Mode-dependent robust and non-fragile finite-time H ∞ control for nonlinear singular Markovian jump systems

Abstract

This paper is concerned with the problem of mode-dependent robust and non-fragile finite-time $H_{\infty}$ control for a class of nonlinear singular Markovian jump systems (NSMJSs) with parameter uncertainties and time-varying norm-bounded disturbance. Some sufficient conditions ensuring the singular stochastic $H_{\infty}$ finite-time boundedness (SS $H_{\infty}$ FTB) are developed for the given system by using the stochastic analysis and linear matrix inequality techniques. Then, a finite-time $H_{\infty}$ state feedback controller is designed, which can guarantee the $H_{\infty}$ finite-time boundedness of the closed-loop systems. Furthermore, a robust and non-fragile finite-time $H_{\infty}$ state feedback controller is also provided to ensure the $H_{\infty}$ finite-time boundedness of the closed-loop systems when the controller gain has an additive perturbation. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

Keywords

Singular Markovian jump systems finite-time boundedness non-fragile control

Introduction

During the past few decades, a lot of attention has been paid to the study of Markovian jump systems because they are very appropriate to model stochastic systems with abrupt changes in their structures and parameters, and they have widely applications in manufacturing systems (Shen and Buscher, 2012), economic systems (Li et al., 2013), electrical systems (Assawinchaichote et al., 2007) and network systems (Kim and Park, 2009). Meanwhile, singular systems, which are also known as descriptor systems, generalized systems or differential-algebraic systems, can describe many physical systems more comprehensively and naturally than the regular systems, and have been extensively applied in the fields of electric circuit systems, socio-economic systems, restraint control systems, chemical systems and network analysis (Dai, 1989; Lewis, 1986). Many control issues on singular systems have been extensively investigated such as stability and stabilization (Xia et al., 2009; Zhong and Yang, 2005), $H_{\infty}$ control (Masubuchi et al., 1997), guaranteed-cost control (Kim, 2009), dissipativity (Wu et al., 2011) and so on. Furthermore, with the development of singular systems and Markovian jump systems, singular Markovian jump systems (SMJSs) have attracted the attention of many scholars due to their practical engineering significance, for example, DC generator model mentioned in Boukas (2008), RLC circuit diagrams and production systems introduced in Wang et al. (2014).

In the aforementioned works, the researches on SMJSs mainly focus on the classical Lyapunov stability in an infinite time interval. However, in some practical industrial applications, it is necessary to consider that the state of the system cannot deviate too much from the equilibrium point in a short or limited period of time, such as missile systems, robot systems, communication systems. Compared with the classical Lyapunov asymptotic stability, for a given initial state, if the system state does not exceed a certain threshold, the system is said to be finite-time stability during a fixed time interval (Amato et al., 2010; Ma et al., 2016). Recently, many appealing results were obtained to ensure finite-time stability (FTS), finite-time stabilization, finite-time $H_{\infty}$ control and finite-time dissipative control of SMJSs. For instance, in Zhang et al. (2012), the robust finite-time stabilization for SMJSs was studied; Li et al. (2016) investigated the finite-time $H_{\infty}$ control for SMJSs; Li and Ma (2018) considered the finite-time dissipative control for SMJSs via quantizing approach and the state feedback controllers were designed to ensure the SMJSs finite-time stabilization.

Moreover, the results mentioned above are all obtained under the condition that the SMJSs are linear systems. Inevitably, the actual systems in real life are not completely linear, such as power systems, production systems and mechanical systems, which usually exhibit strong nonlinear dynamics (Jia, 2000). Thus, when analyzing or modelling these systems, the linear SMJSs model cannot meet the demand. Therefore, it is of great significance to study the nonlinear singular Markovian jump systems (NSMJSs). For example, in Li et al. (2014), the authors designed the fuzzy state-feedback controllers for a class of nonlinear time-delay SMJSs with partly unknown transition rates, Wang et al. (2018) considered the Sliding Mode Control (SMC) for a class of nonlinear Markovian jump singular systems. Furthermore, it is worth mentioning that the non-fragile controller design problem is an attractive subject because it guarantees that the controller parameters can withstand a certain degree of influence or change. The main goal of non-fragile control is to design a feedback controller so that it can tolerate a certain degree of controller gain changes (Xia et al., 2018; Zhang et al., 2016). Up to now, although many scholars have studied the finite time stability of SMJSs, there are few papers considering the problem of mode-dependent robust and non-fragile finite-time $H_{\infty}$ control for nonlinear continuous-time SMJSs.

Motivated by the above discussions, in this paper, we focus on the problem of mode-dependent robust and non-fragile finite-time $H_{\infty}$ control for a class of NSMJSs with parameter uncertainties and time-varying norm-bounded disturbance. The main contributions of this paper include: (i) By using stochastic finite-time stability analysis methods and Lyapunov functional approach, a mode-dependent non-fragile state feedback controller is designed to ensure the resulting closed-loop systems are singular stochastic $H_{\infty}$ finite-time boundedness for all permissible disturbances and uncertainties. (ii) Sufficient conditions that ensure the non-fragile $H_{\infty}$ control performance of the considered systems are given in terms of linear matrix inequalities (LMIs).

Notations: $R^{n}$ represents the set of n-dimensional Euclidean space, $R^{m \times n}$ denotes the set of all $m \times n$ real matrices. $(Ω, P, F)$ is a probability space, $Ω$ is the sample space; $F$ is the algebra of events; $P$ is the probability measure defined on $F$ . $E {.}$ denotes the expectation operator with respect to some probability measure $P$ . I is the identity matrix of appropriate dimension, while $I_{n}$ represents the n-dimensional identity matrix. $diag {. . .}$ denotes a block-diagonal matrix. The superscripts $' T'$ and $' - 1'$ represent the transpose and the inverse of a matrix, respectively. The symbol * denotes the symmetric part of a symmetry matrix.

Problem statement and preliminaries

Consider a class of NSMJSs with Markovian jump parameters on a complete probability space $(Ω, P, F)$ as follows

{\begin{matrix} E (r_{t}) \overset{\cdot}{x} (t) = f (r_{t}) [x (t), u (t), w (t)], \\ z (t) = g (r_{t}) [x (t), u (t), w (t)], \end{matrix}

(1)

where $x (t) \in R^{n}$ is state vector, $u (t) \in R^{m}$ is control input, $z (t) \in R^{p}$ is controlled output, $E (r_{t}) \in R^{n \times n}$ is singular matrix with rank $(E (r_{t})) = r \leq n$ , and $w (t) \in R^{q}$ is external disturbance input satisfies

E {\int_{0}^{\infty} w^{T} (t) w (t) dt} \leq d^{2}, d \geq 0

(2)

In system (1), $f (r_{t}) [x (t)$ , $u (t)$ , $w (t)]$ , and $g (r_{t}) [x (t)$ , $u (t)$ , $w (t)]$ are unknown continuous nonlinear functions described by

‖ v (r_{t}) [x (t), u (t), w (t)] ‖ \leq ‖ A_{e} (r_{t}) x (t) + B_{e} (r_{t}) u (t) + G_{e} (r_{t}) w (t) ‖

(3)

‖ g (r_{t}) [x (t), u (t), w (t)] ‖ \leq ‖ \bar{C} (r_{t}) x (t) + \bar{D} (r_{t}) u (t) + \bar{H} (r_{t}) w (t) ‖

