H ∞ sampled-data control for T-S fuzzy singularly perturbed systems with actuator saturation

Abstract

In this paper, the problem of the sampled-data control for Takagi-Sugeno (T-S) fuzzy singularly perturbed systems (SPSs) with actuator saturation is considered. Based on the Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach, a design method of H_∞ sampled-data controller for T-S fuzzy SPSs with actuator saturation is proposed. Then, a convex optimization problem is formulated to maximize the disturbance tolerance ability. For the case that the systems are not subject to disturbances, the proposed design method is specialized to solve the problems of the asymptotical stabilization and optimization of the basin of attraction. Finally, numerical examples are provided to demonstrate the merits of the obtained results.

Keywords

Takagi-Sugeno (T-S) fuzzy singularly perturbed systems (SPSs)sampled-data control actuator saturation linear matrix inequality (LMI)

1 Introduction

In practical control systems, such as economic models, motor control systems, and power systems, small inductance, electric conductivity, and electric-capacity are often involved. These small time-constants easily lead to high order and ill-conditioned numerical issues in the process of system analysis and controller design. Therefore, singularly perturbed systems (SPSs), which have a small singular perturbation parameter ɛ determining the degree of separation between slow and fast dynamic modes of the systems, are proposed to describe and deal with this kind of systems [1, 2].

It should be noted that the stability and stabilization problems for SPSs are more complex than that for normal systems, and have attracted much attention [3 –5]. The major part of these problems for SPSs is to determine an upper bound ɛ₀ for the singular perturbation parameter ɛ, such that SPSs are stable for all ɛ ∈ (0, ɛ₀) or ɛ ∈ (0, ɛ₀]. In the existing results, by frequency- and time-domains methods, the problems of estimating the stability bound for SPSs are considered [3, 4]. In addition, the stabilization bound problems for SPSs have been discussed in [5] by designing an H_∞ controller. Obviously, the problems of enlarging the upper bound ɛ₀ for nonlinear SPSs are complex and challenging topics, and only a few works have been carried out on the stabilization bound synthesis for nonlinear SPSs [5, 6]. Therefore, obtaining the design method of H_∞ controller which can enlarge the stability bound, and guaranteeing the less conservatism for nonlinear SPSs are expected to be researched, which motivate us for the present study.

Since Takagi-Sugeno (T-S) fuzzy model combines the flexibility of fuzzy logic theory and the rigorous mathematical analysis tools into a unified framework, it has been extensively studied and successfully applied to approximate a wide class of nonlinear control systems [7, 8]. Recently, many researchers have focused on the control problem for both continuous- and discrete-time T-S fuzzy SPSs [9, 10]. And the problem of enlarging the bound of ɛ for T-S fuzzy SPSs is first considered in [5]. It is worth mentioning that the upper bound obtained in [5] is a little small and some nonlinear factors are not involved in the running process of the system. Considering the conservatism and the application range of the system, researching the stabilization bound problem for T-S fuzzy SPSs with saturation factor attracts our attention.

In addition, owing to the fact that actuator saturation is common for the physical systems. Various control problems for the systems with actuator saturation have been extensively studied in the literature [11 –14]. Moreover, the research on SPSs in the presence of actuator saturation has been achieved a lot of outstanding results. To estimate the basin of attraction of SPSs, some methods that depend on decomposing the original SPSs were proposed in [12], and an alternative approach that is independent of system decomposition was proposed in [13] to avoid the possible ill-conditioned numerical problems. And, the research on T-S fuzzy systems with input saturation has been achieved a lot of great results (see [14] and references therein). As we know, the problem of controller design for T-S fuzzy SPSs with actuator saturation has not been considered.

With the development of digital computing technology, digital computers, which are applied to sample a continuous-time measurement signal to produce a discrete-time signal, are usually utilized in industrial applications to control continuous-time systems. Then, the generated discrete-time control input signal is converted back into a continuous-time control input signal by using a zero-order hold. Such control systems are referred as sampled-data systems [15, 16], which include both continuous- and discrete-time signals in the continuous-time framework. Due to the characteristic that the control signals are kept constant between any two consecutive sampling instants and only are changed at each sampling instant, it is more complicated to deal with the sampled-data systems. Recently, two main methods have been developed for the analysis and controller design for sampled-data systems [16 –19]. The first one is to model a sampled-data system as a discrete-time system [16, 17], in which some stability conditions are derived. It is worth noticing that the sampled control for SPSs with input saturation is discussed in [16]. The second one is to model a sampled-data system as a continuous-time system with a delayed control input, which was proposed in [18] and latter used in [19]. More recently, many sampled-data analysis and synthesis results have been reported for T-S fuzzy systems [20, 21]. To the best of our knowledge, the problem of sampled-data control for T-S fuzzy SPSs with input saturation is stillopen.

This paper will consider the problem of H_∞ sampled-date controller design for T-S fuzzy SPSs subject to actuator saturation. First, by an ɛ-dependent Lyapunov-Krasovskii function, LMI conditions which guarantee that the closed-loop systems satisfy H_∞ performance are derived. Then, using the obtained conditions, the method of sampled-data controller design is proposed and the estimation of the largest disturbance tolerance ability is given by solving a convex optimization problem. Especially, for the system without disturbances, sufficient conditions which guarantee that the closed-loop system is asymptotical stable are derived. Furthermore, the estimation of the basin of attraction is obtained. Finally some numerical examples are provided to demonstrate the efficiency of the proposed results.

Notation: For a matrix X, X^-1 and X^T denote the inverse and the transpose of X, respectively. X > 0 (X < 0) means that X is positive definite (negative definite). sym (X) denotes X + X^T. Symmetric elements in the matrix are denoted by *. Matrices, if not explicitly stated, are assumed to have compatible dimensions. And diag{ ⋯ } is a block-diagonal matrix.

2 Problem description and preliminaries

Consider a class of SPSs, which can be described by the following fuzzy model

$\begin{matrix} Plant rule i : \\ IF \\ v_{1} (t) is M_{i 1}, v_{2} (t) is M_{i 2}, \dots, v_{g} (t) is M_{ig} \\ THEN \\ E (ɛ) \dot{x} (t) = A_{i} x (t) + B_{i} sat (u (t)) + B_{ω i} ω (t) \\ z (t) = C_{i} x (t) + D_{i} u (t) \\ for i = 1, 2, \dots, r \end{matrix}$ (1) where r is the number of IF-THEN rules, M_ik, i = 1, …, r, k = 1, …, g, are fuzzy sets, v₁ (t), …, v_g (t) are premise variables. $x (t) \in R^{n}$ is the state vector, $u (t) \in R^{p}$ is the control input, $z (t) \in R^{l}$ is the controlled output, $ω (t) \in R^{q}$ is the disturbance input which belongs to $W_{η} = {ω : R_{+} \to R^{q}, \int_{0}^{\infty} ω^{T} ω dt \leq η}$ for a positive number η, $E (ɛ) = diag {I_{n_{1}}, ɛ I_{n_{2}}} \in R^{n \times n}$ with n₁ + n₂ = n, ɛ is a positive scalar representing the singular perturbation parameter, A_i, B_i, B_ωi, C_i and D_i are known real constant matrices with appropriate dimensions.

