Analysis and Controller Design for Parameter Varying T-S Fuzzy Systems with Markov Jump

Abstract

In this paper, we investigate a novel T-S fuzzy parameter varying system with Markov jump, in which parameters depend not only on a Markov chain but also on linear parameter varying elements that take values in convex polytopic sets. Stable conditions and the gain-scheduling controller design method for this system are obtained. Applying Lyapunov function depending on the operation mode and full block S-procedure lemma, we obtain stochastic stabilization conditions. We find that this novel system has two distinct advantages. On the one hand, it inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects under the frame of T-S fuzzy systems; on the other, it obtains the advantages of dealing with time-varying characteristics from the point of linear parameter varying (LPV) systems. Finally, the theory results are illustrated via numerical simulation.

1. Introduction

The traditional T-S fuzzy system has been studied and applied widely by many researchers due to its advantages in structure and theory. In fact, the T-S fuzzy system is a convex combination of linear systems, and is also a direct generalization of state-space model in terms of LPV theory. In theory, there is a connection between T-S fuzzy systems LPV or qLPV systems and they can transform into each other [1]. In [2], the T-S fuzzy system was proven to be equivalent to the LPV system. For the the modeling problem, it presented two methods for the automated generation of LPV and T-S systems. The references [3] proposed a tensor product (TP) model transformation-based framework requiring minimal human intuition to numerically reconstruct LPV and T-S fuzzy model-based linear parameter varying and quasi-linear parameter varying representations of state-space model. The problem “How to Vary the Input Space of a T-S Fuzzy Model” was solved in [4].

When dealing with a complex nonlinear system, we often use T-S fuzzy system to approximate it, because this method has high precision. Markov jump is a commonly used tool for describing system randomness. Many complex nonlinear systems Markov jump [5,6] are given and investigated under the frame of T-S fuzzy systems. A large number of Markov jump systems are described and investigated applying the fuzzy systems [7,8,9,10,11,12]. The T-S fuzzy system theory has been applied to investigate semi-Markov jump model reliable mixed $H_{\infty }$/passive control problem [13]. In [14], Shen used an event-triggered communication scheme to stabilize the closed-loop system with a prescribed $H_{\infty }$ performance level. Mode-dependent non-fragile observer-based controller [15] is designed for fractional-order T-S fuzzy systems with Markovian jump via non-PDC scheme. The reference [16] examines the stochastic control problem of continuous-time fuzzy systems with Markov jump. On the other hand, many T-S fuzzy systems are investigated in view of Markov jump. For the resilient estimation problem, the T-S fuzzy system theory is generalized to Markov systems under a dynamic event-triggered scheme [17]. Many other continuous-time generalized T-S fuzzy systems with Markov jump are adequately analyzed via different methods [18,19,20,21].

The positive stochastic stabilization issue is investigated for discrete-time T-S fuzzy Markov jump nonlinear systems in [22]. In the point of partial differential T-S fuzzy systems, Ji and Zhang [23] are concerned with a finite-dimensional guaranteed cost sampled-data fuzzy control problem. Song [24] investigated the problem of state estimation for partial differential T-S fuzzy Markov jump delayed neural networks and proposed a memory-based control scheme that contains a constant signal transmission delay.

In practice, the T-S fuzzy system with varying parameters is worth researching for many time-varying cases. The idea behind this paper comes from references [25,26,27,28,29] about T-S fuzzy system, LPV system with Markov jump.

$$\mathrm{IF}{z}_1\left(t\right)\mathrm{is}{\mathcal{F}}_i^1\mathrm{AND}\dots z_J\left(t\right)\mathrm{is}{\mathcal{F}}_i^J,\mathrm{THEN}$$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)=A_i(\gamma \left(t\right),\theta \left(t\right))x\left(t\right)+B_i(\gamma \left(t\right),\theta \left(t\right))u\left(t\right),\hfill \\ y\left(t\right)=C_i(\gamma \left(t\right),\theta \left(t\right))x\left(t\right)+D_i(\gamma \left(t\right),\theta \left(t\right))u\left(t\right),\hfill \end{array}\right.\end{array}$$

(1)

where $\mathcal{Ƶ}={(z_1{z}_2\cdots z_J)}^T$ is the premise variable, ${\mathcal{F}}_i^j$, $i=1,2,\cdots ,I$, $j=1,2,\cdots ,J$, is a fuzzy or linguistic term, which can be quantified by membership functions. The given I and J are positive integers which specify the number of fuzzy rules and premise variables, respectively. $x\left(t\right)\in {\mathbf{R}}^n$ is the system state, $y\left(t\right)\in {\mathbf{R}}^p$ is the measurement output and $u\left(t\right)\in {\mathbf{R}}^q$ is the control input signal. The matrix functions $A_i(\gamma \left(t\right),\theta \left(t\right))\in {\mathbf{R}}^{n\times n}$, $B_i(\gamma \left(t\right),\theta \left(t\right))\in {\mathbf{R}}^{n\times q}$, $C_i(\gamma \left(t\right),\theta \left(t\right))\in {\mathbf{R}}^{p\times n}$ and $D_i(\gamma \left(t\right),\theta \left(t\right))\in {\mathbf{R}}^{p\times q}$ depend on the time-varying parameter $\theta$. The time-varying parameter $\theta \left(t\right)={({\theta }_1\left(t\right)\cdots {\theta }_v\left(t\right))}^T\in \Omega$ is independent from the system state variables and can be measured on-line. For the convenience of the following expression, define ${\theta }_0=1$ and abbreviate $\theta \left(t\right)$, ${\theta }_j\left(t\right)$ and $h_i\left(\mathcal{Ƶ}\right)$ as $\theta$, ${\theta }_j$ and $h_i$. The parameter subset $\Omega ={[-1,1]}^v$ is a bounded closed set and contains $2^v$ vertices at ${\Omega }^v={\{-1,1\}}^v$. The parameter $\gamma \left(t\right)$ takes discrete values in a finite state space $\mathcal{M}=\{1,2,\cdots ,m\}$ which is a right continuous homogeneous Markov process and the jumping of $\gamma$ satisfies the transition probability matrix $\Lambda ={\left\{{\lambda }_{s\mathsf{\ell }}\right\}}_{m\times m}$ determined by

$$\begin{array}{c}\hfill \mathrm{Pr}\{\gamma (t+\Delta t)=\mathsf{\ell }|\gamma \left(t\right)=s\}=\left\{\begin{array}{cc}{\lambda }_{s\mathsf{\ell }}\Delta t+o(\Delta t),\hfill & \mathsf{\ell }\ne s,\hfill \\ 1+{\lambda }_{s\mathsf{\ell }}\Delta t+o(\Delta t),\hfill & \mathsf{\ell }=s,\hfill \end{array}\right.\end{array}$$

(2)

where $\Delta t>0$, ${lim}_{t\to 0}(o(\Delta t)/\Delta t)=0$, ${\lambda }_{s\mathsf{\ell }}\ge 0$ denotes the transition rate from mode s to ℓ at time $t+\Delta t$ if $\mathsf{\ell }\ne s$ and ${\lambda }_{\mathsf{\ell }\mathsf{\ell }}=-{\sum }_{s\in \mathcal{M},\mathsf{\ell }\ne s}{\lambda }_{s\mathsf{\ell }}$.

