Open Access

Finite-time \({H_{\infty}}\) memory state feedback control for uncertain singular TS fuzzy time-delay system under actuator saturation

Advances in Difference Equations20162016:52

https://doi.org/10.1186/s13662-016-0763-0

Received: 25 October 2015

Accepted: 19 January 2016

Published: 16 February 2016

Abstract

In this paper, the problem of finite-time \({H_{\infty}}\) memory feedback control for singular T-S fuzzy systems is addressed. Conditions are obtained to guarantee that the closed-loop system is finite-time bounded with a prescribed \({H_{\infty}}\) performance γ. The considered memory controller can be obtained by solving the LMIs. In addition, the estimation of the largest domain of attraction of the closed-loop system can be solved by solving an optimization problem. Finally, examples illustrate the feasibility of the proposed method.

Keywords

finite time memory state feedback control \({H_{\infty}}\) control singular TS fuzzy system actuator saturation

1 Introduction

In recent years, there has been growing attention to singular systems, because of their extensive applications in many practical systems, for example, electrical circuits, power systems, networks, and other systems [1, 2]. Time delays are frequently encountered in various engineering systems such as aircraft, chemical processes, economics, networks, communication, and biological systems. It has been shown that the existence of time delays is often one of the main causes of instability and poor performance in a system. Therefore, the singular time-delay system has received considerable attention [14].

T-S fuzzy models have been widely studied because they can represent a wide class of nonlinear systems, especially the singular T-S fuzzy model. Many valuable stability analysis and control synthesis results for singular T-S fuzzy systems can be available, for example, memory dissipative control, memory \({H_{\infty}}\) control, and \({H_{\infty}}\) filters were studied in [57], respectively. In [8], the problem of delay-dependent dissipative control was discussed for a class of nonlinear system via a descriptor T-S fuzzy model.

In practice, actuator saturation is very ubiquitous, which is a main cause of poor performance of the closed-loop systems and sometimes it may lead to the system being unstable [911]. Robust \({H_{\infty}}\) static output feedback stabilization and robust stabilization for T-S fuzzy system subject to actuator saturation were discussed in [9] and [10], respectively. Yang and Tong [11] put forward the problem of output feedback robust stabilization of switched fuzzy systems with time delay and actuator saturation. For a singular T-S fuzzy system subject to actuator saturation, the reader may refer to [5, 6]. Control of time-delay fuzzy descriptor systems with actuator saturation was demonstrated in [12]. Furthermore, an external disturbance is often the source of instability and poor performance of systems. The \({H_{\infty}}\) control technique is used to minimize the effects of the external disturbances, \({H_{\infty}}\) control for fuzzy systems is addressed in [3, 4, 6, 7, 9].

In addition, a memory state feedback controller with input constraints yields less conservative sufficient conditions in terms of LMIs and allows for a wider feasible region of numerical optimization [13]. In [14], a new stabilization condition for T-S fuzzy systems with time delay was obtained by the memory state feedback controller. In [15], memory state feedback control for singular systems with multiple internal incommensurate constant point delays was demonstrated. In [16], analysis and synthesis of memory-based fuzzy sliding mode controllers were discussed.

In some practical engineering applications, the finite-time control is of practical significance. If the system state does not exceed a prescribed region during a fixed time interval, it is said to have finite-time stability \((FTB)\). It is well recognized that finite-time stability is different from Lyapunov asymptotical stability [1720]. For a singular system, there are few articles considering finite-time control. See [21], Observer-based finite-time \({H_{\infty}}\) control for discrete singular stochastic systems was discussed. Ma et al. discussed the problem of finite-time \({H_{\infty}}\) control for a class of discrete-time switched singular time-delay systems subject to actuator saturation in [22]. For switched singular linear system, Wang et al. used an average dwell time approach to study the problem of finite-time stabilization in [23]. However, so far, for singular T-S fuzzy time-delay system subject to actuator saturation, one has an open area for the study of finite-time control.

Motivated by the above discussion, in this paper, the problem of finite-time \({H_{\infty}}\) memory feedback control for a singular T-S fuzzy system is demonstrated. The main contributions of this paper can be listed as follows: (1) conditions are obtained to guarantee that the closed-loop system is not only regular, impulse-free, finite-time bounded but also satisfying the presided \({H_{\infty}}\) performance γ; (2) the considered memory controller can be obtained by solving the LMIs; (3) the estimation of the largest domain of attraction of the closed-loop system can be solved by an optimization problem; (4) examples illustrate the feasibility of the proposed method; (5) the domain of attraction is simulated in Figure 2.

Notation

Throughout this paper, \({{\mathrm{R}}^{n}}\) denotes the n-dimensional Euclidean space, and \({{\mathrm{R}}^{n \times m}}\) is the set of real matrices. For \(A \in{{{\mathrm {R}} } ^{n \times m}}\), \({A^{ - 1}}\)and \({A^{\mathrm {T}}}\)denote the matrix inverse and matrix transpose, respectively. \(\lambda ( A )\) means the eigenvalue of A. For a real symmetric matrix \(A \in{{\mathrm{R}}^{n \times n}}\), \(A > 0\) (\(A \ge0\)) means that A is positive definite (positive semi-definite). The symbol means the symmetric term in a symmetric matrix.

2 Preliminaries

Consider the following singular TS fuzzy model:

Plant rule i: IF \({\theta_{1}}(t)\) is \({M_{i1}}\) and \({\theta_{2}}(t)\) is \({M_{i2}}\cdots{\theta_{p}}(t)\) is \({M_{ip}}\), THEN
$$ \begin{aligned} &E\dot{x}(t) = {{\bar{A}}_{i}}x(t) + {{\bar{A}}_{di}}x\bigl(t - d(t)\bigr) + {{\bar{B}}_{i}} \operatorname{sat} \bigl( {u ( t )} \bigr) + {{\bar{B}}_{\omega i}}\omega ( t ), \\ &z ( t ) = {{\bar{C}}_{i}}x ( t ) + {{\bar{C}}_{di}}x \bigl( {t - d(t)} \bigr) + {{\bar{D}}_{i}}\operatorname{sat} \bigl( {u ( t )} \bigr) + {{\bar{D}}_{\omega i}}\omega ( t ) , \\ &x(t) = \phi(t),\quad t \in[ - d,0], \end{aligned} $$
(1)
where \(\theta(t) = { [ {{{\theta_{1}}(t)} \ {{\theta_{2}}(t)} \ {\cdots} \ {{\theta_{p}}(t)} } ]^{\mathrm{T}}}\) is premise variable, \(i \in\Re: = \{ {1,2,\ldots,r} \}\), r is the number of IF-THEN rules, \({M_{ik}}\) (\(i = 1,2,\ldots,r\), \(k = 1,2,\ldots,p\)) is the fuzzy set. \(x ( t ) \in{{\mathrm{R}}^{n}}\) is the state vector, \(\omega ( t ) \in{{\mathrm{R}}^{q}}\) is the disturbance input which belongs to \({L_{2}}[ {0,\infty} )\). \(z ( t ) \in{{\mathrm{R}}^{p}}\) is the control output, \(\phi ( t )\) is the initial condition of the system. \(d(t)\) is a time-varying continuous function that satisfies \(0 \le d(t) \le d\) and \(\dot{d}(t) \le h\), \(h < 1\). E is a constant matrix satisfying \(\operatorname{rank} ( E ) \le n\). \(u ( t ) \in{{\mathrm{R}}^{l}}\) is the control input, and \(\operatorname{sat}:\mathrm{R}^{l} \to\mathrm{R}^{l}\) is the standard saturation function defined as follows:
$$\operatorname{sat} \bigl( {u ( t )} \bigr) = { \bigl[ {\operatorname{sat} \bigl( {{u_{1}} ( t )} \bigr), \ldots, \operatorname{sat} \bigl( {{u_{l}} ( t )} \bigr)} \bigr]^{\mathrm{T}}}, $$
without loss of generality, \(\operatorname{sat} ( {{u_{i}} ( t )} ) = \operatorname{sign} ( {{u_{i}} ( t )} )\min \{ {1,\vert {{u_{i}} ( t )} \vert } \}\). Here the notation of \(\operatorname{sat} ( \cdot )\) is abused to denote the scalar values and the vector valued saturation functions. For a positive scalar b and time scalar T, \(\int_{0}^{{T}} {{\omega^{\mathrm{T}}} ( t )\omega ( t )} \le b\). \({\bar{A}_{i}} = {A_{i}} + \Delta{A_{i}}\), \({\bar{A}_{di}} = {A_{di}} + \Delta {A_{di}}\), \({\bar{B}_{i}} = {B_{i}} + \Delta{B_{i}}\), \({\bar{B}_{\omega i}} = {B_{\omega i}} + \Delta{B_{\omega i}}\), \({\bar{C}_{i}} = {C_{i}} + \Delta{C_{i}}\), \({\bar{C}_{di}} = {C_{di}} + \Delta {C_{di}}\), \({\bar{D}_{i}} = {D_{i}} + \Delta{D_{i}}\), \({\bar{D}_{\omega i}} = {D_{\omega i}} + \Delta{D_{\omega i}}\). \({A_{i}}\), \({A_{di}}\), \({B_{i}}\), \({B_{\omega i}}\), \({C_{i}}\), \({C_{di}}\), \({D_{i}}\), \({D_{\omega i}}\) are known real constant matrices with appropriate dimensions; \(\Delta{A_{i}}\), \(\Delta{A_{di}}\), \(\Delta{B_{i}}\), \(\Delta {B_{\omega i}}\), \(\Delta{C_{i}}\), \(\Delta{C_{di}}\), \(\Delta{D_{i}}\), \(\Delta{D_{\omega i}}\) are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form
$$ \begin{bmatrix} {\Delta{A_{i}}} & {\Delta{A_{di}}} & {\Delta{B_{i}}} & {\Delta {B_{\omega i}}} \\ {\Delta{C_{i}}} & {\Delta{C_{di}}} & {\Delta{D_{i}}} & {\Delta {D_{\omega i}}} \end{bmatrix} = \begin{bmatrix} {{H_{1i}}} \\ {{H_{2i}}} \end{bmatrix} \Delta [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} {{E_{1i}}} & {{E_{2i}}} & {{E_{3i}}} & {{E_{4i}}} \end{array}\displaystyle ], $$
(2)
where \({H_{1i}}\), \({H_{2i}}\), \({E_{1i}}\), \({E_{2i}}\), \({E_{3i}}\), \({E_{4i}}\) are known real constant matrices with appropriate dimensions and Δ is for unknown real and possibly time-varying matrices satisfying \({\Delta^{\mathrm{T}}}\Delta \le I\).
Using a singleton fuzzifier, product inference, and a center-average defuzzifier, the global dynamics of the TS system (1) is described by the convex sum form:
$$ \begin{aligned} &E\dot{x}(t) = \sum_{i = 1}^{r} {{h_{i}}\bigl(\theta(t)\bigr)} \bigl[ {{{\bar{A}}_{i}}x(t) + {{\bar{A}}_{di}}x\bigl(t - d(t)\bigr) + {{\bar{B}}_{i}} \operatorname{sat} \bigl( {u ( t )} \bigr) + {{\bar{B}}_{\omega i}}\omega ( t )} \bigr], \\ &z ( t ) = \sum_{i = 1}^{r} {{h_{i}}\bigl(\theta(t)\bigr)} \bigl[ {{{\bar{C}}_{i}}x ( t ) + {{\bar{C}}_{di}}x \bigl( {t - d(t)} \bigr) + {{\bar{D}}_{i}}\operatorname{sat} \bigl( {u ( t )} \bigr) + {{\bar{D}}_{\omega i}}\omega ( t )} \bigr], \\ &x(t) = \phi(t),\quad t \in[ - d,0], \end{aligned} $$
(3)
where \({h_{i}}(\theta(t)) = {{{\beta_{i}}(\theta(t))} /{\sum_{i = 1}^{r} {{\beta_{i}}(\theta(t))} }}\), \({\beta_{i}}(\theta(t)) = \prod_{j = 1}^{p} {{M_{ij}} ( {{\theta_{j}} ( t )} )} \), and \({M_{ij}} ( {{\theta_{j}} ( t )} )\) is the grade of membership of \({\theta_{j}}(t)\) in \({M_{ij}}\). It is easy to see that \({\beta_{i}}(\theta(t)) \ge0\) and \(\sum_{i = 1}^{r} {{\beta_{i}}(\theta(t)) \ge0} \). Hence, we have \({h_{i}}(\theta(t)) \ge 0\) and \(\sum_{i = 1}^{r} {{h_{i}}(\theta(t))} = 1\). In the sequel, for brevity we use \({h_{i}}\) to denote \({h_{i}}(\theta(t))\).
Consider the memory state feedback fuzzy controller:
$$ u(t) = \sum_{i = 1}^{r} {{h_{i}}\bigl(\theta(t)\bigr) \bigl[ {{K_{i}}x(t) + {K_{di}}x\bigl(t - d ( t )\bigr)} \bigr],} $$
(4)
where the memoryless state feedback gain \({K_{i}}\) and the memory state feedback gain \({K_{di}}\) are matrices to be determined with appropriate dimensions.

