Stabilization of unknown nonlinear systems with T-S fuzzy model and dynamic delay partition

Abstract

In this paper, a new method, namely, the dynamic delay partitioning method, is firstly developed to solve the problems of stability analysis and stabilization for a class of unknown nonlinear systems. To study the system stability and facilitate the design of fuzzy controller, Takagi-Sugeno (T-S) fuzzy models are employed to represent the system dynamics of the unknown nonlinear systems. Different from previous results, the delay interval [0, d (t)] is partitioned into some variable subintervals by employing dynamic delay partitioning method. Thus, new delay-dependent stability criteria for fuzzy time-delay systems is derived, which is less conservative than previous results. By the criteria, the approach to design a fuzzy controller is developed. Two examples are provided to demonstrate the effectiveness for conservatism reduction.

Keywords

Dynamic delay partition nonlinear systems Takagi-Sugeno (T-S) fuzzy model time-varying delay stabilization.

1 Introduction

Nonlinear systems with time-delay are very common in various industrial fields such as chemical processes and communication systems. Time-delay appears commonly in various practical systems such as chemical processes and communication systems, which generally lowers the system performance and even results in instability. In the last a few years, many authors considered the stability analysis and synthesis of time-delay systems as an important issue, and proposed several feasible approaches (see, for instance, [1, 2], and the references therein). Recently, fuzzy logic control based on Takagi-Sugeno (T-S) model becomes more and more popular for nonlinear systems. It is an efficient approach to deal with analysis and synthesis problems for complex nonlinear systems, especially in the presence of incomplete knowledge of the plant, and has many successful applications in industrial processes. The last decade has witnessed a rapidly growing interest in T-S fuzzy systems, and many important results have been reported. To mention a few, the problem of stability analysis is investigated in [3, 4]; stabilizing and H_∞ control designs are reported in [5 –9]; observer problem is addressed in [10 –12].

Today, there are two kinds of useful methods dealing with problems associated with time delay: free-weighting matrix approach [13, 14] and augmented Lyapunov functional method [15, 16]. [13] considered the term $- \int_{t - τ}^{t - d (t)} {\dot{x}}^{T} (s) W \dot{x} (s) ds$ in Lyapunov functional, which has been usually neglected in a previous literature, where d (t) denotes the time-varying delay and τ denotes the upper bound of d (t), i.e., d (t) ∈ [0, τ]. The free-weighting matrix approach is used as a main tool to make the criteria less conservative in the literature, and only the lower and upper bounds of delay function d (t) are considered. [15] used a new augmented Lyapunov functional, which contains a structure more general than the traditional ones in [13, 14] for involving an integral term of the state vector in Lyapunov functional. This new type of Lyapunov functional enables to establish less conservative results. Recently, a novel method was proposed for T-S fuzzy time-delay systems in [17, 19], which was carried out by partitioning the time delay interval [0, τ] into m subintervals with the same size. This method utilizes the information in the interval [0, τ] to achieve the aim of reducing conservativeness.

In the literature for T-S fuzzy systems with time-varying delay d (t), the delay interval [0, d (t)] is usually considered as a single interval. Therefore, how to use the information in [0, d (t)] to further obtain less conservative stability for T-S fuzzy systems with time-varying delay d (t) motivates our present study. Our proposed approach is to partition the delay interval [0, d (t)] into smaller variable subintervals, and study the stability based on these subintervals. In the following, we will introduce the idea of partitioning delay interval [0, d (t)] from the viewpoint of mathematics. We determine multiple points m_ρd (t) , ρ = 1, 2, ⋯ , K, where m_ρ ∈ (0, 1) and m₁ < m₂ < ⋯ < m_K, such that interval [0, d (t)] can be partitioned into K + 1 subintervals [0, m₁d (t)], [m₁d (t), m₂d (t)], ⋯, [m_Kd (t), d (t)]. Different from [17] which partitions the delay interval into the fixed subintervals with the same size, this kind of dynamic mode to partition delay interval is nominated as dynamic delay partitioning method since the points m_id (t) can be chosen arbitrarily in the interval [0, d (t)]. Therefore, parameters m_i are dynamic delay partitioning parameters, and (m₁, m₂, ⋯ , m_K) is a parameter sequence satisfying 0 < m₁ < m₂ < ⋯ < m_K < 1. Meanwhile, [0, m₁d (t)], [m₁d (t), m₂d (t)], ⋯, [m_Kd (t), d (t)]) are called K + 1 variable subintervals, respectively.

Different from previous studies, the dynamic delay partitioning method has the following features:

1) The idea of partitioning delay interval into several subintervals will be utilized in this method. Unlike the previous works, which treat the delay interval [0, d (t)] as one single interval, [0, d (t)] will be partitioned into several subintervals in this paper. This is to say that much more information in the interval [0, d (t)] can be utilized.

2) Different from the fixed subintervals with the same size mode in [17], the dynamic delay partitioning method is the one with dynamic subintervals. In this method the delay interval is partitioned into variable subintervals, i.e., the points m_ρd (t) can be reconfigurable at delay interval [0, d (t)]. Compared to the fixed subintervals, this method has inherent flexibility, and should be more suitable to deal with time-varying delay d (t).

3) The stability results based on dynamic delay partitioning method are related to the number of subintervals, and the size of the variable subintervals or the position of the variable points (the values of parameters m_ρ). When the positions of the variable points are varied, the proposed stability criteria are also different. In order to obtain the optimal parameters m_ρ, we proposed an implementation based on optimization methods.

Therefore, this paper proposes a new dynamic delay partitioning method to deal with the stability of T-S fuzzy systems with time-varying delay, so that the larger allowable upper bound can be obtained by introducing these variable subintervals. Combining the fuzzy Lyapunov function approach [20], a new method is first proposed for the delay-dependent stability analysis of T-S fuzzy time-delay systems. The proposed stability conditions are much less conservative than most of the existing results due to the dynamic delay partitioning method, and they become even less conservative when the partitioning goes finer. Using these results, the problem of stabilization is also solved. Two illustrative examples show that our results are less conservative than those in previous literature.

The rest of this paper is organized as follows. Section 2 illustrates problem formulation and preliminaries of T-S fuzzy time-delay systems. Stability analysis is presented in Section 3. Controller design is presented in Section 4. In Section 5, the detailed implementations are presented to solve the delay partitioning parameters. Illustrative examples are given to demonstrate the effectiveness of the proposed approach in Section 6. Finally, conclusions are drawn in Section 7.