Applying the “algebraic product” operation to quantify the linguistic term “and”, the “product” implication method to infer the local output according to each fuzzy linguistic rule and the center-average defuzzification method, the fuzzy parameter varying system with Markov jump described in the linguistic form (1) could be represented by the following analytical formula:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \dot{x}\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[A_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)x\left(t\right)+B_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)u\left(t\right)],}\hfill \\ {\displaystyle y\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[C_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)x\left(t\right)+D_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)u\left(t\right)],}\hfill \end{array}\right.\end{array}$$

(3)

where

$$\begin{array}{c}\hfill h_i\left(\mathcal{Ƶ}\right)={\displaystyle \frac{{\mathcal{W}}_i\left(\mathcal{Ƶ}\right)}{{\displaystyle \sum _{i=1}^I{\mathcal{W}}_i\left(\mathcal{Ƶ}\right)}}},{\mathcal{W}}_i\left(\mathcal{Ƶ}\right)=\prod _{j=1}^p{\mathcal{F}}_i^j\left(z_j\right)\mathrm{satisfying}\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)=1.\end{array}$$

Under the frame of LPV systems, assume that system matrices of (3) are affine on $\theta$ such that when $\gamma \left(t\right)=\mathsf{\ell }$ the system matrices are affine dependent on $\theta$ as

$$\begin{array}{c}\hfill A_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)=\sum _{j=0}^v{A}_{ij}^{\mathsf{\ell }}{\theta }_j\left(t\right),B_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)=\sum _{j=0}^v{B}_{ij}^{\mathsf{\ell }}{\theta }_j\left(t\right),\\ \hfill C_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)=\sum _{j=0}^v{C}_{ij}^{\mathsf{\ell }}{\theta }_j\left(t\right),D_i^{\mathsf{\ell }}\left(\theta \left(t\right)\right)=\sum _{j=0}^v{D}_{ij}^{\mathsf{\ell }}{\theta }_j\left(t\right).\end{array}$$

The system (3) can describe T-S fuzzy system, LPV system and Markov switch system. Specifically, when the system (3) contains only one fuzzy rule, it degenerates into an LPV (to be precise, it is qLPV) system with Markov jump [30,31]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}\left(t\right)=A^{\mathsf{\ell }}\left(\theta \left(t\right)\right)x\left(t\right)+B^{\mathsf{\ell }}\left(\theta \left(t\right)\right)u\left(t\right),\hfill \\ y\left(t\right)=C^{\mathsf{\ell }}\left(\theta \left(t\right)\right)x\left(t\right)+D^{\mathsf{\ell }}\left(\theta \left(t\right)\right)u\left(t\right);\hfill \end{array}\right.\end{array}$$

(4)

when the system matrices of (3) are not dependent on $\theta$, it degenerates into a T-S fuzzy system with Markov jump [14]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \dot{x}\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[A_i^{\mathsf{\ell }}x\left(t\right)+B_i^{\mathsf{\ell }}u\left(t\right)],}\hfill \\ {\displaystyle y\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[C_i^{\mathsf{\ell }}x\left(t\right)+D_i^{\mathsf{\ell }}u\left(t\right)];}\hfill \end{array}\right.\end{array}$$

(5)

when the system (3) ignores the time-varying parameter and Markov jump, it degenerates into a conventional T-S fuzzy system [32,33]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \dot{x}\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[A_{i}x\left(t\right)+B_{i}u\left(t\right)],}\hfill \\ {\displaystyle y\left(t\right)=\sum _{i=1}^I{h}_i\left(\mathcal{Ƶ}\right)[C_{i}x\left(t\right)+D_{i}u\left(t\right)].}\hfill \end{array}\right.\end{array}$$

(6)

Therefore, the system (3) merges several kinds of common ones. Under the frame of T-S fuzzy system, it is also a universal approximator with arbitrary precision for any nonlinear system [34]. Differently, the system (3) will contain less rules than the conventional T-S fuzzy system when it is used to deal with nonlinear time-varying systems. This makes it convenient to obtain stability conditions and design controllers. The system (3) combined in convex hull is more compendious to model a nonlinear system than LPV system. On the other hand, the system (3) can model more complex nonlinear systems due to consideration of Markov jump. Each of its subsystems is a Markov jump system describing randomness. It is a novel nonlinear time-varying fuzzy system worth studying.

The notations used in the paper are rather standard

Table 1. Symbols.

R	Set of real numbers
${\mathbf{R}}^{m\times n}$	The set of real $m\times n$ matrices
${\mathbf{S}}^n$	Real symmetric $n\times n$ matrices
${\mathbf{S}}_{+}^n$	Positive-definite matrices
$I_n$	The n order identity matrix
★	symmetry element
$P>0$	P is a symmetric positive definite matrix.
$P^{-1}$	The inverse of matrix P
$P^T$	The transposition of matrix P
$\mathbf{He}\left\{M\right\}$	$\mathbf{He}\left\{M\right\}=M+M^T$
${\tilde{A}}_i$	$A_i-A_1$
${\tilde{A}}_{ij}$	$A_{ij}-A_{i1}$
$A_{h\theta }^{\mathsf{\ell }}$	$A_{h\theta }^{\mathsf{\ell }}=A_1^{\mathsf{\ell }}\left(\theta \right)+{\sum }_{i=2}^I{h}_i{\tilde{A}}_i^{\mathsf{\ell }}\left(\theta \right)={\sum }_{j=0}^v{A}_{1j}^{\mathsf{\ell }}{\theta }_j+{\sum }_{i=2}^I{\sum }_{j=0}^v{h}_i{\tilde{A}}_{ij}^{\mathsf{\ell }}{\theta }_j$

. $\mathbf{R}$ stands for the set of real numbers. ${\mathbf{R}}^{m\times n}$ is the set of real $m\times n$ matrices. We use ${\mathbf{S}}^n$ to denote real symmetric $n\times n$ matrices, and ${\mathbf{S}}_{+}^n$ for positive-definite matrices. The n order identity matrix is denoted by $I_n$. In some matrix, the sign “★” means symmetry element. $P>0$ means P is a symmetric positive definite matrix. $P^{-1}$ and $P^T$ are the inverse and the transposition of matrix P, respectively. The parameter ℓ present the system mode, for example, $P^{\mathsf{\ell }},A^{\mathsf{\ell }}$. The sign “$\mathbf{He}$” is defined as $\mathbf{He}\left\{M\right\}=M+M^T$. For any matrices $A_i$ and $A_{ij}$, ${\tilde{A}}_i$ and ${\tilde{A}}_{ij}$ denotes the matrices $A_i-A_1$ and $A_{ij}-A_{i1}$. For convenience of expression and proof, the notation $A_{h\theta }^{\mathsf{\ell }}$ is introduced

$$\begin{array}{c}\hfill A_{h\theta }^{\mathsf{\ell }}:=\sum _{i=1}^I{h}_i\sum _{j=0}^v{A}_{ij}^{\mathsf{\ell }}{\theta }_j.\end{array}$$

(7)

Due to ${\displaystyle \sum _{i=1}^I{h}_i=1}$,

$$A_{h\theta }^{\mathsf{\ell }}=A_1^{\mathsf{\ell }}\left(\theta \right)+\sum _{i=2}^I{h}_i{\tilde{A}}_i^{\mathsf{\ell }}\left(\theta \right)=\sum _{j=0}^v{A}_{1j}^{\mathsf{\ell }}{\theta }_j+\sum _{i=2}^I\sum _{j=0}^v{h}_i{\tilde{A}}_{ij}^{\mathsf{\ell }}{\theta }_j.$$

For the set of membership functions, we define $H={[h_2{h}_3\cdots h_r]}^T\in {\mathbf{R}}^{r-1}$,

$$\begin{array}{c}\hfill H_1=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right],H_2=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right],\cdots ,H_r=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right],\end{array}$$

(8)

with $H_i\in {\mathbf{R}}^{r-1}$, $i=1,2,\cdots ,r$ for convenience of presentation.

In this paper, we also study the system (3) with external disturbance and give the $H_{\infty }$ gain schedule controllers design methods. The system (3) with external disturbance is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=A_{h\theta }^{\mathsf{\ell }}x+B_{h\theta }^{\mathsf{\ell }w}w+B_{h\theta }^{\mathsf{\ell }u}u,\hfill \\ y=C_{h\theta }^{\mathsf{\ell }}x+D_{h\theta }^{\mathsf{\ell }}w,\hfill \end{array}\right.\end{array}$$

(9)

where $w\left(t\right)\in {\mathbf{R}}^{n_w}$ is exogenous input. The superscripts $\mathsf{\ell }w$,$\mathsf{\ell }u$ and $\mathsf{\ell }y$ are just to distinguish different system matrices. The superscript ℓ is the system mode. The definition subscript $h\theta$ is the same as (7). Other nations are the same as them in the system (3).

