Observer-based tracking control using unmeasurable premise variables for time-delay switched fuzzy systems

Abstract

This paper deals with observer-based tracking controller design for time-delay switched fuzzy system with unmeasurable premise variables. A switched fuzzy system, which integrates fuzzy and switching features, can describe continuous and discrete modes and complex nonlinear phenomena of objective reality as well as their coupling and interactions. A fuzzy state observer is investigated for estimating the unmeasured states and unmeasurable premise variables. The stability and the switching control law based on measured state of the dynamic errors system are then analyzed by multiple Lyapunov function approach. In this study, the H_∞ tracking control problem of a time-delay switched fuzzy system is solvable, which some sub time-delay fuzzy systems are allowed to be unstable. The variation-of-constants formula is used to overcome the difficulties caused by the estimation error and exotic disturbance. Finally, a competitive experiment on an R/C hovercraft is given to demonstrate the effectiveness of the proposed control schemes.

Keywords

Switched fuzzy system time-delay unmeasurable premise variables tracking control observer

1 Introduction

Switched systems, as a kind of hybrid systems, arise in engineering practice where several dynamical models are required to represent a system due to various jumping parameters and changing environmental factors. Such a system consists of a series of modes driven by differential/difference equations and a switching logic [6 , 23].

On the other hand, in the past two decades, the fuzzy logic theory has been proven to be a practical and effective way to deal with the analysis and synthesis problems for nonlinear systems [1 , 29]. Recently switched systems have been extended further to encompass fuzzy systems. The idea is firstly putted forward by Palm and Driankov [25]. Also, Tanaka et al. [13, 15], on the basis of TS fuzzy systems introduced new switching fuzzy systems for more complicated real systems such as multiple nonlinear systems, switched nonlinear hybrid systems, and second order nonholonomic systems. This class of the switching fuzzy model in [13, 15] is to switch local Takagi-Sugeno (T-S) fuzzy models represented in each region. The model switching depends on how to divide the fuzzy rule, which obviously affects stability analysis results.

A further and significant step has also been taken to utilize Lyapunov-function based control design techniques to the control synthesis problem [3 , 8]. Consequently, the control of PDC for T-S fuzzy systems and the stability issues have been studied. The so-called parallel distributed compensation (PDC) [9, 10] is one such control design framework that has been proposed and developed over the last few years. The PDC control structure [10] utilizes a nonlinear state feedback controller which mirrors the structure of the associated T-S model. In the PDC concept, each control rule is distributively designed for the corresponding rule of a T-S fuzzy model. Linear control theory can be used to design the consequences of fuzzy control rules because the consequences of T-S fuzzy models are described by linear state equations. The fuzzy controller shares the same fuzzy sets with the fuzzy model in the premises.

Since mathematical modeling of physical systems and processes in many areas of engineering often leads to complex nonlinear systems, which brings severe difficulties to analysis and synthesis, researchers have been seeking more effective methods for the control of nonlinear systems. Recently, we [11] have developed a novel switching-dependent model, called switched fuzzy model, to achieve nonlinear control effectively. The proposed corresponding model of switched fuzzy systems, presented below, differs from the existing ones in the literature cited in the fact that each sub-system is a T-S fuzzy system hence defining a class of switched fuzzy systems. Comparing with the previous work [13, 15], this model of switched fuzzy system is switching between each of the sub fuzzy systems, not depending on region fuzzy rule. It should be noted, this class inherits some essential features of hybrid systems while retaining all the information and knowledge representation capacity of fuzzy systems. The switching law can be designed as any combinational function of variables. Particularly, when the parameters sudden change or discontinuous change, the switching rules can be designed as any combinational function of variables, which make up the insufficiency of the switching depending on the single variable of the switching fuzzy model. It is clear that the field of switched fuzzy systems is becoming very popular. It is possible to find a switching law that renders the switched fuzzy systems stable for instable sub fuzzy systems.

For switched fuzzy systems, because of the complicated behavior caused by the interaction between the continuous dynamics and discrete switching, the problem of time-delay is more difficult to study. Only a few results have been reported in the literature such as the issues on robust control [17]. The importance of the study of tracking control for switched fuzzy systems with time-delay arises from the extensive researches in [12]. More practically, some control and detection generally make use of the switched fuzzy dynamics which are impossible or hard to measure accurately. Indeed, direct measurement of these values is not available or very expensive. To the best of the authors’ knowledge, up to now, the issue of tracking control, which has been well addressed for switched or fuzzy systems with time-delay [27], has been rarely investigated for switched fuzzy systems with time-delay and for the unmeasurable premise variables case in particular. Recently, observer-based control schemes received considerable attention [22]. Particularly, the observer-based tracking control problem for time-delay switched fuzzy system using unmeasurable premise variables has, however, turned out to be much more difficult and challenging.

Motivated by the above considerations, in this paper, the observer based H_∞ model reference tracking control problem of a class of switched fuzzy systems with time-delay and unmeasurable premise variables is solvable based on the multiple Lyapunov functions method. Meanwhile, a switching law depending on the observer state is solved. Moreover, the case of membership functions considering on unmeasurable premise variable is also investigated for observer and controller design.