And $sat : R^{p} \to R^{p}$ is the standard saturation function defined as follows $sat (u (t)) = {[sat (u_{1} (t)), \dots, sat (u_{p} (t))]}^{T}$ where without loss of generality $\begin{matrix} sat (u_{i} (t)) = sign (u_{i} (t)) min {1, | u_{i} (t) |} \\ i = 1, \dots, p \end{matrix}$

Denote $w_{i} (v (t)) = \prod_{k = 1}^{g} M_{ik} (v_{k} (t)),$ i = 1, …, r, where M_ik (v_k (t)) is the grade of membership of v_k (t) in M_ik. In this paper, it is assumed that $w_{i} (v (t)) \geq 0, \sum_{i = 1}^{r} w_{i} (v (t)) > 0 .$

Let $μ_{i} (v (t)) = \frac{w_{i} (v (t))}{\sum_{i = 1}^{r} w_{i} (v (t))},$ then $μ_{i} (v (t)) \geq 0, \sum_{i = 1}^{r} μ_{i} (v (t)) = 1 .$ In the sequel, we denote μ_i (v (t)) by μ_i.

Using singleton fuzzifier, product inference, and center-average defuzzifier, the global dynamics of the T-S fuzzy system (1) is described as follows $\begin{matrix} E (ɛ) \dot{x} (t) & = & \sum_{i = 1}^{r} μ_{i} [A_{i} x (t) + B_{i} sat (u (t)) \\ + B_{ω i} ω (t)] \\ z (t) & = & \sum_{i = 1}^{r} μ_{i} [C_{i} x (t) + D_{i} u (t)] \end{matrix}$ (2)

Remark 1. In the control system design, an inverted pendulum is a classical nonlinear system model [8]. By combining IF-THEN rules, the inverted pendulum nonlinear system can be approximated by the T-S fuzzy model. SPSs are applied in convection-diffusion systems, diffusion-drift motion systems, telecommunication systems, which often occurs the multiple time-scales phenomena. In addition, T-S fuzzy SPSs have been applied in many fields, such as a tunnel diode circuit, an inverted pendulum controlled by a motor via a gear train, and so on. Moreover, the phenomenon of input saturation is commonly appeared in the process of practical engineering control. It is necessary to consider the saturation factor in the above mentioned systems.

The sampled-data control law is described by the following fuzzy model $\begin{matrix} Controller rule i : \\ IF v_{1} (t) is M_{i 1}, v_{2} (t) is M_{i 2}, \dots, v_{g} (t) is M_{ig} \\ THEN \\ u (t) = K_{i} (ɛ) x (t_{k}), t_{k} \leq t < t_{k + 1} \\ for i = 1, 2, \dots, r \end{matrix}$ where t_k (k = 0, 1, …) is the sampling instant,t₀ ≥ 0, $lim_{k \to \infty} t_{k} = \infty .$

Because the controller rules are the same as the plant rules, the fuzzy sampled-data controller is given as follows $u (t) = \sum_{i = 1}^{r} μ_{i} (v (t_{k})) [K_{i} (ɛ) x (t_{k})]$ (3)

We assume that 0 < t_k+1 - t_k ≤ τ, k = 0, 1, 2, …, where τ denotes the upper bound of the interval between two consecutive sampling instants. Denote τ (t) = t - t_k, t ∈ [t_k, t_k+1). It is very explicit that 0 < τ (t) < τ, because of τ (t) < t_k+1 - t_k. It is different from the general time-delay systems, the time-varying delay τ (t) is piecewise continuous with $\dot{τ} (t) = 1$ for t ≠ t_k, and the derivative does not exist at the sampling instant.

Remark 2. The sampled control problem is considered in [15, 16], which is only suitable for the periodic sampling case. That is, all these papers consider the case with a constant sampling distance. However, in most practical applications, the precise periodic sampling is obtained difficultly, owing to the overfull interrupt request signals for the microprocessor and the irregular disturbances. Therefore, the problem of stabilization for T-S fuzzy systems with nonuniform uncertain sampling is considered in this paper.

The fuzzy sampled-data controller (3) can be reconstructed as follows $u (t) = \sum_{i = 1}^{r} μ_{i} (v (t - τ (t))) [K_{i} (ɛ) x (t - τ (t))]$ (4)

Applying Equation (4) to the system (2), the closed-loop system is obtained as follows

$\begin{matrix} E (ɛ) \dot{x} (t) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} μ_{j} [A_{i} x (t) + B_{i} sat (K_{j} (ɛ) \\ x (t - τ (t))) + B_{ω i} ω (t)] \\ z (t) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} μ_{i} μ_{j} [C_{i} x (t) + D_{i} K_{j} (ɛ) \\ x (t - τ (t))] \end{matrix}$ (5)

Definition 1. [11] For a matrix $H_{i} \in R^{p \times n},$ define the hth row of H_i as H_i(h) and define $L (H_{i})$ as $\begin{matrix} L (H_{i}) = {x (t) \in R^{n} : | H_{i (h)} x | \leq 1 h \in [1, p]}, \\ i = 1, \dots, r \end{matrix}$

In fact, if $x \in L (H_{i})$ for any i = 1, …, r, then $x \in L (\sum_{i = 1}^{r} μ_{i} H_{i}) .$

Definition 2. [11] Let P be a positive-defined symmetric matrix and ρ is a positive scalar, denote Ω (P, ρ) by the following set $Ω (P, ρ) = {x (t) \in R^{n} : x {(t)}^{T} Px (t) \leq ρ}$

Let $D$ be the set of p × p diagonal matrices whose diagonal elements are either I or 0. Suppose each element of $D$ is labelled as D_s, s = 1, 2, …, 2^p, and denote $D_{s}^{-} = I - D_{s}$ . Clearly, if $D_{s} \in D$ , then ${D_{s}}^{-} \in D$ .