In the paper, we investigate a novel system, which merges the T-S fuzzy parameter varying system and Markov jump. We obtain its stable conditions and gain-scheduling controller design method. Applying Lyapunov function depending on the operation mode and full block S-procedure Lemma, we obtain stochastic stabilization conditions. It inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects. This study expands the scope of application of T-S fuzzy system theory.

The remainder of this article is organized as follows: In Section 3, the stability analysis of open-loop fuzzy parameter varying system with Markov jump is provided based on a Lyapunov function that depends on the operating mode. In Section 4, state feedback fuzzy gain schedule controllers are designed using the full block S-procedure lemma and Lyapunov stability theory. At the end of Section 4, the $H_{\infty }$ controller will be provided in the form of corollary. A numerical example is utilized to validate our the controller synthesis conditions in Section 5. Section 6 concludes the article.

2. Preliminary

In the remainder of this article, the full block S-procedure lemma will be extensively and continually examined. It plays an important role in the proof of our main results. To improve readability of this article, this lemma is introduced as below.

Lemma 1

(Full Block S-procedure [35]). For a given linear functional transformation (LFT) expression

$$\begin{array}{c}\hfill G(\Delta )=G_{22}+G_{21}\Delta {(I-G_{11}\Delta )}^{-1}{G}_{12}\end{array}$$

(10)

with $\Delta \in \Omega$ and $\left\{0\right\}\in \Omega$, the inequality $G^T(\Delta )MG(\Delta )<0$ is equivalent to the existence of a full block multiplier $\Pi =\left[\begin{array}{cc}{\Pi }_{11}& {\Pi }_{12}\\ {\Pi }_{12}^T& {\Pi }_{22}\end{array}\right]$ satisfying

$$\begin{array}{c}\hfill {\left[\begin{array}{cc}{G}_{11}& G_{12}\\ I& 0\\ G_{21}& G_{22}\end{array}\right]}^T\left[\begin{array}{ccc}{\Pi }_{11}& {\Pi }_{12}& 0\\ {\Pi }_{12}^T& {\Pi }_{22}& 0\\ 0& 0& M\end{array}\right]\left[\begin{array}{cc}{G}_{11}& G_{12}\\ I& 0\\ G_{21}& G_{22}\end{array}\right]<0,\\ \hfill {\left[\begin{array}{c}I\\ \Delta \end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{11}& {\Pi }_{12}\\ {\Pi }_{12}^T& {\Pi }_{22}\end{array}\right]\left[\begin{array}{c}I\\ \Delta \end{array}\right]\ge 0\end{array}$$

for all $\Delta \in \Omega$.

This lemma has been used in robust control researches [36,37,38] to convert the inequality containing uncertainty matrix $\Delta$ and the nominal part of the system decomposes into two separate inequalities. According to Lemma 1, the uncertain $\Delta$ can be separated from the original matrix inequality $G^T(\Delta )MG(\Delta )<0$. However, $\Delta$ is not simply treated as uncertain in the paper, but a time-varying parameter measured on line. Note that the second inequality contains quadratic terms of $\Delta$ with indefinite sign. It is reasonable to believe that the restriction ${\Pi }_{22}<0$ could render some conservatism. Nevertheless, the comparison examples in reference [27] have shown that the proposed relaxation based on full block S-procedure is much better than existing relaxation methods. In our future research, we would like to remove the requirement of ${\Pi }_{22}<0$ for further improvement of the full block S-procedure relaxation. Generally speaking, infinite number of inequalities should be checked to ensure this inequality over the set $\Omega$. Nevertheless for a convex polytope $\Omega$, a condition with finite LMIs can be derived by enforcing the sub-matrix ${\Pi }_{22}<0$.

Remark 1.

In Lemma 1, let $\Delta \in {\mathbf{R}}^{n_1\times n_2}$ and $G(\Delta )={\left[I{\Delta }^T\right]}^T$, then $G(\Delta )$ can be expressed as a LFT, namely,

$$\begin{array}{c}\hfill G(\Delta )=G_{22}+G_{21}\Delta {(I-G_{11}\Delta )}^{-1}{G}_{12},\end{array}$$

(11)

where $G_{11}=0,G_{12}=I_{n_2},G_{21}=\left[\begin{array}{cc}0& I_{n_1}\end{array}\right]\in {\mathbf{R}}^{(n_1+n_2)\times n_1},G_{22}=\left[\begin{array}{cc}{I}_{n_2}& 0\end{array}\right]\in {\mathbf{R}}^{(n_1+n_2)\times n_2}$.

In Section 4, the parameter $\Delta$ in Remark 1 will be replaced by some other matrix functions depending on time-varying parameters or premise variables in the fuzzy rules. Consequently, the state feedback fuzzy gain schedule controller will be derived by using the full block S-procedure lemma.

Lemma 2

(Bound Real Lemma [13]). For a system G, the expression form of state space is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=Ax+Bw,\hfill \\ y=Cx+Dw,\hfill \end{array}\right.\end{array}$$

(12)

where x and w are the state and external disturbance. The system matrices are A, B, C and D, then the following two conditions are equal,

1. The systems is exponentially stable and ${\Vert D+C(sI-A)}^{-1}{B\Vert }_{\infty }<\mu$.

2. There is a positive definite matrix P satisfying

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{A}^{T}P+PA& PB& C^T\\ B^{T}P& -\mu I& D^T\\ C& D& -\mu I\end{array}\right]<0.\end{array}$$

Definition 1

([15]). The system (9) is said to robustly stochastically stable if for any finite initial condition $x\left(0\right)\in {\mathbf{R}}^n$ and $\gamma \left(0\right)$, one has $E[\parallel x\left(t\right){\parallel }^2]\to 0$ as $t\to \infty$.

3. Stability Analysis of Fuzzy Parameter Varying System with Markov Jump

The issue of stability analysis is extremely important in the field of control theory. In this section, we will therefore first analyze the stability of the open-loop fuzzy parameter varying system with Markov jump based on Lyapunov function that depends on the operating mode. Consider the system (3) without the control input

$$\begin{array}{c}\hfill \dot{x}=\sum _{i=1}^I{h}_i{A}_i^{\mathsf{\ell }}\left(\theta \right)x.\end{array}$$

(13)

Theorem 1.

If there exist matrices $P^{\mathsf{\ell }}\in {\mathbf{S}}_{+}^n$, $\mathsf{\ell }=1,2,\cdots ,m$ such that

$$\begin{array}{c}\hfill \mathbf{He}\left\{P^{\mathsf{\ell }}{A}_i^{\mathsf{\ell }}\left(\Theta \right)\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0\end{array}$$

(14)

for any $\Theta \in {\Omega }^v$ and $i=1,2,\cdots ,I$, $\mathsf{\ell }=1,2,\cdots ,m$, then the fuzzy parameter varying system (13) with Markov jump is stochastically stable at the origin.

Proof. 