2 Time-delay switched fuzzy model description

The idea of switched fuzzy model which consists of sub T-S fuzzy models is broadened by introducing the fuzzy model construction based on the sector nonlinearity concept. Many nonlinear systems can be expressed as switched fuzzy systems.

Here, consider the time-delay switched fuzzy model including N_σ(t) pieces of rules as follows, which each subsystem is a time-delay fuzzy system, namely, sub time-delay fuzzy system.

$\begin{matrix} R_{σ (t)}^{l} : if z_{σ (t) 1} (t) is M_{σ (t) 1}^{l} \dots and \\ z_{σ (t) p} (t) is M_{σ (t) p}^{l}, then \\ {\begin{matrix} \dot{x} (t) = A_{σ (t) l} x (t) + D_{σ (t) l} x (t - τ) \\ + B_{σ (t) l} u_{σ (t)} (t) + ω (t), l = 1, 2, \dots, N_{σ (t)} \\ y (t) = C_{σ (t) l} x (t), t \in [0, \infty), x (0) = 0, \\ φ_{σ (t)} (θ) = x (t + θ), θ \in [t_{j} - τ, t_{j}], j = 0, 1, \dots . \end{matrix} \end{matrix}$ (1) with $σ (t) : M = {1, 2, \dots, m} .$ (2)

Where is a piecewise constant function, called a switching signal. It can be characterized by $\begin{matrix} \sum = {x_{0}; (i_{0}, t_{0}), (i_{1}, t_{1}), \dots, (i_{j}, t_{j}) | \\ i_{j} \in M, j = 0, 1, \dots} . \end{matrix}$

Moreover, σ (t) = i implies that the i-th subsystem (A_i, D_i, B_i, C_i) is active. $R_{σ (t)}^{l}$ denotes the lth fuzzy inference rule, N_σ(t) is the number of inference rules, $u_{σ (t) l} (t) \in ℝ^{q}$ is the control input, $ω (t) \in ℝ^{n}$ is the bounded exogenous disturbance, τ is the constant bounded time delay in the state, φ_σ(t) (t) is the continuous vector valued function.

It is readily seen that the i-th sub time-delay fuzzy system can be represented as follows:

$\begin{matrix} R_{i}^{l} : if z_{i 1} (t) is M_{i 1}^{l} \dots and z_{ip} (t) is M_{ip}^{l}, then \\ {\begin{matrix} \dot{x} (t) = A_{il} x (t) + D_{il} x (t - τ) + B_{il} u_{i} (t) + ω (t) \\ y (t) = C_{il} x (t), t \in [0, \infty), x (0) = 0 \end{matrix} \\ l = 1, 2, \dots, N_{i}, i = 1, 2, \dots, m . \end{matrix}$ (3)

Therefore the global model of the i-th sub time-delay fuzzy system is described by ${\begin{matrix} \dot{x} (t) = \sum_{l = 1}^{N_{i}} η_{il} (z (t)) [A_{il} x (t) + D_{il} x (t - τ) \\ + B_{il} u_{i} (t) + ω (t)] \\ y (t) = \sum_{l = 1}^{N_{i}} η_{il} C_{il} x (t) \end{matrix}$ (4) along with $0 \leq η_{il} (z (t)) \leq 1, \sum_{l = 1}^{N_{i}} η_{il} (z (t)) = 1 .$ (5) $w_{il} (z (t)) = \prod_{ρ = 1}^{p} M_{i ρ}^{l} (z_{ρ} (t)), η_{il} (z (t)) = \frac{w_{il} (z (t))}{\sum_{l = 1}^{N_{i}} w_{il} (z (t))},$ where $M_{i ρ}^{l} (z_{ρ} (t))$ denotes the membership function which z_ρ (t) belongs to the fuzzy set $M_{i ρ}^{l}$ .

Given a reference model ${\dot{x}}_{r} (t) = A_{r} x_{r} (t) + r (t), x_{r} (0) = 0$ (6) and performance index $\int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) dt < {\tilde{γ}}^{2} \int_{0}^{\infty} {\tilde{ω}}^{T} (t) \tilde{ω} (t) dt$ (7) where $x_{r} (t) \in ℝ^{n}$ is reference state, A_r is a Hurwitz matrix, r (t) is bounded reference input; e_r (t) = x (t) - x_r (t) denotes the error between the real state of the time-delay switched fuzzy system (1) and the reference state, $\tilde{ω} (t) = {(ω^{T} (t) r^{T} (t))}^{T}$ , $\tilde{γ} > 0$ is a prescribed attenuation level.

3 State observer and controller design

3.1 Problem statement

Let us quote that two cases are distinguished: (i) the scheduling vector z (t) does not depend on the estimation states, i.e., $\hat{z} (t) = z (t)$ and (ii) z (t) depends on completely or partially on estimated states. In the following, we assume that the scheduling vector depends on the estimated states.