The aim of this paper is to design an H_∞ sampled-data controller, such that for any ɛ ∈ (0, ɛ₀], we have

the state trajectories of the closed-loop system (5) which start from the origin will remain inside Ω (E (ɛ) Z^-1 (ɛ), γ²η).

the closed-loop system (5) satisfies H_∞ performance γ within Ω (E (ɛ) Z^-1 (ɛ), γ²η), which means, under the zero initial condition, the closed-loop system (5) satisfies $J_{ω} = \int_{0}^{\infty} (z^{T} (t) z (t) - γ^{2} ω^{T} (t) ω (t)) dt < 0$ for any nonzero ω (t) ∈ W_η.

Lemma 1. [11] Let $K \in R^{p \times n},$ $H \in R^{p \times n},$ then for any $x (t) \in L (H)$ , we have $sat (Kx (t)) \in co {D_{s} Kx (t) + {D_{s}}^{-} Hx (t)}$ or equivalently $sat (Kx (t)) = \sum_{s = 1}^{2^{p}} α_{s} (D_{s} K + D_{s}^{-} H) x (t)$ where co stands for the convex hull, α_s for s = 1, 2, …, 2^p are some scalars which satisfy 0 ≤ α_s ≤ 1 and $\sum_{s = 1}^{2^{p}} α_{s} = 1$ .

Lemma 2. [6] For a positive scalar ɛ₀ and symmetric matrices S₁, S₂ and S₃ with appropriate dimensions, if $\begin{matrix} S_{1} \geq 0 \\ S_{1} + ɛ_{0} S_{2} > 0 \\ S_{1} + ɛ_{0} S_{2} + ɛ_{0}^{2} S_{3} > 0 \end{matrix}$ hold, then $S_{1} + ɛ S_{2} + ɛ^{2} S_{3} > 0, \forall ɛ \in (0, ɛ_{0}]$

Lemma 3. [6] If there exist matrices Z_i (i = 1, 2, …, 5) with $Z_{i} = Z_{i}^{T} (i = 1, 2, 3, 4)$ satisfying $Z_{1} > 0$ (6) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & ɛ_{0} Z_{5}^{T} \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} \end{matrix}] > 0$ (7) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & ɛ_{0} Z_{5}^{T} \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} + ɛ_{0}^{2} Z_{4} \end{matrix}] > 0$ (8) then $E (ɛ) Z (ɛ) = Z^{T} (ɛ) E (ɛ) > 0, \forall ɛ \in (0, ɛ_{0}]$ (9) where $Z (ɛ) = [\begin{matrix} Z_{1} + ɛ Z_{3} & ɛ Z_{5}^{T} \\ Z_{5} & Z_{2} + ɛ Z_{4} \end{matrix}] .$

Lemma 4. [25] Given real matrices T₁, T₂ with appropriate dimensions, for any positive definite matrix G, we can get $T_{1} T_{2} + {T_{2}}^{T} {T_{1}}^{T} \leq T_{1} G {T_{1}}^{T} + {T_{2}}^{T} G^{- 1} T_{2}$

3 Main results

In this section, we concentrate our attention on the problem of designing an H_∞ sampled-data controller.

Theorem 1. For a given H_∞ performance bound γ, ɛ-bound ɛ₀ > 0 and scalars τ > 0, η > 0, if there exist matrices Z_l (l = 1, …, 5), with $Z_{l} = Z_{l}^{T} (l = 1, 2, 3, 4),$ matrices Y_j, H_1j, H_2j, and positive-defined symmetric matrices R₁, R₂, such that LMIs (6)– (8) and the following inequalities hold $[\begin{matrix} Z_{1} & * \\ H_{1 j (h)} & \frac{1}{γ^{2} η} \end{matrix}] \geq 0$ (10) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & * & * \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} & * \\ H_{1 j (h)} & ɛ_{0} H_{2 j (h)} & \frac{1}{γ^{2} η} \end{matrix}] \geq 0$ (11) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & * & * \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} + ɛ_{0}^{2} Z_{4} & * \\ H_{1 j (h)} & ɛ_{0} H_{2 j (h)} & \frac{1}{γ^{2} η} \end{matrix}] \geq 0$ (12) $\begin{matrix} h \in [1, p], j = 1, \dots, r \\ Ψ_{ii} (0) < 0, i = 1, \dots, r \\ Ψ_{ij} (0) + Ψ_{ji} (0) < 0, 1 \leq i < j \leq r \\ Ψ_{ii} (ɛ_{0}) < 0, i = 1, \dots, r \end{matrix}$ (13) $Ψ_{ij} (ɛ_{0}) + Ψ_{ji} (ɛ_{0}) < 0, 1 \leq i < j \leq r$ where $\begin{matrix} Ψ_{ij} (0) = [\begin{matrix} sym (A_{i} Z (0)) + Ξ (0) & * \\ {(B_{i} {\tilde{K}}_{j})}^{T} + {\tilde{R}}_{2}^{T} E (0) & - 2 {\tilde{R}}_{2} \\ 0 & {\tilde{R}}_{2}^{T} \\ B_{ω i}^{T} & 0 \\ τ A_{i} Z (0) & τ B_{i} {\tilde{K}}_{j} \\ C_{i} Z (0) & D_{i} Y_{j} \end{matrix} \\ \begin{matrix} * & * & * & * \\ * & * & * & * \\ - {\tilde{R}}_{1} - {\tilde{R}}_{2} & * & * & * \\ 0 & - γ^{2} I & * & * \\ 0 & τ B_{ω i} & {\tilde{R}}_{2} - sym (Z (0)) & * \\ 0 & 0 & 0 & - I \end{matrix}] \\ Ψ_{ij} (ɛ_{0}) = [\begin{matrix} sym (A_{i} Z (ɛ_{0})) + Ξ (ɛ_{0}) & * \\ {(B_{i} {\tilde{K}}_{j})}^{T} + {\tilde{R}}_{2}^{T} E (ɛ_{0}) & - 2 {\tilde{R}}_{2} \\ 0 & {\tilde{R}}_{2}^{T} \\ B_{ω i}^{T} & 0 \\ τ A_{i} Z (ɛ_{0}) & τ B_{i} {\tilde{K}}_{j} \\ C_{i} Z (ɛ_{0}) & D_{i} Y_{j} \end{matrix} \\ \begin{matrix} * & * & * & * \\ * & * & * & * \\ - {\tilde{R}}_{1} - {\tilde{R}}_{2} & * & * & * \\ 0 & - γ^{2} I & * & * \\ 0 & τ B_{ω i} & {\tilde{R}}_{2} - sym (Z (ɛ_{0})) & * \\ 0 & 0 & 0 & - I \end{matrix}] \\ Z (ɛ) = [\begin{matrix} Z_{1} + ɛ Z_{3} & ɛ Z_{5}^{T} \\ Z_{5} & Z_{2} + ɛ Z_{4} \end{matrix}], \\ {\tilde{K}}_{j} = D_{s} Y_{j} + D_{s}^{-} H_{j}, \\ H_{j} (ɛ) = H_{j} E (ɛ) = [\begin{matrix} H_{1 j} & H_{2 j} \end{matrix}] E (ɛ), \\ {\tilde{R}}_{1} = Z (ɛ) R_{1} Z^{T} (ɛ), {\tilde{R}}_{2} = Z (ɛ) R_{2} Z^{T} (ɛ), \\ Ξ (0) = E (0) {\tilde{R}}_{1} E (0) + E (0) {\tilde{R}}_{2} E (0) \end{matrix}$