Take the Lyapunov function candidate dependent on system mode

$$V(x,\mathsf{\ell })=x^T{P}^{\mathsf{\ell }}x.$$

Directly compute the derivative with time along the trajectory of the system (13), we have

$$\begin{array}{c}\hfill LV(x,\mathsf{\ell })=x^T\left(\mathbf{He}\left\{P^{\mathsf{\ell }}{A}_{h\theta }^{\mathsf{\ell }}\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s\right)x\end{array}$$

for any $\theta \in \Omega$. The inequality $LV(x,\mathsf{\ell })<0$ is ensured if

$$\begin{array}{c}\hfill \mathbf{He}\left\{P^{\mathsf{\ell }}{A}_{h\theta }^{\mathsf{\ell }}\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0\end{array}$$

(15)

for any $\theta \in \Omega$, $i=1,2,\cdots ,I$ and $\mathsf{\ell }=1,2,\cdots ,m$. From the definition of $A_{h\theta }^{\mathsf{\ell }}$ in (7), $A_{h\theta }^{\mathsf{\ell }}$ is linearly dependent on h and $\theta$. Therefore, according to the extreme theorem [39], the inequality (15) holds if and only if it holds at $H_i$ and $\Theta \in {\Omega }^v$, that is

$$\begin{array}{c}\hfill \mathbf{He}\left\{P^{\mathsf{\ell }}{A}_i^{\mathsf{\ell }}\left(\Theta \right)\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0.\end{array}$$

The proof is then completed. □

Remark 2.

Since any ${\theta }_j\in [-1,1]$ in this paper, the inequality (14) holds at ${\Theta }_j=1$ and ${\Theta }_j=-1$, $j=1,2,\cdots ,v$. When $I=2$ and $v=1$, the inequality (14) contains 4 LMIs for system mode ℓ:

$$\begin{array}{c}{\displaystyle \mathbf{He}\left\{P^{\mathsf{\ell }}(A_{10}^{\mathsf{\ell }}+A_{11}^{\mathsf{\ell }})\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0,}\hfill \\ {\displaystyle \mathbf{He}\left\{P^{\mathsf{\ell }}(A_{10}^{\mathsf{\ell }}-A_{11}^{\mathsf{\ell }})\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0,}\hfill \end{array}$$

(16)

$$\begin{array}{c}{\displaystyle \mathbf{He}\left\{P^{\mathsf{\ell }}(A_{20}^{\mathsf{\ell }}+A_{21}^{\mathsf{\ell }})\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0,}\hfill \\ {\displaystyle \mathbf{He}\left\{P^{\mathsf{\ell }}(A_{20}^{\mathsf{\ell }}-A_{21}^{\mathsf{\ell }})\right\}+\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0.}\hfill \end{array}$$

(17)

Then $\mathbf{He}\left\{P^{\mathsf{\ell }}{A}_{h\theta }^{\mathsf{\ell }}\right\}+{\sum }_{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s<0$ can be ensured by LMIs (16)–(17). It is a linear combination of (16)–(17) with positive coefficients $h_1(1+\theta )/2$, $h_1(1-\theta )/2$, $h_2(1+\theta )/2$ and $h_2(1-\theta )/2$.

Remark 3.

From the point of inequality (14), the term $\mathbf{He}\left\{P_{\mathsf{\ell }}{A}_i^{\mathsf{\ell }}\left(\Theta \right)\right\}$ plays a decisive role for stability analysis. It seems that the term ${\sum }_{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s$ has the opposite effect for stability conditions because of $P^s>0$, but transition rate ${\lambda }_{\mathsf{\ell }s}$ are not all positive, especially ${\lambda }_{\mathsf{\ell }\mathsf{\ell }}$. On the other hand, from the point of Markov jump system, the positive definite matrix $P^{\mathsf{\ell }}$ is common in the same system mode for different fuzzy rules. However, those subsystems are combined by membership functions.

Remark 4.

If take all $P^{\mathsf{\ell }}$ as the same one in the inequality (14), $P^1=P^2=\cdots =P^m$, and apply the transition rate condition ${\lambda }_{\mathsf{\ell }\mathsf{\ell }}=-{\sum }_{s\in \mathcal{M},\mathsf{\ell }\ne s}{\lambda }_{s\mathsf{\ell }}$, then the Theorem 1 degenerates into the result by using the common quadratic Lyapunov function.

Remark 5.

Rotondo [2] thinks the fuzzy parameter varying system with Markov jump is a kind of special linear parameter varying system as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=A^{\mathsf{\ell }}(H,\theta )x+B^{\mathsf{\ell }}(H,\theta )u,\hfill \\ y=C^{\mathsf{\ell }}(H,\theta )x+D^{\mathsf{\ell }}(H,\theta )u.\hfill \end{array}\right.\end{array}$$

(18)

However, the parameters have certain internal connections in the system (18). The parameter H is always dependent on system states and the sum of nonnegative $h_i$ is less than 1. Therefore, many traditional methods of LPV theory are not applied to the system (13) directly unless ignore their relationships and treat them just as bounded parameters. This will greatly increase the conservatism of stability conditions and creates difficulties for controller design.

4. Full States Feedback Fuzzy Gain Schedule Controller Design

The full states feedback controller is the most direct control method in control theory when all system states are measurable. Inspired by T-S fuzzy parallel compensation control method and gain schedule control of LPV theory, full states feedback fuzzy gain schedule controller is investigated. However, the quadratic terms of normalized fuzzy weighting functions, e.g., ${\sum }_{i=1}^I{\sum }_{j=1}^I{h}_i{h}_j{B}_i^{\mathsf{\ell }}{K}_j^{\mathsf{\ell }}$, present a great challenge in the process of recasting the controller synthesis condition into LMIs conditions. Traditional methods admit every sub-item is negative shielding membership functions. To solve the problem, many various relaxation techniques have been studied in [26,27]. More difficultly, the quadratic terms ${\sum }_{i=1}^I{\sum }_{j=1}^I{h}_i{h}_j{B}_i^{\mathsf{\ell }}{K}_j^{\mathsf{\ell }}$ include not only parameters dependent on fuzzy weighting functions but also varying parameters $\theta$. In this section, we apply twice full block S-procedure lemma to separate H and $\theta$, respectively.

According to the idea of PDC controller design [40], we state a traditional result based on common Lyapunov function to design full states feedback gain-schedule fuzzy controller. Since the proof is very simple, we directly give the result.

Theorem 2.

If there exist $P\in {\mathbf{S}}_{+}^n$ and matrix functions $K_j^{\mathsf{\ell }}\left(\theta \right)\in {\mathbf{R}}^{q\times n},j=1,2,\cdots ,I$ satisfying the following matrix inequalities

$$\begin{array}{c}\hfill \mathbf{He}\left\{P(A_i^{\mathsf{\ell }}\left(\theta \right)+B_i^{\mathsf{\ell }}\left(\theta \right)K_j^{\mathsf{\ell }}\left(\theta \right))\right\}<0\end{array}$$

(19)

for any $\theta \in \Omega$, then the system (3) is stabilized by the state feedback fuzzy gain schedule controller defined in (20)

$$\begin{array}{c}\hfill u=\sum _{j=1}^I{h}_j{K}_j^{\mathsf{\ell }}\left(\theta \right)x,\end{array}$$

(20)

where, $K_j^{\mathsf{\ell }}\left(\theta \right)=K_{j0}^{\mathsf{\ell }}+K_{j1}^{\mathsf{\ell }}{\theta }_1+\cdots +K_{jv}^{\mathsf{\ell }}{\theta }_v.$

Remark 6.

The inequality (19) in Theorem 2 is not linear about unknown variable matrices P and K. However, it can be converted to a LMI of $P^{-1}$.

Remark 7.

Though the inequality (19) can converted to a LMI, the quadratic term of θ needs to be dealt with via the grid method of LPV theory. It will also require a lot conservatism [41]. So it is a challenge for fuzzy parameter varying system with Markov jump.

Remark 8.