Suppose that the state observer is with the form

$\begin{matrix} R_{σ (t)}^{l} : if {\hat{z}}_{σ (t) 1} (t) is M_{σ (t) 1}^{l} \dots \\ and {\hat{z}}_{σ (t) p} (t) is M_{σ (t) p}^{l}, then \\ {\begin{matrix} \dot{\hat{x}} (t) = A_{σ (t) l} \hat{x} (t) + D_{σ (t) l} \hat{x} (t - τ) + B_{σ (t) l} u_{σ (t)} (t) \\ + L_{σ (t) l} (y (t) - \hat{y} (t)) \\ \hat{y} (t) = C_{σ (t) l} \hat{x} (t) \end{matrix} \end{matrix}$ (8) in which y (t) and σ (t) are measurable output and switching signal of systems (1), respectively. The matrices $L_{σ (t) l} \in ℝ^{n \times s}$ are to be determined later.

Then the global model of the i-th sub fuzzy observer is described ${\begin{matrix} \dot{\hat{x}} (t) = \sum_{l = 1}^{N_{i}} η_{il} (\hat{z} (t)) \\ [\begin{matrix} A_{il} \hat{x} (t) + D_{il} \hat{x} (t - τ) + B_{il} u_{i} (t) \\ + L_{il} (y (t) - \hat{y} (t)) \end{matrix}] \\ \hat{y} (t) = \sum_{l = 1}^{N_{i}} η_{il} (\hat{z} (t)) C_{il} \hat{x} (t) \end{matrix}$ (9)

Let us respectively define the observer state and the reference state estimation error, the reference state, the real state and the observer state estimation error, the observer state, and the output estimationerror as: $[\begin{matrix} {\hat{e}}_{r} (t) \\ x_{r} (t) \\ e (t) \\ \hat{x} (t) \\ e_{y} (t) \end{matrix}] = [\begin{matrix} \hat{x} (t) - x_{r} (t) \\ x_{r} (t) \\ x (t) - \hat{x} (t) \\ \hat{x} (t) \\ y (t) - \hat{y} (t) \end{matrix}]$ (10)

The aim is to design a controller u = u (t) and a switching law ensuring the observer based H_∞ model reference tracking performance of the system (1).

The controller is given by the structure: $u_{i} (t) = \sum_{l = 1}^{N_{i}} η_{il} (\hat{z} (t)) K_{il} {\hat{e}}_{r} (t)$ (11)

Subtracting (9) from (4) give the error switched system

$\begin{matrix} \dot{e} (t) & = & \sum_{l = 1}^{N_{i}} η_{il} (z) \sum_{j = 1}^{N_{i}} η_{ij} (\hat{z}) [\begin{matrix} A_{il} x (t) + D_{il} x (t - τ) \\ + B_{il} K_{ij} {\hat{e}}_{r} (t) + ω (t) \end{matrix}] \\ - \sum_{θ = 1}^{N_{i}} η_{i θ} (\hat{z}) \sum_{n = 1}^{N_{i}} η_{in} (\hat{z}) \\ [\begin{matrix} A_{i θ} \hat{x} (t) + D_{i θ} \hat{x} (t - τ) + B_{i θ} K_{in} {\hat{e}}_{r} (t) \\ + L_{i θ} (y (t) - \hat{y} (t)) \end{matrix}] \end{matrix}$ (12)

$\begin{matrix} {\dot{\hat{e}}}_{r} (t) \\ = \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) A_{il} {\hat{e}}_{r} (t) + \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) A_{il} x_{r} (t) \\ + \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) D_{il} {\hat{e}}_{r} (t - τ) + \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) D_{il} x_{r} (t - τ) \\ + \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) \sum_{j = 1}^{N_{i}} η_{ij} (\hat{z}) B_{il} K_{ij} {\hat{e}}_{r} (t) \\ + \sum_{l = 1}^{N_{i}} η_{il} (\hat{z}) L_{il} [y (t) - \hat{y} (t)] - A_{r} x_{r} (t) - r (t) \end{matrix}$ (13) Then ${\tilde{x}}^{T} = [\begin{matrix} {\hat{e}}_{r} (t) & x_{r} (t) & e (t) & \hat{x} (t) & e_{y} (t) \end{matrix}]$ .

Thus, the closed loop dynamics can be expressed as:

$\begin{matrix} \dot{\tilde{x}} (t) & = & \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} \\ [{\tilde{A}}_{ilhj θ n} \tilde{x} (t) + {\tilde{D}}_{ilh θ} \tilde{x} (t - τ) + \tilde{f} (t)] \end{matrix}$ (14)