Notation. The notations that are used throughout this paper are fairly standard. A^T stands for transpose of matrix A; the notation P > 0 (≥0) means that P is real symmetric and positive (semipositive); $R^{n}$ denotes the n dimensional Euclidean vector space; diag{⋯} stands for a block-diagonal matrix; and in symmetric block matrices or long matrix expressions, we use an asterisk “∗" to represent a term that is induced by symmetry. sym(A) is defined as A + A^T. If not explicitly stated, all matrices are assumed to have compatible dimensions.

2 Preliminaries

2.1 A Takagi-Sugeno fuzzy control system

Consider the following nonlinear system $\dot{x} (t) = f (x (t)) + f_{d} (x (t - d (t))) + g (x (t)) u (t),$ (1) where $x (t) \in R^{n}$ is the state vector and $u (t) \in R^{p}$ is the input vector. We assume that f (x (t)), f_d (x (t)) and g (x (t)) are sufficiently smooth on a domain $𝔻 \subset R^{n}$ , and f (0) = f_d (0) =0, where “sufficiently smooth" means that all the partial derivatives are defined and continuous. The time-delay d (t) is a time-varying delay satisfying the inequalities below:

$0 \leq d (t) \leq τ, 0 \leq \dot{d} (t) \leq μ,$ (2) where τ > 0 and μ are two scalars. The initial condition of the system (1) is given by

$x (t) = φ (t), t \in [- τ, 0],$ (3)φ (t) is a continuous vector-valued initial function defined on [-τ,0].

Throughout this paper, we assume that systems (1) and (3) are controlled and the system state is available for feedback.

From [21], the nonlinear system (1) can be represented by some simple local linear dynamic systems with their linguistic description as

Plant Rule i: IF z₁ (t) is M_i1 and z₂ (t) is M_i2 and ⋯ and z_g (t) is M_ig, THEN

$\dot{x} (t) = A_{i} x (t) + A_{di} x (t - d (t)) + B_{i} u (t),$ (4) where i = 1, 2, ⋯ , r and r is the number of IF-THEN rules, z₁ (t) , z₂ (t) , ⋯ , z_g (t) are the premise variables of (4) and M_ij (i = 1, 2, ⋯ , r, j = 1, 2, ⋯ , g) are the fuzzy sets corresponding to z_j (t) and the plant rules, and A_i, A_di and B_i are known parameter matrices of appropriate dimensions.

By using a center average defuzzifier, product inference, and a singleton fuzzifier, the global dynamics of the T-S fuzzy systems (4) are described by

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} λ_{i} (z (t)) [A_{i} x (t) + A_{di} x (t - d (t)) \\ + B_{i} u (t)], \end{matrix}$ (5) where

$\begin{matrix} λ_{i} (z (t)) = 1, λ_{i} (z (t)) = \frac{w_{i} (z (t))}{\sum_{i = 1}^{r} w_{i} (z (t))} \geq 0, \\ w_{i} (z (t)) = \prod_{j = 1}^{g} M_{ij} (z_{j} (t)), \end{matrix}$ with M_ij (z_j (t)) representing the grade of membership of z_j (t) in M_ij. Then, it can be seen that, for all t,

$0, i = 1, 2, \dots, r, \sum_{i = 1}^{r} w_{i} (z (t)) > 0 .$

From (1) and (5), we have

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} λ_{i} (z (t)) [A_{i} x (t) + A_{di} x (t - d (t)) + B_{i} u (t)] \\ + Δ f + Δ f_{d} + Δ g, \end{matrix}$ where

$\begin{matrix} Δ f & = & f (x (t)) - \sum_{i = 1}^{r} λ_{i} (z (t)) A_{i} x (t), \\ Δ f_{d} & = & f_{d} (x (t)) - \sum_{i = 1}^{r} λ_{i} (z (t)) A_{di} x (t - d (t)), \\ Δ g & = & g (x (t)) u (t) - \sum_{i = 1}^{r} λ_{i} (z (t)) B_{i} u (t), \end{matrix}$ which denote the bounded modeling errors between the nonlinear plant (1) and the global fuzzy model (5).

In the following, a T-S fuzzy-model-based controller will be designed via parallel distribution compensation (PDC) [22], in which the fuzzy sets with the fuzzy model are shared for the designed fuzzy controller in the premise parts.

Controller Rule i: IF z₁ (t) is M_i1 and z₂ (t) is M_i2 and ⋯ and z_g (t) is M_ig, THEN

$u (t) = G_{i} x (t),$ (6) and G_i (i = 1, 2, ⋯ , r) are the fuzzy controller gain matrices to be determined.

The state feedback control law inferred is

$u (t) = \sum_{i = 1}^{r} λ_{i} (z (t)) G_{i} x (t) = G (t) x (t) .$ (7)

For the sake of simplicity, λ_i (z (t)) is denoted by λ_i (t) in the following.

Substituting (7) into (8) yields the closed-loop global fuzzy system of (8).

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} λ_{i} (t) λ_{j} (t) [(A_{i} + B_{i} G_{j}) x (t) \\ + A_{di} x (t - d (t))] + Δ f + Δ f_{d} + Δ g . \end{matrix}$ (8)

Suppose that there exist known real constant matrices D_ai, D_di, D_bi, N_ai, N_di, and N_bi (i = 1, 2, ⋯ , r) of appropriate dimensions such that

$\begin{matrix} Δ f & = & \sum_{i = 1}^{r} λ_{i} (t) D_{ai} F (t) N_{ai} x (t), \\ Δ f_{d} & = & \sum_{i = 1}^{r} λ_{i} (t) D_{di} F (t) N_{di} x (t - d (t)), \\ Δ g & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} λ_{i} (t) λ_{j} (t) D_{bi} F (t) N_{bi} G_{j} x (t), \end{matrix}$ (9) where F (t) is an unknown real time-varying matrix with Lebesgue measurable elements bounded by

$F^{T} (t) F (t) \leq I .$ (10)

Then, the closed-loop system (8) can be rewritten as

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} λ_{i} (t) λ_{j} (t) [(A_{i} (t) + B_{i} (t) G_{j}) x (t) \\ + A_{di} (t) x (t - d (t))], \end{matrix}$ (11) where

$\begin{matrix} A_{i} (t) & = & A_{i} + ▵ A_{i} (t), A_{di} (t) = A_{di} + ▵ A_{di} (t), \\ B_{i} (t) & = & B_{i} + ▵ B_{i} (t), \end{matrix}$ (12) and i = 1, 2, ⋯ , r, ΔA_i (t), ΔA_di (t), and ΔB_i (t) are unknown real matrices of appropriate dimensions representing time-varying parameter uncertainties of the system (11) and satisfying

$\begin{matrix} Δ A_{i} (t) = D_{ai} F (t) N_{ai}, Δ A_{di} (t) = D_{di} F (t) N_{di}, \\ Δ B_{i} (t) = D_{bi} F (t) N_{bi} . \end{matrix}$ (13)