Then, for any ɛ ∈ (0, ɛ₀], the closed-loop system (5) with H_∞ sampled-data controller (3) K_j (ɛ) = Y_jE (ɛ) Z^-1 (ɛ) satisfies the following conditions

the state trajectories of the closed-loop system (5) which start from the origin will remain inside Ω (E (ɛ) Z^-1 (ɛ), γ²η).

under the zero initial condition, the closed-loop system (5) satisfies H_∞ performance γ within Ω (E (ɛ) Z^-1 (ɛ), γ²η), for any nonzero ω (t) ∈ W_η.

Proof. From Lemma 2, for any ɛ ∈ (0, ɛ₀], LMIs (10)–(12) imply $\begin{matrix} [\begin{matrix} Z^{T} (ɛ) E (ɛ) & * \\ H_{j (h)} (ɛ) & \frac{1}{γ^{2} η} \end{matrix}] \geq 0, h \in [1, p], \\ j = 1, \dots, r, \end{matrix}$ (14) whereH_j(h) (ɛ) denotes the hth row of H_j (ɛ).

Pre- and post-multiplying inequality (14) by $diag {\begin{matrix} Z^{- T} (ɛ) & I \end{matrix}}$ and its transpose, respectively, we have $\begin{matrix} [\begin{matrix} E (ɛ) Z^{- 1} (ɛ) & * \\ H_{j (h)} (ɛ) Z^{- 1} (ɛ) & \frac{1}{γ^{2} η} \end{matrix}] \geq 0, h \in [1, p], \\ j = 1, \dots, r, \end{matrix}$ which implies $\begin{matrix} Z^{- T} (ɛ) H_{j (h)}^{T} (ɛ) H_{j (h)} (ɛ) Z^{- 1} (ɛ) \\ \leq \frac{1}{γ^{2} η} E (ɛ) Z^{- 1} (ɛ) \end{matrix}$

Then for any x (t) ∈ Ω (E (ɛ) Z^-1 (ɛ), γ²η), it holds that $\begin{matrix} x^{T} (t) Z^{- T} (ɛ) H_{j (h)}^{T} (ɛ) H_{j (h)} (ɛ) Z^{- 1} (ɛ) x (t) \leq 1 \\ h \in [1, p], j = 1, \dots, r, \end{matrix}$ which implies that $Ω (E (ɛ) Z^{- 1} (ɛ), γ^{2} η) \subseteq ⋂_{j = 1}^{r} L (H_{j} (ɛ) Z^{- 1} (ɛ))$

And, it can be derived that $\begin{matrix} | \sum_{j = 1}^{r} μ_{j} H_{j (h)} (ɛ) Z^{- 1} (ɛ) x | \\ \leq \sum_{j = 1}^{r} | μ_{j} | | H_{j (h)} (ɛ) Z^{- 1} (ɛ) x | \leq \sum_{j = 1}^{r} μ_{j} = 1 \end{matrix}$

The above inequality shows that $x (t) \in L (\sum_{j = 1}^{r} μ_{j} H_{j} (ɛ) Z^{- 1} (ɛ))$

Let P (ɛ) = Z^-1 (ɛ), we can obtain that $Ω (E (ɛ) Z^{- 1} (ɛ), γ^{2} η) \subseteq L (\sum_{j = 1}^{r} μ_{j} H_{j} (ɛ) P (ɛ)) .$

As a result, by Lemma 1, we have

$\begin{matrix} A_{i} x (t) + B_{i} sat (K_{j} (ɛ) x (t - τ (t))) + B_{ω i} ω (t) \\ = \sum_{s = 1}^{2^{p}} α_{s} [A_{i} x (t) + B_{i} (D_{s} K_{j} (ɛ) \\ + D_{s}^{-} H_{j} (ɛ) P (ɛ)) x (t - τ (t)) + B_{ω i} ω (t)] \end{matrix}$ (15) for any x (t) ∈ Ω (E (ɛ) Z^-1 (ɛ), γ²η).

Using Lemma 2 again, for any ɛ ∈ (0, ɛ₀], LMI (13) implies that $\begin{matrix} [\begin{matrix} sym (A_{i} Z (ɛ)) + Ξ (ɛ) & * & * \\ {(B_{i} {\tilde{K}}_{j})}^{T} + {\tilde{R}}_{2}^{T} E (ɛ) & - 2 {\tilde{R}}_{2} & * \\ 0 & {\tilde{R}}_{2}^{T} & - {\tilde{R}}_{1} - {\tilde{R}}_{2} \\ B_{ω i}^{T} & 0 & 0 \\ τ A_{i} Z (ɛ) & τ B_{i} {\tilde{K}}_{j} & 0 \\ C_{i} Z (ɛ) & D_{i} Y_{j} & 0 \end{matrix} \\ \begin{matrix} * & * & * \\ * & * & * \\ * & * & * \\ - γ^{2} I & * & * \\ τ B_{ω i} & - sym (Z (ɛ)) + {\tilde{R}}_{2} & * \\ 0 & 0 & - I \end{matrix}] < 0 \end{matrix}$ (16)

By Lemma 4, it is obtained that $\begin{matrix} - sym (Z (ɛ)) + {\tilde{R}}_{2} \\ = - sym (Z (ɛ)) + Z (ɛ) R_{2} Z^{T} (ɛ) \geq - R_{2}^{- 1} \end{matrix}$

