Π_{q = 1}^{N_{\tilde{σ}} (t_{0}, t)} μ_{\tilde{σ} (£_{q})} e^{- ℶ_{\tilde{σ} (£_{q})} (t - £_{q}) - Σ_{q = 0}^{N_{\tilde{σ}} (t_{0}, t) - 1} ℶ_{\tilde{σ} (£_{q})} (£_{q + 1} - £_{q})} V_{\tilde{σ} (t_{0})} (t_{0}) \\ + \int_{£_{q}}^{t} e^{- ℶ_{\tilde{σ} (£_{q})} (t - s)} γ (| | \overset{⌣}{ν} | |) ds \\ + e^{- ℶ_{\tilde{σ} (£_{q})} (t - £_{q})} Σ_{q = 0}^{N_{\tilde{σ}} (t_{0}, t) - 1} \int_{£_{q}}^{£_{q + 1}} \overset{⌣}{ℶ} ds \\ \leq Π_{i = 1}^{m} μ_{i}^{2 N_{σ i} (t_{0}, t) + 2 N_{σ' i} (t_{0}, t)} T (t_{0}, t) V_{\tilde{σ} (t_{0})} (t_{0}) \\ + \int_{t_{0}}^{t} Π_{i = 1}^{m} μ_{i}^{2 N_{σ i} (s, t) + 2 N_{σ' i} (s, t)} T (s, t) γ (| | \overset{⌣}{ν} | |) ds \end{matrix}

where

\begin{matrix} \overset{⌣}{ℶ} = Π_{q = N_{\tilde{σ}} (t_{0}, t) + 1 - N_{\tilde{σ}} (s, t)}^{N_{\tilde{σ}} (t_{0}, t)} μ_{\tilde{σ} (£_{q})} e^{\hat{ℶ}} γ (| | \overset{⌣}{ν} | |) \\ \hat{ℶ} = - ℶ_{\tilde{σ} (£_{q})} (£_{q + 1} - s) - Σ_{g = 1}^{N_{\tilde{σ}} (s, t) - 1} ℶ_{\tilde{σ} (£_{q - g})} (£_{q - g + 1} - £_{q - g}) \\ T (t_{0}, t) = \exp (\sum_{i \in M} - ℶ_{s} T_{si} (t_{0}, t) + ℶ_{a} T_{ui} (t_{0}, t)) \\ T (s, t) = \exp (\sum_{i \in M} - ℶ_{s} T_{si} (s, t) + ℶ_{a} T_{ui} (s, t)) \end{matrix}

From equation (14), it is true that there exist $ϰ_{i}$ with $4 \ln μ_{i} / τ_{ai} < ϰ_{i} < ℶ_{s} - τ_{d} (ℶ_{s} + ℶ_{a}) / τ_{ai}$ , which deduce that $τ_{d} (ℶ_{s} + ℶ_{a}) \leq (ℶ_{s} - ϰ_{i}) τ_{ai}$ . With the help of Lemmas 4 and 5, we conclude

V_{\tilde{σ} (t)} \leq ε e^{δ (t - t_{0})} V_{\tilde{σ} (t_{0})} (t_{0}) - \frac{ε}{δ} γ (∥ \overset{⌣}{ν} ∥_{[t_{0}, t]})

(31)

where $ε = \exp (\sum_{i = 1}^{m} 4 N_{0 i} \ln μ_{i} + 2 τ_{d} / τ_{ai} \ln μ_{i} + c_{i})$ , $δ = max_{i \in M} (4 \ln μ_{i} / τ_{ai} - ϰ_{i})$ , and $c_{i} = (ℶ_{s} + ℶ_{a}) N_{0 i} τ_{d}$ .

Moreover, it is evident to us that

ς_{1} | | η | |^{2} \leq V_{\tilde{σ}} \leq ς_{2} | η_{t} |_{cl}^{2}

(32)

where

\begin{matrix} ς_{1} = min_{i, j \in M} {λ_{min} (P_{ij})} \\ ς_{2} = max_{i, j \in M} {λ_{max} (P_{ij})} + \frac{1}{ℶ_{ij}} max_{i, j \in M} {λ_{max} (Q_{ij})} + \frac{y}{ℶ_{ij}} max_{i, j \in M} {λ_{max} (R_{ij})} \end{matrix}

Then, from equations (31) and (32), it can be highlighted that system (18) is ISS with regard to $ω$ .□