The closed-loop system (8) for the nominal case is given by

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} λ_{i} (t) λ_{j} (t) [(A_{i} + B_{i} G_{j}) x (t) \\ + A_{di} x (t - d (t))] \\ = & [A (t) + B (t) G (t)] x (t) + A_{d} (t) x (t - d (t)), \end{matrix}$ (14) where

$\begin{matrix} A (t) & = & \sum_{i = 1}^{r} λ_{i} (t) A_{i}, A_{d} (t) = \sum_{i = 1}^{r} λ_{i} (t) A_{di}, \\ B (t) & = & \sum_{i = 1}^{r} λ_{i} (t) B_{i} . \end{matrix}$

2.2 Fuzzy Lyapunov function candidate

To find the fuzzy Lyapunov function which does not associate with the time-derivatives of membership functions in the stability analysis, consider a line-integral function given as in [23],

$V_{Γ} (t) = 2 \int_{Γ (0, x)} f (ψ) \cdot d ψ,$ (15) where Γ (0, x) is a path from the origin 0 to the current state x, $ψ \in R^{n}$ is a dummy vector for the integral, $f (x) \in R^{n}$ is a vector which is a function of the state x, (·) denotes an inner product, and $d ψ \in R^{n}$ is an infinitesimal displacement vector.

Consider the fuzzy vector f (x) which shares the same fuzzy rules as in the plant rules.

Plant Rule i: IF z₁ (t) is M_i1 and z₂ (t) is M_i2 and ⋯ and z_g (t) is M_ig, THEN

$f (x (t)) = {\hat{P}}_{i} x (t), i = 1, 2, \dots, r,$ (16) where ${\hat{P}}_{i} \in R^{n \times n}$ is a positive definite symmetric matrix satisfying

${\hat{P}}_{i} = \bar{P} + D_{i},$ (17) where

$\begin{matrix} \bar{P} = [\begin{matrix} 0 & p_{12} & \dots & p_{1 n} \\ p_{12} & 0 & \dots & p_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{1 n} & p_{2 n} & \dots & 0 \end{matrix}], D_{i} = [\begin{matrix} d_{11}^{i} & 0 & \dots & 0 \\ 0 & d_{22}^{i} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & d_{nn}^{i} \end{matrix}] . \end{matrix}$ (18)

Therefore, the off-diagonal elements of ${\hat{P}}_{i}$ are same, but the diagonal elements are different according to the fuzzy sets in the premise parts of the fuzzy rules.

The above fuzzy vector can be rewritten simply as

$f (x (t)) = \sum_{i = 1}^{r} λ_{i} (t) {\hat{P}}_{i} x (t) ≜ \hat{P} (t) x (t) .$ (19)

Before presenting the main results of this paper, we first introduce the following definition and lemma for the fuzzy system above, which will be essential for our derivation.

Definition 1. (Lie derivative) [24]: Let h: $R^{n} \to R$ be a smooth scalar function, and g: $R^{n} \to R^{n}$ be a smooth vector field on $R^{n}$ , then the Lie derivative of h with respect to g is a scalar function defined by

$L_{g} h = \nabla hg,$

where

$\nabla h = \partial h / \partial x .$ Similarly, if V is a Lyapunov function candidate for the system, its derivative $\dot{V}$ can be written as L_gV where $\dot{x} = g (x)$ . That is,

$\dot{V} = L_{g} V = \frac{\partial V}{\partial x} \dot{x} = \frac{\partial V}{\partial x} g (x) .$

Lemma 1. [25]: Given appropriate dimension matrices M, E and F satisfying F^TF ≤ I, for any real scalar ɛ > 0, the following result holds

$MFE + E^{T} F^{T} M^{T} \leq ɛ M M^{T} + ɛ^{- 1} E^{T} E .$

Lemma 2. (Jensen’s Inequality): For any constant matrix Ω > 0, vector function χ (t) with appropriate dimensions, and function $σ (t) \in R$ satisfying 0 < σ (t) ≤ δ, we have

$\begin{matrix} [\int_{t - σ (t)}^{t} χ (s) ds]^{T} Ω [\int_{t - σ (t)}^{t} χ (s) ds] \\ \leq σ (t) \int_{t - σ (t)}^{t} χ^{T} (s) Ω χ (s) ds \\ \leq δ \int_{t - δ}^{t} χ^{T} (s) Ω χ (s) ds . \end{matrix}$

3 Stability Analysis

In this section, the dynamic delay partitioning approach to the analysis and synthesis of the T-S fuzzy time-delay systems is presented.

Theorem 1. Consider the nominal fuzzy time-delay system in (14) and suppose that the controller gain matrices in (7) are known. Given parameters m_ρ (ρ = 1, 2, ⋯ , K) satisfying 0 < m₁ < m₂ < ⋯ < m_K < 1, if there exist matrices ${\hat{P}}_{i} > 0$ defined in (17) (i = 1, 2, ⋯ , r), and

$\begin{matrix} P = P^{T} > 0, Q_{hi} = Q_{hi}^{T} > 0, R_{i} = R_{i}^{T} > 0, \\ Z = Z^{T} > 0, i = 1, 2, \dots, r, h = 1, 2, \dots, K + 1, \end{matrix}$ and K is a positive integer, such that the following LMIs hold: $\begin{matrix} Ω_{{iilb}_{1} \dots b_{K + 1}} < 0, i, l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (20) $\begin{matrix} Ω_{{ijlb}_{1} \dots b_{K + 1}} + Ω_{{jilb}_{1} \dots b_{K + 1}} < 0, 1 \leq i < j \leq r, \\ l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (21) where

$Ω_{{ijlb}_{1} \dots b_{K + 1}} = Λ_{ijl} + Q_{b_{1} \dots b_{K + 1}},$ (22)

$Λ_{ijl} = [\begin{matrix} Θ_{1 ij} & {\hat{Θ}}_{1} & 0 & \dots & 0 & ϝ_{1 i} & 0 & (A_{i} + B_{i} G_{j})^{T} \\ * & Θ_{2} & {\hat{Θ}}_{2} & \dots & 0 & 0 & 0 & 0 \\ \dots & \dots \\ * & * & * & \dots & Θ_{K + 1} & {\hat{Θ}}_{K + 1} & 0 & 0 \\ * & * & * & \dots & * & Θ_{K + 2} & {\hat{Θ}}_{K + 2} & A_{di}^{T} \\ * & * & * & \dots & * & * & - R_{l} - Z & 0 \\ * & * & * & \dots & * & * & * & - \frac{1}{τ^{2}} Z^{- 1} \end{matrix}],$