Where $\begin{matrix} {\tilde{A}}_{ilhj θ n} \\ = [\begin{matrix} A_{ih} + B_{ih} K_{ij} & A_{ih} - A_{r} & 0 & 0 & L_{ih} \\ 0 & A_{r} & 0 & 0 & 0 \\ B_{il} K_{ij} - B_{i θ} K_{in} & 0 & A_{il} & A_{il} - A_{i θ} & - L_{i θ} \\ B_{ih} K_{ij} & 0 & 0 & A_{ih} & L_{ih} \\ 0 & 0 & C_{il} & C_{il} - C_{ih} & 0 \end{matrix}], \end{matrix}$ $\begin{matrix} {\tilde{D}}_{ilh θ} & = & [\begin{matrix} D_{ih} & D_{ih} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & D_{ih} & D_{il} - D_{i θ} & 0 \\ 0 & 0 & 0 & D_{ih} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ \tilde{f} (t) & = & [\begin{matrix} - r (t) \\ r (t) \\ ω (t) \\ 0 \\ 0 \end{matrix}] . \end{matrix}$

The corresponding nominal system of (14) is

Remark 1. When observer-based tracking control for time-delay switched fuzzy system is designed, the state-space-partition-based switching law should not include the nonmeasurable state information, whereas the tracking performance index is based on the error between the real state and the reference state. Therefore, some existing observer-based tracking control schemes (see e.g. [7]) can not be applied to state-dependent switching control. In order to conquer the difficulties caused by the estimation error and exotic disturbance, the variation-of-constants formula in the following part is introduced for the design of switching tracking control law.

3.2 Stabilization of Switched Fuzzy control model

Theorem 1. For system (14), Suppose there exist constants β_ij $(i, j \in \bar{M})$ (either all nonnegative or all non-positive), and positive definite symmetric matrixes P_i, S_i and constants δ > 0, such that the following matrix inequalities are satisfied, then the observer based H_∞ model reference tracking performance in (1) is guaranteed via the state-feedback controller (11):

$\begin{matrix} [\begin{matrix} ϒ_{ilhj θ n} + \sum_{j = 1}^{m} β_{ij} (P_{j} - P_{i}) & P_{i} {\tilde{D}}_{ilh θ} & P_{i} \\ * & - e^{- δ τ} S_{i} & 0 \\ * & * & - γ^{2} I \end{matrix}] < 0 \\ l, h, j, θ, n = 1, 2 \dots, N_{i}, i = 1, 2, \dots, m . \end{matrix}$ (16)

And, the switching law is designed as: $σ = σ (\hat{x} (t)) = i : [0, + \infty) \to \bar{M} = {1, 2, \dots, m}$ (17) where $ϒ_{ilhj θ n} = {\tilde{A}}_{ilhj θ n}^{T} P_{i} + P_{i} {\tilde{A}}_{ilhj θ n} + δ P_{i} + S_{i} + Q$ .

Proof. Without loss of generality, suppose β_ij ≥ 0. For any $i \in \bar{M}$ , if ${\hat{x}}^{T} (t) (P_{j} - P_{i}) \hat{x} (t) \geq 0$ , $\forall j \in \bar{M}$ . Form (16), we have $[\begin{matrix} ϒ_{ilhj θ n} & P_{i} {\tilde{D}}_{ilh θ} & P_{i} \\ * & - e^{- δ τ} S_{i} & 0 \\ * & * & - γ^{2} I \end{matrix}] < 0$ (18)

Obviously, for $\forall \hat{x} (t) \in ℝ^{n} ∖ {0}$ , there certainly is an $i \in \bar{M}$ such that ${\hat{x}}^{T} (t) (P_{j} - P_{i}) \hat{x} (t) \geq 0, \forall j \in \bar{M}$ (19)

For arbitrary $i \in \bar{M}$ , let $Ω_{i} = {\hat{x} (t) \in R^{n} ∖ {0} | {\hat{x}}^{T} (t) (P_{j} - P_{i}) \hat{x} (t) \geq 0, \forall j \in \bar{M}}$

Then $⋃_{i = 1}^{m} Ω_{i} = ℝ^{n} ∖ {0}$ . Further, let construct the sets ${\bar{Ω}}_{1} = Ω_{1}, \dots, {\bar{Ω}}_{i} = Ω_{i} - ⋃_{j = 1}^{i - 1} {\bar{Ω}}_{j}, \dots, {\bar{Ω}}_{m} = Ω_{m} - ⋃_{j = 1}^{m - 1} {\bar{Ω}}_{j}$ . Obviously, we have $⋃_{i = 1}^{m} Ω_{i} = ℝ^{n} ∖ {0}$ , and ${\bar{Ω}}_{i} \cap {\bar{Ω}}_{j} = Φ$ , i ≠ j.

By Schur complement lemma, the condition (18) is equivalent to the following inequality $Ξ^{T} (t) [\begin{matrix} ϒ_{ilhj θ n} + γ^{- 2} P_{i} P_{i} & P_{i} {\tilde{D}}_{ilh θ} \\ {\tilde{D}}_{ilh θ}^{T} P_{i} & - e^{- δ τ} S_{i} \end{matrix}] Ξ (t) < 0$ (20)

When $\hat{x} (t) \in {\bar{Ω}}_{i}$ , σ (t) = i. Now choose multiple Lyapunov functional as $V_{i} (\tilde{x} (t)) = {\tilde{x}}^{T} (t) P_{i} \tilde{x} (t) + \int_{t - τ}^{t} {\tilde{x}}^{T} (θ) e^{- δ (t - θ)} S_{i} \tilde{x} (θ) d θ$ (21) which is positive definite since P_i and S_i are positive definite matrices.