(4)

where $v (r_{t}) [x (t), u (t), w (t)] = f (r_{t}) [x (t), u (t), w (t)] - [\bar{A} (r_{t}) x (t) + \bar{B} (r_{t}) u (t) + \bar{G} (r_{t}) w (t)]$ . For notational brevity, define $v (r_{t}) [x (t), u (t), w (t)] \overset{Δ}{=} v (r_{t})$ . And $\bar{A} (r_{t})$ , $\bar{B} (r_{t})$ , $\bar{C} (r_{t})$ , $\bar{D} (r_{t})$ , $\bar{G} (r_{t})$ and $\bar{H} (r_{t})$ are uncertain matrices and satisfy the following constrains

[\begin{matrix} \bar{A} (r_{t}) & \bar{B} (r_{t}) & \bar{G} (r_{t}) \\ \bar{C} (r_{t}) & \bar{D} (r_{t}) & \bar{H} (r_{t}) \end{matrix}]

= [\begin{matrix} A (r_{t}) + Δ_{A} (r_{t}) & B (r_{t}) + Δ_{B} (r_{t}) & G (r_{t}) + Δ_{G} (r_{t}) \\ C (r_{t}) + Δ_{C} (r_{t}) & D (r_{t}) + Δ_{D} (r_{t}) & H (r_{t}) + Δ_{H} (r_{t}) \end{matrix}]

= [\begin{matrix} A (r_{t}) & B (r_{t}) & G (r_{t}) \\ C (r_{t}) & D (r_{t}) & H (r_{t}) \end{matrix}] + Δ (r_{t}) F (r_{t}) [\begin{matrix} μ_{1} (r_{t}) & μ_{2} (r_{t}) & μ_{3} (r_{t}) \\ μ_{4} (r_{t})] & μ_{5} (r_{t}) & μ_{6} (r_{t}) \end{matrix}]

(5)

where $A (r_{t})$ , $B (r_{t})$ , $G (r_{t})$ , $C (r_{t})$ , $D (r_{t})$ , $H (r_{t})$ , $Δ (r_{t})$ , $μ_{1} (r_{t})$ , $μ_{2} (r_{t})$ , $μ_{3} (r_{t})$ , $μ_{4} (r_{t})$ , $μ_{5} (r_{t})$ , $μ_{6} (r_{t})$ are known matrices for the NSMJSs (1), and $F (r_{t})$ is an unknown matrix function satisfying $F^{T} (r_{t}) F (r_{t}) \leq I .$

In system (1), the mode jumping process ${r_{t}, t \geq 0}$ is a right-continuous Markovian stochastic process taking values in a finite space $S : = {1, 2, 3, . . ., y}$ with the following mode transition probabilities

\Pr {r_{r + Δ t} = j | r_{t} = i} = {\begin{matrix} π_{ij} Δ t + o (Δ t), & if i \neq j, \\ 1 + π_{ij} Δ t + o (Δ t), & if i = j, \end{matrix}

(6)

where $\lim_{Δ t \to 0} o (Δ t) / Δ t = 0$ , $π_{ij}$ satisfies $π_{ij} \geq 0 (i \neq j)$ and $π_{ii} = - \sum_{j = 1, j \neq i}^{y} π_{ij} .$

Consider the following non-fragile state feedback controller

u (t) = (K (r_{t}) + Δ_{K} (r_{t})) x (t)

(7)

where $K (r_{t})$ is the controller gain matrix, $Δ_{K} (r_{t}) = \hat{Δ} (r_{t}) F (r_{t}) μ_{7} (r_{t})$ represents the controller gain perturbation, $\hat{Δ} (r_{t})$ and $μ_{7} (r_{t})$ are known matrices.

Remark 1. When the controller is affected by ambient temperature or aging of the equipment, it will destroy the system performance and even make the system unable to work normally. Therefore, when designing the controller, the controller gain disturbance should be considered to ensure that the designed controller parameters can withstand a certain degree of influence or change. Here, these disturbances in the controller gain are modeled as uncertain gains that depend on uncertain parameters.

From system (1) and controller (7), the closed-loop dynamic systems can be written as

{\begin{matrix} E (r_{t}) \overset{\cdot}{x} (t) = {\bar{A}}_{c} (r_{t}) x (t) + \bar{G} (r_{t}) w (t) + v (r_{t}), \\ z (t) = g (r_{t}) (x (t), u (t), w (t)), \end{matrix}

(8)

where ${\bar{A}}_{c} (r_{t}) = \bar{A} (r_{t}) + \bar{B} (r_{t}) \bar{K} (r_{t})$ , $\bar{K} (r_{t}) = K (r_{t}) + Δ_{K} (r_{t})$

Definition 1. (Xu and Lam, 2006)

The system (8) is said to be regular in the time interval [0, T], if the characteristic polynomial $\det (sE - \bar{A} (r_{t}))$ is not identically zero for all $t \in$ [0, T].

The system (8) with is said to be impulse free in the time interval [0, T], if $\deg (\det (sE - \bar{A} (r_{t}))) = rank (E)$ for all $t \in$ [0, T].

Definition 2 (Zhang et al., 2012)

The system (8) with $w (t) = 0$ is said to be singular stochastic finite-time stability (SSFTS) with respect to $(c_{1}$ , $c_{2}$ , $T$ , $R (r_{t}))$ , with 0 $< c_{1} < c_{2}$ and $R (r_{t}) > 0$ , if the system is regular and impulse free in time interval $[0, T]$ and satisfies

\begin{matrix} E {x^{T} (0) E^{T} R (r_{t}) Ex (0)} \leq c_{1}^{2} \Rightarrow E {x^{T} (t) E^{T} R (r_{t}) Ex (t)} \\ < c_{2}^{2}, \forall t \in [0, T] . \end{matrix}

(9)

The system (8) satisfying (2) is said to be singular stochastic finite-time boundedness (SSFTB) with respect to $(c_{1}, c_{2}, T, R (r_{t})$ , $d)$ , with 0 $< c_{1} < c_{2}$ and $R (r_{t}) > 0$ , if the system is regular and impulse free in time interval [0, T] and the condition (9) holds.

The system (8) is said to be singular stochastic $H_{\infty}$ finite-time boundedness (SS $H_{\infty}$ FTB) with respect to $(c_{1}, c_{2}, T, R (r_{t}), γ, d)$ , if the system is SSFTB with respect to $(c_{1}, c_{2}, T, R (r_{t})$ , $d)$ , and under the zero-initial condition the controlled output $z (t)$ satisfies

E {\int_{0}^{T} z^{T} (t) z (t))} < γ^{2} E {\int_{0}^{T} w^{T} (t) w (t) dt} .

(10)

For simplicity, in the sequel, for each possible $r_{t} = i$ , $i \in S$ , the matrices $E (r_{t})$ , $A_{e} (r_{t})$ and $Δ_{A} (r_{t})$ will be denoted by $E_{i}$ , $A_{ei}$ and $Δ_{A_{i}}$ respectively, and so forth.

Lemma 1. (Ma et al., 2015) Let $U$ and $V$ be real matrices with appropriate dimensions. For any scalar $μ > 0$ , then

U^{T} V + V^{T} U \leq μ U^{T} U + μ^{- 1} V^{T} V .