In Theorem 2, the inequality (19) is conservatively simplified from

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^I\sum _{j=1}^I{h}_i{h}_j\mathbf{He}\left\{P(A_i^{\mathsf{\ell }}\left(\theta \right)+B_i^{\mathsf{\ell }}\left(\theta \right)K_j^{\mathsf{\ell }}\left(\theta \right))\right\}<0.}\end{array}$$

(21)

The inequality (21) is equal to

$$\begin{array}{c}\hfill {\left[\begin{array}{c}{h}_1{I}_n\\ ⋮\\ h_I{I}_n\end{array}\right]}^T\left[\begin{array}{ccc}{G}_{11}& \cdots & G_{1I}\\ ⋮& & ⋮\\ G_{I1}& \cdots & G_{II}\end{array}\right]\left[\begin{array}{c}{h}_1{I}_n\\ ⋮\\ h_I{I}_n\end{array}\right]<0,\end{array}$$

(22)

where $G_{ij}=\mathbf{He}\left\{P(A_i^{\mathsf{\ell }}\left(\theta \right)+B_i^{\mathsf{\ell }}\left(\theta \right)K_j^{\mathsf{\ell }}\left(\theta \right))\right\}$. The inequality (19) is equal to every entry $G_{ij}<0$ of block core matrix. The simplification from (22) to (19) has some conservatism. The conservatism of such a relaxation procedure can be revealed by a simple example: The inequality $\left[\begin{array}{cc}{h}_1& h_2\end{array}\right]\left[\begin{array}{cc}-1& 0.7\\ 0.7& -0.5\end{array}\right]\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]<0$ clearly holds for any non-negative $h_i$ satisfying $h_1+h_2=1$, but not every entry of core matrix is negative.

In the following result, we will overcome some conservatism as far as we can based on Lyapunov function dependent on the system mode and full block S-procedure lemma. A relaxation technique proposed in this section can be used in effectively solving controller synthesis problem of fuzzy parameter varying system with Markov jump.

Theorem 3.

If there exist positive definite matrices $P^{\mathsf{\ell }}\in {\mathbf{S}}_{+}^n$, some constant matrices $U_{jk}^{\mathsf{\ell }}\in {\mathbf{R}}^{q\times n}$, $\mathsf{\ell }=1,2,\cdots ,m$, $j=1,2,\cdots ,I$ and $k=0,1,\cdots ,v$, and two symmetric matrices ${\Pi }_1^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]$, ${\Pi }_2^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]$ with appropriate dimensions satisfying, for any $\Theta \in {\Omega }^v$,

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{M}^{\mathsf{\ell }}+\left[\begin{array}{cc}{\Pi }_1^{\mathsf{\ell }}+{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]& \begin{array}{c}0\\ P_c^{\mathsf{\ell }}\end{array}\\ \begin{array}{cc}★& ★\end{array}& {\widehat{P}}_D^{\mathsf{\ell }}\end{array}\right]<0,\end{array}$$

(23)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}I\\ H_i\otimes I_n\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}I\\ H_i\otimes I_n\end{array}\right]\ge 0,{\Pi }_{1,22}^{\mathsf{\ell }}<0,\end{array}$$

(24)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}I\\ \Theta \otimes I_{In}\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}I\\ \Theta \otimes I_{In}\end{array}\right]\ge 0,{\Pi }_{2,22}^{\mathsf{\ell }}<0,\end{array}$$

(25)

where ${\widehat{P}}_D^{\mathsf{\ell }}=\mathrm{diag}\{{\lambda }_{\mathsf{\ell }1}{P}^1,\cdots ,{\lambda }_{\mathsf{\ell }(\mathsf{\ell }-1)}{P}^{\mathsf{\ell }-1},{\lambda }_{\mathsf{\ell }(\mathsf{\ell }+1)}{P}^{\mathsf{\ell }+1},\cdots ,{\lambda }_{\mathsf{\ell }{n}_{\gamma }}{P}^m\}$, $P_c^{\mathsf{\ell }}=\left[{\underbrace{P^{\mathsf{\ell }},P^{\mathsf{\ell }},\cdots ,P^{\mathsf{\ell }}}}_{m-1}\right]$,

$$\begin{array}{c}\hfill M^{\mathsf{\ell }}=\mathbf{He}\left\{\left[\begin{array}{cccc}{\overline{M}}_{00}^{\mathsf{\ell }}+{\widehat{M}}_0^{\mathsf{\ell }}-{\displaystyle \frac{{\lambda }_{\mathsf{\ell }\mathsf{\ell }}}{2}}{P}_{\mathsf{\ell }}& {\overline{M}}_{01}^{\mathsf{\ell }}+{\widehat{M}}_1^{\mathsf{\ell }}& \cdots & {\overline{M}}_{0n_{\theta }}^{\mathsf{\ell }}+{\widehat{M}}_v^{\mathsf{\ell }}\\ {\overline{M}}_{10}^{\mathsf{\ell }}& {\overline{M}}_{11}^{\mathsf{\ell }}& \cdots & {\overline{M}}_{1n_{\theta }}^{\mathsf{\ell }}\\ ⋮& ⋮& & ⋮\\ {\overline{M}}_{n_{\theta }0}^{\mathsf{\ell }}& {\overline{M}}_{n_{\theta }1}^{\mathsf{\ell }}& \cdots & {\overline{M}}_{n_{\theta }{n}_{\theta }}^{\mathsf{\ell }}\end{array}\right]\right\},\end{array}$$

$$\begin{array}{c}\hfill {\widehat{M}}_j^{\mathsf{\ell }}=\left[\begin{array}{cccc}{A}_{1j}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}& {\tilde{A}}_{2j}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}& \cdots & {\tilde{A}}_{Ij}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \\ 0& 0& \cdots & 0\end{array}\right],{\overline{M}}_{ij}^{\mathsf{\ell }}=\left[\begin{array}{cccc}{B}_{1i}^{\mathsf{\ell }}{U}_{1j}^{\mathsf{\ell }}& B_{1i}^{\mathsf{\ell }}{\tilde{U}}_{2j}^{\mathsf{\ell }}& \cdots & B_{1i}^{\mathsf{\ell }}{\tilde{U}}_{Ij}^{\mathsf{\ell }}\\ {\tilde{B}}_{2i}^{\mathsf{\ell }}{U}_{1j}^{\mathsf{\ell }}& {\tilde{B}}_{2i}^{\mathsf{\ell }}{\tilde{U}}_{2j}^{\mathsf{\ell }}& \cdots & {\tilde{B}}_{2i}^{\mathsf{\ell }}{\tilde{U}}_{Ij}^{\mathsf{\ell }}\\ ⋮& ⋮& & ⋮\\ {\tilde{B}}_{Ii}^{\mathsf{\ell }}{U}_{1j}^{\mathsf{\ell }}& {\tilde{B}}_{Ii}^{\mathsf{\ell }}{\tilde{U}}_{2j}^{\mathsf{\ell }}& \cdots & {\tilde{B}}_{Ii}^{\mathsf{\ell }}{\tilde{U}}_{Ij}^{\mathsf{\ell }}\end{array}\right]\end{array}$$

with ${\tilde{A}}_{ij}^{\mathsf{\ell }}=A_{ij}^{\mathsf{\ell }}-A_{1j}^{\mathsf{\ell }}$, ${\tilde{B}}_{ij}^{\mathsf{\ell }}=B_{ij}^{\mathsf{\ell }}-B_{1j}^{\mathsf{\ell }}$, ${\tilde{U}}_{ij}^{\mathsf{\ell }}=U_{ij}^{\mathsf{\ell }}-U_{1j}^{\mathsf{\ell }}$, then the closed-loop fuzzy parameter varying system with Markov jump (3) is stabilized by the state-feedback fuzzy gain schedule controller in the form of

$$u(x,\mathsf{\ell })=\sum _{k=1}^I\sum _{j=0}^v{h}_k{U}_{kj}^{\mathsf{\ell }}{Q}^{\mathsf{\ell }}{\theta }_{j}x,$$

where $Q^{\mathsf{\ell }}={\left(P^{\mathsf{\ell }}\right)}^{-1}$.

Proof. 