First, we will prove that the system (14) is asymptotically stable while $\tilde{ω} (t) \equiv 0$ . ${\dot{V}}_{i} (t)$ for (14) is

$\begin{matrix} {\dot{V}}_{i} (\tilde{x} (t)) + δ V_{i} (\tilde{x} (t)) \\ = \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} \\ Ξ^{T} (t) [\begin{matrix} {\tilde{A}}_{ilhj θ n}^{T} P_{i} + P_{i} {\tilde{A}}_{ilhj θ n} + δ P_{i} + S_{i} & P_{i} {\tilde{D}}_{ilh θ} \\ {\tilde{D}}_{ilh θ}^{T} P_{i} & - e^{- δ τ} S_{i} \end{matrix}] Ξ (t) \end{matrix}$ (22)

Where $Ξ (t) = [\begin{matrix} \tilde{x} (t) \\ \tilde{x} (t - τ) \end{matrix}]$ . Then from (20) and $Q = [\begin{matrix} I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$ , it holds true that $Ξ^{T} (t) {\begin{matrix} [\begin{matrix} \begin{matrix} {\tilde{A}}_{ilhj θ n}^{T} P_{i} + P_{i} {\tilde{A}}_{ilhj θ n} \\ + δ P_{i} + S_{i} \end{matrix} & P_{i} {\tilde{D}}_{ilh θ} \\ {\tilde{D}}_{ilh θ}^{T} P_{i} & - e^{- δ τ} S_{i} \end{matrix}] \\ + [\begin{matrix} Q + γ^{- 2} P_{i} P_{i} & 0 \\ 0 & 0 \end{matrix}] \end{matrix}} Ξ (t) < 0$ (23) in which i = σ (t) given by (17). Taking (5), (20) and (23) into account we deduce that ${\dot{V}}_{i} (\tilde{x} (t)) + δ V_{i} (\tilde{x} (t)) < 0, Ξ (t) \neq 0$ . So $V_{i} (\tilde{x} (t)) \leq e^{- δ (t - t_{0})} V_{i} (\tilde{x} (t_{0}))$

Then we have ${∥ \tilde{x} (t) ∥}^{2} \leq \frac{λ_{M} (P_{i}) + τ λ_{M} (S_{i})}{λ_{m} (P_{i})} e^{- δ (t - t_{0})} {∥ \tilde{x} (t_{0}) ∥}^{2}$ (24) where λ_m (·) (λ_M (·)) denotes the minimum (maximum) eigenvalue of a symmetric matrix, which implies exponential stability of the nominal systems of (14), i.e., the system (15). Therefore, by the variation-of-constants method in [24], and considering (24), there are constant 0 < κ ≤ 1 and α > 0 such that $∥ {\tilde{x}}_{t} (t, t_{j}, φ_{i}, \tilde{f}) ∥ \leq \int_{t_{0}}^{t} κ e^{- α (t - s)} ∥ \tilde{f} (s) ∥ ds, t \in [t_{j}, t_{j + 1})$ (25)

When $\tilde{ω} (t) = 0$ , note that $\tilde{f} (t) \to 0 (t \to \infty)$ . So it follows form [24] that $\tilde{x} (t) \to 0$ as t → 0. This in turn gives rise to that the time-delay switched fuzzy system (14) is asymptotically stable of the non-perturbed.

Second, we will prove that under zero initial condition with $\tilde{ω} (t) = 0$ that $\int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) dt < {\tilde{γ}}^{2} \int_{0}^{\infty} {\tilde{ω}}^{T} (t) \tilde{ω} (t) dt$ . Again, there holds σ (t) = i, t ∈ [t_j, t_j+1). Calculating ${\dot{V}}_{i} (\tilde{x} (t))$ along the trajectory of the system (14) with $\tilde{ω} (t) \neq 0$ , we have

$\begin{matrix} {\dot{V}}_{i} (\tilde{x} (t)) \\ \leq \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} \\ Ξ^{T} (t) [\begin{matrix} {\tilde{A}}_{ilhj θ n}^{T} P_{i} + P_{i} {\tilde{A}}_{ilhj θ n} + S_{i} & P_{i} {\tilde{D}}_{ilh θ} \\ {\tilde{D}}_{ilh θ}^{T} P_{i} & - e^{- δ τ} S_{i} \end{matrix}] Ξ (t) \\ + \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \\ \sum_{n = 1}^{N_{i}} η_{in} γ^{- 2} {\tilde{x}}^{T} (t) P_{i} P_{i} {\tilde{x}}^{T} (t) \\ + \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} γ^{2} {\tilde{f}}^{T} (t) \tilde{f} (t) \end{matrix}$ (26)

In view of the condition (23) and the structure of Q yields, it follows $\begin{matrix} \dot{V} (\tilde{x} (t)) \\ \leq \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} Ξ^{T} (t) [\begin{matrix} - Q & 0 \\ 0 & 0 \end{matrix}] \\ Ξ (t) + \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} γ^{2} {\tilde{f}}^{T} (t) \tilde{f} (t) \end{matrix}$