Define the following subsets of \({{\mathrm{R}}^{n}}\).

Let \(P \in{{\mathrm{R}}^{n \times n}}\) be a symmetric matrix, ρ be a scalar. Denote
$$\varepsilon \bigl( {{E^{\mathrm{T}}}PE,\rho} \bigr) = \bigl\{ {x ( t ) \in{{ \mathrm{R}}^{n}}:{x^{\mathrm{T}}} ( t ){E^{\mathrm{T}}}PEx ( t ) \le \rho} \bigr\} . $$
For matrices \({H_{i}}\), \({H_{di}}\), \({h_{ik}}\), \({h_{dik}}\) are the kth row of the matrix \({H_{i}}\) and \({H_{di}}\), respectively, we define
$$L ( {{H_{i}}, {H_{di}}} ) = \bigl\{ {x ( t ) \in {{ \mathrm{R}}^{n}}:\bigl\vert {{h_{ik}}x ( t ) + {h_{dik}}x \bigl( {t - \tau ( t )} \bigr)} \bigr\vert \le1, k \in [ {1,l} ]} \bigr\} . $$
Thus \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} )\) is an ellipsoid and \(L ( {{H_{i}}, {H_{di}}} )\) is a polyhedral consisting of states for which the saturation does not occur.

Let D be the set of \(l \times l\) diagonal matrices whose diagonal elements are either 1 or 0. Suppose each element of D is labeled as \({E_{s}}\), \(s = 1,2, \ldots, \eta = {2^{l}}\), and denote \(E_{s}^{-} = I - {E_{s}}\). Clearly, if \({E_{s}} \in D\), then \({E_{s}}^{-} \in D\).

Lemma 1

([24])

Let \(F,H \in{{\mathrm{R}}^{p \times n}}\). Then for any \(x ( t ) \in L ( H )\),
$$\operatorname{sat} \bigl( {Fx ( t )} \bigr) \in \operatorname{co} \bigl\{ {{E_{s}}Fx ( t ) + {E_{s}}^{-} Hx ( t ), s = 1,2, \ldots,\eta} \bigr\} ; $$
or, equivalently,
$$\operatorname{sat} \bigl( {Fx ( t )} \bigr) = \sum_{s = 1}^{\eta}{{\alpha_{s}} \bigl( {{E_{s}}F + E_{s}^{-} H} \bigr)} x ( t ), $$
where co stands for the convex hull, \({\alpha_{s}}\) for \(s =1,2, \ldots,\eta\) are some scalars which satisfy \(0 \le{\alpha_{s}} \le1\) and \(\sum_{s = 1}^{\eta}{{\alpha_{s}}} = 1\).

Lemma 2

For any constant matrices \({N_{1}},{N_{2}} \in {{\mathrm{R}}^{n \times n}}\), \(L \in{{\mathrm{R}}^{n \times p}}\), positive-definite symmetric matrix \(Z \in{{\mathrm{R}}^{n \times n}}\), and time-varying delay \(d ( t )\), we have
$$ - \int_{t - d ( t )}^{t} {{{\dot{x}}^{\mathrm{T}}} ( s ){E^{\mathrm{T}}}ZE\dot{x} ( s )}\,ds \le{\xi ^{\mathrm{T}}} ( t ) \bigl\{ {\Pi + d ( t ){Y^{\mathrm{T}}} {Z^{ - 1}}Y} \bigr\} \xi ( t ), $$
(5)
where
$$\begin{aligned}& Y = [ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} {{N_{1}}} & {{N_{2}}} & L \end{array}\displaystyle } ], \qquad {\xi^{\mathrm{T}}} ( t ) = \bigl[ { \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} {{x^{\mathrm{T}}} ( t )} & {{x^{\mathrm{T}}} ( {t - d ( t )} )} & {{\omega^{\mathrm{T}}} ( t )} \end{array}\displaystyle } \bigr], \\& \Pi = \begin{bmatrix} {N_{1}^{\mathrm{T}}E + {E^{\mathrm{T}}}{N_{1}}} & { - N_{1}^{\mathrm{T}}E - {E^{\mathrm{T}}}{N_{1}}} & {{E^{\mathrm{T}}}L} \\ {*} & { - N_{2}^{\mathrm{T}}E - {E^{\mathrm{T}}}{N_{2}}} & { - {E^{\mathrm {T}}}L} \\ {*} & * & 0 \end{bmatrix}. \end{aligned}$$

Proof

Let \(C = \bigl[ {\scriptsize\begin{matrix}{} {{Z^{{1 /2}}}} & {{Z^{{1/ 2}}}Y} \cr 0 & 0 \end{matrix}} \bigr]\), then \({C^{\mathrm{T}}}C = \bigl[ {\scriptsize\begin{matrix}{} Z & Y \cr {{Y^{\mathrm{T}}}} & {{Y^{\mathrm{T}}}{Z^{ - 1}}Y} \end{matrix}} \bigr] \ge0\). It follows that
$$ { \int_{t - d ( t )}^{t} \left .\begin{bmatrix} {E\dot{x} ( s )} \\ {\xi ( t )} \end{bmatrix} \right .^{\mathrm{T}}} \begin{bmatrix} Z & Y \\ {{Y^{\mathrm{T}}}} & {{Y^{\mathrm{T}}}{Z^{ - 1}}Y} \end{bmatrix} \begin{bmatrix} {E\dot{x} ( s )} \\ {\xi ( t )} \end{bmatrix}\,ds \ge0. $$
(6)
Notice that \(2\int_{t - d ( t )}^{t} {{\xi ^{\mathrm{T}}} ( t )Y\dot{x} ( s )\,ds = 2{\xi ^{\mathrm{T}}} ( t ){Y^{\mathrm{T}}} [ E \ { - E} \ 0 ]\xi ( t )} \), rearranging (6) yield (5). □

Remark 1

Lemma 2 will play a key role in decreasing the conservatism, which can be seen from Example 1.