(11)

Lemma 2. (Jiang et al., 2019) Given symmetric matrix $S = [\begin{matrix} S_{11} & S_{12} \\ S_{12}^{T} & S_{22} \end{matrix}]$ , then $S < 0$ is equivalent to $S_{22} < 0$ and $S_{11} - S_{12} S_{22}^{- 1} S_{12}^{T} < 0$ .

The main purpose of this article is to design a non-fragile finite-time $H_{\infty}$ state feedback controller of the form (7) to ensure that the closed-loop systems (8) is singular stochastic $H_{\infty}$ finite-time boundedness by using the stochastic finite-time stability analysis methods and linear matrix inequality techniques (Jia, 2003). The problem encountered in the analysis of this paper is how to deal with nonlinear terms in the derivation process so that all the conditions obtained are expressed in terms of LMIs.

Main results

In this section, the mode-dependent robust and non-fragile finite-time $H_{\infty}$ control problem will be investigated for the NSMJSs (8). Firstly, some sufficient conditions ensuring the singular stochastic $H_{\infty}$ finite-time boundedness are developed for systems (8). Then, based on these obtained conditions, a finite-time $H_{\infty}$ state feedback controller is designed, which can guarantee the $H_{\infty}$ finite-time boundedness of the closed-loop systems. Moreover, a robust and non-fragile finite-time $H_{\infty}$ state feedback controller is provided.

Theorem 1. Given scalars $T > 0$ , $d > 0$ , $α \geq 0$ , $c_{1} > 0$ and matrices $R_{i} > 0$ , the system (8) is SS $H_{\infty}$ FTB with respect to $(c_{1}, c_{2}, T, R_{i}, γ, d)$ , if there exist nonsingular matrices $P_{i}$ , $Q_{1 i}$ and positive scalars $γ$ , $c_{2}$ , $ε_{i}$ such that the following constraints hold for each mode $i \in S$

[\begin{matrix} β_{i} & P_{i}^{T} {\bar{G}}_{i} & A_{cei}^{T} & {\bar{C}}_{ki}^{T} \\ * & - γ^{2} e^{- α T} I & G_{ei}^{T} & {\bar{H}}_{i}^{T} \\ * & * & - ε_{i} I & 0 \\ * & * & * & - I \end{matrix}] < 0

(12)

E_{i}^{T} P_{i} = P_{i}^{T} E_{i} \geq 0

(13)

E_{i}^{T} P_{i} = E_{i}^{T} R_{i}^{\frac{1}{2}} Q_{1 i} R_{i}^{\frac{1}{2}} E_{i}

(14)

λ_{\max} (Q_{1 i}) c_{1}^{2} + γ^{2} e^{- α T} d^{2} < c_{2}^{2} e^{- α T} λ_{\min} (Q_{1 i})

(15)

where

β_{i} = {\bar{A}}_{ci}^{T} P_{i} + P_{i}^{T} {\bar{A}}_{ci} + \sum_{j = 1}^{y} π_{ij} E_{j}^{T} P_{j} + ε_{i} P_{i}^{T} P_{i} - α E_{i}^{T} P_{i},

A_{cei} = A_{ei} + B_{ei} {\bar{K}}_{i},

{\bar{C}}_{ki} = {\bar{C}}_{i} + {\bar{D}}_{i} {\bar{K}}_{i}

Proof. Firstly, we shall prove the regularity and the absence of impulse of system (8) in time interval [0, T]. From Lemma 2 and (12), it can be obtained that

{\bar{A}}_{ci}^{T} P_{i} + P_{i}^{T} {\bar{A}}_{ci} + \sum_{j = 1}^{y} π_{ij} E_{j}^{T} P_{j} - α E_{i}^{T} P_{i} < 0

(16)

which implies

{\bar{A}}_{c i}^{T} P_{i} + P_{i}^{T} {\bar{A}}_{c i} + (π_{i i} - α) E_{i}^{T} P_{i} < - \sum_{j = 1, j \neq i}^{y} π_{i j} E_{j}^{T} P_{j} \leq 0

(17)

Since the matrix $E_{i}$ is singular, there must exist nonsingular matrices $M_{Li}$ and $M_{Ri}$ such that $M_{Li} E_{i} M_{Ri} = [\begin{matrix} I_{r} & 0 \\ 0 & 0 \end{matrix}]$ . Let

M_{Li} {\bar{A}}_{ci} M_{Ri} = [\begin{matrix} {\hat{A}}_{1 i} & {\hat{A}}_{2 i} \\ {\hat{A}}_{3 i} & {\hat{A}}_{4 i} \end{matrix}], M_{Li}^{- T} P_{i} M_{Ri} = [\begin{matrix} {\hat{P}}_{1 i} & {\hat{P}}_{2 i} \\ {\hat{P}}_{3 i} & {\hat{P}}_{4 i} \end{matrix}] .

(18)

Pre- and post- multiplying (13) and (17) by $M_{Ri}^{T}$ and $M_{Ri}$ , it is not difficult to get ${\hat{P}}_{2 i} = 0$ and ${\hat{P}}_{4 i}^{T} {\hat{A}}_{4 i} + {\hat{A}}_{4 i}^{T} {\hat{P}}_{4 i} < 0$ , therefore ${\hat{A}}_{4 i}$ is nonsingular for every $i \in S$ . In this case, one can get

\det (sE - {\bar{A}}_{ci}) = \det (s I_{r} - {\hat{A}}_{1 i} + {\hat{A}}_{2 i} {\hat{A}}_{4 i}^{- 1} {\hat{A}}_{3 i}) * \det (- {\hat{A}}_{4 i})

(19)

when choosing the scalar s as some value which is not equal to any eigenvalue of the matrix ${\hat{A}}_{1 i} - {\hat{A}}_{2 i} {\hat{A}}_{4 i}^{- 1} {\hat{A}}_{3 i}$ , one have $\det (s I_{r} - {\hat{A}}_{1 i} + {\hat{A}}_{2 i} {\hat{A}}_{4 i}^{- 1} {\hat{A}}_{3 i}) \neq 0$ , $\det (sE - {\bar{A}}_{ci}) \neq 0$ and it follows from (19) that $\deg (\det (sE - {\bar{A}}_{ci})) = r = rank (E)$ , which implies that system (8) is regular and impulse free in time interval [0, T].

Now, we prove the SS $H_{\infty}$ FTB of system (8). Consider the Lyapunov function candidate as $V (x (t), i, t) = x^{T} (t) E_{i}^{T} P_{i} x (t)$ . Calculating the weak infinitesimal operator $Γ$ of $V (x (t), i, t)$ along the solution of NSMJSs (8), yields that

\begin{matrix} Γ V (x (t), i, t) = {\overset{\cdot}{x}}^{T} (t) E_{i}^{T} P_{i} x (t) + x^{T} (t) E_{i}^{T} P_{i} \overset{\cdot}{x} (t) + \sum_{j = 1}^{y} \\ π_{ij} x^{T} (t) E_{j}^{T} P_{j} x (t) \\ = {({\bar{A}}_{ci} x (t) + {\bar{G}}_{i} w (t) + v_{i})}^{T} P_{i} x (t) + x^{T} (t) P_{i}^{T} ({\bar{A}}_{ci} x (t) \\ + {\bar{G}}_{i} w (t) + v_{i}) + \sum_{j = 1}^{y} π_{ij} x^{T} (t) E_{j}^{T} P_{j} x (t) \\ = x^{T} (t) [{\bar{A}}_{ci}^{T} P_{i} + P_{i}^{T} {\bar{A}}_{ci} + \sum_{j = 1}^{y} π_{ij} E_{j}^{T} P_{j}] x (t) \\ + w^{T} (t) {\bar{G}}_{i}^{T} P_{i} x (t) + x^{T} (t) P_{i}^{T} {\bar{G}}_{i} w (t) \\ + v_{i}^{T} (t) P_{i} x (t) + x^{T} (t) P_{i}^{T} v_{i} (t) . \end{matrix}