Theorem 3. For given matrices $L_{ir}$ , $i \in O$ , $r \in J$ , and known constants $ι_{ij} > 0$ , $d > 0$ , $ℶ_{s} > 0$ , $ℶ_{a} > 0$ , $μ_{i} \geq 1$ , and $0 < {\tilde{ξ}}_{j} < 1$ , if the existence of matrices ${{}^{‵}P}_{ij} > 0, {{}^{‵}Q}_{ij} > 0, {{}^{‵}R}_{j} > 0$ and ${\hat{\tilde{℧}}}_{j} > 0$ , ${\hat{G}}_{j}$ and ${\overset{⌣}{G}}_{j}$ make the following conditions right for $\forall i, j \in O$ , $r \in J$

{\overset{`}{P}}_{jj} \leq μ_{i} {\overset{`}{P}}_{ij}, e^{(ℶ_{s} + ℶ_{a}) d} {\overset{`}{Q}}_{jj} \leq μ_{i} {\overset{`}{Q}}_{ij}, e^{(ℶ_{s} + ℶ_{a}) d} {\overset{`}{R}}_{j} \leq μ_{i} {\overset{`}{R}}_{j}

(33)

{\overset{`}{P}}_{ij} \leq μ_{i} {\overset{`}{P}}_{ii}, {\overset{`}{R}}_{j} \leq μ_{i} {\overset{`}{R}}_{i}

(34)

[\begin{matrix} ℵ_{ijr} & B_{ir} G_{j} F + Λ_{ij} {\overset{`}{P}}_{ij} & - B_{ir} G_{j} & {\overset{`}{P}}_{ij} {\overset{⌣}{A}}_{ir}^{T} & {\overset{`}{P}}_{ij} \\ * & - Λ_{ij} {\overset{`}{R}}_{j} + {\tilde{ξ}}_{j} F^{T} {\hat{\tilde{℧}}}_{j} F & - {\tilde{ξ}}_{j} F^{T} {\hat{\tilde{℧}}}_{j} & F^{T} G_{j}^{T} B_{ir}^{T} & 0 \\ * & * & ({\tilde{ξ}}_{j} - 1) {\hat{\tilde{℧}}}_{j} & - G_{j}^{T} B_{ir}^{T} & 0 \\ * & * & * & - \frac{1}{2 d} {\overset{`}{R}}_{j} & 0 \\ * & * & * & * & - {\overset{`}{Q}}_{ij} \end{matrix}] < 0

(35)

where

\begin{matrix} ℵ_{i j r} = {{}^{‵}P}_{i j} {\overset{⌣}{A}}_{i r}^{T} + {\overset{⌣}{A}}_{i r} {{}^{‵}P}_{i j} + (ℶ_{i j} + 1) {{}^{‵}P}_{i j} + Λ_{i j} ι_{i j}^{2} {{}^{‵}ℛ}_{j} - 2 Λ_{i j} ι_{i j} {}^{‵}P \\ {{}^{‵}P}_{i j} = diag {{{}^{‵}P}_{i j}, {{}^{‵}P}_{i j}, {{}^{‵}P}_{i j}, {{}^{‵}P}_{i j}}, {{}^{‵}Q}_{i j} = diag {{{}^{‵}Q}_{i j}, {{}^{‵}Q}_{i j}, {{}^{‵}Q}_{i j}, {{}^{‵}Q}_{i j}} \\ {{}^{‵}ℛ}_{j} = diag {Ŕ_{j}, Ŕ_{j}, Ŕ_{j}, Ŕ_{j}}, ℶ_{i j} = \partial_{i j} ℶ_{s} + (\partial_{i j} - 1) ℶ_{a} \\ ℬ_{i r} = diag {B_{i r}, B_{i r}, 0, 0}, Λ_{i j} = {\begin{array}{l} \frac{1}{d}, & i \neq j \\ \frac{e^{- ℶ_{s} d}}{d}, & i = j \end{array} \\ {\overset{⌣}{A}}_{i r} = [\begin{matrix} {\tilde{A}}_{i r} & {\tilde{C}}_{i r} \\ 0 & {\bar{A}}_{i r} \end{matrix}], {\tilde{A}}_{i r} = [\begin{matrix} A_{i r} & 0 \\ 0 & A_{i r} \end{matrix}] \\ {\bar{A}}_{i r} = [\begin{matrix} A_{i r} - L_{i r} C_{i} & 0 \\ 0 & A_{i r} - L_{i r} C_{i} \end{matrix}], {\tilde{C}}_{i r} = [\begin{matrix} - L_{i r} C_{i} & 0 \\ 0 & L_{i r} C_{i} \end{matrix}] \\ F = [\begin{matrix} I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \end{matrix}], G_{j} = [\begin{matrix} {\hat{G}}_{j} \\ 0 \end{matrix}], {\hat{G}}_{j} = [\begin{matrix} {\hat{G}}_{j} & {\overset{⌣}{G}}_{j} \\ {\hat{G}}_{j} & {\overset{⌣}{G}}_{j} \end{matrix}] \end{matrix}

(36)

Then, for arbitrary switching signal whose MDADT meets the following inequality

τ_{ai} > τ_{ai}^{★} = \frac{4 \ln μ_{i} + τ_{d} (ℶ_{s} + ℶ_{a})}{ℶ_{s}}

(37)

the system (1) with the controller (7) is ISS.