Pre- and post-multiplying inequality (16) by $diag {Z^{- T} (ɛ), Z^{- T} (ɛ) E (ɛ), Z^{- T} (ɛ) E (ɛ), I, I, I}$ and its transpose, respectively, we have $\begin{matrix} [\begin{matrix} sym (P^{T} (ɛ) A_{i}) + {\bar{R}}_{1} + {\bar{R}}_{2} & * \\ {(B_{i} {\bar{K}}_{j})}^{T} + {\bar{R}}_{2}^{T} & - 2 {\bar{R}}_{2} \\ 0 & {\bar{R}}_{2}^{T} \\ B_{ω i}^{T} P (ɛ) & 0 \\ τ A_{i} & τ B_{i} {\bar{K}}_{j} \\ C_{i} & D_{i} Y_{j} E (ɛ) P (ɛ) \end{matrix} \\ \begin{matrix} * & * & * & * \\ * & * & * & * \\ - {\bar{R}}_{1} - {\bar{R}}_{2} & * & * & * \\ 0 & - γ^{2} I & * & * \\ 0 & τ B_{ω i} & - R_{2}^{- 1} & * \\ 0 & 0 & 0 & - I \end{matrix}] < 0 \end{matrix}$ where $\begin{matrix} {\bar{K}}_{j} = D_{s} Y_{j} E (ɛ) P (ɛ) + D_{s}^{-} H_{j} (ɛ) P (ɛ), \\ {\bar{R}}_{1} (ɛ) = E^{T} (ɛ) R_{1} E (ɛ), {\bar{R}}_{2} = E^{T} (ɛ) R_{2} E (ɛ) . \end{matrix}$

Let K_j (ɛ) = Y_jE (ɛ) P (ɛ). According to Schur complement, we have

$\begin{matrix} {\tilde{Ψ}}_{ij} + τ^{2} {[E (ɛ) \dot{x} (t)]}^{T} R_{2} E (ɛ) \dot{x} (t) \\ + z^{T} (t) z (t) - γ^{2} ω^{T} (t) ω (t) < 0 \end{matrix}$ (17) where ${\tilde{Ψ}}_{ij} = [\begin{matrix} sym (P^{T} (ɛ) A_{i}) + {\bar{R}}_{1} + {\bar{R}}_{2} & * & * \\ {(B_{i} {\bar{K}}_{j})}^{T} + {\bar{R}}_{2}^{T} & - 2 {\bar{R}}_{2} & * \\ 0 & {\bar{R}}_{2}^{T} & \begin{matrix} - {\bar{R}}_{1} \\ - {\bar{R}}_{2} \end{matrix} \end{matrix}] .$

By Lemma 3, LMIs (6)–(8) guarantee that Equation (9) holds, which implies $E (ɛ) P (ɛ) = P^{T} (ɛ) E (ɛ) > 0, \forall ɛ \in (0, ɛ_{0}]$

Choose an ɛ-dependent Lyapunov-Krasovskii functional $\begin{matrix} V (x (t)) \\ = x^{T} (t) E (ɛ) P (ɛ) x (t) \\ + \int_{t - τ}^{t} x^{T} (s) E^{T} (ɛ) R_{1} E (ɛ) x (s) ds \\ + τ \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) E^{T} (ɛ) R_{2} E (ɛ) \dot{x} (s) ds d θ \end{matrix}$

Taking the time derivative of V (x_t) along the trajectories of the system (5) yields $\begin{matrix} \dot{V} (x (t)) = {[E (ɛ) \dot{x} (t)]}^{T} P (ɛ) x (t) \\ + x^{T} (t) P^{T} (ɛ) [E (ɛ) \dot{x} (t)] \\ + x^{T} (t) E^{T} (ɛ) R_{1} E (ɛ) x (t) \\ - x^{T} (t - τ) E^{T} (ɛ) R_{1} E (ɛ) x (t - τ) \\ + τ^{2} {\dot{x}}^{T} (t) E^{T} (ɛ) R_{2} E (ɛ) \dot{x} (t) \\ - τ \int_{t - τ}^{t} {\dot{x}}^{T} (s) E^{T} (ɛ) R_{2} E (ɛ) \dot{x} (s) ds \end{matrix}$

Based on the Jensen inequality, we can get $\begin{matrix} - τ \int_{t - τ}^{t} {\dot{x}}^{T} (s) E^{T} (ɛ) R_{2} E (ɛ) \dot{x} (s) ds \\ \leq {[\begin{matrix} x (t) \\ x (t - τ (t)) \\ x (t - τ) \end{matrix}]}^{T} Π [\begin{matrix} x (t) \\ x (t - τ (t)) \\ x (t - τ) \end{matrix}] \end{matrix}$ where $Π = [\begin{matrix} - {\bar{R}}_{2} & {\bar{R}}_{2} & 0 \\ * & - 2 {\bar{R}}_{2} & {\bar{R}}_{2} \\ * & * & - {\bar{R}}_{2} \end{matrix}] .$

Then, taking into account Equations (15) and (17), we have $\begin{matrix} \dot{V} (x (t)) < {[E (ɛ) \dot{x} (t)]}^{T} P (ɛ) x (t) \\ + x^{T} (t) P^{T} (ɛ) [E (ɛ) \dot{x} (t)] \\ + x^{T} (t) E^{T} (ɛ) R_{1} E (ɛ) x (t) \\ - x^{T} (t - τ) E^{T} (ɛ) R_{1} E (ɛ) x (t - τ) \\ + τ^{2} {\dot{x}}^{T} (t) E^{T} (ɛ) R_{2} E (ɛ) \dot{x} (t) \\ + {[\begin{matrix} x (t) \\ x (t - τ (t)) \\ x (t - τ) \end{matrix}]}^{T} Π [\begin{matrix} x (t) \\ x (t - τ (t)) \\ x (t - τ) \end{matrix}] \\ \leq \sum_{i = 1}^{r} \sum_{j = 1}^{r} \sum_{s = 1}^{2^{p}} μ_{i} μ_{j} α_{s} (ζ^{T} (t) ({\bar{Ψ}}_{ij}) ζ (t) \\ + τ^{2} {[E (ɛ) \dot{x} (t)]}^{T} R_{2} E (ɛ) \dot{x} (t)) \\ < - z^{T} (t) z (t) + γ^{2} ω^{T} (t) ω (t) \end{matrix}$ where $ζ^{T} (t) = [\begin{matrix} x^{T} (t) & x^{T} (t - τ (t)) & x^{T} (t - τ) \end{matrix}]$ which shows $\dot{V} (x (t)) < - z^{T} (t) z (t) + γ^{2} ω^{T} (t) ω (t)$

When ω (t) ∈ W_η, integrating both sides of the above inequality from 0 to ∞ yields $\begin{matrix} V (x (t)) + \int_{0}^{\infty} z^{T} (t) z (t) dt \\ < V (x (0)) + γ^{2} \int_{0}^{\infty} ω^{T} (t) ω (t) dt \\ < V (x (0)) + γ^{2} η \end{matrix}$

Under the zero initial condition, it is obtained that V (x (t)) < γ²η, which implies that the state trajectories of system (5) which start from the origin will remain inside Ω (E (ɛ) Z^-1 (ɛ), γ²η). Noting that V (x (t)) > 0, it is also obtained that $J_{ω} = \int_{0}^{\infty} (z^{T} (t) z (t) - γ^{2} ω^{T} (t) ω (t)) dt < 0$

Thus, the closed-loop system (5) satisfies the H_∞ performance for any ɛ ∈ (0, ɛ₀]. This completes the proof.