(20)

Let $A_{cei} = A_{ei} + B_{ei} {\bar{K}}_{i}$ , it follows from Lemma 1 and (3) that

\begin{matrix} v_{i}^{T} (t) P_{i} x (t) + x^{T} (t) P_{i}^{T} v_{i} (t) \\ \leq ε_{i} x^{T} (t) P_{i}^{T} P_{i} x (t) + ε_{i}^{- 1} v_{i}^{T} (t) v_{i} (t) \\ \leq ε_{i} x^{T} (t) P_{i}^{T} P_{i} x (t) \\ + ε_{i}^{- 1} {[A_{cei} x (t) + G_{ei} w (t)]}^{T} [A_{cei} x (t) + G_{ei} w (t)] . \end{matrix}

(21)

Hence

\begin{matrix} Γ V (x (t), i, t) \\ \leq x^{T} (t) [{\bar{A}}_{ci}^{T} P_{i} + P_{i}^{T} {\bar{A}}_{ci} + \sum_{j = 1}^{y} π_{ij} E_{j}^{T} P_{j}] x (t) \\ + w^{T} (t) {\bar{G}}_{i}^{T} P_{i} x (t) + x^{T} (t) P_{i}^{T} {\bar{G}}_{i} w (t) + ε_{i} x^{T} (t) P_{i}^{T} P_{i} x (t) \\ + ε^{- 1} {[A_{cei} x (t) + G_{ei} w (t)]}^{T} [A_{cei} x (t) + G_{ei} w (t)] \\ = {[\begin{matrix} x (t) \\ w (t) \end{matrix}]}^{T} [\begin{matrix} β_{i} + α E_{i}^{T} P_{i} + ε_{i}^{- 1} A_{cei}^{T} A_{cei} & P_{i}^{T} {\bar{G}}_{i} + ε_{i}^{- 1} A_{cei}^{T} G_{ei} \\ * & ε_{i}^{- 1} G_{ei}^{T} G_{ei} \end{matrix}] \\ [\begin{matrix} x (t) \\ w (t) \end{matrix}], \end{matrix}

(22)

then, it follows from (4), (12), (22) and Lemma 2 that

\begin{matrix} Γ V (x (t), i, t) \\ < α V (x (t), i, t) + γ^{2} e^{- α T} w^{T} (t) w (t) \\ - {({\bar{C}}_{i} x_{i} + {\bar{D}}_{i} u (i) + {\bar{H}}_{i} w (i))}^{T} ({\bar{C}}_{i} x_{i} + {\bar{D}}_{i} u (i) + {\bar{H}}_{i} w (i)) \\ < α V (x (t), i, t) + γ^{2} e^{- α T} w^{T} (t) w (t) - z^{T} (t) z (t) . \end{matrix}

(23)

On one hand, from (23), it is easy to obtain that

Γ V (x (t), i, t) < α V (x (t), i, t) + γ^{2} e^{- α T} w^{T} (t) w (t),

(24)

which infers

Γ [e^{- α t} V (x (t), i, t)] < e^{- α t} γ^{2} e^{- α T} w^{T} (t) w (t)

(25)

Integrating (25) from 0 to t, and taking the mathematical expectation of the obtained formula, it can be derived that

\begin{matrix} e^{- α t} E {V (x (t), i, t)} < E {V (x (0), r_{0}, 0)} \\ + E {\int_{0}^{t} e^{- α τ} γ^{2} e^{- α T} w^{T} (τ) w (τ)] d τ} . \end{matrix}

(26)

Then, given $E {x^{T} (0) E_{i}^{T} R_{i} E_{i} x (0)} \leq c_{1}^{2}$ and combining (14), (26) can be rewritten as

\begin{matrix} E {V (x (t), i, t)} = E {x^{T} (t) E_{i}^{T} P_{i} x (t)} \\ < e^{α t} E {V (x (0), r_{0}, 0)} + e^{α t} E \\ {\int_{0}^{t} e^{- α τ} γ^{2} e^{- α T} w^{T} (τ) w (τ)] d τ} \\ < e^{α t} [λ_{\max} (Q_{1 i}) c_{1}^{2} + γ^{2} e^{- α T} d^{2}] . \end{matrix}

(27)

According to (14), we have

\begin{matrix} E {x^{T} (t) E_{i}^{T} P_{i} x (t)} = E {x^{T} (t) E_{i}^{T} R_{i}^{\frac{1}{2}} Q_{1 i} R_{i}^{\frac{1}{2}} E_{i} x (t)} \\ \geq λ_{\min} (Q_{1 i}) E {x^{T} (t) E_{i}^{T} R_{i} E_{i} x (t)}, \end{matrix}

(28)

thus, for all $t \in [0, T]$ , it can be deduced from (15), (27) and (28) that

\begin{matrix} E {x^{T} (t) E_{i}^{T} R_{i} E_{i} x (t)} < \frac{e^{α T} [λ_{\max} (Q_{1 i}) c_{1}^{2} + γ^{2} e^{- α T} d^{2}]}{λ_{\min} (Q_{1 i})} \\ < c_{2}^{2}, \end{matrix}

(29)

so far, we have proved that the system (8) is SSFTB.

On the other hand, we can obtain from (23) that

Γ [e^{- α t} V (x (t), i, t)] < e^{- α t} γ^{2} e^{- α T} w^{T} (t) w (t) - e^{- α t} z^{T} (t) z (t),

(30)

under zero initial condition, integrating (30) from 0 to T, we have

\begin{matrix} \int_{0}^{T} e^{- α t} [z^{T} (t) z (t) - γ^{2} e^{- α T} w^{T} (t) w (t)] dt \\ < - \int_{0}^{T} Γ [e^{- α t} V (x (t), i, t)] dt \\ < - e^{- α T} V (x (T), i, T) + V (x (0), r_{0}, 0) \\ < V (x (0), r_{0}, 0) = 0 . \end{matrix}

(31)

Further, it can be easily seen that

\begin{matrix} E {\int_{0}^{T} z^{T} (t) z (t) dt} \\ < e^{α T} E {γ^{2} \int_{0}^{T} e^{- α (T + t)} w^{T} (t) w (t) dt} \\ < γ^{2} E {\int_{0}^{T} w^{T} (t) w (t) dt} . \end{matrix}

(32)

Therefore, the NSMJS (8) is SS $H_{\infty}$ FTB. This completes the proof.

Remark 2. Constraint (3) indicates that the unknown nonlinear function $f (r_{t}) [x (t), u (t), w (t)]$ is located in the n-dimensional hypersphere, its center is linear subsystem $\bar{A} (r_{t}) x (t) + \bar{B} (r_{t}) u (t) + \bar{G} (r_{t}) w (t)$ with uncertain matrices and its radius is bounded by the norm $‖ A_{e} (r_{t}) x (t) + B_{e} (r_{t}) u (t) + G_{e} (r_{t}) w (t) ‖$ . The analysis method in this paper can also be used to solve the NSMJSs with similar structures (Ren and Zong, 2017; Song and He, 2015).