Lemma 3

([25])

Let ϒ, Γ, and Λ be real matrices of appropriate dimensions with Λ satisfying \(\Lambda{\Lambda^{\mathrm{T}}} \le I\). Then the following inequality holds for any constant \(\varepsilon > 0\):
$$\Upsilon\Lambda \Gamma + {\Gamma^{\mathrm{T}}} {\Lambda^{\mathrm{T}}} { \Upsilon ^{\mathrm{T}}} \le\varepsilon\Upsilon{\Upsilon^{\mathrm{T}}} + { \varepsilon^{ - 1}} {\Gamma^{\mathrm{T}}}\Gamma. $$

Lemma 4

([26])

For given matrices \(E, X > 0\), Y, if \(( {{E^{\mathrm{T}}}X +Y{\Gamma^{\mathrm{T}}}} )\) is nonsingular, then there exist matrices \(S > 0\), I, such that \(ES + {\mathrm {I}}{{\mathrm {K}}^{\mathrm{T}}} = { ( {{E^{\mathrm {T}}}X + Y{\Gamma^{\mathrm{T}}}} )^{ - 1}}\), where \(X,S \in {{\mathrm{R}}^{n \times n}}\), \(Y,{\mathrm {I}} \in{R^{n \times ( {n - r} )}}\), and \(\Gamma,{\mathrm {K}} \in{R^{n \times ( {n - r} )}}\) are any matrices with full column rank satisfying \({E^{\mathrm{T}}}\Gamma = 0\), \(E{\mathrm {K}} = 0\).

From Lemma 1, for any \(x ( t ) \in L ( {{H_{j}}, {H_{dj}}} )\), denoting , and , then
(7)
then the closed-loop system can be obtained
$$ \begin{aligned} &E\dot{x}(t) = \tilde{A}x(t) + {{\tilde{A}}_{d}}x\bigl(t - d(t)\bigr) + {{\tilde{B}}_{\omega}}\omega ( t ), \\ &z ( t ) = \tilde{C}x ( t ) + {{\tilde{C}}_{d}}x \bigl( {t - d(t)} \bigr) + {{\tilde{D}}_{\omega}}\omega ( t ) , \\ &x(t) = \phi(t),\quad t \in[ - d,0], \end{aligned} $$
(8)
where

Definition 1

([18])

For some positive constants, \({c_{1}}\), b, T and symmetric positive matrix \({R_{c}}\), the closed-loop system (8) is finite-time bounded \({FTB}\) subject to \(( {{c_{1}}\ {c_{2}}\ b\ T\ {R_{c}}} )\), if there exists scalar \({c_{2}} > {c_{1}}\), such that
$$\begin{aligned} &\sup_{ - d \le\theta \le0} \bigl\{ {{x^{\mathrm {T}}} ( \theta ){E^{\mathrm{T}}} {R_{c}}Ex ( \theta ),{{\dot{x}}^{\mathrm{T}}} ( \theta ){E^{\mathrm {T}}} {R_{c}}E\dot{x} ( \theta )} \bigr\} \le{c_{1}} \\ &\quad\Rightarrow\quad {x^{\mathrm{T}}} ( t ){E^{\mathrm {T}}} {R_{c}}Ex ( t ) \le{c_{2}},\quad\forall t_{0} \in [ - d, 0 ], t \in [ 0, T ]. \end{aligned}$$

Definition 2

([18])

For some positive constants, \({c_{1}}\), b, T and symmetric positive matrix \({R_{c}}\), the closed-loop system (8) is finite-time bounded \(({{FT}}{H_{\infty}}{\mathrm{B}})\) subject to \(( {{c_{1}}\ {c_{2}}\ b\ T\ {R_{c}}} )\), if (8) is \({{FTB}}\) with respect to \(( {{c_{1}}\ {c_{2}}\ b\ T\ {R_{c}}} )\) and under the zero-initial condition such that
$$ \int_{0}^{T} {{z^{\mathrm{T}}} ( t )z ( t )\,dt} < { \gamma^{2}} \int_{0}^{T} {{\omega^{\mathrm{T}}} ( t )\omega ( t )\,dt}. $$
(9)

Definition 3

([27])

  1. (i)

    When \(\omega ( t ) = 0\), the continuous-time SMJS (8) is said to be regular in time interval \([0,T]\), if the characteristic polynomial \(\det ({sE - \tilde{A}} )\) is not identically zero for all \(t\in [ {0,T} ]\).

     
  2. (ii)

    When \(\omega ( t ) = 0\), the continuous-time SMJS (8) is said to be impulse-free in time interval \([0,T]\), if \(\deg ( {\det ( {sE - \tilde{A}} )} ) = \operatorname{rank} ( E )\) for all \(t \in [ {0,T} ]\).

     

3 Main results

Theorem 1

For positive constants \({c_{1}}\), b, T, δ and positive definite matrix \({R_{c}}\), the closed-loop system (8) is \(FTB\) subject to \(( {{c_{1}}\ {c_{2}}\ b\ T\ {R_{c}}} )\) at the origin with \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} )\) contained in the domain of attraction, if there exist a constant \({c_{2}} > 0\), positive definite matrices P, \({Q_{1}}\), \({Q_{2}}\) and any matrices \({N_{1}}\), \({N_{2}}\), L with appropriate dimensions, matrix S for \(i,j \in\Re\), and \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} ) \subset L ( {{H_{i}},{H_{di}}} )\) such that
$$\begin{aligned}& \begin{bmatrix} {\Theta + \Pi} & {\Gamma_{1}^{\mathrm{T}}} & {d{Y^{\mathrm{T}}}} \\ {*} & { - {{ ( {d{Q_{2}}} )}^{ - 1}}} & 0 \\ {*} & * & { - {Q_{2}}} \end{bmatrix} < 0, \end{aligned}$$
(10)
$$\begin{aligned}& \biggl( {{\lambda_{2}} + {\lambda_{3}}d + {\lambda _{4}}\frac{{{d^{2}}}}{2}} \biggr){c_{1}} + { \lambda_{5}}b < {\lambda _{1}} {c_{2}} {e^{ - \delta T}}, \end{aligned}$$
(11)
where Π, Y are defined in Lemma 2, and
$$\begin{aligned}& \Theta = \begin{bmatrix} {{\varpi_{11}}} & {P{{\tilde{A}}_{d}}} & {P{{\tilde{B}}_{\omega}}} \\ {*} & { - ( {1 - h} ){Q_{1}}} & 0 \\ {*} & * & { - Q} \end{bmatrix}, \\& {\varpi_{11}} = {\hat{P}^{\mathrm{T}}}\tilde{A} + {\tilde{A}^{\mathrm {T}}}\hat{P} - \delta{E^{\mathrm{T}}}\hat{P} + {Q_{1}}, \qquad {\Gamma_{1}} = [ {\tilde{A}\quad {{\tilde{A}}_{d}}\quad {{\tilde{B}}_{\omega}}} ], \\& {\lambda_{1}} = {\lambda_{\min}} ( {\bar{P}} ),\qquad{ \lambda_{2}} = {\lambda_{\max}} ( {\bar{P}} ),\qquad{ \lambda_{3}} = {\lambda _{\max}} ( {{{\bar{Q}}_{1}}} ),\qquad{\lambda_{4}} = {\lambda_{\max }} ( {{{\bar{Q}}_{2}}} ), \\& {\lambda_{5}} = {\lambda_{\max}} ( Q ), \qquad\hat{P} = { \bigl( {{E^{\mathrm{T}}}P + S{R^{\mathrm{T}}}} \bigr)^{\mathrm {T}}},\qquad {E^{\mathrm{T}}}\hat{P} = {E^{\mathrm{T}}}R_{c}^{{1/2}}\bar{P}R_{c}^{{1 /2}}E, \\& {Q_{1}} = R_{c}^{{1 /2}}{\bar{Q}_{1}}R_{c}^{{1 /2}}, \qquad{Q_{2}} = R_{c}^{{1/2}}{\bar{Q}_{2}}R_{c}^{{1/2}}, \end{aligned}$$
\(R \in{{\mathrm{R}}^{n \times ( {n - r} )}}\) is any matrix with full column rank satisfying \({E^{\mathrm{T}}}R = 0\).

Proof

Firstly, we proof the system (8) with \(w ( t ) = 0\) is regular, impulse-free.