Remark 3. T-S fuzzy SPSs with input saturation are studied in this paper, and an ɛ-dependent Lyapunov functions is designed to analyze the stability, and to design the controller. The main idea of designing a sampled-data controller is to translate input vector into a time-delay part. Therefore, the results can be generalized to solve the nonlinear systems under the network-based with time-delay, uncertainties, and so on [22]. It is worth noting that the nonlinear problem caused by the uncertainties can be solved by adopting proper matrix inequalities and Schur complement. In addition, the existing of singular perturbation parameter and saturation factor causes the computation complexity in the process of controller design. Considering the method in [23 –26], our results can be extended to construct an ɛ-dependent fuzzy/piecewise Lyapunov functions by modifying the matrix Z (ɛ). Certainly, the number of decision variables will be increased, but it may be an alternative way to reduce the conservatism of the system. We will consider these problems in our future works.

Under the conditions of Theorem 1 and the above discussion, for any ω (t) ∈ W_η and scalar γ > 0, the problem of estimating the largest of η, which is called the disturbance tolerance of the closed-loop system, can be formulated as follows

$\begin{matrix} max_{H_{j}, Y_{j}, Z_{i}} η \\ s . t . (6) - (8) and (10) - (13) \end{matrix}$ (18)

Remark 4. If x (0) ∈ Ω (E (ɛ) Z^-1 (ɛ), 1), considering Theorem 1, it is obtained that V (x (t)) < 1 + γ²η. For ω (t) ∈ W_η, the state trajectories of the closed-loop system (5) which start from Ω (E (ɛ) Z^-1 (ɛ), 1) will remain inside Ω (E (ɛ) Z^-1 (ɛ), 1 + γ²η).

If ω (t) = 0, according to Theorem 1, the stability condition for the closed-loop system (5) with ω (t) = 0 can be obtained.

Corollary 1. For a given scalar τ > 0 and ɛ-bound ɛ₀ > 0, if there exist matrices Z_l (l = 1, …, 5), with $Z_{l} = Z_{l}^{T} (l = 1, 2, 3, 4),$ matrices Y_j, H_1j, H_2j, and positive-defined symmetric matrices R₁, R₂, such that LMIs (6)–(8) and the following inequalities hold $[\begin{matrix} Z_{1} & * \\ H_{1 j (h)} & 1 \end{matrix}] \geq 0$ (19) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & * & * \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} & * \\ H_{1 j (h)} & ɛ_{0} H_{2 j (h)} & 1 \end{matrix}] \geq 0$ (20) $[\begin{matrix} Z_{1} + ɛ_{0} Z_{3} & * & * \\ ɛ_{0} Z_{5} & ɛ_{0} Z_{2} + ɛ_{0}^{2} Z_{4} & * \\ H_{1 j (h)} & ɛ_{0} H_{2 j (h)} & 1 \end{matrix}] \geq 0$ (21) $\begin{matrix} h \in [1, p], j = 1, \dots, r \\ {\bar{Ψ}}_{ii} (0) < 0, i = 1, \dots, r \\ {\bar{Ψ}}_{ij} (0) + {\bar{Ψ}}_{ji} (0) < 0, 1 \leq i < j \leq r \\ {\bar{Ψ}}_{ii} (ɛ_{0}) < 0, i = 1, \dots, r \\ {\bar{Ψ}}_{ij} (ɛ_{0}) + {\bar{Ψ}}_{ji} (ɛ_{0}) < 0, 1 \leq i < j \leq r \end{matrix}$ (22) where $\begin{matrix} {\bar{Ψ}}_{ij} (0) & = & [\begin{matrix} sym (A_{i} Z (0)) + Ξ (0) & * \\ {(B_{i} {\tilde{K}}_{j})}^{T} + {\tilde{R}}_{2}^{T} E (0) & - 2 {\tilde{R}}_{2} \\ 0 & {\tilde{R}}_{2}^{T} \\ τ A_{i} Z (0) & τ B_{i} {\tilde{K}}_{j} \end{matrix} \\ \begin{matrix} * & * \\ * & * \\ - {\tilde{R}}_{1} - {\tilde{R}}_{2} & * \\ 0 & {\tilde{R}}_{2} - sym (Z (0)) \end{matrix}] \\ {\bar{Ψ}}_{ij} (ɛ_{0}) & = & [\begin{matrix} sym (A_{i} Z (ɛ_{0})) + Ξ (ɛ_{0}) & * \\ {(B_{i} {\tilde{K}}_{j})}^{T} + {\tilde{R}}_{2}^{T} E (ɛ_{0}) & - 2 {\tilde{R}}_{2} \\ 0 & {\tilde{R}}_{2}^{T} \\ τ A_{i} Z (ɛ_{0}) & τ B_{i} {\tilde{K}}_{j} \end{matrix} \\ \begin{matrix} * & * \\ * & * \\ - {\tilde{R}}_{1} - {\tilde{R}}_{2} & * \\ 0 & {\tilde{R}}_{2} - sym (Z (ɛ_{0})) \end{matrix}] \end{matrix}$

Then, for any ɛ ∈ (0, ɛ₀], the closed-loop system (5) with the fuzzy sampled-data controller (3) K_j (ɛ) = Y_jE (ɛ) Z^-1 (ɛ) is asymptotically stable within $\begin{matrix} Ω (E (ɛ) Z^{- 1} (ɛ), 1) \\ = {x | x^{T} E (ɛ) Z^{- 1} (ɛ) x \leq 1} . \end{matrix}$

In addition, the ellipsoid Ω (E (ɛ) Z^-1 (ɛ), 1) is a basin of attraction of the closed-loop system (5) with ω (t) = 0.

By using the method in Theorem 1, the proof of this corollary is similar to Theorem 1. Hence, we omit it here.

Remark 5. Let X₀ be a set of initial conditions. The design objective is to find the controller gain K_j (ɛ), such that all trajectories of the closed-loop system (5) starting from X₀ will remain inside it, that is to say, X₀ is an invariant set for the closed-loop system (5). In the current work, X₀ is considered as an ellipsoid Ω (E (ɛ) Z^-1 (ɛ), 1).