Remark 3. Note that in the study of finite-time $H_{\infty}$ control for SMJSs, most of the existing articles chose singular matrices $E_{i} = E$ . In this article, we consider the mode-dependent singular matrices $E_{i}$ , which are more general than those articles.

Next, based on Theorem 1, a robust finite-time $H_{\infty}$ state feedback controller will be designed to guarantee the finite-time $H_{\infty}$ control of NSMJS (8).

Theorem 2. Given scalars $T > 0, d > 0, α \geq 0$ , $c_{1} > 0$ and matrices $R_{i} > 0$ , the closed-loop NSMJS (8) controlled by a state feedback controller with $K_{i} = L_{i} X_{i}^{- 1}$ is SS $H_{\infty}$ FTB with respect to $(c_{1}, c_{2}, T, R_{i},$ $γ, d)$ , if there exist nonsingular matrices $X_{i}$ , $Q_{1 i}$ , $M_{Li}$ and $M_{Ri}$ , positive symmetric matrices $W_{2 i}$ , $matrices L_{i}$ , positive scalars $γ$ , $c_{2}$ , $ε_{i}$ , $Σ_{i}$ , $η_{i}$ , $φ_{i}$ and $ζ_{i}$ , such that the following constraints hold for each mode $i \in S$

0 \leq X_{i}^{T} E_{i}^{T} = E_{i} X_{i} \leq φ_{i} I

(33)

ζ_{i} I < R_{i}^{\frac{1}{2}} M_{Li}^{- 1} diag {W_{1 i}, W_{2 i}} M_{Li}^{- T} R_{i}^{\frac{1}{2}} < I

(34)

\frac{1}{ζ_{i}} c_{1}^{2} + γ^{2} e^{- α T} d^{2} - c_{2}^{2} e^{- α T} < 0

(35)

Ξ_{i} = [\begin{matrix} Ξ_{11 i} & Ξ_{12 i} & Ξ_{13 i} \\ * & - η_{i} I & 0 \\ * & * & - Σ_{i} I \end{matrix}] < 0

(36)

where

\begin{matrix} Ξ_{11 i} = [\begin{matrix} Ξ_{11 i}^{11} & G_{i} & Ξ_{11 i}^{13} & Ξ_{11 i}^{14} & Ξ_{11 i}^{15} \\ * & Ξ_{11 i}^{22} & G_{ei}^{T} & H_{i}^{T} & 0 \\ * & * & - ε_{i} I & 0 & 0 \\ * & * & * & Ξ_{11 i}^{44} & 0 \\ * & * & * & * & - {\hat{H}}_{i} \end{matrix}], \\ Ξ_{12 i} = {[\begin{matrix} μ_{1} (i) X_{i} + μ_{2} (i) L_{i} & μ_{3} (i) & 0 & 0 & 0 \end{matrix}]}^{T}, \\ Ξ_{13 i} = {[\begin{matrix} μ_{4} (i) X_{i} + μ_{5} (i) L_{i} & μ_{6} (i) & 0 & 0 & 0 \end{matrix}]}^{T}, \\ Ξ_{11 i}^{11} = A_{i} X_{i} + B_{i} L_{i} + X_{i}^{T} A_{i}^{T} + L_{i}^{T} B_{i}^{T} + ε_{i} I \\ + (π_{ii} - α) E_{i} X_{i} + η_{i} Δ_{i} Δ_{i}^{T} \\ Ξ_{11 i}^{13} = (A_{ei} X_{i} + B_{ei} L_{i})^{T}, \\ Ξ_{11 i}^{14} = (C_{i} X_{i} + D_{i} L_{i})^{T}, \\ Ξ_{11 i}^{15} = [\sqrt{π_{i 1}} X_{i}^{T} \dots \sqrt{π_{i (i - 1)}} X_{i}^{T} \sqrt{π_{i (i + 1)}} X_{i}^{T} \dots \sqrt{π_{iy}} X_{i}^{T}], \\ Ξ_{11 i}^{22} = - γ^{2} e^{- α T} I, \\ Ξ_{11 i}^{44} = Σ_{i} Δ_{i} Δ_{i}^{T} - I, \end{matrix}

\begin{matrix} {\hat{H}}_{i} = diag {X_{1} + X_{1}^{T} - φ_{1} I, \dots, X_{i - 1} + X_{i - 1}^{T} - φ_{i - 1} I, \\ X_{i + 1} + X_{i + 1}^{T} - φ_{i + 1} I, \dots, X_{y} + X_{y}^{T} - φ_{y} I}, \end{matrix}

W_{1 i} = [\begin{matrix} I_{r} & 0 \end{matrix}] M_{Li} E_{i} X_{i} M_{Li}^{T} {[\begin{matrix} I_{r} & 0 \end{matrix}]}^{T}

Proof. Firstly, pre- and post-multiplying (12) by $diag {P_{i}^{- T}, I, I, I}$ and $diag {P_{i}^{- 1}, I, I, I}$ respectively, we have

[\begin{matrix} P_{i}^{- T} {\bar{A}}_{ci}^{T} + {\bar{A}}_{ci} P_{i}^{- 1} + \sum_{j = 1}^{y} π_{ij} P_{i}^{- T} E_{j}^{T} P_{j} P_{i}^{- 1} + ε_{i} I - α P_{i}^{- T} E_{i}^{T} & {\bar{G}}_{i} & P_{i}^{- T} A_{cei}^{T} & P_{i}^{- T} {\bar{C}}_{ki}^{T} \\ * & - γ^{2} e^{- α T} I & G_{ei}^{T} & {\bar{H}}_{i}^{T} \\ * & * & - ε_{i} I & 0 \\ * & * & * & - I \end{matrix}] < 0 .

(37)

Let $X_{i} = P_{i}^{- 1}$ , (37) can be rewritten as

\begin{matrix} [\begin{matrix} X_{i}^{T} {\bar{A}}_{ci}^{T} + {\bar{A}}_{ci} X_{i} + \sum_{j = 1}^{y} π_{ij} X_{i}^{T} E_{j}^{T} X_{j}^{- 1} X_{i} + ε_{i} I - α X_{i}^{T} E_{i}^{T} & {\bar{G}}_{i} & X_{i}^{T} A_{cei}^{T} & X_{i}^{T} {\bar{C}}_{ki}^{T} \\ * & - γ^{2} e^{- α T} I & G_{ei}^{T} & {\bar{H}}_{i}^{T} \\ * & * & - ε_{i} I & 0 \\ * & * & * & - I \end{matrix}] < 0 \end{matrix}

(38)

Similarly, pre- and post-multiplying (33) by $X_{i}^{- T}$ and $X_{i}^{- 1}$ , respectively, we have $E_{i}^{T} X_{i}^{- 1} = X_{i}^{- T} E_{i} \leq φ_{i} X_{i}^{- T} X_{i}^{- 1} .$ Thus

\sum_{j = 1, j \neq i}^{y} π_{ij} X_{i}^{T} E_{j}^{T} X_{j}^{- 1} X_{i} \leq \sum_{j = 1, j \neq i}^{y} π_{ij} X_{i}^{T} φ_{j} X_{j}^{- T} X_{j}^{- 1} X_{i}

(39)

It can be derived from Lemma 1 that

φ_{j}^{- 1} X_{j} X_{j}^{T} = (φ_{j} X_{j}^{- T} X_{j}^{- 1})^{- 1} \geq X_{j}^{T} + X_{j} - φ_{j} I