Since \(\operatorname{rank}E = r < n\), there must exist two invertible matrices G and \(H \in{{\mathrm{R}}^{n \times n}}\), then R can be rewritten as \(R = {G^{\mathrm{T}}} \bigl[ {\scriptsize\begin{matrix}{} 0 \cr \Phi \end{matrix}} \bigr]\), where \(\Phi \in{{\mathrm{R}}^{ ( {n - r} ) \times ( {n - r} )}}\). Denote
$$ \begin{aligned} &GEH = \begin{bmatrix} {{I_{r}}} & 0 \\ 0 & 0 \end{bmatrix}, \qquad G\tilde{A}H = \begin{bmatrix} {{A_{1}}} & {{A_{2}}} \\ {{A_{3}}} & {{A_{4}}} \end{bmatrix}, \qquad {G^{ - {\mathrm{T}}}}P{G^{ - 1}} = \begin{bmatrix} {{P_{1}}} & {{P_{2}}} \\ {{P_{3}}} & {{P_{4}}} \end{bmatrix}, \\ &{H^{\mathrm{T}}}S = \begin{bmatrix} {S_{1}} \\ {S_{2}} \end{bmatrix},\qquad {H_{i}}H = [ {{H_{i1}}\quad {H_{i2}}} ],\qquad {H_{di}}H = [ {{H_{di1}}\quad {H_{di2}}} ], \\ &x ( t ) = H \begin{bmatrix} {x_{1}}(t) \\ {x_{2}}(t) \end{bmatrix}. \end{aligned} $$
(12)
Pre- and post-multiplying \({\varpi_{11}} < 0\) by \({H^{\mathrm{T}}}\) and H, respectively, we can get \(A_{4}^{\mathrm{T}}\Phi S_{2}^{\mathrm{T}} + {S_{2}}{\Phi^{\mathrm{T}}}{A_{4}} < 0\), which implies \({A_{4}}\) is nonsingular and thus the pair \((E,\tilde{A})\) is regular and impulse-free. From Definition 3, system (8) is regular and impulse-free.
Choose the Lyapunov function as follows:
$$\begin{aligned} V\bigl(x(t)\bigr) ={}& {x^{\mathrm{T}}} ( t ){E^{\mathrm{T}}}PEx ( t ) + \int_{t - d ( t )}^{t} {{e^{\delta ( {t - s} )}} {x^{\mathrm{T}}} ( s ){Q_{1}}x ( s )}\,ds \\ &{}+ \int_{ - d}^{0} { \int_{t + \theta}^{t} {{e^{\delta ( {t - s} )}} {{\dot{x}}^{\mathrm{T}}} ( s ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( s )} } \,ds\,d\theta. \end{aligned}$$
(13)
Along the trajectories of system (8), the corresponding time derivation of (13) is given by
$$\begin{aligned} \dot{V} \bigl( {x ( t )} \bigr) ={}& 2x^{\mathrm{T}} ( t ){\hat{P}}^{\mathrm{T}}E\dot{x} ( t ) + {x^{\mathrm {T}}}(t){Q_{1}}x(t) \\ &{}- \bigl(1 - \dot{d} ( t )\bigr){e^{\delta d ( t )}} {x^{\mathrm {T}}}\bigl(t - d(t)\bigr){Q_{1}}x\bigl(t - d(t)\bigr) \\ &{}+ d{{\dot{x}}^{\mathrm{T}}} ( t ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( t ) - d \int_{ t - d}^{ t} {{{\dot{x}}^{\mathrm{T}}} ( s ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( s )}\,ds \\ \le{}&\delta V \bigl( {x ( t )} \bigr) + 2{x^{\mathrm{T}}} ( t ){\hat{P}}^{\mathrm{T}}E\dot{x} ( t ) + {x^{\mathrm {T}}}(t){Q_{1}}x(t) \\ &{}- (1 - h){x^{\mathrm{T}}}\bigl(t - d(t)\bigr){Q_{1}}x\bigl(t - d(t)\bigr) \\ &{}+ d{{\dot{x}}^{\mathrm{T}}} ( t ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( t ) - d \int_{ t - d}^{ t} {{{\dot{x}}^{\mathrm{T}}} ( s ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( s )}\,ds - \delta {x^{\mathrm{T}}} ( t ){{\hat{P}}^{\mathrm{T}}}Ex ( t ). \end{aligned}$$
Then, via Lemma 2,
$$- d \int_{ t - d}^{ t} {{{\dot{x}}^{\mathrm{T}}} ( s ){E^{\mathrm{T}}} {Q_{2}}E\dot{x} ( s )}\,ds \le{\xi^{\mathrm{T}}} ( t ) \bigl\{ {\Pi + d{Y^{\mathrm{T}}}Q_{2}^{ - 1}Y} \bigr\} \xi ( t ), $$
we have
$$\dot{V} \bigl( {x ( t )} \bigr) - \delta V \bigl( {x ( t )} \bigr) - { \omega^{\mathrm{T}}} ( t )Q\omega ( t ) = {\xi^{\mathrm{T}}} ( t ) \bigl( { \Theta + d\Gamma_{1}^{\mathrm{T}}{Q_{2}} { \Gamma_{1}} + \Pi + d{Y^{\mathrm{T}}}Q_{2}^{ - 1}Y} \bigr)\xi ( t ). $$
Considering (11) and the Schur complement, it yields
$$ \dot{V} \bigl( {x ( t )} \bigr) - \delta V \bigl( {x ( t )} \bigr) - {\omega^{\mathrm{T}}} ( t )Q\omega ( t ) < 0, $$
(14)
pre- and post-multiplying (14) by \({e^{ - \delta t}}\), and integrating it from 0 to t (\({\forall t \in [0 ,T ]} \)), it follows that
$$V \bigl( {x ( t )} \bigr) < {e^{\delta t}} \biggl[ {V \bigl( {x ( 0 )} \bigr) + \int_{0}^{t} {{e^{ - \delta s}} {\omega ^{\mathrm{T}}} ( s )Q\omega ( s )}\,ds} \biggr]. $$
From these,
$$\begin{aligned} &V \bigl( {x ( 0 )} \bigr) + \int_{0}^{t} {{e^{ - \delta s}} { \omega^{\mathrm{T}}} ( s )Q\omega ( s )}\,ds \\ &\quad\le \biggl( {{\lambda_{2}} + {\lambda_{3}}d + { \lambda_{4}}\frac {{{d^{2}}}}{2}} \biggr)\sup_{ - d \le\theta \le0} \bigl\{ {{x^{\mathrm{T}}} ( \theta ){E^{{\mathrm {T}} } } {R_{c}}Ex ( \theta ),{{\dot{x}}^{\mathrm{T}}} ( \theta ){E^{\mathrm {T}} } {R_{c}}E\dot{x} ( \theta )} \bigr\} + {\lambda_{5}}b \\ &\quad\le \biggl( {{\lambda_{2}} + {\lambda_{3}}d + { \lambda_{4}}\frac {{{d^{2}}}}{2}} \biggr){c_{1}} + { \lambda_{5}}b , \end{aligned}$$
then
$$V ( {{x_{t}}} ) \le{e^{\delta T}} \biggl[ { \biggl( {{ \lambda_{2}} + {\lambda_{3}}d + {\lambda_{4}} \frac{{{d^{2}}}}{2}} \biggr){c_{1}} + {\lambda_{5}}b} \biggr], $$
considering \(V ( {{x_{t}}} ) \ge{\lambda_{1}}{x^{\mathrm{T}}} ( t ){E^{{\mathrm {T}} } }{R_{c}}Ex ( t )\), from condition (11), we have
$${x^{\mathrm{T}}} ( t ){E^{{\mathrm {T}} } } {R_{c}}Ex ( t ) < {c_{2}}. $$

From Definition 1, the closed-loop system (8) is \({FTB}\). This completes the proof. □

Remark 2

In Theorem 1, sufficient conditions are obtained to guarantee that the closed-loop is finite-time bounded. Then Theorem 2 will give finite-time dissipative conditions.

Theorem 2

For positive constants \({c_{1}}\), b, T, δ, positive definite matrix \({R_{c}}\), closed-loop system (8) is \({{FT}}{H_{\infty}}{{B}}\) with respect to \(( {{c_{1}}\ {c_{2}}\ b\ T {R_{c}}} )\) at the origin with \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} )\) contained in the domain of attraction, if there exist constant \({c_{2}} > 0\), and positive definite matrices P, \({Q_{1}}\), \({Q_{2}}\) and any matrices \({N_{1}}\), \({N_{2}}\), L with appropriate dimensions, matrix S, for \(i,j \in\Re\) and \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} ) \subset L ( {{H_{i}},{H_{di}}} )\) such that the following conditions hold:
$$\begin{aligned}& \Psi = \begin{bmatrix} {\Lambda + \Pi} & {\Gamma_{1}^{\mathrm{T}}} & {d{Y^{\mathrm{T}}}} & {\Gamma_{2}^{\mathrm{T}}} \\ {*} & { - {{ ( {d{Q_{2}}} )}^{ - 1}}} & 0 & 0 \\ {*} & * & { - {Q_{2}}} & 0 \\ {*} & * & * & { - I} \end{bmatrix} < 0, \end{aligned}$$
(15)
$$\begin{aligned}& \biggl( {{\lambda_{2}} + {\lambda_{3}}d + {\lambda _{4}}\frac{{{d^{2}}}}{2}} \biggr){c_{1}} + { \gamma^{2}} {e^{ - \delta T}}b < {\lambda_{1}} {c_{2}} {e^{ - \delta T}}, \end{aligned}$$
(16)
where Π, Y, \({\Gamma_{1}}\), and \({\varpi_{11}}\) are defined in Theorem 1, and
$$\Lambda = \begin{bmatrix} {{\varpi_{11}}} & {P{{\tilde{A}}_{d}}} & {P{{\tilde{B}}_{\omega}}} \\ {*} & { - ( {1 - h} ){Q_{1}}} & 0 \\ {*} & * & { - {\gamma^{2}}{e^{ - \delta T}}} \end{bmatrix}, \qquad {\Gamma_{2}} = [ {\tilde{C}} \quad {{{\tilde{C}}_{d}}} \quad {{{\tilde{D}}_{\omega}}} ]. $$