Remark 6. The problem of the stabilization bound for T-S fuzzy SPSs with input saturation is considered in this paper. By constructing an appropriate Lyapunov-Krasovskii functional, we get the stabilization conditions and the design method of fuzzy sampled-data controller. Similar to the method in [10], the result of this paper can be extended to deal with the problem of analysis and synthesis for SPSs with time-varying delay.

Considering Corollary 1, which gives conditions for an ellipsoid to be inside the domain of attraction. Now let x₀ is a given initial state. To maximize the volume of the basin of attraction with least degree of conservatism, the issue can be formulated as the following optimization problem

\begin{matrix} max_{H_{j}, Y_{j}, Z_{i}} α \\ s . t . (a) α x_{0} \in Ω (E (ɛ) Z^{- 1} (ɛ), 1) \end{matrix}

(23)

\begin{matrix} and α > 0 \\ (b) (6) - (8) and (19) - (22) \end{matrix}

It can be seen that the condition (a) is equivalent to $[\begin{matrix} \frac{1}{α^{2}} & x_{0}^{T} E (ɛ) \\ E (ɛ) x_{0} & Z^{T} (ɛ) E (ɛ) \end{matrix}] \geq 0$

By Lemma 2 and $x_{0}^{T} = [\begin{matrix} x_{01}^{T} & x_{02}^{T} \end{matrix}],$ the above inequality is guaranteed by $[\begin{matrix} \frac{1}{α^{2}} & * \\ x_{01} & Z_{1} \end{matrix}] \geq 0$ (24) $[\begin{matrix} \frac{1}{α^{2}} & * & * \\ x_{01} & Z_{1} + ɛ_{0} Z_{3} & * \\ x_{02} ɛ_{0} & ɛ_{0} Z_{5} & ɛ_{0} Z_{2} \end{matrix}] \geq 0$ (25) $[\begin{matrix} \frac{1}{α^{2}} & * & * \\ x_{01} & Z_{1} + ɛ_{0} Z_{3} & * \\ x_{02} ɛ_{0} & ɛ_{0} Z_{5} & ɛ_{0} Z_{2} + ɛ_{0}^{2} Z_{4} \end{matrix}] \geq 0$ (26)

Then, let $λ = \frac{1}{α^{2}},$ the optimization problem (23) can be reformulated as the following convex optimization problem

$\begin{matrix} min_{H_{j}, Y_{j}, Z_{i}} λ \\ s . t . (6) - (8), (19) - (22) and (24) - (26) \end{matrix}$ (27)

Remark 7. The basin of attraction is usually described by the associated Lyapunov-Krasovskii functional. Based on this fact, we formulate convex optimization problems by using corresponding stabilization conditions. Owing that some mature methods for normal systems to optimize the basin of attraction have been obtained [11]. This paper extended the classical method to the method of constructing anɛ-dependent basin of attraction for T-S fuzzy SPSs. The singular perturbation structure of T-S fuzzy SPSs is fully considered. Certainly, the convex optimization problem for maximizing the basin of attraction of T-S fuzzy SPSs is quite different from the normal systems and even more complex than the normal systems.

Remark 8. It is noted that LMI conditions can be solved in polynomial time by specialized algorithms with complexity proportional to $C = B^{3} A,$ where $B$ and $C$ denote the numbers of lines and decision variables of the LMI, respectively. In Table 1, the numbers of lines and decision variables of LMI problem (LMIP) (18) and LMIP (27) are presented.

Table 1

Number of decision variables and lines in LMIP (18) and LMIP (27)

	Decision variables ( $B$ )	Lines ( $C$ )
LMIP (18)	n₁n₂ + 2n (n + 1) + 2prn + n (n + 1)	$\begin{matrix} (2 n + n_{1}) + pr (2 n + n_{1} + 3) \\ + 2 (4 n + q + l) (r + \frac{r (r - 1)}{2}) 2^{p} \end{matrix}$
LMIP (27)	n₁n₂ + 2n (n + 1) + 2prn + n (n + 1)	$\begin{matrix} (1 + pr) (2 n + n_{1} + 3) \\ + (2 n + n_{1} + 8 n \frac{r + (r - 1)}{2}) 2^{p} \end{matrix}$

4 Numerical examples

In this section, we give some examples to illustrate the effectiveness of the proposed conditions.

Example 1. Consider SPSs with $\begin{matrix} E (ɛ) & = & [\begin{matrix} 1 & 0 & 0 \\ 0 & ɛ & 0 \\ 0 & 0 & ɛ \end{matrix}], A = [\begin{matrix} - 1 & - 1 & - 1 \\ 0 & 0.1 & 0 \\ 1 & 0 & - 0.1 \end{matrix}], \\ B & = & [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}], B_{ω} = [\begin{matrix} 0.1 \\ 0.2 \\ 0.1 \end{matrix}], C^{T} = [\begin{matrix} 0.1 \\ 0.1 \\ 0.1 \end{matrix}], \\ D & = & 0 . \end{matrix}$

For given τ = 1.5, when ɛ ≤ 0.1, we obtain that the closed-loop system can satisfies H_∞ performance with γ = 2.5 by applying Theorem 1. However, when ɛ = 0.05, the method of [27] can not achieve the H_∞ performance level for γ < 4.7. Moreover, when ɛ = 0.1, the controller of [27] can not guarantee the stability of the system. And, the results of comparing with the existing literature [27] are given in Table 2.

Table 2
Comparisons of minimum allowed γ for Example 1

ɛ 0.01 0.05 0.1

literature [27] 0.23 > 4.7 -

Theorem 1 2.05 2.5 2.7

ɛ	0.01	0.05	0.1
literature [27]	0.23	> 4.7	-
Theorem 1	2.05	2.5	2.7

Example 2. Consider an inverted pendulum system controlled by a DC motor via a gear train, which was first proposed in [28]. The system is described by $\begin{matrix} {\dot{x}}_{1} (t) & = & x_{2} (t) + 0.1 ω (t) \\ {\dot{x}}_{2} (t) & = & \frac{g}{l} sin x_{1} (t) + \frac{N K_{m}}{m l^{2}} x_{3} (t) \\ L_{a} {\dot{x}}_{3} (t) & = & - K_{b} N x_{2} (t) - R_{a} x_{3} (t) \\ + u (t) + ω (t) \\ z (t) & = & 0.1 x_{1} (t) + 0.1 u (t) \end{matrix}$ (28) where $x_{1} (t) = θ_{p} (t), x_{2} (t) = {\dot{θ}}_{p} (t), x_{3} (t) = I_{a} (t),$ u (t) is the control input, ω (t) is the disturbance input, K_m is the motor torque constant, K_b is the back emf constant, and N is the gear ratio. The parameters for the plant are given as g = 9.8 m/s², l = 1 m, m = 1 kg, N = 10, K_m = 0.1 Nm/A, K_b = 0.1 Vs/rad, R_a = 1Ω and L_a = ɛ mH and the input voltage is required to satisfy |u| ≤ 1. Note that the inductance L_a represents the small parameter in the system.