(40)

From (39) and (40), one can obtain that

\begin{matrix} \sum_{j = 1, j \neq i}^{y} π_{ij} X_{i}^{T} E_{j}^{T} X_{j}^{- 1} X_{i} \leq \sum_{j = 1, j \neq i}^{y} π_{ij} X_{i}^{T} (X_{j}^{T} + X_{j} - φ_{j} I)^{- 1} X_{i} \\ = Ξ_{11 i}^{15} {\hat{H}}_{i}^{- 1} (Ξ_{11 i}^{15})^{T} . \end{matrix}

(41)

Based on Lemma 2 and (41), it is easy to see that (38) is equivalent to

[\begin{matrix} X_{i}^{T} {\bar{A}}_{ci}^{T} + {\bar{A}}_{ci} X_{i} + ε_{i} I + (π_{ii} - α) X_{i}^{T} E_{i}^{T} & {\bar{G}}_{i} & X_{i}^{T} A_{cei}^{T} & X_{i}^{T} {\bar{C}}_{ki}^{T} & Ξ_{11 i}^{15} \\ * & - γ^{2} e^{- α T} I & G_{ei}^{T} & {\bar{H}}_{i}^{T} & 0 \\ * & * & - ε_{i} I & 0 & 0 \\ * & * & * & - I & 0 \\ * & * & * & * & - {\hat{H}}_{i} \end{matrix}] \begin{matrix} < 0 \end{matrix}

(42)

Let $L_{i} = K_{i} X_{i}$ , and expanding the special form of matrices ${\bar{A}}_{ci}, A_{cei}, {\bar{C}}_{ki}, {\bar{G}}_{i}, {\bar{H}}_{i}$ and ${\bar{B}}_{i}$ , (42) can be rewritten by

Ξ_{1 i} + Ξ_{2 i} < 0

(43)

where

Ξ_{1 i} = [\begin{matrix} {\bar{β}}_{i} & G_{i} & X_{i}^{T} A_{ei}^{T} + L_{i}^{T} B_{ei}^{T} & X_{i}^{T} C_{i}^{T} + L_{i}^{T} D_{i}^{T} & Ξ_{11 i}^{15} \\ * & - γ^{2} e^{- α T} I & G_{ei}^{T} & H_{i}^{T} & 0 \\ * & * & - ε_{i} I & 0 & 0 \\ * & * & * & - I & 0 \\ * & * & * & * & - {\hat{H}}_{i} \end{matrix}],

Ξ_{2 i} = [\begin{matrix} X_{i}^{T} Δ_{A_{i}}^{T} + Δ_{A_{i}} X_{i} + L_{i}^{T} Δ_{B_{i}}^{T} + Δ_{B_{i}} L_{i} & Δ_{G_{i}} & 0 & X_{i}^{T} Δ_{C_{i}}^{T} + L_{i}^{T} Δ_{D_{i}}^{T} & 0 \\ * & 0 & 0 & Δ_{H_{i}}^{T} & 0 \\ * & * & 0 & 0 & 0 \\ * & * & * & 0 & 0 \\ * & * & * & * & 0 \end{matrix}],

{\bar{β}}_{i} = X_{i}^{T} A_{i}^{T} + A_{i} X_{i} + L_{i}^{T} B_{i}^{T} + B_{i} L_{i} + ε_{i} I + (π_{ii} - α) X_{i}^{T} E_{i}^{T}

It can be deduced from Lemma 1 and (5) that

\begin{matrix} Ξ_{2 i} = [\begin{matrix} Δ_{i} \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] F_{i} [\begin{matrix} μ_{1} (i) X_{i} + μ_{2} (i) L_{i} & μ_{3} (i) & 0 & 0 & 0 \end{matrix}] \\ + {[\begin{matrix} μ_{1} (i) X_{i} + μ_{2} (i) L_{i} & μ_{3} (i) & 0 & 0 & 0 \end{matrix}]}^{T} F_{i}^{T} {[\begin{matrix} Δ_{i} \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \end{matrix}

\begin{matrix} + [\begin{matrix} 0 \\ 0 \\ 0 \\ Δ_{i} \\ 0 \end{matrix}] F_{i} [\begin{matrix} μ_{4} (i) X_{i} + μ_{5} (i) L_{i} & μ_{6} (i) & 0 & 0 & 0 \end{matrix}] \\ + {[\begin{matrix} μ_{4} (i) X_{i} + μ_{5} (i) L_{i} & μ_{6} (i) & 0 & 0 & 0 \end{matrix}]}^{T} F_{i}^{T} {[\begin{matrix} 0 \\ 0 \\ 0 \\ Δ_{i} \\ 0 \end{matrix}]}^{T} \end{matrix}

\begin{matrix} \leq η_{i} [\begin{matrix} Δ_{i} Δ_{i}^{T} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] + η_{i}^{- 1} [\begin{matrix} X_{i}^{T} μ_{1}^{T} (i) + L_{i}^{T} μ_{2}^{T} (i) \\ μ_{3}^{T} (i) \\ 0 \\ 0 \\ 0 \end{matrix}] \end{matrix} [\begin{matrix} μ_{1} (i) X_{i} + μ_{2} (i) L_{i} & μ_{3} (i) & 0 & 0 & 0 \end{matrix}]

\begin{matrix} + Σ_{i} [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & Δ_{i} Δ_{i}^{T} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] + Σ_{i}^{- 1} [\begin{matrix} X_{i}^{T} μ_{4}^{T} (i) + L_{i}^{T} μ_{5}^{T} (i) \\ μ_{6}^{T} (i) \\ 0 \\ 0 \\ 0 \end{matrix}] \\ [\begin{matrix} μ_{4} (i) X_{i} + μ_{5} (i) L_{i} & μ_{6} (i) & 0 & 0 & 0 \end{matrix}] \end{matrix}

\begin{matrix} = [\begin{matrix} η_{i} Δ_{i} Δ_{i}^{T} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & Σ_{i} Δ_{i} Δ_{i}^{T} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ + η_{i}^{- 1} [\begin{matrix} X_{i}^{T} μ_{1}^{T} (i) + L_{i}^{T} μ_{2}^{T} (i) \\ μ_{3}^{T} (i) \\ 0 \\ 0 \\ 0 \end{matrix}] \\ [\begin{matrix} μ_{1} (i) X_{i} + μ_{2} (i) L_{i} & μ_{3} (i) & 0 & 0 & 0 \end{matrix}] \end{matrix}

\begin{matrix} + Σ_{i}^{- 1} [\begin{matrix} X_{i}^{T} μ_{4}^{T} (i) + L_{i}^{T} μ_{5}^{T} (i) \\ μ_{6}^{T} (i) \\ 0 \\ 0 \\ 0 \end{matrix}] \\ [\begin{matrix} μ_{4} (i) X_{i} + μ_{5} (i) L_{i} & μ_{6} (i) & 0 & 0 & 0 \end{matrix}] \end{matrix}

= {\hat{Ξ}}_{2 i}

(44)

Hence, the following inequality implies that (43) holds

Ξ_{1 i} + {\hat{Ξ}}_{2 i} < 0 .

(45)

Then, according to Lemma 2, it can be easily seen that the resulting (45) is equivalent to (36), that is, (36) can guarantee that (12) in Theorem 1 holds.