In addition, the controller gains and event-triggered parameters are ${\bar{K}}_{j} = {\hat{G}}_{j} Ŕ_{j}^{- 1}$ , ${\underline{K}}_{j} = {\hat{G}}_{j} {\overset{`}{R}}_{j}^{- 1}$ , and ${\tilde{℧}}_{j} = diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}}^{- 1} {\hat{\tilde{℧}}}_{j} diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}}^{- 1}$ , respectively.

Proof. Let $R_{j} = R_{ij}$ , ${\overset{`}{P}}_{ij} = P_{ij}^{- 1}$ , ${\overset{`}{Q}}_{ij} = Q_{ij}^{- 1}$ , ${\overset{`}{R}}_{j} = R_{j}^{- 1}$ , $\forall i, j \in O$ , multiply the matrices $diag {{\overset{`}{P}}_{ij}, {\overset{`}{R}}_{j}, diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}}, I}$ on the left and right of equation (12), we derive

[\begin{matrix} ℵ_{ijr}' & B_{ir} G_{j} F + Λ_{ij} {\overset{`}{P}}_{ij} & - B_{ir} G_{j} & {\overset{`}{P}}_{ij} \overset{⌣}{A}_{ir}^{T} \\ * & - Λ_{ij} {\overset{`}{R}}_{j} + {\tilde{ξ}}_{j} F^{T} {\hat{\tilde{℧}}}_{j} F & - {\tilde{ξ}}_{j} F^{T} {\hat{\tilde{℧}}}_{j} & F^{T} G_{j}^{T} B_{ir}^{T} \\ * & * & ({\tilde{ξ}}_{j} - 1) {\hat{\tilde{℧}}}_{j} & - G_{j}^{T} B_{ir}^{T} \\ * & * & * & - \frac{1}{2 d} {\overset{`}{R}}_{j} \end{matrix}] < 0

(38)

where

\begin{matrix} {ℵ'}_{ijr} = {\overset{`}{P}}_{ij} \overset{⌣}{A}_{ir}^{T} + {\overset{⌣}{A}}_{ir} {\overset{`}{P}}_{ij} + {\overset{`}{P}}_{ij} Q_{ij} {\overset{`}{P}}_{ij} + (ℶ_{ij} + 1) {\overset{`}{P}}_{ij} - Λ_{ij} {\overset{`}{P}}_{ij} R_{j} {\overset{`}{P}}_{ij} \\ {\hat{\tilde{℧}}}_{j} = diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}} {\tilde{℧}}_{j} diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}}, G_{j} = K_{j} diag {{\overset{`}{R}}_{j}, {\overset{`}{R}}_{j}} \end{matrix}

One can get $- Λ_{ij} {\overset{`}{P}}_{ij} R_{j} {\overset{`}{P}}_{ij} \leq Λ_{ij} ι_{ij}^{2} {\overset{`}{R}}_{j} - 2 Λ_{ij} ι_{ij} {\overset{`}{P}}_{ij}$ , $\forall ι_{ij} > 0$ with Lemma 3. Then, in the light of the Schur complement, obviously, equation (35) is established.

Moreover, for $P_{ij} \leq μ_{i} P_{jj}$ , according to the Schur complement lemma, we get

[\begin{matrix} - μ_{i} P_{jj} & I \\ I & - P_{ij}^{- 1} \end{matrix}] \leq 0

then use another equivalent condition of the Schur complement lemma, we can obtain $- P_{ij}^{- 1} + 1 / μ_{i} P_{jj}^{- 1} \leq 0$ , which is ${\overset{`}{P}}_{jj} \leq μ_{i} {\overset{`}{P}}_{ij}$ . So, equations (33) and (34) are equal to equations (10) and (11), respectively.

This ends the proof.□

Examples

An example is given in this portion to present the applicability of the previous method. Consider switched LPV system (1) with two subsystems