$\begin{matrix} Θ_{1 ij} & = & sym [(P + {\hat{P}}_{i}) (A_{i} + B_{i} G_{j})] + R_{i} - {\hat{Θ}}_{1}, \\ {\hat{Θ}}_{1} & = & \frac{1}{m_{1}} Z, \\ ϝ_{1 i} & = & (P + {\hat{P}}_{i}) A_{di}, \\ Θ_{2} & = & - \frac{1}{m_{1}} Z - {\hat{Θ}}_{2}, \\ {\hat{Θ}}_{2} & = & \frac{1}{m_{2} - m_{1}} Z, \dots, \\ Θ_{K + 1} & = & - \frac{1}{m_{K} - m_{K - 1}} Z - {\hat{Θ}}_{K + 1}, \\ {\hat{Θ}}_{K + 1} & = & \frac{1}{1 - m_{K}} Z, \\ Θ_{K + 2} & = & - \frac{1}{1 - m_{K}} Z - {\hat{Θ}}_{K + 2}, {\hat{Θ}}_{K + 2} = Z, \\ Q_{b_{1} \dots b_{K + 1}} & = & diag [Q_{1 b_{1}}, - (1 - m_{1} μ) Q_{1 b_{1}} + Q_{2 b_{2}}, \\ - (1 - m_{2} μ) Q_{2 b_{2}} + Q_{3 b_{3}}, \dots, \\ - (1 - m_{K} μ) Q_{K, b_{K}} + Q_{K + 1, b_{K + 1}}, \\ - (1 - μ) Q_{K + 1, b_{K + 1}}, 0, 0], \\ b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ then the fuzzy system in (14) is asymptoticallystable.

Proof. Construct the following augmented Lyapunov-Krasovskii functional: $V (t) = V_{Γ} (t) + V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t),$ (23) where V_Γ (t) is the same as (15), and

$\begin{matrix} V_{1} (t) = x^{T} (t) Px (t), \\ V_{2} (t) = \sum_{h = 1}^{K + 1} \int_{t - m_{h} d (t)}^{t - m_{h - 1} d (t)} x^{T} (s) {\bar{Q}}_{h} (s) x (s) ds, \\ V_{3} (t) = \int_{t - τ}^{t} x^{T} (s) R (s) x (s) ds, \\ V_{4} (t) = τ \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) dsd θ, \end{matrix}$ with

$\begin{matrix} {{\bar{Q}}_{h} (s), R (s)} = \sum_{i = 1}^{r} λ_{i} (s) {Q_{hi}, R_{i}}, \\ m_{0} = 0, m_{K + 1} = 1, h = 1, 2, \dots, K + 1 . \end{matrix}$

According to $0 \leq \dot{d} (t) \leq μ$ and Newton-Leibniz formula, calculating the derivatives of V (t) with respect to t along the trajectories of (14)yields

${\dot{V}}_{1} (t) = 2 x^{T} (t) P \dot{x} (t),$ (24)

$\begin{matrix} {\dot{V}}_{2} (t) \\ \leq & \sum_{h = 1}^{K + 1} x^{T} (t - m_{h - 1} d (t)) {\bar{Q}}_{h} (t - m_{h - 1} d (t)) \\ \times x (t - m_{h - 1} d (t)) \\ μ) x^{T} (t - m_{h} d (t)) {\bar{Q}}_{h} (t - m_{h} d (t)) \\ \times x (t - m_{h} d (t)), \end{matrix}$ (25)

$\begin{matrix} {\dot{V}}_{3} (t) = x^{T} (t) & R (t) x (t) - x^{T} (t - τ) R (t - τ) x (t - τ), \end{matrix}$ (26)

$\begin{matrix} {\dot{V}}_{4} (t) \\ = & τ^{2} {\dot{x}}^{T} (t) Z \dot{x} (t) - τ \int_{t - τ}^{t} {\dot{x}}^{T} (s) Z \dot{x} (s) ds \\ \leq & τ^{2} {\dot{x}}^{T} (t) Z \dot{x} (t) - \int_{t - τ}^{t - d (t)} {\dot{x}}^{T} (s) dsZ \int_{t - τ}^{t - d (t)} \dot{x} (s) ds \\ - \frac{1}{1 - m_{K}} \int_{t - d (t)}^{t - m_{K} d (t)} {\dot{x}}^{T} (s) dsZ \int_{t - d (t)}^{t - m_{K} d (t)} \dot{x} (s) ds \\ - \frac{1}{m_{K} - m_{K - 1}} \int_{t - m_{K} d (t)}^{t - m_{K - 1} d (t)} {\dot{x}}^{T} (s) dsZ \\ \times \int_{t - m_{K} d (t)}^{t - m_{K - 1} d (t)} \dot{x} (s) ds \\ - \dots - \frac{1}{m_{2} - m_{1}} \int_{t - m_{2} d (t)}^{t - m_{1} d (t)} {\dot{x}}^{T} (s) dsZ \\ \times \int_{t - m_{2} d (t)}^{t - m_{1} d (t)} \dot{x} (s) ds \\ - \frac{1}{m_{1}} \int_{t - m_{1} d (t)}^{t} {\dot{x}}^{T} (s) dsZ \int_{t - m_{1} d (t)}^{t} \dot{x} (s) ds \\ = & τ^{2} {\dot{x}}^{T} (t) Z \dot{x} (t) + w^{T} (t) \tilde{Z} w (t), \end{matrix}$ (27) where

$\begin{matrix} w^{T} (t) & = & [x^{T} (t) x^{T} (t - m_{1} d (t)) \dots \\ x^{T} (t - m_{K} d (t)) x^{T} (t - d (t)) x^{T} (t - τ)], \end{matrix}$

$\tilde{Z} = [\begin{matrix} - \frac{1}{m_{1}} & \frac{1}{m_{1}} & 0 & \dots & 0 & 0 & 0 \\ * & {\tilde{Z}}_{2} & \frac{1}{m_{2} - m_{1}} & \dots & 0 & 0 & 0 \\ \dots & \dots \\ * & * & * & * & {\tilde{Z}}_{K + 1} & \frac{1}{1 - m_{K}} & 0 \\ * & * & * & * & * & - \frac{1}{1 - m_{K}} - 1 & 1 \\ * & * & * & * & * & * & - 1 \end{matrix}] Z,$

$\begin{matrix} {\tilde{Z}}_{2} & = & - \frac{1}{m_{1}} - \frac{1}{m_{2} - m_{1}}, \dots, \\ {\tilde{Z}}_{K + 1} & = & - \frac{1}{m_{K} - m_{K - 1}} - \frac{1}{1 - m_{K}} . \end{matrix}$

Therefore, the following holds:

$\dot{V} (t) \leq w^{T} (t) (Θ (t) + τ^{2} {\tilde{A}}^{T} (t) Z \tilde{A} (t)) w (t),$

where

$\begin{matrix} Θ (t) \\ = & [\begin{matrix} Θ_{1} (t) & {\hat{Θ}}_{1} (t) & 0 & \dots & 0 & ϝ_{1} (t) & 0 \\ * & Θ_{2} (t) & {\hat{Θ}}_{2} (t) & \dots & 0 & 0 & 0 \\ \dots & \dots \\ * & * & * & * & Θ_{K + 1} & {\hat{Θ}}_{K + 1} & 0 \\ * & * & * & * & * & Θ_{K + 2} & {\hat{Θ}}_{K + 2} \\ * & * & * & * & * & * & - R (t - τ) - Z \end{matrix}], \end{matrix}$

$\begin{matrix} Θ_{1} (t) & = & sym [(P + \hat{P} (t)) (A (t) + B (t) G (t))] \\ + R (t) + {\bar{Q}}_{1} (t) - {\hat{Θ}}_{1} (t), \\ {\hat{Θ}}_{1} (t) & = & \frac{1}{m_{1}} Z, \\ ϝ_{1} (t) & = & (P + \hat{P} (t)) A_{d} (t), \\ Θ_{2} (t) & = & - (1 - m_{1} μ) {\bar{Q}}_{1} (t - m_{1} d (t)) \\ + {\bar{Q}}_{2} (t - m_{1} d (t)) - \frac{1}{m_{1}} Z - {\hat{Θ}}_{2} (t), \\ {\hat{Θ}}_{2} (t) & = & \frac{1}{m_{2} - m_{1}} Z, \dots, \\ Θ_{K + 1} (t) & = & - (1 - m_{K} μ) {\bar{Q}}_{K} (t - m_{K} d (t)) \\ + {\bar{Q}}_{K + 1} (t - m_{K} d (t)) \\ - \frac{1}{m_{K} - m_{K - 1}} Z - {\hat{Θ}}_{K + 1} (t), \\ {\hat{Θ}}_{K + 1} (t) & = & \frac{1}{1 - m_{K}} Z, \\ Θ_{K + 2} (t) & = & - (1 - μ) {\bar{Q}}_{K + 1} (t - d (t)) \\ - \frac{1}{1 - m_{K}} Z - {\hat{Θ}}_{K + 2} (t), \\ {\hat{Θ}}_{K + 2} (t) & = & Z, \end{matrix}$

$\tilde{A} (t) = [A (t) + B (t) G (t) 0 \dots 0 A_{d} (t) 0] .$

By the Schur complement, $Θ (t) + τ^{2} {\tilde{A}}^{T} (t)$ $Z \tilde{A} (t) < 0$ is equivalent to

$[\begin{matrix} Θ (t) & {\tilde{A}}^{T} (t) \\ * & - \frac{1}{τ^{2}} Z^{- 1} \end{matrix}] < 0 .$ (28)

Note that (28) can be written as

$\begin{matrix} \sum_{i = 1}^{r} λ_{i} (t) \sum_{j = 1}^{r} λ_{j} (t) \sum_{l = 1}^{r} λ_{l} (t - τ) \\ \times \sum_{b_{1} = 1}^{r} λ_{b_{1}} (t - m_{b_{1}} d (t)) \dots \\ \times \sum_{b_{K + 1} = 1}^{r} λ_{b_{K + 1}} (t - m_{b_{K + 1}} d (t)) Ω_{{ijlb}_{1} \dots b_{K + 1}} \\ = & \sum_{i = 1}^{r} λ_{i}^{2} (t) \sum_{l = 1}^{r} λ_{l} (t - τ) \sum_{b_{1} = 1}^{r} λ_{b_{1}} (t - m_{b_{1}} d (t)) \dots \\ \times \sum_{b_{K + 1} = 1}^{r} λ_{b_{K + 1}} (t - m_{b_{K + 1}} d (t)) Ω_{{iilb}_{1} \dots b_{K + 1}} \\ + 2 \sum_{i = 1}^{r - 1} λ_{i} (t) \sum_{j > i}^{r} λ_{j} (t) \sum_{l = 1}^{r} λ_{l} (t - τ) \\ \times \sum_{b_{1} = 1}^{r} λ_{b_{1}} (t - m_{b_{1}} d (t)) \dots \\ \times \sum_{b_{K + 1} = 1}^{r} λ_{b_{K + 1}} (t - m_{b_{K + 1}} d (t)) \\ \times \frac{Ω_{{ijlb}_{1} \dots b_{K + 1}} + Ω_{{jilb}_{1} \dots b_{K + 1}}}{2} < 0 . \end{matrix}$ (29)

Obviously, the conditions (20)-(21) establish the sufficient conditions for (29) to hold. Thus, the closed-loop system in (14) is asymptotically stable.

The proof is completed.

Remark 1. Obviously, Theorem 1 is the delay-partitioning-parameter-dependent criterion, and it depends on not only the value of parameter sequence (m₁, m₂, ⋯ , m_K), but also the number ofsubintervals, i.e., the delay partitioning number K. When the delay partitioning number K and the value of delay partitioning parameter sequence are set with different values, the stability results in Theorem 1 are also changed, and the less conservative stability results can be obtained.

Remark 2. When the number K becomes larger, the conservatism of the results is further reduced, while the computational cost increases.

4 Controller Design

Based on Theorem 1, a state-feedback controller is designed guaranteeing the asymptotic stability of the closed-loop nominal fuzzy system in (14).

Theorem 2. Consider the nominal fuzzy time-delay system in (14). Given parameters ɛ, ɛ_i, m_ρ (ρ = 1, 2, ⋯ , K) satisfying 0 < m₁ < m₂ < ⋯ < m_K < 1, the closed-loop system in (14) is asymptotically stable if there exist matrices X, M_i, ${\hat{Q}}_{hi} = {\hat{Q}}_{hi}^{T} > 0$ , ${\hat{R}}_{i} = {\hat{R}}_{i}^{T} > 0$ , $\hat{Z} = {\hat{Z}}^{T} > 0$ , i = 1, 2, ⋯ , r, h = 1, 2, ⋯ , K + 1, and K is a positive integer, such that $\begin{matrix} {\hat{Ω}}_{{iilb}_{1} \dots b_{K + 1}} < 0, \\ i, l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (30) $\begin{matrix} {\hat{Ω}}_{{ijlb}_{1} \dots b_{K + 1}} + {\hat{Ω}}_{{jilb}_{1} \dots b_{K + 1}} < 0, 1 \leq i < j \leq r, \\ l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (31) where