The next is to prove that the conditions (14) and (15) in Theorem 1 can be guaranteed by (34) and (35). Note that $P_{i} = X_{i}^{- 1}$ , (14) is equivalent to

X_{i}^{T} E_{i}^{T} = X_{i}^{T} E_{i}^{T} R_{i}^{\frac{1}{2}} Q_{1 i} R_{i}^{\frac{1}{2}} E_{i} X_{i}

(46)

Therefore, we only need to find the suitable matrices $Q_{1 i} > 0$ to satisfy (46). Letting $M_{Ri}^{- 1} X_{i} M_{Li}^{T} = [\begin{matrix} {\hat{X}}_{1 i} & {\hat{X}}_{2 i} \\ {\hat{X}}_{3 i} & {\hat{X}}_{4 i} \end{matrix}]$ , and taking into account $0 \leq X_{i}^{T} E_{i}^{T} = E_{i} X_{i}$ , one can calculate ${\hat{X}}_{1 i} = {\hat{X}}_{1 i}^{T} \geq 0$ and ${\hat{X}}_{2 i} = 0$ . Thus, (46) holds when making $Q_{1 i} = R_{i}^{- \frac{1}{2}} M_{Li}^{T} [\begin{matrix} {\hat{X}}_{1 i} & 0 \\ 0 & W_{2 i} \end{matrix}] {{}^{- 1}M}_{Li} R_{i}^{- \frac{1}{2}}$ with the arbitrary positive definite matrices $W_{2 i}$ . Moreover, ${\hat{X}}_{1 i}$ are expressed as the following form

\begin{matrix} W_{1 i} = {\hat{X}}_{1 i} = [\begin{matrix} I_{r} & 0 \end{matrix}] M_{Ri}^{- 1} X_{i} M_{Li}^{T} {[\begin{matrix} I_{r} & 0 \end{matrix}]}^{T} \\ = [\begin{matrix} I_{r} & 0 \end{matrix}] M_{Li} E_{i} X_{i} M_{Li}^{T} {[\begin{matrix} I_{r} & 0 \end{matrix}]}^{T} \end{matrix}

(47)

Hence, the condition (14) is satisfied. On the other hand, it can be seen from (34) that $I < Q_{1 i} < {ζ_{i}}^{- 1}$ I, which together with (35) can result in (15). This completes the proof.

Next, based on Theorem 2, we aim to design a robust and non-fragile finite-time $H_{\infty}$ state feedback controller to guarantee the finite-time $H_{\infty}$ performance of NSMJS (8).

Theorem 3. Given scalars $T > 0, d > 0, α \geq 0$ , $c_{1} > 0$ and matrices $R_{i} > 0$ , the closed-loop NSMJS (8) controlled by a non-fragile state feedback controller with ${\bar{K}}_{i} = L_{i} X_{i}^{- 1} + Δ_{ki}$ is SS $H_{\infty}$ FTB with respect to $(c_{1}, c_{2}, T, R_{i}, γ, d)$ , if there exist nonsingular matrices $X_{i}$ , $Q_{1 i}$ , $M_{Li}$ and $M_{Ri}$ , positive symmetric matrices $W_{2 i}$ , $matrices L_{i}$ , positive scalars $γ$ , $c_{2}$ , $ε_{i}$ , $Σ_{i}$ , $η_{i}$ , $φ_{i}$ , $ζ_{i}$ and $δ_{i}$ , such that conditions (33), (34) and (35) and the following constraints hold for each mode $i \in S$

[\begin{matrix} Ξ_{i} + δ_{i} Ψ_{i}^{T} Ψ_{i} & Θ_{i}^{T} \\ * & - δ_{i} I \end{matrix}]

(48)

where

Ψ_{i} = {\hat{Δ}}_{i}^{T} [\begin{matrix} B_{i}^{T} & 0 & B_{ei}^{T} & D_{i}^{T} & 0 & μ_{2}^{T} (i) & μ_{5}^{T} (i) \end{matrix}],

Θ_{i} = μ_{7 i} [\begin{matrix} X_{i} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

Proof: Replacing $L_{i}$ in Theorem 2 with $L_{i} + Δ_{Ki} X_{i}$ , and noting that $Δ_{Ki} = {\hat{Δ}}_{i} F_{i} μ_{7} (i)$ , condition (36) can be rewritten as

Ξ_{i} + Ψ_{i}^{T} Ω_{i} + Ω_{i}^{T} Ψ_{i} \leq 0

(49)

where $Ω_{i} = F_{i} μ_{7 i} [\begin{matrix} X_{i} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$

According to Lemma 1, it yields

Ψ_{i}^{T} Ω_{i} + Ω_{i}^{T} Ψ_{i} \leq δ_{i} Ψ_{i}^{T} Ψ_{i} + δ_{i}^{- 1} Ω_{i}^{T} Ω_{i}

\leq δ_{i} Ψ_{i}^{T} Ψ_{i} + δ_{i}^{- 1} Θ_{i}^{T} Θ_{i},

(50)

which implies that the following inequality can guarantee that (49) holds

Ξ_{i} + δ_{i} Ψ_{i}^{T} Ψ_{i} + δ_{i}^{- 1} Θ_{i}^{T} Θ_{i} \leq 0

(51)

It follows from Lemma 2 that (51) is equivalent to (48). The proof of Theorem 3 is completed.

Numerical examples

In this section, two numerical examples are provided to illustrate the effectiveness of the obtained results.

Example 1. In order to compare Theorem 2 with the existing results of Zhang et al. (2012) and Zhao et al. (2019). Consider the systems (8) without nonlinear as follows

\begin{matrix} E_{1} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], A_{1} = [\begin{matrix} - 0.8 & 1.5 \\ 2 & 3 \end{matrix}], B_{1} = [\begin{matrix} - 1 & 0.2 \\ 0.5 & - 0.1 \end{matrix}], \\ G_{1} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}], Δ_{1} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.1 \end{matrix}] \end{matrix}

\begin{matrix} E_{2} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], A_{2} = [\begin{matrix} - 2 & 1.2 \\ 1 & 4 \end{matrix}], B_{2} = [\begin{matrix} - 1 & 1 \\ 0.5 & - 2 \end{matrix}], \\ G_{1} = [\begin{matrix} 0.2 \\ 0.3 \end{matrix}], Δ_{1} = [\begin{matrix} 0.1 & 0.02 \\ 0 & 0.1 \end{matrix}] \end{matrix}

C_{1} = C_{2} = [\begin{matrix} 1 & 1 \end{matrix}], D_{1} = D_{2} = [\begin{matrix} 0.1 & 0.2 \end{matrix}], H_{1} = H_{2} = [\begin{matrix} 0.1 \end{matrix}]

The transition rate matrix is given by $Π = [\begin{matrix} - 1.2 & 1.2 \\ 1 & - 1 \end{matrix}]$ .

Choose $R_{1} = R_{2} = I_{2}$ , $T = 2$ , $c_{1} = 1$ , $d = 1$ and $α = 2.8$ . By applying the Matlab Toolbox, solving the optimization problem without considering the existence of nonlinear terms in the systems (8) in Theorem 2, we can get the optimal values $c_{2} = 8.1991$ .

Table 1 presents the optimal value $c_{2}$ of different methods, which shows that the result obtained by Theorem 2 is smaller than those in Zhang et al. (2012) and in Corollary 18 of Zhao et al. (2019). Therefore, it can be seen that our obtained results are much better than existing articles (Zhang et al., 2012; Zhao et al., 2019).