\begin{matrix} A_{11} = [\begin{matrix} - 5 & 0 \\ 0.04 & - 2.4 \end{matrix}], A_{12} = [\begin{matrix} - 4 & - 0.1 \\ 0 & - 2 \end{matrix}], C_{1} = [\begin{matrix} 1 & 2.1 \end{matrix}] \\ A_{21} = [\begin{matrix} - 2 & 0 \\ 0.11 & - 1.5 \end{matrix}], A_{22} = [\begin{matrix} - 1.9 & 0.3 \\ 0.04 & - 2 \end{matrix}], C_{2} = [\begin{matrix} 0.1 & 0.9 \end{matrix}] \\ B_{11} = [\begin{matrix} 0.1 \\ 5.2 \end{matrix}], B_{12} = [\begin{matrix} 4.2 \\ 1.3 \end{matrix}], B_{21} = [\begin{matrix} - 9 \\ 1 \end{matrix}], B_{22} = [\begin{matrix} - 1 \\ - 1.6 \end{matrix}] \\ D_{11} = [\begin{matrix} 0.1 \\ 3.6 \end{matrix}], D_{12} = [\begin{matrix} 0.13 \\ 2.9 \end{matrix}], D_{21} = [\begin{matrix} 7.2 \\ 0.01 \end{matrix}], D_{22} = [\begin{matrix} 5.3 \\ 0.01 \end{matrix}] \end{matrix},

$v_{1} = v_{2} = 0.3 e^{- t}$ with $\bar{v} = 0.3$ , $ω_{1} = ω_{2} = \sin (t) / (5 + t)^{2}$ with $\bar{ω} = - \underline{ω} = 1 / 5$ .

Take into account the Theorem 1 with $ζ_{1} = ζ_{2} = 1$ and $g_{1} (ρ) = 1 + e^{- t} / 3$ , $g_{2} (ρ) = 2 - e^{- t} / 3$ , we can acquire the observer gains

\begin{matrix} L_{11} = [\begin{matrix} - 2.0727 \\ - 0.0565 \end{matrix}], L_{12} = [\begin{matrix} - 1.6097 \\ - 0.0226 \end{matrix}] \\ L_{21} = [\begin{matrix} - 0.0281 \\ - 0.5468 \end{matrix}], L_{22} = [\begin{matrix} 0.2952 \\ - 1.1019 \end{matrix}] \end{matrix}

and $A_{ir} - L_{ir} C_{i} (i \in M, r \in J)$ are Metzler obviously.

Parameters involved in the theorem are listed here, $ℶ_{s} = 0.5$ , $ℶ_{a} = 0.02$ , $τ_{d} = 0.9$ , $μ_{1} = 1.01$ , $μ_{2} = 1.03$ , $d = 0.015$ , $ι_{ij} = 0.1 (i, j \in {1, 2})$ , ${\tilde{ξ}}_{1} = 0.01$ , and ${\tilde{ξ}}_{2} = 0.01$ , by solving the linear matrix inequalities (LMIs) of Theorem 3, we obtain the MDADT satisfying $τ_{a 1}^{★} = 1.0156$ and $τ_{a 2}^{★} = 1.1725$ . Furthermore, the controller gains and the parameters related to event triggering can also be acquired through the calculation program, they are as follows

\begin{matrix} {\bar{K}}_{1} = [\begin{matrix} - 0.0153 & - 0.0447 \end{matrix}], {\underline{K}}_{1} = [\begin{matrix} - 0.0153 & - 0.0447 \end{matrix}] \\ {\bar{K}}_{2} = [\begin{matrix} 0.0117 & 0.0166 \end{matrix}], {\underline{K}}_{2} = [\begin{matrix} 0.0117 & 0.0166 \end{matrix}] \\ {\tilde{℧}}_{1} = [\begin{matrix} 1.1001 & 0.5515 & 0 & 0 \\ 0.5515 & 9.9305 & 0 & 0 \\ 0 & 0 & 1.1001 & 0.5515 \\ 0 & 0 & 0.5515 & 9.9305 \end{matrix}] \times 10^{5} \\ {\tilde{℧}}_{2} = [\begin{matrix} 1.134 & 0.5627 & 0 & 0 \\ 0.5627 & 9.8838 & 0 & 0 \\ 0 & 0 & 1.134 & 0.5627 \\ 0 & 0 & 0.5627 & 9.8838 \end{matrix}] \times 10^{5} \end{matrix}

Figure 2 shows the state response of the closed-loop system. The state estimation is displayed in Figures 3 and 4. Figure 5 is devoted to the event-triggered information.

Conclusion

This paper has investigated the interval observer–based ETC of switched LPV systems with unknown external disturbance and measurement noise. We have established the event-triggered mechanism and the controllers based on state estimations. After that, by modeling the considered systems as switched time-delay systems and resorting to the merging switching signal method and Lyapunov–Krasovskii functional, for different cases of synchronization and asynchronization between the controller switching signal and the system switching signal, we have derived sufficient conditions to ensure the stability of the switched LPV systems, and we have also drawn the controller gains contenting the MDADT conditions. Finally, an example has been delivered to confirm the correctness of the results. For further work, we will think about more uncertain factors in the switched LPV systems.

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 by the National Natural Science Foundation of China under grants 62273218 and the National Key R&D Program of China under grant 2018YFB1700100.

ORCID iD

Jiaxiao Lv

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