${\hat{Ω}}_{{ijlb}_{1} \dots b_{K + 1}} = {\hat{Λ}}_{ijl} + {\hat{Q}}_{b_{1} \dots b_{K + 1}},$ (32) $\begin{matrix} {\hat{Λ}}_{ijl} = [\begin{matrix} Φ_{1 ij} & {\hat{Φ}}_{1} & 0 & \dots & 0 & {\hat{ϝ}}_{1 i} & 0 & {\hat{ϝ}}_{1 ij} \\ * & Φ_{2} & {\hat{Φ}}_{2} & \dots & 0 & 0 & 0 & 0 \\ \dots & \dots \\ * & * & * & \dots & Φ_{K + 1} & {\hat{Φ}}_{K + 1} & 0 & 0 \\ * & * & * & \dots & * & Φ_{K + 2} & {\hat{Φ}}_{K + 2} & X^{T} A_{di}^{T} \\ * & * & * & \dots & * & * & - {\hat{R}}_{l} - \hat{Z} & 0 \\ * & * & * & \dots & * & * & * & Φ_{K + 4} \end{matrix}], \end{matrix}$

$\begin{matrix} Φ_{1 ij} = (ɛ + ɛ_{i}) sym (A_{i} X + B_{i} M_{j}) + {\hat{R}}_{i} - {\hat{Φ}}_{1}, \\ {\hat{Φ}}_{1} = \frac{1}{m_{1}} \hat{Z}, \\ {\hat{ϝ}}_{1 i} = (ɛ + ɛ_{i}) A_{di} X, \\ {\hat{ϝ}}_{1 ij} = (A_{i} X + B_{i} M_{j})^{T}, \\ Φ_{2} = - \frac{1}{m_{1}} \hat{Z} - {\hat{Φ}}_{2}, {\hat{Φ}}_{2} = \frac{1}{m_{2} - m_{1}} \hat{Z}, \\ Φ_{K + 1} = - \frac{1}{m_{K} - m_{K - 1}} \hat{Z} - {\hat{Φ}}_{K + 1}, \\ {\hat{Φ}}_{K + 1} = \frac{1}{1 - m_{K}} \hat{Z}, \\ Φ_{K + 2} = - \frac{1}{1 - m_{K}} \hat{Z} - {\hat{Φ}}_{K + 2}, {\hat{Φ}}_{K + 2} = \hat{Z}, \\ Φ_{K + 4} = \frac{1}{τ^{2}} (X + X^{T} - \hat{Z}), \end{matrix}$

$\begin{matrix} {\hat{Q}}_{b_{1} \dots b_{K + 1}} & = & diag [{\hat{Q}}_{1 b_{1}}, - (1 - m_{1} μ) {\hat{Q}}_{1 b_{1}} + {\hat{Q}}_{2 b_{2}}, \\ - (1 - m_{2} μ) {\hat{Q}}_{2 b_{2}} + {\hat{Q}}_{3 b_{3}}, \dots, \\ - (1 - m_{K} μ) {\hat{Q}}_{K, b_{K}} + {\hat{Q}}_{K + 1, b_{K + 1}}, \\ - (1 - μ) {\hat{Q}}_{K + 1, b_{K + 1}}, 0, 0], \\ b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r . \end{matrix}$ Moreover, the fuzzy state-feedback controller can be chosen as (6), and the matrix gains of the controller are given by

$G_{i} = M_{i} X^{- 1}, i = 1, 2, \dots, r .$ (33)

Proof. The proof is based on the conditions of Theorem 1. We assume that X is invertible. Define Y = X^-T, $\bar{Y} = diag [Y, Y, \dots, Y, \hat{Z}]$ , and $\bar{Z} = diag [I, I, \dots, Y, {\hat{Z}}^{- 1}]$ . Pre- and post-multiplying (30) and (31) with $\bar{Y}$ and ${\bar{Y}}^{T}$ , $\bar{Z}$ and ${\bar{Z}}^{T}$ , respectively. Defining P = ɛY, ${\hat{P}}_{i} = ɛ_{i} Y$ , $Z = Y \hat{Z} Y^{T}$ , $R_{i} = Y {\hat{R}}_{i} Y^{T}$ , $Q_{hi} = Y {\hat{Q}}_{hi} Y^{T}$ (i = 1, 2, ⋯ , r, h = 1, 2, ⋯ , K + 1), we arrive at (20) and (21). The proof is completed.

In this following, we discuss the problem of state-feedback controller design for the fuzzy delay systems in (11).

Theorem 3. Consider the fuzzy time-delay system in (11). Given parameters ɛ, ɛ_i, m_ρ (ρ = 1, 2, ⋯ , K) satisfying 0 < m₁ < m₂ < ⋯ < m_K < 1, the closed-loop system in (11) is asymptotically stable if there exist matrices X, M_i, ${\hat{Q}}_{hi} = {\hat{Q}}_{hi}^{T} > 0$ , ${\hat{R}}_{i} = {\hat{R}}_{i}^{T} > 0$ , $\hat{Z} = {\hat{Z}}^{T} > 0 (i = 1, 2, \dots, r, h = 1, 2, \dots, K + 1)$ , scalars ɛ_{ijlb₁⋯b_K+1} > 0, $ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{'} > 0$ , $ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{″} > 0$ , and K is a positive integer, such that $\begin{matrix} Φ_{{iilb}_{1} \dots b_{K + 1}} < 0, \\ i, l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (34) $\begin{matrix} Φ_{{ijlb}_{1} \dots b_{K + 1}} + Φ_{{jilb}_{1} \dots b_{K + 1}} < 0, 1 \leq i < j \leq r, \\ l, b_{1}, \dots, b_{K + 1} = 1, 2, \dots, r, \end{matrix}$ (35) where

$\begin{matrix} Φ_{{ijlb}_{1} \dots b_{K + 1}} = \\ [\begin{matrix} δ_{{ijlb}_{1} \dots b_{K + 1}} & \tilde{P} D_{ai} & \tilde{P} D_{bi} & \tilde{P} D_{di} \\ * & - ɛ_{{ijlb}_{1} \dots b_{K + 1}} I & 0 & 0 \\ * & * & - ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{'} I & 0 \\ * & * & * & - ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{″} I \end{matrix}], \end{matrix}$

$\begin{matrix} δ_{{ijlb}_{1} \dots b_{K + 1}} = {\hat{Ω}}_{{ijlb}_{1} \dots b_{K + 1}} + ɛ_{{ijlb}_{1} \dots b_{K + 1}} {\tilde{N}}_{Ai}^{T} {\tilde{N}}_{Ai} \\ + ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{'} {\tilde{N}}_{B_{ij}}^{T} {\tilde{N}}_{B_{ij}} + ɛ_{{ijlb}_{1} \dots b_{K + 1}}^{″} {\tilde{N}}_{Di}^{T} {\tilde{N}}_{Di}, \end{matrix}$