Where

$\begin{matrix} {\tilde{x}}^{T} (t) Q \tilde{x} (t) & = & {[\begin{matrix} {\hat{e}}_{r} (t) \\ x_{r} (t) \\ e (t) \\ \hat{x} (t) \\ e_{y} (t) \end{matrix}]}^{T} [\begin{matrix} I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} {\hat{e}}_{r} (t) \\ x_{r} (t) \\ e (t) \\ \hat{x} (t) \\ e_{y} (t) \end{matrix}] \\ = & {\hat{e}}_{r}^{T} (t) {\hat{e}}_{r} (t) \end{matrix}$ (27)

Substituting (27) into (26) results in

$\begin{matrix} \dot{V} (\tilde{x} (t)) \\ \leq \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} (- {\hat{e}}_{r}^{T} (t) {\hat{e}}_{r} (t)) \\ + \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} γ^{2} {\tilde{f}}^{T} (t) \tilde{f} (t) \end{matrix}$ (28)

Note that ${\hat{e}}_{r} (t) = e_{r} (t) - e (t)$ , with $E = diag {\frac{1}{2}, \dots, \frac{1}{2}}$ gives

$\begin{matrix} - {\hat{e}}_{r}^{T} (t) {\hat{e}}_{r} (t) = - e_{r}^{T} (t) e_{r} (t) - e^{T} (t) e (t) + 2 e_{r}^{T} (t) e (t) \\ \leq - e_{r}^{T} (t) e_{r} (t) - e^{T} (t) e (t) + e_{r}^{T} (t) E e_{r} (t) + e^{T} (t) E^{- 1} e (t) \\ = - \frac{1}{2} e_{r}^{T} (t) e_{r} (t) + {∥ e (t) ∥}^{2} \end{matrix}$ (29)

Let e_t (t, t_j, φ_i, ω) denote the solution of (12) with initial condition (t_j, φ_i). It holds that $∥ e_{t} (t, t_{j}, φ_{i}, ω) ∥ \leq \int_{t_{0}}^{t} κ e^{- α (t - s)} ∥ ω (s) ∥ ds$ (30)

Considering Cauchy-Schwartz Inequality, we have

$\begin{matrix} {∥ e (t) ∥}^{2} & \leq & \int_{t_{0}}^{t} κ^{2} e - α (t - s) ds • \int_{t_{0}}^{t} e^{- α (t - s)} {∥ ω (s) ∥}^{2} ds \\ \leq & \frac{κ^{2}}{α} \int_{t_{0}}^{t} e^{- α (t - s)} {∥ ω (s) ∥}^{2} ds \end{matrix}$ (31)

Therefore, when $\tilde{ω} (t) \neq 0$ , there has ${\tilde{f}}^{T} (t) \tilde{f} (t) = 2 r^{T} r + ω^{T} ω = 2 {∥ r (t) ∥}^{2} + {∥ ω (t) ∥}^{2}$ (32)

Combining (29), (31), (32) with (28) gives

$\begin{matrix} \dot{V} (\tilde{x} (t)) \leq \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} \\ [\begin{matrix} - \frac{1}{2} e_{r}^{T} (t) e_{r} (t) + \frac{κ^{2}}{δ} \int_{t_{0}}^{t} e^{- α (t - s)} {∥ ω (s) ∥}^{2} ds \\ + 2 γ^{2} {∥ r (t) ∥}^{2} + γ^{2} {∥ ω (t) ∥}^{2} \end{matrix}] \end{matrix}$ (33)

Integrating (33) from zero to ∞ and with $\frac{κ}{α} < γ$ , it holds that $\begin{matrix} \int_{0}^{\infty} \sum_{i_{j} \in M} \dot{V} (\tilde{x} (t)) dt < \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \sum_{j = 1}^{N_{i}} η_{ij} \sum_{θ = 1}^{N_{i}} η_{i θ} \sum_{n = 1}^{N_{i}} η_{in} \\ [\begin{matrix} - \frac{1}{2} \int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) + 2 γ^{2} \int_{0}^{\infty} {∥ r (t) ∥}^{2} dt \\ + \frac{κ^{2}}{α} \int_{0}^{\infty} \int_{t_{0}}^{t} e^{- α (t - s)} {∥ ω (s) ∥}^{2} dsdt + γ^{2} \int_{0}^{\infty} {∥ ω (t) ∥}^{2} dt \end{matrix}] \\ < \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} \\ [- \frac{1}{2} \int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) dt + 2 γ^{2} \int_{0}^{\infty} {\tilde{ω}}^{T} (t) \tilde{ω} (t) dt] \end{matrix}$

Let $2 γ^{2} = \frac{1}{2} {\tilde{γ}}^{2}$ , There has $\begin{matrix} \int_{0}^{\infty} \sum_{i_{j} \in M} \dot{V} (\tilde{x} (t)) dt \\ < \sum_{l = 1}^{N_{i}} η_{il} \sum_{h = 1}^{N_{i}} η_{ih} [\begin{matrix} - \frac{1}{2} \int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) dt \\ + \frac{1}{2} {\tilde{γ}}^{2} \int_{0}^{\infty} {\tilde{ω}}^{T} (t) \tilde{ω} (t) dt \end{matrix}] \end{matrix}$

It is easy to derive $\int_{0}^{\infty} e_{r}^{T} (t) e_{r} (t) dt < {\tilde{γ}}^{2} \int_{0}^{\infty} {\tilde{ω}}^{T} (t) \tilde{ω} (t) dt$ (34)

That is (7) and the H_∞ control performance is achieved with a prescribed ${\tilde{γ}}^{2}$ .