Consider closed-loop system (3) with the full states feedback fuzzy gain schedule controller

$$\begin{array}{c}\hfill \dot{x}=\sum _{i=1}^I{h}_i\left[A_i^{\mathsf{\ell }}\left(\theta \right)x+B_i^{\mathsf{\ell }}\left(\theta \right)\sum _{k=1}^I{h}_k{U}_k^{\mathsf{\ell }}\left(\theta \right)Q^{\mathsf{\ell }}x\right],\end{array}$$

(26)

where $U_k^{\mathsf{\ell }}\left(\theta \right)={\sum }_{j=0}^v{h}_k{U}_{kj}^{\mathsf{\ell }}{\theta }_j$. Because the membership functions satisfy $h_1=1-{\sum }_{i=2}^I{h}_i$, then the system (26) is rewritten as

$$\begin{array}{c}\hfill \dot{x}=A_1^{\mathsf{\ell }}\left(\theta \right)x+\sum _{i=2}^I{h}_i{\tilde{A}}_i^{\mathsf{\ell }}\left(\theta \right)x+\left(B_1^{\mathsf{\ell }}\left(\theta \right)+\sum _{i=2}^I{h}_i{\tilde{B}}_i^{\mathsf{\ell }}\left(\theta \right)\right)\left(U_1^{\mathsf{\ell }}\left(\theta \right)Q^{\mathsf{\ell }}+\sum _{k=2}^I{h}_k{\tilde{U}}_k^{\mathsf{\ell }}\left(\theta \right)Q^{\mathsf{\ell }}\right)\end{array}$$

(27)

Take the Lyapunov function candidate dependent on system mode $V(x,\mathsf{\ell })=x^T{Q}^{\mathsf{\ell }}x$, where ${\left(Q^{\mathsf{\ell }}\right)}^{-1}=P^{\mathsf{\ell }}$. Directly computing the derivative with respect to the time along the trajectory of the system (27), the following stability condition can be obtained

$$\begin{array}{c}\hfill {\displaystyle \mathbf{He}\{A_{h\theta }^{\mathsf{\ell }}{P}^{\mathsf{\ell }}+B_{h\theta }^{\mathsf{\ell }}{U}_{h\theta }^{\mathsf{\ell }}\}+P^{\mathsf{\ell }}\sum _{s=1}^m{\lambda }_{\mathsf{\ell }s}{P}^s{P}^{\mathsf{\ell }}<0.}\end{array}$$

(28)

Applying Schurer’s lemma, the inequality (28) is equal to

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\mathbf{He}\{A_{h\theta }^{\mathsf{\ell }}{P}^{\mathsf{\ell }}+B_{h\theta }^{\mathsf{\ell }}{U}_{h\theta }^{\mathsf{\ell }}\}+{\lambda }_{\mathsf{\ell }\mathsf{\ell }}{P}^{\mathsf{\ell }}& P_c^{\mathsf{\ell }}\\ ★& P_D^{\mathsf{\ell }}\end{array}\right]<0.\end{array}$$

(29)

It is easy to see that the inequality has the following decomposition regard H as argument

$$\begin{array}{c}\hfill {\left[\begin{array}{cc}{I}_n& 0\\ H\otimes I_n& 0\\ I_n& 0\\ 0& I_{mn}\end{array}\right]}^T\left[\begin{array}{cc}{M}_s^{\mathsf{\ell }}& 0\\ 0& {\widehat{P}}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{cc}{I}_n& 0\\ H\otimes I_n& 0\\ I_n& 0\\ 0& I_{mn}\end{array}\right]<0,\end{array}$$

(30)

where $M_s^{\mathsf{\ell }}$ is a matrix respecting to $\theta$ and system mode ℓ,

$$\begin{array}{cc}\hfill & M_s^{\mathsf{\ell }}=\hfill \\ \hfill & \mathbf{He}\left\{\left[\begin{array}{cc}{\displaystyle (A_1^{\mathsf{\ell }}\left(\theta \right)+\frac{{\lambda }_{\mathsf{\ell }\mathsf{\ell }}I}{2})P^{\mathsf{\ell }}+B_1^{\mathsf{\ell }}\left(\theta \right)U_1^{\mathsf{\ell }}\left(\theta \right)}& {\tilde{A}}_2^{\mathsf{\ell }}\left(\theta \right)P^{\mathsf{\ell }}+B_1^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_2^{\mathsf{\ell }}\left(\theta \right)\cdots {\tilde{A}}_I^{\mathsf{\ell }}\left(\theta \right)P^{\mathsf{\ell }}+B_1^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_I^{\mathsf{\ell }}\left(\theta \right)\\ {\tilde{B}}_2^{\mathsf{\ell }}\left(\theta \right)U_1^{\mathsf{\ell }}\left(\theta \right)& {\tilde{B}}_2^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_2^{\mathsf{\ell }}\left(\theta \right)\cdots {\tilde{B}}_2^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_I^{\mathsf{\ell }}\left(\theta \right)\\ ⋮& ⋮⋮\\ {\tilde{B}}_I^{\mathsf{\ell }}\left(\theta \right)U_1^{\mathsf{\ell }}\left(\theta \right)& {\tilde{B}}_I^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_2^{\mathsf{\ell }}\left(\theta \right)\cdots {\tilde{B}}_I^{\mathsf{\ell }}\left(\theta \right){\tilde{U}}_I^{\mathsf{\ell }}\left(\theta \right)\end{array}\right]\right\}.\hfill \end{array}$$

According to the full block S-procedure lemma, the inequality (30) is guaranteed if there is a full-block multiplier ${\Pi }_1^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]$ satisfying

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{M}_s^{\mathsf{\ell }}+{\Pi }_1^{\mathsf{\ell }}& \begin{array}{c}0\\ P_c^{\mathsf{\ell }}\end{array}\\ \begin{array}{cc}★& ★\end{array}& {\widehat{P}}_D^{\mathsf{\ell }}\end{array}\right]<0,\end{array}$$

(31)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}{I}_n\\ H\otimes I_n\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}{I}_n\\ H\otimes I_n\end{array}\right]\ge 0.\end{array}$$

(32)

where ${\widehat{P}}^{\mathsf{\ell }}=\left[\begin{array}{cc}0& P_c^{\mathsf{\ell }}\\ ★& P_D^{\mathsf{\ell }}\end{array}\right].$ Notice that the inequality (31) depends on the parameter $\theta$ and it is observed that $M_s^{\mathsf{\ell }}$ has the following decomposition

$$\begin{array}{c}\hfill M_s^{\mathsf{\ell }}={\left[\begin{array}{c}{I}_{In}\\ {\theta }_1{I}_{In}\\ ⋮\\ {\theta }_v{I}_{In}\end{array}\right]}^T{M}^{\mathsf{\ell }}\left[\begin{array}{c}{I}_{In}\\ {\theta }_1{I}_{In}\\ ⋮\\ {\theta }_v{I}_{In}\end{array}\right].\end{array}$$

Then the inequality (31) can be rewritten as

$$\begin{array}{c}\hfill {\left[\begin{array}{cc}{I}_{In}& 0\\ \theta \otimes I_{In}& 0\\ I_{In}& 0\\ 0& I_{Imn}\end{array}\right]}^T\left[\begin{array}{cc}\begin{array}{cc}{\Pi }_1^{\mathsf{\ell }}& 0\\ ★& M^{\mathsf{\ell }}\end{array}& \begin{array}{c}0\\ P_c^{\mathsf{\ell }}\end{array}\\ \begin{array}{cc}★& ★\end{array}& {\widehat{P}}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{cc}{I}_{In}& 0\\ \theta \otimes I_{In}& 0\\ I_{In}& 0\\ 0& I_{Imn}\end{array}\right]<0,\end{array}$$

(33)

It is in the form of $G^T(\Delta )MG(\Delta )<0$. Therefore, we can apply the full block S-procedure lemma again for the inequality (33). That is to say that the existence of a full-block multiplier ${\Pi }_2^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]$ satisfies

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{M}^{\mathsf{\ell }}+\left[\begin{array}{cc}{\Pi }_1^{\mathsf{\ell }}+{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]& \begin{array}{c}0\\ P_c^{\mathsf{\ell }}\end{array}\\ \begin{array}{cc}★& ★\end{array}& {\widehat{P}}_D^{\mathsf{\ell }}\end{array}\right]<0,\end{array}$$

(34)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}I\\ \theta \otimes I_{In}\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}I\\ \theta \otimes I_{In}\end{array}\right]\ge 0,\end{array}$$

(35)

Because of the negative definitive ${\Pi }_{1,22}^{\mathsf{\ell }}<0$ and ${\Pi }_{2,22}^{\mathsf{\ell }}<0$, it is adequate to verify the inequalities (32) and (35) at vertices $H_i$ and $\Theta$, which are the same as LMIs (24) and (25). Therefore it follows from the Lyapunov stability theory that the system (27) or (3) are stable at the origin, which concludes the proof. □

Remark 9.