Substituting the parameters into (28), we have $\begin{matrix} {\dot{x}}_{1} (t) & = & x_{2} (t) + 0.1 ω (t) \\ {\dot{x}}_{2} (t) & = & 9.8 sin x_{1} (t) + x_{3} (t) \\ ɛ {\dot{x}}_{3} (t) & = & - x_{2} (t) - x_{3} (t) \\ + sat (u (t)) + ω (t) \\ z (t) & = & 0.1 x_{1} (t) + 0.1 u (t) \end{matrix}$ (29)

As in [5], the membership functions of the fuzzy sets are chosen as follows $M_{1} (x_{1} (t)) = 1 - \frac{| x_{1} (t) |}{π}, M_{2} (x_{1} (t)) = \frac{| x_{1} (t) |}{π} .$

Then, the following T-S fuzzy model can exactly represent the dynamics of nonlinear SPSs (29) under -π ≤ x₁ (t) ≤ π $\begin{matrix} Plant rule 1 : \\ IF x_{1} (t) is M_{1} (x_{1} (t)), THEN \\ E (ɛ) \dot{x} (t) = A_{1} x (t) + B_{1} sat (u (t)) + B_{ω 1} ω (t) \\ z (t) = C_{1} x (t) + D_{1} u (t) \\ Plant rule 2 : \\ IF x_{1} (t) is M_{2} (x_{1} (t)), THEN \\ E (ɛ) \dot{x} (t) = A_{2} x (t) + B_{2} sat (u (t)) + B_{ω 2} ω (t) \\ z (t) = C_{2} x (t) + D_{2} u (t) \end{matrix}$ where $\begin{matrix} A_{1} & = & [\begin{matrix} 0 & 1 & 0 \\ 9.8 & 0 & 1 \\ 0 & - 1 & - 1 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & - 1 \end{matrix}], \\ E (ɛ) & = & [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ɛ \end{matrix}], B_{1} = B_{2} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], \\ D_{1} & = & D_{2} = 0.1, \\ B_{ω 1} & = & B_{ω 2} = [\begin{matrix} 0.1 \\ 0 \\ 1 \end{matrix}], C_{1} = C_{2} = {[\begin{matrix} 0.1 \\ 0 \\ 0 \end{matrix}]}^{T} . \end{matrix}$

Solving the optimization problem (18) with τ = 0.3, γ = 0.3, we can obtain that η_max = 1.9, and upper bound ɛ₀ = 0.9.

In addition, the control law is obtained that $\begin{matrix} K_{1} & = & [\begin{matrix} - 0.662 & - 0.4123 & - 0.3575 \end{matrix}], \\ K_{2} & = & [\begin{matrix} - 0.6718 & - 0.4257 & - 0.4124 \end{matrix}] . \end{matrix}$

In this case, by Theorem 1, the upper bounds of ɛ subject to satisfying the H_∞ performance level are shown in Table 3. The problem of H_∞ control for T-S fuzzy SPSs was studied in [5] and the results of the upper bounds for ɛ were given in Table 3. It can be seen that the upper bounds obtained by Theorem 1 are bigger than those given in [5].

Table 3

The upper bounds of ɛ by Theorem 1

Methods	literature [5]	Theorem 1
ɛ ₀	0.3078	0.9

And the state trajectories of T-S fuzzy SPSs with input saturation and disturbance are shown in Fig. 1.

Fig.1

Trajectories of states.

Then, solving the optimization problem (27) with ɛ₀ = 0.15, τ = 0.17, λ = 0.2, α = 2.2, we can get the controller gain $\begin{matrix} K_{1} & = & [\begin{matrix} - 1.8925 & - 1.1780 & - 0.1045 \end{matrix}], \\ K_{2} & = & [\begin{matrix} - 1.2438 & - 1.1333 & - 0.0814 \end{matrix}] . \end{matrix}$

For the case ɛ = 0.15, the state dynamic system are shown in Fig. 2. From the figure, our proposed method can guarantee that T-S fuzzy SPSs are well stabilized. The basin of attraction of the system under the controller of (3) and the trajectory starting from $x_{0} = {[\begin{matrix} - 0.5 & 1 & 3 \end{matrix}]}^{T}$ are shown in Fig. 3. It can be seen that the trajectory starting from $x_{0} = {[\begin{matrix} - 0.5 & 1 & 3 \end{matrix}]}^{T} \in Ω (E (ɛ) Z^{- 1} (ɛ), 1)$ remains inside Ω (E (ɛ) Z^-1 (ɛ), 1) and converges to the equilibrium point of the system.

Fig.2

Trajectories of states when ω (t) = 0.

Fig.3

Basin of attraction of the system with ɛ = 0.15, ω (t) = 0 and the converging trajectory starting from ${[\begin{matrix} - 0.5 & 1 & 3 \end{matrix}]}^{T}$ .

5 Conclusion

In this paper, the problem of H_∞ sampled-data controller design for T-S fuzzy SPSs subject to actuator saturation is of concern. First, we proposed a method of the fuzzy sampled-data stabilization controller design and constructed an optimization algorithm to maximize the disturbance tolerance ability. Then, by considering the special situation of the system without the disturbances, we give the corresponding LMI and the optimization problem, such that the stability of the system is ensured and an ɛ-dependent basin of attraction of the closed-loop system is maximized. Finally, numerical examples and simulation results are given to demonstrate the validity of the proposed method.

It should be noted that the factors of uncertainties and packet dropouts are often encountered in the practical engineering. Therefore, our future work will generalize the proposed results to deal with system uncertainties and network communication characteristics.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61374043, 61603392), and the China Postdoctoral Science Foundation funded project (2013M530278, 2014T70558). The authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions improved the quality of this paper.

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H ∞ sampled-data control for T-S fuzzy singularly perturbed systems with actuator saturation

Abstract

Keywords

1 Introduction

2 Problem description and preliminaries

Table 2 Comparisons of minimum allowed γ for Example 1 ɛ 0.01 0.05 0.1 literature [27] 0.23 > 4.7 - Theorem 1 2.05 2.5 2.7

Footnotes

Acknowledgments

References

Table 2
Comparisons of minimum allowed γ for Example 1

ɛ 0.01 0.05 0.1

literature [27] 0.23 > 4.7 -

Theorem 1 2.05 2.5 2.7