Remark 2. It is noted that Theorem 1, expressed in the form of matrix inequalities, cannot be directly implemented for the stability analysis. Our next objective is to convert the inequalities to some finite LMIs, which can be readily solved using standard numerical software.

We have the following theorem.

Theorem 2. For system (14), Suppose there exist constants β_ij $(i, j \in \bar{M})$ (either all nonnegative or all non-positive), and positive definite symmetric matrixes P_i, S_i and constants δ > 0, such that the following matrix inequalities are satisfied, then the observer based H_∞ model reference tracking performance in (1) is guaranteed via the state-feedback controller (11): $\begin{matrix} [\begin{matrix} {\tilde{ϒ}}_{ilhj θ n} + Π_{ij} & {\tilde{D}}_{ilh θ} & X_{i} \\ * & - e^{- δ τ} S_{i} & 0 \\ * & * & - (S_{i}^{- 1} + Q^{- 1}) \end{matrix}] < 0 \\ l, h, j, θ, n = 1, 2 \dots, N_{i}, i = 1, 2, \dots, m . \end{matrix}$ (35)

And, the switching law is designed as: $σ = σ (\hat{x} (t)) = i : [0, + \infty) \to \bar{M} = {1, 2, \dots, m}$ (36) where ${\tilde{ϒ}}_{ilhj θ n} = X_{i} {\tilde{A}}_{ilhj θ n}^{T} + {\tilde{A}}_{ilhj θ n} X_{i} + δ X_{i} + γ^{2} I$ , $Π_{ij} = \sum_{j = 1}^{m} β_{ij} (X_{i} X_{j}^{- 1} X_{i} - X_{i})$ , $X_{i} = P_{i}^{- 1}$ .

Proof. The proof is similar to that of Theorem 1. Multiplying both sides of (19) by the symmetric matrix $diag (P_{i}^{- 1}, I)$ , applying Schur complement lemma we know that (35) is equivalent to the inequality (19). Here it is omitted.

Strictly speaking, Theorem 2 is not a linear matrix inequality (LMI) since it contains cross products of free weighting matrices X_i in which the gain matrices K_il, K_in, L_ih, L_iθ are involved.

By choosing the structure of the matrix X_i and S_i as follows: $X_{i} = diag (\begin{matrix} {\bar{X}}_{i} & {\bar{X}}_{i} & {\bar{X}}_{i} & {\bar{X}}_{i} & {\bar{X}}_{i} \end{matrix})$ (37) $S_{i} = diag (\begin{matrix} S_{i 1} & S_{i 2} & S_{i 3} & S_{i 4} & S_{i 5} \end{matrix})$ (38)

Hence, the observer and the controller gains are obtained respectively by solving the LMI constraints given by the following theorem.

Theorem 3. For system (14), Suppose there exist constants β_ij $(i, j \in \bar{M})$ (either all nonnegative or all non-positive), and positive definite symmetric matrixes ${\bar{X}}_{i}$ , S_i1, S_i2, S_i3, S_i4, S_i5 and constants δ > 0, such that the following matrix inequalities are satisfied, then the observer based H_∞ model reference tracking performance in (1) is guaranteed via the state-feedback controller (11): $[\begin{matrix} ℘_{ilhj θ n} & (*) \\ ℵ_{i} & ℑ \end{matrix}] < 0$ (39) $l, h, j, θ, n = 1, 2 \dots, N_{i}, i = 1, 2, \dots, m$