Proof

It is clear that \(\Gamma_{2}^{\mathrm{T}}{\Gamma_{2}} > 0\), then via the Schur complement, we can get from (15) that
$$ \begin{bmatrix} {\Lambda + \Pi} & {d\Gamma_{1}^{\mathrm{T}}} & {d{Y^{\mathrm{T}}}} \\ {*} & { - Q_{2}^{ - 1}} & 0 \\ {*} & * & { - {Q_{2}}} \end{bmatrix} < 0. $$
(17)
Let \(Q = - {\gamma^{2}}{e^{ - \delta T}}I\), by Theorem 1, combing (15) and (17), the closed-loop system (8) is \(FTB\) with respect to \(( {{c_{1}}\ {c_{2}}\ b\ T\ {R_{c}}} )\).
On the other hand, select the same Lyapunov function candidate as Theorem 1 and define the following function:
$$J = \dot{V} ( {{x_{t}}} ) - \delta V ( {{x_{t}}} ) + {z^{\mathrm{T}}} ( t )z ( t ) - {\gamma^{2}} {e^{ - \delta T}} { \omega^{\mathrm{T}}} ( t )\omega ( t ), $$
using the Schur complement, it can be seen from (15) that
$$J = {\xi^{\mathrm{T}}} ( t ) \bigl[ {\Lambda + d\Gamma _{1}^{\mathrm{T}}{Q_{2}} {\Gamma_{1}} + \Pi + d{Y^{\mathrm{T}}}Q_{2}^{ - 1}Y + \Gamma_{2}^{\mathrm{T}}{\Gamma_{2}}} \bigr]\xi ( t ) < 0, $$
similar to the handling method in Theorem 1 and considering the zero initial condition, it is clear that
$$0 < V ( {{x_{t}}} ){e^{ - \delta T}} < \int_{0}^{T} {{e^{ - \delta t}} \bigl[ {{ \gamma^{2}} {e^{ - \delta T}} {\omega^{\mathrm{T}}} ( t )\omega ( t ) - {z^{\mathrm{T}}} ( t )z ( t )} \bigr]\,dt} , $$
then we have
$$\int_{0}^{T} {{z^{\mathrm{T}}} ( t )z ( t )\,dt} < { \gamma^{2}} {e^{ - \delta T}} \int_{0}^{T} {{\omega^{\mathrm{T}}} ( t )\omega ( t )\,dt} , $$
from Definition 2, the closed-loop system (8) is \({{FT}}{H_{\infty}}{{B}}\), and the \({H_{\infty}}\) performance index is \(\bar{\gamma}= {\gamma^{2}}{e^{ - \delta T}}\). This completes the proof. □

Remark 3

Theorem 2 gives the sufficient conditions for the \({{FT}}{H_{\infty}}{{B}}\) of the closed-loop system. However, the conditions (15) and (16) are nonlinear matrix inequalities, which will be transformed into LMIs in Theorem 3.

Theorem 3

For positive constants \({c_{1}}\), \({c_{2}}\), b, T, α, positive definite matrix \({R_{c}}\), if there exist positive definite matrices X, \({\tilde{Q}_{1}}\), \({\tilde{Q}_{2}}\), Ψ, any matrices \({\tilde{N}_{1}}\), \({\tilde{N}_{2}}\), , with appropriate dimensions, constants and \(\mu, {\eta_{\mathrm{1}}} > 0\), \({\eta_{\mathrm{2}}} > 0\), \({\eta_{\mathrm{3}}} > 0\), \({\varepsilon _{ij}} > 0\), \({\chi_{ij}} > 0\) for \(i,j \in\Re\), such that
$$\begin{aligned}& {\Phi_{iis}} < 0, \end{aligned}$$
(18)
$$\begin{aligned}& {\Phi_{ijs}} + {\Phi_{jis}} < 0, \end{aligned}$$
(19)
$$\begin{aligned}& {\eta_{1}} {I_{n}} < R_{c}^{{1/2}}{G^{ - 1}} \begin{bmatrix} \left ( [ {{I_{r}}\quad 0} ]GEX{G^{\mathrm{T}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix} \right ) & {\mathrm{0}} \\ {\mathrm{0}} & \Psi \end{bmatrix} {G^{ - {\mathrm{T}}}}R_{c}^{{1/2}} < {I_{n}}, \end{aligned}$$
(20)
$$\begin{aligned}& {M_{1}} > \eta_{2}^{ - 1}R_{c}^{ - 1}, \end{aligned}$$
(21)
$$\begin{aligned}& {M_{2}} > \eta_{3}^{ - 1}R_{c}^{ - 1}, \end{aligned}$$
(22)
$$\begin{aligned}& \begin{bmatrix} { ( {{\eta_{2}}d + \frac{{{\eta_{3}}{d^{2}}}}{2}} ){c_{1}} + ( {{r^{2}}b - {c_{2}}} ){e^{ - \delta T}}} & {\sqrt{{c_{1}}} } \\ {*} & { - {\eta_{1}}} \end{bmatrix} < 0, \end{aligned}$$
(23)
$$\begin{aligned}& \begin{bmatrix} { - {\rho^{ - 1}}} & {{w_{i}}} & {{w_{di}}} \\ {*} & { - {E^{\mathrm{T}}}XE} & 0 \\ {*} & * & { - {E^{\mathrm{T}}}XE} \end{bmatrix} \le0, \end{aligned}$$
(24)
where
$$\begin{aligned} &{\Phi _{ijs}} = \left [\textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} {{\psi _{11}}} & {{\psi _{12}}} & {{\psi _{13}}} & {{\psi _{14}}} & {d\tilde{N}_{1}^{\mathrm{T}}} & {{\psi _{16}}} & {{H_{1i}}} & {{\psi _{18}}} & {{X^{\mathrm{T}}}} & 0 & 0 \\ {*} & {{\psi _{22}}} & { - \tilde{L}} & {{\psi _{24}}} & {d\tilde{N}_{2}^{\mathrm{T}}} & {{\psi _{26}}} & 0 & {{\psi _{28}}} & 0 & {{X^{\mathrm{T}}}} & 0 \\ {*} & * & {{\psi _{33}}} & {B_{\omega i}^{\mathrm{T}}} & {d\tilde{L}_{1}^{\mathrm{T}}} & {D_{\omega i}^{\mathrm{T}}} & 0 & {E_{4i}^{\mathrm{T}}} & 0 & 0 & 0 \\ {*} & * & * & {{\psi _{44}}} & 0 & 0 & {{H_{1i}}} & 0 & 0 & 0 & {{X^{\mathrm{T}}}} \\ {*} & * & * & * & {{\psi _{55}}} & 0 & 0 & 0 & 0 & 0 & 0 \\ {*} & * & * & * & * & { - I} & {{H_{2i}}} & 0 & 0 & 0 & 0 \\ {*} & * & * & * & * & * & { - {\chi _{ij}}} & 0 & 0 & 0 & 0 \\ {*} & * & * & * & * & * & * & { - {\varepsilon _{ij}}} & 0 & 0 & 0 \\ {*} & * & * & * & * & * & * & * & { - {M_{1}}} & 0 & 0 \\ {*} & * & * & * & * & * & * & * & * & {\frac{{ - {M_{1}}}}{{( {h - 1} )}}} & 0 \\ {*} & * & * & * & * & * & * & * & * & * & { - {M_{2}}} \end{array}\displaystyle \right ], \\ &{\psi _{11}} = {A_{i}}X + {B_{i}}{\hbar _{sj}} + {( {{A_{i}}X + {B_{i}}{\hbar _{sj}}} )^{\mathrm{T}}} - \delta X + \tilde{N}_{1}^{\mathrm{T}} + {{\tilde{N}}_{1}}, \\ &{\psi _{12}} = {A_{di}}X + {B_{i}}{\hbar _{dsj}} - \tilde{N}_{1}^{\mathrm{T}} - {{\tilde{N}}_{2}}, \qquad{\psi _{13}} = {B_{\omega i}} + \tilde{L}, \\ &{\psi _{14}} = \hbar _{sj}^{\mathrm{T}}B_{i}^{\mathrm{T}} + XA_{i}^{\mathrm{T}},\qquad {\psi _{16}} = XC_{i}^{\mathrm{T}} + \hbar _{sj}^{\mathrm{T}}D_{i}^{\mathrm{T}},\qquad {\psi _{18}} = XE_{1i}^{\mathrm{T}} + \hbar _{sj}^{\mathrm{T}}E_{3i}^{\mathrm{T}}, \\ &{\psi _{22}} = - \tilde{N}_{2}^{\mathrm{T}} - {{\tilde{N}}_{2}},\qquad {\psi _{24}} = XA_{di}^{\mathrm{T}} + \hbar _{dsj}^{\mathrm{T}}B_{i}^{\mathrm{T}},\qquad {\psi _{26}} = XC_{di}^{\mathrm{T}} + \hbar _{dsj}^{\mathrm{T}}D_{i}^{\mathrm{T}}, \\ &{\psi _{28}} = XE_{2i}^{\mathrm{T}} + \hbar _{dsj}^{\mathrm{T}}E_{3i}^{\mathrm{T}},\qquad {\psi _{33}} = - {\gamma ^{2}}{e^{ - \delta T}}I,\qquad {\psi _{44}} = {{ - {M_{2}}} / d}, \\ &{\psi _{55}} = - X - {X^{\mathrm{T}}} + {M_{2}},\qquad {\hbar _{sj}} = {E_{s}}{Y_{j}} + E_{s}^{-} {W_{j}},\qquad {\hbar _{dsj}} = {E_{s}}{Y_{dj}} + E_{s}^{-} {W_{dj}}, \end{aligned}$$
G, H are nonsingular matrices that make \(GEH = \bigl[ {\scriptsize\begin{matrix}{} {{I_{r}}} & 0 \cr 0 & 0 \end{matrix}} \bigr]\), then closed-loop system (8) is \({{FT}}{H_{\infty}}{{B}}\) with \({H_{\infty}}\) performance index \(\bar{\gamma}= {\gamma^{2}}{e^{ - \delta T}}\), and the controller feedback gains are given by
$${K_{j}} = {Y_{j}} {X^{ - 1}},\qquad {K_{dj}} = {Y_{dj}} {X^{ - 1}}. $$