$\begin{matrix} {\tilde{P}}^{T} = [ɛ + ɛ_{i} 0 \dots 0 I], \\ {\tilde{N}}_{Ai} = [N_{ai} X 0 \dots 0], \\ {\tilde{N}}_{B_{ij}} = [N_{bi} M_{j} 0 \dots 0], \\ {\tilde{N}}_{Di} = [0 \dots 0 N_{di} X 0 0], \end{matrix}$ and ${\hat{Ω}}_{{ijlb}_{1} \dots b_{K + 1}}$ is defined in (32). Moreover, the fuzzy state-feedback controller can be chosen as (6), and the matrix gains of the controller are given by

$G_{i} = M_{i} X^{- 1}, i = 1, 2, \dots, r .$

Proof. Replacing A_i, B_i, and A_di in (30) with A_i + ΔA_i (t), B_i + ΔB_i (t), and A_di + ΔA_di (t), and from (13), we have

$\begin{matrix} {\hat{Ω}}_{{iilb}_{1} \dots b_{K + 1}} + sym (\tilde{P} D_{ai} F (t) {\tilde{N}}_{Ai} + \tilde{P} D_{bi} F (t) {\tilde{N}}_{B_{ii}} \\ + \tilde{P} D_{di} F (t) {\tilde{N}}_{Di}) < 0 . \end{matrix}$ (36)

According to Lemma 1, the inequality (36) holds if

$\begin{array}{l} {\hat{Ω}}_{i i l b_{1} \dots b_{K + 1}} + ε_{i i l b_{1} \dots b_{K + 1}}^{- 1} \tilde{P} D_{a i} D_{a i}^{T} {\tilde{P}}^{T} \\ + ε_{i i l b_{1} \dots b_{K + 1}}^{' - 1} \tilde{P} D_{a i} D_{a i}^{T} {\tilde{P}}^{T} \\ + ε_{i i l b_{1} \dots b_{K + 1}}^{" - 1} \tilde{P} D_{a i} D_{a i}^{T} {\tilde{P}}^{T} \\ + ε_{i i l b_{1} \dots b_{K + 1}} {\tilde{N}}_{A i}^{T} {\tilde{N}}_{A i} + ε_{i i l b_{1} \dots b_{K + 1}}^{'} {\tilde{N}}_{B i i}^{T} {\tilde{N}}_{B i i} \\ + ε_{i i l b_{1} \dots b_{K + 1}}^{"} {\tilde{N}}_{D i}^{T} {\tilde{N}}_{D i} < 0, \end{array}$

which, by the Schur complement, is equivalent to the inequality in (34). By following similar lines, the inequality (35) can be obtained. The proof iscompleted.

Remark 3. The sizes of subintervals are variable by altering parameter sequence (m₁, m₂, ⋯ , m_K) satisfying 0 < m₁ < m₂ < ⋯ < m_K < 1. Obviously, parameter sequence (m₁, m₂, ⋯ , m_K) is more general and flexile than the parameter sequence (1/m, 2/m, ⋯ , (m - 1)/m) in [17].

Remark 4. It is a typical tradeoff between conservativeness and complexity. Our results are less conservative but are with more parameters. That is to say, by increasing the useful parameters, the aim of reducing conservativeness will be achieved. Meanwhile, the stability conditions are expressed in the form of matrix inequalities, which can be checked easily by using the interior point algorithm. Thus, the overall computation complexity looks similar to the previous methods.

Remark 5. Some studies on simultaneously reducing the complexity and conservativeness still need to be further carried out for the dynamic delay partitioning method.

5 The Implementation of Optimal Delay Partitioning Parameters

The key problem in this paper is how to choose the values of delay partitioning parameters m_ρ, where ρ = 1, 2, ⋯ , K. Take Theorem 1 for example, the optimization process is listed as follows to find the optimal parameter sequence (m₁, m₂, ⋯ , m_K). It consists of three steps.

1) Define δ₁ = m₁, δ₂ = m₂ - m₁, ⋯ , δ_K = m_K - m_K-1. According to the range of m_ρ, we can know that δ_i satisfies the following conditions: $\sum_{i = 1}^{K} δ_{i} < 1,$ (37) $δ_{i} > 0 .$ (38)

2) Given μ, K, and (δ₁, δ₂, ⋯ , δ_K), we can obtain maximum allowable τ based on LMI ToolBox of MATLAB, which subjects to matrix inequality (20) and (21), and other restrictions in Theorem 1.

3) Choose the K + 1 groups of parameter sequence $Δ^{j} = (δ_{1}^{j}, δ_{2}^{j}, \dots, δ_{K}^{j})$ , where $δ_{ρ}^{j}$ satisfying (37) and (38), ρ = 1, 2, ⋯ , K and j = 1, 2, ⋯ , K + 1. Based on these data and step 2), applying N-M simplex method proposed in [26], the optimal parameter sequence and the corresponding maximum allowable τ can be obtained for Theorem 1.

6 Illustrative Example

In this section, two examples are provided to illustrate the effectiveness of the proposed methods.

6.1 Example 1 (Controller Design)

Consider the following fuzzy time-delay system with uncertainties:

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{2} λ_{i} (t) [(A_{i} + Δ A_{i} (t)) x (t) \\ + (A_{di} + Δ A_{di} (t)) x (t - d (t)) \\ + (B_{i} + Δ B_{i} (t)) u (t)], \end{matrix}$ (39) where

$\begin{matrix} A_{1} = [\begin{matrix} 0 & 0.6 \\ 0 & 1 \end{matrix}], A_{2} = [\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}], A_{d 1} = [\begin{matrix} 0.5 & 0.9 \\ 0 & 2 \end{matrix}], \\ A_{d 2} = [\begin{matrix} 0.9 & 0 \\ 1 & 1.6 \end{matrix}], B_{i} = [\begin{matrix} 1 \\ 1 \end{matrix}], \end{matrix}$ and

$\begin{matrix} Δ A_{i} (t) = D_{ai} F (t) N_{ai}, Δ A_{di} (t) = D_{di} F (t) N_{di}, \\ Δ B_{i} (t) = D_{bi} F (t) N_{bi}, F (t) = \sin (t), i = 1, 2, \end{matrix}$ where

$\begin{matrix} D_{ai} = D_{di} = D_{bi} = [\begin{matrix} 1 \\ 0 \end{matrix}], N_{a 1} = [0 0], \\ N_{a 2} = [- 0.05 0], N_{di} = [0 0], N_{bi} = 0.03, \\ i = 1, 2 . \end{matrix}$

We consider robust control for the fuzzy system in (39). The results by applying the methods in [17 –19] and Theorem 3 are listed in Table 1. The reduction of conservatism given by our approach is clearlyshown.