The full block S-procedure lemma is used twice in the proof Theorem 3. To make it easier for the reader to understand, we omit some complex calculations. In the first time for the inequality (30), the constant matrices $G_{11},G_{12},G_{21}$ and $G_{22}$ are given as the same idea as Remark 1. In the second time for the inequality (34), the constant matrices are a little different from the first time because $M^{\mathsf{\ell }}$ needs to be divided into $M_{11}^{\mathsf{\ell }},M_{12}^{\mathsf{\ell }},M_{21}^{\mathsf{\ell }}$ and $M_{22}^{\mathsf{\ell }}$.

Remark 10.

The inequality (23) plays an important role in Theorem 3. It contains that the key matrix $M^{\mathsf{\ell }}$ may be not negative definite because of ${\Pi }_1^{\mathsf{\ell }}$. If we take the matrix ${\Pi }_1^{\mathsf{\ell }}=0$, Theorem 3 will degenerate to the result based on the common quadratic Lyapunov function.

Remark 11.

The core matrix $M^{\mathsf{\ell }}$ in Theorem 3 contains the old information of quadratic Lyapunov function method, for example, $A_{1j}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}+B_{1j}^{\mathsf{\ell }}{U}_{1j}^{\mathsf{\ell }}$. The terms ${\displaystyle \frac{{\lambda }_{\mathsf{\ell }\mathsf{\ell }}}{2}}{P}^{\mathsf{\ell }}$, $P_c^{\mathsf{\ell }}$ and ${\widehat{P}}_D^{\mathsf{\ell }}$ are the new information from the Lyapunov function depending on the operating mode. They have contributed to reducing conservativeness for our method.

Remark 12.

Notice that the inequality (24) contains the quadratic terms of H. Honestly speaking, infinite number of matrices inequalities should be held to ensure (32) even if the space containing H is convex. However, the inequality (24) will be convex with respect to H with enforcing the matrix ${\Pi }_{1,22}^{\mathsf{\ell }}<0$. Therefore, if the space containing H is a convex polytope containing zero, the inequality (32) only needs to be checked at the vertices of H, that is the inequality (24). This would lead researchers to check the finite number of linear matrix inequalities. A similar situation also occurs in inequality (25).

Remark 13.

The controller design method in Theorem 3 is effective for system matrices uncertainty problem regarding θ as uncertainty. The controller will be independent of θ and be a fuzzy gain-schedule controller with respect to the membership function $h_i$. It only needs the matrices $U_{ij}=0,j=1,2,\cdots ,v$ in Theorem 3.

In the actual system, the external disturbance is inevitable and is worth studying in depth. Fortunately, the design method in Theorem 3 is also used to design fuzzy gain-schedule $H_{\infty }$ controllers for the system (9). The method of Theorem 3 can be transplanted to the system (9) in parallel applying Lemma 2. We give the result directly without the proof because it is very similar to the proof of Theorem 3.

Corollary 1.

For a given $\mu >0$, if there exist a positive definite matrix $P^{\mathsf{\ell }}\in {\mathbf{S}}^n$, some constant matrices $U_{ij}\in {\mathbf{R}}^{q\times n},\mathsf{\ell }=1,2,\cdots ,m,i=1,\cdots ,r$ and $j=0,1,\cdots ,m$, and two full block multipliers ${\Pi }_1^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]$, ${\Pi }_2^{\mathsf{\ell }}=\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]$ satisfying for any $\Theta \in {\Omega }^v$, $i=1,2,\cdots ,r$,

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{M}_{\mu }^{\mathsf{\ell }}+\left[\begin{array}{cc}{\Pi }_1^{\mathsf{\ell }}+{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]& \begin{array}{c}0\\ P_c^{\mathsf{\ell }}\end{array}\\ \begin{array}{cc}★& ★\end{array}& {\widehat{P}}_D^{\mathsf{\ell }}\end{array}\right]<0,\end{array}$$

(36)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}{I}_n\\ H_i\otimes I_n\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{1,11}^{\mathsf{\ell }}& {\Pi }_{1,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{1,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}{I}_n\\ H_i\otimes I_n\end{array}\right]\ge 0,{\Pi }_{1,22}^{\mathsf{\ell }}<0,\end{array}$$

(37)

$$\begin{array}{c}\hfill {\left[\begin{array}{c}{I}_{In}\\ \Theta \otimes I_{In}\end{array}\right]}^T\left[\begin{array}{cc}{\Pi }_{2,11}^{\mathsf{\ell }}& {\Pi }_{2,12}^{\mathsf{\ell }}\\ ★& {\Pi }_{2,22}^{\mathsf{\ell }}\end{array}\right]\left[\begin{array}{c}{I}_{In}\\ \Theta \otimes I_{In}\end{array}\right]\ge 0,{\Pi }_{2,22}^{\mathsf{\ell }}<0,\end{array}$$

(38)

where $n=n+n_w+p$, the signal ⊗ represents the Kronecker product, $M_{\mu }^{\mathsf{\ell }}$ is defined as

$$\begin{array}{c}\hfill M_{\mu }^{\mathsf{\ell }}=\mathbf{He}\left\{\left[\begin{array}{ccc}{\widehat{M}}_1^{\mathsf{\ell }\mu }& \cdots & {\widehat{M}}_v^{\mathsf{\ell }\mu }\\ 0& \cdots & 0\\ ⋮& & ⋮\\ 0& \cdots & 0\end{array}\right]+\left[\begin{array}{ccc}{\widehat{M}}_{00}^{\mathsf{\ell }\mu }& \cdots & {\widehat{M}}_{0v}^{\mathsf{\ell }\mu }\\ {\widehat{M}}_{10}^{\mathsf{\ell }\mu }& \cdots & {\widehat{M}}_{1v}^{\mathsf{\ell }\mu }\\ ⋮& & ⋮\\ {\widehat{M}}_{v0}^{\mathsf{\ell }\mu }& \cdots & {\widehat{M}}_{vv}^{\mathsf{\ell }\mu }\end{array}\right]\right\}-\left[\begin{array}{ccccc}0& 0& 0& \cdots & 0\\ 0& \mu I_q& 0& \cdots & 0\\ 0& 0& \mu I_p& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right],\end{array}$$

$${\widehat{M}}_i^{\mathsf{\ell }\mu }=\left[\begin{array}{ccccccc}{A}_{1i}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}& B_{1i}^{\mathsf{\ell }w}& 0& \cdots & {\tilde{A}}_{Ii}^{\mathsf{\ell }}{P}^{\mathsf{\ell }}& {\tilde{B}}_{Ii}^{\mathsf{\ell }w}& 0\\ 0& 0& 0& \cdots & 0& 0& 0\\ C_{1i}^{\mathsf{\ell }y}{P}^{\mathsf{\ell }}& D_{1i}^{\mathsf{\ell }w}& 0& \cdots & {\tilde{C}}_{Ii}^{\mathsf{\ell }y}{P}^{\mathsf{\ell }}& {\tilde{D}}_{Ii}^{\mathsf{\ell }w}& 0\end{array}\right],$$