Proof

From Theorem 2, considering (2), we can get the following relation according to matrix inequality (15):
$$ \Psi = \sum_{s = 1}^{\eta}{\sum _{i = 1}^{r} {\sum _{j = 1}^{r} {{h_{i}} {h_{j}}} } } {\alpha_{s}} ( {{\Psi _{ijs}} + \Delta{\Psi_{ijs}}} ) < 0, $$
(25)
where
Noticing (2) and Lemma 3, there exists a constant \({\chi_{ij}} > 0\), such that
$$\Delta{\Psi_{ijs}} = {\Upsilon_{1}}\Delta{ \Upsilon_{2}} + \Upsilon _{2}^{\mathrm{T}}{ \Delta^{\mathrm{T}}}\Upsilon_{1}^{\mathrm{T}} \le{\chi _{ij}} {\Upsilon_{1}}\Upsilon_{1}^{\mathrm{T}} + - \chi_{ij}^{ - 1}\Upsilon_{2}^{\mathrm{T}}{ \Upsilon_{2}}, $$
where
Then via the Schur complement, (25) is equivalent to
$$ \sum_{s = 1}^{\eta}{\sum _{i = 1}^{r} {{h_{i}} {\alpha _{s}} {\Xi_{iis}}} } + \sum_{s = 1}^{\eta}{\sum_{i = 1}^{r} {\sum _{j = 1}^{r} {{h_{i}} {h_{j}}} } } {\alpha_{s}} ( {{\Xi_{ijs}} + {\Xi_{jis}}} ) < 0, $$
(26)
where
$${\Xi_{ijs}} = \begin{bmatrix} {{\Psi_{ijs}}} & {{\Upsilon_{1}}} & {\Upsilon_{2}^{\mathrm{T}}} \\ {*} & { - {\chi_{ij}}} & 0 \\ {*} & * & { - \chi_{ij}^{\mathrm{T}}} \end{bmatrix}. $$
From Theorem 1, \({\varpi_{11}} < 0\), we can get \(\hat{P} = { ( {{E^{T}}P + S{R^{\mathrm{T}}}} )^{\mathrm{T}}}\) is nonsingular. Using Lemma 4, as the deal method in [26] there exists \(X = {\hat{P}^{ - 1}} = { ( {E\tilde{P} + \tilde{S}{{\tilde{R}}^{\mathrm{T}}}} )^{\mathrm{T}}}\), where \(\tilde{P} > 0\) and \(\tilde{R} \in {{\mathrm{R}}^{n \times ( {n - r} )}}\) is any matrix with full column rank and satisfies \(E\tilde{R} = 0\). It is easy to see that
$$ EX = {X^{\mathrm{T}}} {E^{\mathrm{T}}} = E\tilde{P}{E^{\mathrm{T}}} \ge0. $$
(27)
Denoting \({H^{ - 1}}X{G^{\mathrm{T}}} = \bigl[ {\scriptsize\begin{matrix}{} {{X_{11}}} & {{X_{12}}} \cr {{X_{21}}} & {{X_{22}}} \end{matrix}} \bigr]\), from (27), it is easy to obtain \({X_{12}} = 0\), and \({X_{11}}\) is symmetric, then we have \({H^{ - 1}}X{G^{\mathrm{T}}} = \bigl[ {\scriptsize\begin{matrix}{} {{X_{11}}} & 0 \cr {{X_{21}}} & {{X_{22}}} \end{matrix}} \bigr]\), so \({X_{11}}\) and \({X_{22}}\) are nonsingular. There, it can be concluded that \({G^{ - {\mathrm{T}}}}{X^{ - 1}}H = \bigl[ {\scriptsize\begin{matrix}{} {X_{11}^{ - 1}} & 0 \cr { - X_{22}^{ - 1}{X_{21}}X_{11}^{ - 1}} & {X_{22}^{ - 1}} \end{matrix}} \bigr]\) and \([ {{I_{r}}\ 0} ]GEX{G^{\mathrm{T}}} \bigl[ {\scriptsize\begin{matrix}{} {I_{r}} \cr 0 \end{matrix}} \bigr] = {X_{11}}\) are nonsingular.
Then we have
$$\begin{aligned}& {H^{ - {\mathrm{T}}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix} { \left ( { [ {{I_{r}}\quad 0} ]GEX{G^{\mathrm{T}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix}} \right )^{ - 1}} [ {{I_{r}}\quad 0} ]{H^{ - 1}} \\& \quad= {H^{ - {\mathrm{T}}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix} X_{11}^{ - 1} [ {{I_{r}}\quad 0} ]{H^{ - 1}} = {H^{ - {\mathrm {T}}}} \begin{bmatrix} {X_{11}^{ - 1}} & 0 \\ 0 & 0 \end{bmatrix} {H^{ - 1}} \\& \quad= {H^{ - {\mathrm{T}}}} \bigl( {{H^{\mathrm{T}}} {E^{\mathrm {T}}} {G^{\mathrm{T}}}} \bigr) \bigl( {{G^{ - {\mathrm{T}}}} {X^{ - 1}}H} \bigr){H^{ - 1}} \\& \quad= {E^{\mathrm{T}}} {X^{ - 1}} = {E^{\mathrm{T}}}\hat{P}, \end{aligned}$$
pre- and post-multiply (18) and (19) \(\operatorname{diag} \{ {{{\hat{P}}^{\mathrm{T}}},{{\hat{P}}^{\mathrm{T}}},I,I,{{\hat{P}}^{\mathrm{T}}},I,I,I,I,I,I} \}\), and denote \({\hat{P}^{\mathrm{T}}}{\tilde{N}_{1}}\hat{P} = {N_{1}}\), \({\hat{P}^{\mathrm {T}}}{\tilde{N}_{2}}\hat{P} = {N_{2}}\), \({\hat{P}^{\mathrm{T}}}\tilde{L} = L\), \({Y_{j}}\hat{P} = {X_{j}}\), \({Y_{dj}}\hat{P} = {K_{dj}}\), \({W_{j}}\hat{P} = {H_{j}}\), \({W_{dj}}\hat{P} = {H_{dj}}\), \({\varepsilon_{ij}} = \chi_{ij}^{ - 1}\), \({M_{1}} = Q_{1}^{ - 1}\), \({M_{2}} = Q_{2}^{ - 1}\), using the Schur complement, (18) and (19) are the sufficient conditions for (22) to hold.
Let
$$\bar{P} = R_{c}^{{1 /2}}{G^{\mathrm{T}}} \begin{bmatrix} {{{ \left ( { [ {{I_{r}}\quad 0} ]GEX{G^{\mathrm{T}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix}} \right )}^{ - 1}}} & {\mathrm{0}} \\ {\mathrm{0}} & \Psi \end{bmatrix} GR_{c}^{{1/2}}, $$
then we have
$$\begin{aligned} {E^{\mathrm{T}}}R_{c}^{{1/ 2}}\bar{P}R_{c}^{{1 /2}}E &= {H^{ - {\mathrm{T}}}}HR_{c}^{{1/2}}{G^{\mathrm{T}}} \begin{bmatrix} {{{ \left ( { [ {{I_{r}}\quad 0} ]GEX{G^{\mathrm{T}}} \begin{bmatrix} {I_{r}} \\ 0 \end{bmatrix}} \right )}^{ - 1}}} & {\mathrm{0}} \\ {\mathrm{0}} & {{\Psi^{ - 1}}} \end{bmatrix} GR_{c}^{{1/ 2}}H{H^{ - 1}} \\ &= {H^{ - {\mathrm{T}}}} \begin{bmatrix} {{X_{11}}} & 0 \\ 0 & 0 \end{bmatrix} {H^{ - 1}} = {E^{\mathrm{T}}}X. \end{aligned}$$

From (20), we have \({I_{n}} < \bar{P} < \frac{1}{{{\eta_{1}}}}{I_{n}}\), and then \({\lambda_{1}} > 1\), \({\lambda _{2}} < {1 /{{\eta_{1}}}}\). It can be seen from (21) that \({\bar{Q}_{1}} = R_{c}^{{{ - 1}/2}}{Q_{1}}R_{c}^{{{ - 1} /2}} < {\eta_{2}}I\), which means that \({\lambda_{3}} < {\eta_{2}}\). Similarly, we can obtained \({\lambda_{4}} < {\eta_{3}}\) from (21). Then using the Schur complement, we can get (16) from (23).