Table 1.

Comparison of optimal value $c_{2}$ .

(Zhang et al., 2012)	(Corollary 18 of Zhao et al., 2019)	Theorem 2
$18.3111$	$11.3967$	$8.1991$

Example 2. Consider a practical circuit model containing a Chua’s diode as shown in Figure 1, which can be described as follows

{\begin{matrix} ω (t) + u (t) = u_{R} (t) + u_{c} (t) + u_{L} (t), \\ u_{R} (t) = i (t) R, \\ u_{L} (t) = L_{i} \frac{di (t)}{dt}, \\ {\bar{α}}_{i} i_{c} (t) = \frac{1}{c_{i}} i_{c} (t) = \frac{d u_{c} (t)}{dt}, \\ i (t) = i_{c} (t) + f (u_{c}), \\ f (u_{c}) = b u_{c} + 0.5 (a - b) {| u_{c} + 1 | - | u_{c} - 1 |} . \end{matrix}

Taking into account the uncertainty of parameters, the above system can be converted to a two-mode NSMJS

{\begin{matrix} \begin{matrix} E_{i} \overset{\cdot}{x} (t) = {\bar{A}}_{i} x (t) + {\bar{B}}_{i} u (t) + {\bar{G}}_{i} w (t) + v (r_{t}), \\ z (t) = {\bar{C}}_{i} x (t) + {\bar{D}}_{i} u (t) + {\bar{H}}_{i} w (t), \end{matrix} \end{matrix}

where x(t)= ${[\begin{matrix} i (t) & u_{c} (t) & u_{L} (t) \end{matrix}]}^{T}$ , ${\bar{A}}_{i}$ , ${\bar{B}}_{i}$ , ${\bar{G}}_{i}$ , ${\bar{C}}_{i}$ , ${\bar{D}}_{i}$ , ${\bar{H}}_{i}$ are given by (5), and the other parameters are chosen as follows

\begin{matrix} E_{i} = [\begin{matrix} L_{i} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], A_{i} = [\begin{matrix} 0 & 0 & 1 \\ {\bar{α}}_{i} & 0 & 0 \\ R & 1 & 1 \end{matrix}], B_{i} = B = [\begin{matrix} 0 \\ 0 \\ - 1 \end{matrix}], \\ G_{i} = G = [\begin{matrix} 0 \\ 0 \\ - 1 \end{matrix}], ϑ_{i} = {[\begin{matrix} 0 & - \frac{1}{c_{i}} f^{T} (u_{c}) & 0 \end{matrix}]}^{T}, \end{matrix}

\begin{matrix} C_{1} = [\begin{matrix} - 1 & 0 & 0.75 \\ 0 & - 1 & 0.75 \\ 0 & 0 & - 1 \end{matrix}], D_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], H_{1} = [\begin{matrix} 0.1 \\ 0.1 \\ 0.1 \end{matrix}], \\ C_{2} = [\begin{matrix} - 1 & 0.5 & - 1 \\ 0 & - 0.75 & 0 \\ 0 & - 1 & 0 \end{matrix}], D_{2} = [\begin{matrix} 0 \\ 0.5 \\ 0 \end{matrix}], H_{2} = [\begin{matrix} 0.2 \\ 0.2 \\ 0.1 \end{matrix}], \end{matrix}

\begin{matrix} {\hat{Δ}}_{1} = {\hat{Δ}}_{2} = [- \begin{matrix} 0.1 & 0.5 & 0.2 \end{matrix}], μ_{7} (1) = μ_{7} (2) \\ = [\begin{matrix} 1.5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix}], F_{1} = F_{2} = \sin (t) \end{matrix}

According to (3), choose

\begin{matrix} A_{e 1} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0.4 & 0 \\ 0 & 0 & 0 \end{matrix}], A_{e 2} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0 \end{matrix}], \\ B_{e 1} = B_{e 2} = G_{e 1} = G_{e 2} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T} \end{matrix}

In addition, use ${\bar{α}}_{1} = 1$ , ${\bar{α}}_{2} = 0.5$ , $L_{1} = 1$ , $L_{2} = 2$ , $R = 1$ , $a = - 1$ , $b = - 0.2$ , $c_{1} = 1$ , $c_{2} = 2$ , the transition rate matrix is given by $Π = [\begin{matrix} - 3 & 3 \\ 2 & - 2 \end{matrix}]$ .

Let $R_{1} = R_{2} = I_{3}$ , $T = 2$ , $c_{1} = 1$ , $d = 1$ and $α = 0 .$ 1. By applying the Matlab LMI toolbox, solving the optimization problem in Theorem 3, we can get $c_{2} = 2.8117$ , the $H_{\infty}$ performance index $γ = 0.9840$ , and

\begin{matrix} X_{1} = [\begin{matrix} 0.3269 & 0 & 0 \\ 0 & 0.3269 & 0 \\ - 0.6832 & 0.3642 & - 1.0544 \end{matrix}], \\ X_{2} = [\begin{matrix} 0.2151 & 0 & 0 \\ 0 & 0.2151 & 0 \\ - 1.2723 & 0.1619 & - 2.2953 \end{matrix}], \end{matrix}

\begin{matrix} L_{1} = [\begin{matrix} 0.8362 & - 0.2710 & 1.7540 \end{matrix}], \\ L_{2} = [\begin{matrix} - 0.1514 & 0.3458 & 4.0304 \end{matrix}] \end{matrix}

Thus, the designed robust state feedback controller gains can be obtained as follows

\begin{matrix} K_{1} = [- 0.9287 1.0243 - 1.6634], \\ K_{2} = [- 11.0887 2.9289 - 1.7559] \end{matrix}

Now, given initial mode $r_{0} = 1$ and initial state $x (0) = {[\begin{matrix} - 0.6 & 0.8 & 0.63 \end{matrix}]}^{T}$ . Figures 2 –5 show the jumping mode, the history of $E {x^{T} (t) E_{i}^{T} R_{i} E_{i} x (t)}$ , the state response and the output response of the closed-loop NSMJSs (8), respectively, which illustrate the system (8) is SS $H_{\infty}$ FTB with respect to $(1, 2.8117, 2, I_{3}, 0.9840, 1)$ .

Conclusions

In this paper, the robust and non-fragile finite-time $H_{\infty}$ control problem for a class of mode-dependent NSMJSs with parameter uncertainties and time-varying norm-bounded disturbance have been illustrated. The systems under consideration are more general than the system with mode-independent singular matrix, which can be seen as special cases of the ones investigated here. Two examples are given to illustrate the effective of the proposed results in this paper. The problem studied in this paper is based on the fact that the transition probabilities of Markovian jump system are fully known. However, in practice, owing to all kinds of complex factors, the exact values of transition probabilities cannot be completely available. For example, due to the packet loss and channel delays in the network control system, it may be more expensive to obtain complete known transition probabilities. Therefore, the study of the $H_{\infty}$ control problem of NSMJSs with unknown transition probabilities becomes more important. In the future, we will apply the method proposed in this paper to study more general systems such as NSMJSs with partly unknown transition probabilities (Cheng et al., 2015; Li and Zhang, 2016) and generally uncertain transition probabilities (Gao et al., 2017; Kao et al., 2014; Yang et al., 2019).

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 (No. 61673277).

ORCID iD

Lin Li

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