Table 1

Comparisons of MAUB τ of the uncertain system (Example 1)

Method	MAUB τ
Theorem 4 of [17]	0.3991
Theorem 4 of [18]	1.1351
Theorem 2 of [19]	1.2151
Theorem 3 of this paper (K = 2)	1.2549

The state feedback gains are given by

$\begin{matrix} G_{1} = [14.0424 - 40.6272], \\ G_{2} = [14.2811 - 45.9233] . \end{matrix}$ (40)

In the simulation, we utilize the following fuzzy membership function:

$\begin{matrix} λ_{1} (t) = \frac{1}{(1 + \exp (- 2 x_{1} (t) + 0.5)}, \\ λ_{2} (t) = 1 - λ_{1} (t) . \end{matrix}$

6.2 Example 2 (Application to Truck-Trailer Control)

Consider the following modified truck-trailer model with time-delay formulated in [17]:

$\begin{matrix} {\dot{x}}_{1} (t) & = & - a \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t) - (1 - a) \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t - σ (t)) \\ + \frac{v \bar{t}}{{lt}_{0}} u (t), \\ {\dot{x}}_{2} (t) & = & a \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t) + (1 - a) \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t - σ (t)), \\ {\dot{x}}_{3} (t) & = & \frac{v \bar{t}}{t_{0}} \sin (x_{2} (t) + a \frac{v \bar{t}}{2 L} x_{1} (t) \\ + (1 - a) \frac{v \bar{t}}{2 L} x_{1} (t - σ (t))), \end{matrix}$ (41) where x₁ (t) is the angle difference between truck and trailer, x₂ (t) is the angle of trailer, x₃ (t) is the vertical position of rear end of trailer. The model parameters are given as l = 2.8, L = 5.5, v = -1.0, $\bar{t} = 2.0$ , and t₀ = 0.5. The constant a is the retarded coefficient, which satisfies the conditions a ∈ [0, 1]. The limits 1 and 0 correspond to no delay term and to a complete delay term, respectively. In this example, we assume a = 0.7. Let $θ (t) = x_{2} (t) + a (v \bar{t} / 2 L) x_{1} (t) + (1 - a) (v \bar{t} / 2 L) x_{1} (t - d (t))$ . The T-S fuzzy model that represents the nonlinear system is as follows:

Plant Rule 1: IF θ (t) is 0 rad, THEN

$\begin{matrix} \dot{x} (t) & = & (A_{1} + Δ A_{1} (t)) x (t) \\ + (A_{d 1} + Δ A_{d 1} (t)) x (t - d (t)) \\ + (B_{1} + Δ B_{1} (t)) u (t), \end{matrix}$

Plant Rule 2: IF θ (t) is -π rad or π rad, THEN

$\begin{matrix} \dot{x} (t) & = & (A_{2} + Δ A_{2} (t)) x (t) \\ + (A_{d 2} + Δ A_{d 2} (t)) x (t - d (t)) \\ + (B_{2} + Δ B_{2} (t)) u (t), \end{matrix}$ where x (t) = [x₁ (t) x₂ (t) x₃ (t)], and

$\begin{matrix} A_{1} = [\begin{matrix} - a \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ a \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ a \frac{v^{2} {\bar{t}}^{2}}{2 {Lt}_{0}} & \frac{v \bar{t}}{t_{0}} & 0 \end{matrix}], A_{d 1} = [\begin{matrix} - (1 - a) \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ (1 - a) \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ (1 - a) \frac{v^{2} {\bar{t}}^{2}}{2 {Lt}_{0}} & 0 & 0 \end{matrix}], \\ A_{2} = [\begin{matrix} - a \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ a \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ a \frac{\bar{g} v^{2} {\bar{t}}^{2}}{2 {Lt}_{0}} & \frac{\bar{g} v \bar{t}}{t_{0}} & 0 \end{matrix}], A_{d 2} = [\begin{matrix} - (1 - a) \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ (1 - a) \frac{v \bar{t}}{{Lt}_{0}} & 0 & 0 \\ (1 - a) \frac{\bar{g} v^{2} {\bar{t}}^{2}}{2 {Lt}_{0}} & 0 & 0 \end{matrix}], \\ B_{1} = [\begin{matrix} \frac{v \bar{t}}{{lt}_{0}} \\ 0 \\ 0 \end{matrix}], B_{2} = [\begin{matrix} \frac{v \bar{t}}{{lt}_{0}} \\ 0 \\ 0 \end{matrix}] . \end{matrix}$

We assume that the uncertainties in the system are modeled as that in [17], where F (t) = sin (2t), and $\begin{array}{l} D_{a 1} = D_{a 2} = D_{d 1} = D_{d 2} = {[\begin{array}{l} 0.255 & 0.255 & 0.255 \end{array}]}^{T}, \\ D_{b 1} = D_{b 2} = {[\begin{array}{l} 0.1790 & 0 & 0 \end{array}]}^{T}, \\ N_{a 1} = N_{a 2} = N_{d 1} = N_{d 2} = [\begin{array}{l} 0.1 & 0 & 0 \end{array}], \\ \begin{array}{l} N_{b 1} = 0.05, & N_{b 2} = 0.15. \end{array} \end{array}$

By using Theorem 3, we found that the fuzzy system is robustly stabilizable with the maximum time-delay value τ = 103.4389, and the fuzzy state-feedback controller gains are given by

$\begin{matrix} G_{1} = [13.8301 - 87.9634 10.2711], \\ G_{2} = [13.8649 - 37.8973 10.2724] . \end{matrix}$ (42)

Suppose the initial condition φ (t) = [0.5π 0.75π - 5] ^T. The state responses of the closed-loop system under the above controllers are shown in Fig. 1.

Fig.1

State response of the uncertain system.

7 Conclusions

This paper has investigated the problem of fuzzy controller design for a class of unknown nonlinear systems with T-S fuzzy model and dynamic delay partition. Compared with previous results, several negative definite terms with dynamic delay partitioning parameters will be added to criteria, which leads to the less conservative stability results. It is clear that dynamic delay partitioning method by applying the variable subintervals with delay d (t) has inherent flexibility, and is more suitable to deal with stability problem with time-varying delay. The results above have been further utilized to solve the stabilization problem. Two examples have been provided to show the effectiveness and advantage of the obtained results. The dynamic delay partitioning idea has been well demonstrated to be efficient for reducing conservatism and could be further extended to solve other related problems. But when the delay partitioning number K gets bigger, there will be a computational problem. Therefore, the study on simultaneously reducing the complexity and conservativeness need to be further carried out for the dynamic delay partitioning method in the future.

Footnotes

Acknowledgment

This work is supported by the National Natural Science Foundation of China (61374124, 6147306, 61627809).

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