Then
$$\begin{aligned}& \begin{aligned} &{x^{\mathrm{T}}} ( t ){E^{\mathrm{T}}}PEx ( t ) = x_{1}^{\mathrm{T}} ( t ){P_{1}} {x_{1}} ( t ), \\ &{H_{i}}x ( t ) = {H_{i1}} {x_{1}} ( t ),\qquad {H_{di}}x ( t ) = {H_{di1}} {x_{1}} ( t ) , \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned}& {x^{\mathrm{T}}} ( t ){E^{\mathrm{T}}}PEx ( t ) \le \rho,\qquad {x^{\mathrm{T}}} \bigl( {t - d ( t )} \bigr){E^{\mathrm{T}}}PEx \bigl( {t - d ( t )} \bigr) \le \rho. \end{aligned}$$
(29)
So, condition \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} ) \subset L ( {{H_{i}},{H_{di}}} )\) can be guaranteed by
$$[ {{h_{i1k}}} \quad {{h_{\tau i1k}}} ]{ \left .\begin{bmatrix} {{P_{1}}} & 0 \\ 0 & {{P_{1}}} \end{bmatrix} \right .^{ - 1}} { [ {{h_{i1k}}} \quad {{h_{\tau i1k}}} ]^{\mathrm{T}}} \le\frac{1}{\rho},\quad k = 1,2, \ldots,l. $$
Via the Schur complement, it can be transformed to
$$ \begin{bmatrix} { - \frac{1}{\rho}} & { [ {{h_{i1k}}} \quad {{h_{di1k}}} ]} \\ {{{ [ {{h_{i1k}}} \quad {{h_{di1k}}} ]}^{\mathrm{T}}}} & { - \begin{bmatrix} {{P_{1}}} & 0 \\ 0 & {{P_{1}}} \end{bmatrix}} \end{bmatrix} \le0,\quad k = 1,2, \ldots,l, $$
or
$$ \begin{bmatrix} { - \frac{1}{\rho}} & [{{h_{i1k}}} \quad 0 ] & { [{{h_{di1k}}} \quad 0 ]} \\ {*} & { - {\mathrm {I}}\tilde{P}{\mathrm {I}}} & 0 \\ {*} & * & { - {\mathrm {I}}\tilde{P}{\mathrm {I}}} \end{bmatrix} \le0,\quad k = 1,2, \ldots,l, $$
(30)
where \({\mathrm {I}}\tilde{P}{\mathrm {I}} = \bigl[ {\scriptsize\begin{matrix}{} {{I_{r}}} & 0 \cr 0 & 0 \end{matrix}} \bigr] \bigl[ {\scriptsize\begin{matrix}{} {{P_{1}}} & {{P_{2}}} \cr {{P_{3}}} & {{P_{4}}} \end{matrix}} \bigr] \bigl[{\scriptsize\begin{matrix}{} {{I_{r}}} & 0 \cr 0 & 0 \end{matrix}} \bigr]\), \({h_{i1k}}\), \({h_{di1k}}\), is the kth row of \({H_{i1}}\) and \({H_{di1}}\), respectively.

Pre- and post-multiplying (30) by \(\operatorname{diag} \{ {1,{X^{\mathrm{T}}},{X^{\mathrm{T}}}} \}\) and its transpose, considering \({H_{i}}X = {W_{i}}\), \({H_{di}}X = {W_{di}}\), \({w_{ik}}\) is the kth row of \({W_{i}}\), \({w_{dik}}\) is the kth row of \({W_{di}}\). Considering (12), then we can obtain (24). This completes the proof. □

Remark 4

Theorem 3 gives a LMI condition for the region \(\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} )\) to be inside the domain of attraction for the closed-loop system (8) under the memory state feedback controller.

Remark 5

With all the ellipsoids satisfying the set invariance condition of Theorem 3, we may choose the largest one to obtain the least conservative estimate of the domain of attraction.

Let \({{\mathrm {X}}_{R}} \in{{\mathrm{R}}^{n}}\) be a prescribed bounded convex set containing the origin, which can be represented as \({{\mathrm {X}}_{R}} = {\operatorname{co}} \{ {x_{0}^{1},x_{0}^{2}, \ldots,x_{0}^{l}} \}\), where \(x_{0}^{1},x_{0}^{2}, \ldots ,x_{0}^{l}\) are a priori given initial states in \({{\mathrm {R}}^{n}}\). With Theorem 3, an exact invariant set with least degree of conservativeness can be formulated as
$$ \textstyle\begin{array}{@{}l} \max \alpha \\ \mbox{s.t. } \left \{ \textstyle\begin{array}{@{}l} ( \mathrm{a} )\quad \alpha{{\mathrm {X}}_{R}} \subset\varepsilon ( {{E^{\mathrm{T}}}PE,\rho} ), \\ ( \mathrm{b} )\quad \mbox{inequality (18)-(24)}. \end{array}\displaystyle \right . \end{array} $$
(31)
Using the Schur complement, constraint (a) is equivalent to
$$ x_{0}^{\mathrm{T}}{E^{\mathrm{T}}}PE{x_{0}} \le \frac{\rho}{{{\alpha^{2}}}} \quad\Leftrightarrow\quad \begin{bmatrix} { - \beta} & {x_{0}^{\mathrm{T}}{E^{\mathrm{T}}}} \\ {{Ex_{0}}} & { - X} \end{bmatrix} \le0, $$
(32)
where \(\beta = {\rho /{{\alpha^{2}}}}\).
From the above discussion, (27) can be transformed to the following LMI optimization problem:
$$ \textstyle\begin{array}{@{}l} \min \beta \\ \mbox{s.t. inequality (18)-(24) and (32)}. \end{array} $$
(33)

Remark 6

Theorem 3 gives the sufficient conditions for designing the finite-time memory controller for TS fuzzy system with time-varying delay. It can be observed that the conditions (18) and (19) are not strict LMIs, once we fix the parameter δ, the conditions can be turned into LMIs-based feasibility problem. Then the conditions in Theorem 3 can be turned into the following LMIs-based feasibility problem with a fixed parameter δ:
$$\textstyle\begin{array}{@{}l} \min {c_{2}} + {\gamma^{2}} \\ X,{{\tilde{Q}}_{1}},{{\tilde{Q}}_{2}},{{\tilde{N}}_{1}},{{\tilde{N}}_{2}},\tilde{L},b,d,\mu,\delta \\ \mbox{s.t. (18)-(24)}. \end{array} $$

4 Numerical value examples

Example 1

Consider the nonlinear system with time delay [28]:
$$\begin{aligned}& \begin{aligned}[b] {{\dot{x}}_{1}} ( t ) ={}& {-} a\frac{{v\bar{t}}}{{ ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} ( t ) - ( {1 - a} )\frac{{v\bar{t}}}{{ ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} \bigl( {t - \tau ( t )} \bigr)\\ &{}+ \frac{{v\bar{t}}}{{ ( {l + \Delta l ( t )} ){t_{0}}}}\operatorname{sat} \bigl( {u ( t )} \bigr), \end{aligned}\\& {{\dot{x}}_{2}} ( t ) = a\frac{{v\bar{t}}}{{ ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} ( t ) + ( {1 - a} )\frac{{v\bar{t}}}{{ ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} \bigl( {t - \tau ( t )} \bigr), \\& \begin{aligned}[b] {{\dot{x}}_{3}} ( t ) ={}& \frac{{v\bar{t}}}{{{t_{0}}}}\sin \biggl[ {x_{2}} ( t ) + a\frac{{v\bar{t}}}{{2 ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} ( t ) \\ &{}+ ( {1 - a} )\frac{{v\bar{t}}}{{2 ( {L + \Delta L ( t )} ){t_{0}}}}{x_{1}} \bigl( {t - \tau ( t )} \bigr) \biggr], \end{aligned} \end{aligned}$$
where \({x_{1}} ( t )\) is the angle difference between truck and trailer, \({x_{2}} ( t )\) is the angle of trailer, \({x_{3}} ( t )\) is the vertical position of rear end of trailer. The model parameters are given as \(l = 2.8\), \(L = 5.5\), \(v = - 1.0\), \(\bar{t} = 2.0\), \(\bar{t} = 2.0\), \({t_{0}} = 0.5\), \(d = {{10{t_{0}}} / \pi}\) and \(a = 0.7\). Then the model is expressed by the following T-S fuzzy system:
$$\dot{x} ( t ) = \sum_{i = 1}^{2} {{h_{i}} ( t ) \bigl[ ( {{A_{1i}} + \Delta{A_{1i}}} )x ( t ) + ( {{A_{2i}} + \Delta{A_{2i}}} )x \bigl( {t - \tau ( t )} \bigr) + ( {{B_{i}} + \Delta{B_{i}}} ) \operatorname{sat} \bigl( {u ( t )} \bigr) \bigr]}, $$
where
$$\begin{aligned}& {A_{11}} = \begin{bmatrix} { - a{{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ {a{{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ {a{{{v^{2}}{{\bar{t}}^{2}}} / { ( {2L{t_{0}}} )}}} & {{{v\bar{t}}/ {{t_{0}}}}} & 0 \end{bmatrix}, \qquad {A_{21}} = \begin{bmatrix} { - ( {1 - a} ){{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ { ( {1 - a} ){{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ { ( {1 - a} ){{{v^{2}}{{\bar{t}}^{2}}} / { ( {2L{t_{0}}} )}}} & {{{v\bar{t}} / {{t_{0}}}}} & 0 \end{bmatrix}, \\& {B_{1}} = \begin{bmatrix} {{{v\bar{t}} / { ( {l{t_{0}}} )}}} \\ 0 \\ 0 \end{bmatrix}, \qquad {A_{12}} = \begin{bmatrix} { - a{{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ {a{{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ {a{{d{v^{2}}{{\bar{t}}^{2}}} / { ( {2L{t_{0}}} )}}} & {{{dv\bar{t}} / {{t_{0}}}}} & 0 \end{bmatrix}, \\& {A_{22}} = \begin{bmatrix} { - ( {1 - a} ){{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ { ( {1 - a} ){{v\bar{t}} / { ( {L{t_{0}}} )}}} & 0 & 0 \\ { ( {1 - a} ){{d{v^{2}}{{\bar{t}}^{2}}} / { ( {2L{t_{0}}} )}}} & {{{v\bar{t}} / {{t_{0}}}}} & 0 \end{bmatrix},\qquad {B_{2}} = \begin{bmatrix} {{{v\bar{t}} /{ ( {l{t_{0}}} )}}} \\ 0 \\ 0 \end{bmatrix}, \\& \Delta{A_{11}} = 0.05\delta ( t ) \begin{bmatrix} {0.5091} & 0 & 0 \\ { - 0.5091} & 0 & 0 \\ {0.5091} & 0 & 0 \end{bmatrix},\qquad \Delta{A_{21}} = 0.05\delta ( t ) \begin{bmatrix} {0.2182} & 0 & 0 \\ { - 0.2182} & 0 & 0 \\ {0.2182} & 0 & 0 \end{bmatrix}, \\& \Delta{B_{1}} = 0.05\delta ( t ) \begin{bmatrix} { - 0.3517} \\ 0 \\ 0 \end{bmatrix},\qquad \Delta{A_{12}} = 0.05\delta ( t ) \begin{bmatrix} {0.5091} & 0 & 0 \\ { - 0.5091} & 0 & 0 \\ {0.8107} & 0 & 0 \end{bmatrix}, \\& \Delta{A_{22}} = 0.05\delta ( t ) \begin{bmatrix} {0.2182} & 0 & 0 \\ { - 0.2182} & 0 & 0 \\ {0.3474} & 0 & 0 \end{bmatrix},\qquad \Delta{B_{1}} = 0.05\delta ( t ) \begin{bmatrix} { - 0.3517} \\ 0 \\ 0 \end{bmatrix}, \end{aligned}$$
where \(\vert {\delta ( t )} \vert < 1\).
On the basis of [28], the saturating constraint is ignored, and we give \(\mu = 0\), \(d = 0.01\), \({c_{1}} = 1\), \({c_{2}} = 10\), \(T = 10\), \(\delta = 0.01\), \({R_{c}} = {I_{3}}\). Solving the LMIs (18)-(24), we can get the memory state feedback controller gain is
$$\begin{aligned}& {K_{11}} = [ {1.1380} \quad { - 1.5257} \quad { - 0.0453} ],\qquad {K_{21}} = [ {0.1589} \quad { - 0.0702} \quad { - 0.1001} ], \\& {K_{12}} = [ {1.2383} \quad { - 2.1297} \quad{0.1456} ],\qquad {K_{22}} = [ {0.1957} \quad { - 0.0052} \quad { - 0.0101} ]. \end{aligned}$$
For simulation, we choose the fuzzy weighting function to be
$${h_{1}} ( t ) = {1 / {1 + \exp\bigl(0.5{x_{1}} ( {t + 1} ) \bigr)}},\qquad {h_{2}} ( t ) = 1 - {h_{1}} ( t ), $$
and the initial condition \({\phi^{\mathrm{T}}} ( t ) = { [ {0.5\pi} \ {0.75\pi} \ { - 5} ]^{\mathrm{T}}}\), \(t \in [ { - 0.01,0} ]\). Figure 1 shows the response of states of the closed-loop systems.
Figure 1