$${\overline{M}}_{ij}^{\mathsf{\ell }\mu }=\left[\begin{array}{ccccccc}{B}_{1i}^{\mathsf{\ell }u}{U}_{1j}^{\mathsf{\ell }}& 0& 0& \cdots & B_{1i}^{\mathsf{\ell }u}{\tilde{U}}_{Ij}^{\mathsf{\ell }}& 0& 0\\ 0& 0& 0& \cdots & 0& 0& 0\\ D_{1i}^{\mathsf{\ell }u}{U}_{1j}^{\mathsf{\ell }}& 0& 0& \cdots & D_{1i}^{\mathsf{\ell }u}{\tilde{U}}_{Ij}^{\mathsf{\ell }}& 0& 0\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮& \\ {\tilde{B}}_{Ii}^{\mathsf{\ell }u}{U}_{1j}^{\mathsf{\ell }}& 0& 0& \cdots & {\tilde{B}}_{Ii}^{\mathsf{\ell }u}{\tilde{U}}_{Ij}^{\mathsf{\ell }}& 0& 0\\ 0& 0& 0& \cdots & 0& 0& 0& \\ {\tilde{D}}_{Ii}^{\mathsf{\ell }u}{U}_{1j}^{\mathsf{\ell }}& 0& 0& \cdots & {\tilde{D}}_{Ii}^{\mathsf{\ell }u}{\tilde{U}}_{Ij}^{\mathsf{\ell }}& 0& 0\end{array}\right],$$

then the closed-loop fuzzy parameter varying system (3) with Markov jump is stabilized by the fuzzy gain schedule controller in the form of

$$u(x,\mathsf{\ell })=\sum _{k=1}^I\sum _{j=0}^v{h}_k{U}_{kj}^{\mathsf{\ell }}{Q}^{\mathsf{\ell }}{\theta }_{j}x,$$

with $Q^{\mathsf{\ell }}={\left(P^{\mathsf{\ell }}\right)}^{-1}$, which renders the ${\mathcal{L}}_2$ gain of the closed-loop system less than μ for any disturbance $w\in {\mathcal{L}}_2$.

5. Simulation

In this section, a numerical example is provided to illustrate the advantages and efficiency of the controller design methods. It demonstrates the usage of Theorem 3 and Corollary 1. The full state feedback fuzzy gain schedule controllers will be synthesized. We are going to compare our design methods with the conventional T-S fuzzy model based the numerical example.

For simplicity, we assume that there are only two fuzzy rules in the rule base and the local time-varying system is of second-order and has two system modes $\mathcal{M}=\{1,2\}$ Figure 1, i.e., $I=2$, $m=2$. Therefore, the analytical formula of the FPV system to be controlled is described as below

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \dot{x}=\sum _{i=1}^2{h}_i(A_i^{\mathsf{\ell }}\left(\theta \right)x+B_i^{\mathsf{\ell }w}\left(\theta \right)w+B_i^{\mathsf{\ell }u}\left(\theta \right)u),}\hfill \\ {\displaystyle y=\sum _{i=1}^2{h}_i(C_i^{\mathsf{\ell }}\left(\theta \right)x+D_i^{\mathsf{\ell }}\left(\theta \right)w),}\hfill \end{array}\right.\end{array}$$

(39)

where $x={\left[x_1{x}_2\right]}^T$ is the system state, y is performance output. $w\in {\mathcal{L}}_2$ is the disturbance signal satisfying ${\parallel w\parallel }_2\le 1$ showed in Figure 2. For convenience, only one state variable, i.e., $x_1$ is utilized as the scheduling variable and the membership functions (seeing Figure 3) are selected as $h_1\left(x_1\right)=0.5(1-sin{x}_1),h_2\left(x_1\right)=1-h\left(x_1\right)$. For simulation purposes, the varying parameter $\theta$ is chosen to be a sinusoidal function with respect to time t, i.e., $\theta =sint$. The system parameter matrices for the two local linear varying systems with Markov jump are listed in Appendix A and set two parameters $a,b\in [0,10]$ in system matrices to compare conservatism with other design methods. The transfer matrix is characterized by

$$\Pi =\left[\begin{array}{cc}-0.8& 0.8\\ 0.3& -0.3\end{array}\right].$$

Firstly, we compare the design method Theorem 3 with the T-S method with less conservatism provided in [26]. The method of Theorem 3 is simpler in algorithm complexity. When $a,b$ are integers and vary from 0 to 10, it is found that the system (39) is stabilized by Theorem 3 for all $(a,b)$ but the method provided in [26] not (see Figure 4). It implies that Theorem 3 we provide has less conservatism than reference [26]. In addition, Corollary 1 is used to obtain optimized ${\mathcal{L}}_2$ gain compared with the method provided in [26]. In

Table 2. Optimal $\mu$ with different a for state feedback case and $b=2$.

a	0	1	2	3	4	5	6	7	8	9	10
Corollary 1	0.04	0.04	0.01	0.07	0.02	0.15	0.3	0.3	0.3	0.4	0.4
Reference [13]	0.10	0.11	0.2	1.1	2.0	3.1	5.5	NaN	NaN	NaN	NaN
Reference [26]	1.1	1.6	1.9	2.5	2.8	6.1	7.1	NaN	NaN	NaN	NaN

, we can find that the optimal $\mu$ obtained by Theorem 3 is less than the references [13,26]. Specially, we simulate the system (39) when $a=1$ and $b=1$ by the LMI toolbox and Simulink in MATLAB(2018a) to solve the inequalities (36)–(38) in Corollary 3 (or (23)–(25) in Theorem 3) and obtain the gain matrices of full state feedback $H_{\infty }$ fuzzy gain schedule controllers,

$${\displaystyle u(x,\mathsf{\ell })=\sum _{i=1}^2\sum _{j=0}^1{h}_i{U}_{ij}^{\mathsf{\ell }}\left(\theta \right)P^{\mathsf{\ell }}x,}$$

where

$U_{10}^1=\left[\begin{array}{cc}10.84& -9.45\end{array}\right],U_{11}^1=\left[\begin{array}{cc}1.24& 6.66\end{array}\right],U_{20}^1=\left[\begin{array}{cc}12.25& -1.47\end{array}\right],U_{21}^1=\left[\begin{array}{cc}-1.62& 2.54\end{array}\right].$

$U_{10}^2=\left[\begin{array}{cc}9.27& 4.45\end{array}\right],U_{11}^2=\left[\begin{array}{cc}-2.61& 9.98\end{array}\right],U_{20}^2=\left[\begin{array}{cc}3.35& -1.55\end{array}\right],U_{21}^2=\left[\begin{array}{cc}1.92& -6.18\end{array}\right].$

$P^1=\left[\begin{array}{cc}10.75& -9.90\\ -9.90& 12.98\end{array}\right],P^2=\left[\begin{array}{cc}6.11& 5.26\\ 5.26& 8.60\end{array}\right].$

It can be observed from these simulations that the designed full state feedback fuzzy gain schedule controller keeps the closed-loop system asymptotically stable (see Figure 5) and succeeds in attenuating the disturbance (see Figure 6).

6. Conclusions

In this article, for the novel nonlinear time-varying model termed as the fuzzy parameter varying system with Markov jump, the stability and $H_{\infty }$ performance conditions of the closed-loop FPV system are derived. Moreover, using the convex optimization and the full block S-procedure technique, the full state feedback fuzzy gain-scheduling $H_{\infty }$ controller is synthesized to stabilize the closed-loop system. Numerical examples are provided to demonstrate the application of our approaches to design fuzzy gain schedule controllers and validate the effectiveness of our results. Although the derivations and the results seem complicated, the conditions can be solved effectively by some well-developed semi-definite programming algorithms.