State response of the closed-loop system (Example  1 ).

Example 2

Consider the TS fuzzy system subject to actuator saturation (1) with two fuzzy rules and the following parameters:
$$\begin{aligned}& E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},\qquad {A_{1}} = \begin{bmatrix} { - 1} & 1 & 0 \\ 1 & { - 2} & 0 \\ 1 & 0 & 0 \end{bmatrix},\qquad {A_{d1}} = \begin{bmatrix} {0.1} & 0 & 0 \\ 0 & {0.1} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \\& {B_{1}} = \begin{bmatrix} 0 \\ {0.1} \\ {0.1} \end{bmatrix}, \qquad {C_{1}} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & {0.1} \end{bmatrix},\qquad {C_{d1}} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & {0.1} & 0 \end{bmatrix},\qquad {D_{1}} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \\& {B_{\omega1}} = \begin{bmatrix} {0.1} & 0 & 0 \\ 0 & 0 & {0.1} \\ 0 & 0 & 0 \end{bmatrix},\qquad {D_{\omega1}} = \begin{bmatrix} {0.1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},\qquad {A_{2}} = \begin{bmatrix} 0 & 1 & 0 \\ {0.1} & {0.5} & 0 \\ 1 & 0 & 0 \end{bmatrix}, \\& {A_{d2}} = \begin{bmatrix} 1 & 0 & 0 \\ {0.1} & { - 0.2} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \qquad {B_{2}} = \begin{bmatrix} 1 \\ - 0.1 \\ 0 \end{bmatrix},\qquad {C_{2}} = \begin{bmatrix} {0.5} & 0 & 0 \\ 0 & {0.5} & 0 \\ 0 & {0.1} & 0 \end{bmatrix}, \\& {C_{d2}} = \begin{bmatrix} {0.5} & 0 & 0 \\ 0 & {0.5} & 0 \\ 0 & 0 & 0 \end{bmatrix},\qquad {D_{2}} = \begin{bmatrix} 0 \\ 0.5 \\ 0 \end{bmatrix}, \qquad {B_{\omega2}} = {D_{\omega2}} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & {0.01} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \\& {H_{ij}} = \begin{bmatrix} { - 0.03} \\ 0.03 \\ 0.03 \end{bmatrix},\qquad {E_{1i}} = [ {0.02} \quad {0 0} ],\qquad {E_{2i}} = [ { - 0.35} \quad { - 0.45 0} ], \\& {E_{3i}} = - 0.15,\qquad {E_{4i}} = [ { - 0.5} \quad { - 0.4 0} ]\quad (i,j = 1,2). \end{aligned}$$
For given \({c_{1}} = 1\), \({c_{2}} = 10\), \(T = 10\), \(b = 0.5\), \(\delta = 0.01\), \(d = 0.1\), \(\mu = 0.1\), \(r = 0.5\), \({R_{c}} = {I_{2}}\), the disturbance input is \(\omega ( t ) = {e^{ - t}}\sin ( { - t} )\), and the membership functions are \({h_{1}} ( t ) = {1 /{ [ {1 + \exp(0.5{x_{1}} ( {t + 1} ))} ]}}\), \({h_{2}} ( t ) = 1 - {h_{1}} ( t )\), using the Matlab toolbox, we can get
$$P = \begin{bmatrix} {5.2496} & {3.2171} & {0.1908} \\ {3.2171} & {9.8628} & { - 0.8331} \\ {0.1908} & { - 0.8331} & {20.3667} \end{bmatrix}, $$
the controller gain can be obtained:
$$\begin{aligned}& \begin{aligned}[b] &{K_{1}} = [ { - 9.6766} \quad { - 6.2899 \quad- 1.3006} ],\qquad {K_{d1}} = [ {6.9720} \quad {5.4183 \quad0.1899} ],\\ &{K_{2}} = [ { - 5.1128} \quad { - 2.5372 \quad- 0.5392} ], \qquad {K_{d2}} = [ { - 16.6810} \quad { - 12.4518 \quad- 0.4143} ]. \end{aligned} \end{aligned}$$

For given the initial condition \({x^{\mathrm{T}}} ( 0 ) = { [ { - 1\ 1\ 0} ]^{\mathrm{T}}}\), by solving the optimization problem (33), we can get \({\beta^{\min}} = 5.25\).

Then, using the above controller gain, Figure 2 plots the estimation of the domain of attraction and the response of the closed-loop system can be seen from Figure 3. It can be seen from Figure 3 that the closed-loop system is \(FTB\) subject to the memory controller. Figure 4 plots the evolution of \({x^{\mathrm{T}}} ( t ){E^{\mathrm{T}}}{R_{c}}Ex ( t )\). It can be seen from Figure 4 that the TS fuzzy system (1) is finite-time bounded with respect to \(( {1,10,{I_{2}},10} )\) via the finite-time fuzzy memory controller.
Figure 2

The estimation of the domain of attraction (Example 2 ).

Figure 3

State response of the closed-loop system (Example  2 ).

Figure 4

The evolution of \(\pmb{{x^{\mathrm{T}}}(t){E^{\mathrm{T}}}{R_{c}}Ex( t )}\) (Example 2 ).

For demonstration of the superiority of the memory state feedback controller presented in this paper, we give the memoryless controller as follows for comparison:
$$ u ( t ) = \sum_{i = 1}^{r} {{h_{i}}\bigl(\theta(t)\bigr)} {K_{i}}x ( t ) . $$
(34)
We can obtain the maximum allowable d for different h in Table 1.
Table 1

Comparison of maximum d for different h (Example 2 )

h

0.01

0.02

0.05

0.07

0.09

(34)

0.00264

0.00239

0.00164

0.00111

0.00053

(4)

0.00275

0.00245

0.00167

0.00113

0.00053

From the comparison in Table 1, it is obvious that the memory state feedback controller presented in this paper is less conservative.

5 Conclusion

In this paper, the problem of finite-time \({H_{\infty}}\) memory feedback of the singular T-S fuzzy system has been studied. Based on the finite-time stability theory, conditions were obtained, which can guarantee that the closed-loop system is finite-time \({H_{\infty}}\) bounded with a presided \({H_{\infty}}\) performance. The memory feedback controller problem can be solved by solving the LMIs. An optimization problem was given to deal with the largest domain of attraction of the closed-loop system. In the end, the examples were given to illustrate the feasibility of the method.

Declarations

Acknowledgements

This work was supported by the Natural Science Foundation of Hebei province No. F2015203362.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Electrical Engineering, Yanshan University
(2)
College of Science, Yanshan University

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© Guan and Liu 2016