With $℘_{ilhj θ n} = [\begin{matrix} ℘_{ilhj θ n}^{(1, 1)} & * & * \\ {\tilde{D}}_{ilh θ}^{T} & ℘_{i}^{(2, 2)} & * \\ X_{i} & 0 & ℘_{i}^{(3, 3)} \end{matrix}] < 0,$ $\begin{matrix} ℘_{ilhj θ n}^{(1, 1)} \\ = [\begin{matrix} Δ_{ihj} & A_{ih} {\bar{X}}_{i} - A_{r} {\bar{X}}_{i} \\ {\bar{X}}_{i} A_{ih}^{T} - {\bar{X}}_{i} A_{r}^{T} & {\bar{X}}_{i} A_{r}^{T} + A_{r} {\bar{X}}_{i} + Ξ_{ij} \\ B_{il} W_{ij} - B_{i θ} W_{in} & 0 \\ B_{ih} W_{ij} & 0 \\ G_{ih} & 0 \end{matrix} \\ \begin{matrix} * \\ * \\ {\bar{X}}_{i} A_{il}^{T} + A_{il} {\bar{X}}_{i} + Ξ_{ij} \\ {\bar{X}}_{i} A_{il}^{T} - {\bar{X}}_{i} A_{i θ}^{T} \\ - G_{i θ} + C_{il} {\bar{X}}_{i} \end{matrix} \begin{matrix} * & * \\ * & * \\ * & * \\ {\bar{X}}_{i} A_{ih}^{T} + A_{ih} {\bar{X}}_{i} + Ξ_{ij} & * \\ G_{ih} + C_{il} {\bar{X}}_{i} - C_{ih} {\bar{X}}_{i} & Ξ_{ij} \end{matrix}], \end{matrix}$ $\begin{matrix} Δ_{ihj} = {\bar{X}}_{i} A_{ih}^{T} + W_{ij}^{T} B_{ih}^{T} + A_{ih} {\bar{X}}_{i} + B_{ih} W_{ij} + Ξ_{ij}, \\ Ξ_{ij} = δ {\bar{X}}_{i} + γ^{2} I - \sum_{j = 1, i \neq j}^{m} β_{ij} {\bar{X}}_{i}, \\ W_{ij} = K_{ij} {\bar{X}}_{i}, G_{ih} = {\bar{X}}_{i} L_{ih}^{T}, \\ ℘_{i}^{(2, 2)} = - diag (\begin{matrix} e^{δ τ} S_{i 1} & e^{δ τ} S_{i 2} & e^{δ τ} S_{i 3} & e^{δ τ} S_{i 4} & e^{δ τ} S_{i 5} \end{matrix}), \\ ℘_{i}^{(3, 3)} = - diag (\begin{matrix} S_{i 1}^{- 1} + I & S_{i 2}^{- 1} & S_{i 3}^{- 1} & S_{i 4}^{- 1} & S_{i 5}^{- 1} \end{matrix}), \\ ℵ_{i} = [\begin{matrix} ℵ_{i 1} & 0 & 0 \\ ⋮ & ⋮ & ⋮ \\ ℵ_{im} & 0 & 0 \end{matrix}], \\ ℵ_{i 1} = diag \\ (\begin{matrix} \sqrt{β_{i 1}} {\bar{X}}_{i} & \sqrt{β_{i 1}} {\bar{X}}_{i} & \sqrt{β_{i 1}} {\bar{X}}_{i} & \sqrt{β_{i 1}} {\bar{X}}_{i} & \sqrt{β_{i 1}} {\bar{X}}_{i} \end{matrix}), \\ ℵ_{im} = diag \\ (\begin{matrix} \sqrt{β_{im}} {\bar{X}}_{i} & \sqrt{β_{im}} {\bar{X}}_{i} & \sqrt{β_{im}} {\bar{X}}_{i} & \sqrt{β_{im}} {\bar{X}}_{i} & \sqrt{β_{im}} {\bar{X}}_{i} \end{matrix}), \\ ℑ = - diag (\begin{matrix} ℑ_{1} & \dots & ℑ_{m} \end{matrix}), \\ ℑ_{1} = diag (\begin{matrix} {\bar{X}}_{1} & {\bar{X}}_{1} & {\bar{X}}_{1} & {\bar{X}}_{1} & {\bar{X}}_{1} \end{matrix}), \\ ℑ_{m} = diag (\begin{matrix} {\bar{X}}_{m} & {\bar{X}}_{m} & {\bar{X}}_{m} & {\bar{X}}_{m} & {\bar{X}}_{m} \end{matrix}) . \end{matrix}$

And, the switching law is designed as: $σ = σ (\hat{x} (t)) = i : [0, + \infty) \to \bar{M} = {1, 2, \dots, m}$ (40)

Proof. Following the same steps of the proof of Theorem 2 of (35), (39) proposed in Theorem 3.

Remark 3. It can be seen from Theorem 1–3 that the stability property of time-delay switched fuzzy system is the case of sub time-delay fuzzy system with switching. Thus, the stability result can be seen as the extension of the results of fuzzy system, but it is not a simple extension due to the interaction between time-delay and switching.

Remark 4. We note that the behaviour of the sub time-delay fuzzy system in other parts of the state space has no influence on the time-delay switched fuzzy system. This allows some subsystems to be unstable, but the time-delay switched fuzzy system is stable. In comparison, fuzzy systems without switching scheme do not exhibit this property.

4 Simulation results

To illustrate this approach, we analyze the stability of a room air regulating system [30]. The state equation of the system is: ${\ddot{T}}_{n} = - (\frac{1}{T_{1}} + \frac{1}{T_{2}}) {\dot{T}}_{n} - \frac{1}{T_{1} T_{2}} T_{n} + \frac{k_{1} k_{2}}{T_{1} T_{2}} u$ where T_n is the air temperature variable of the air-condition room (^oC). ${\dot{T}}_{n}$ is the air temperature variable of the air-condition room (^oC/min). T₁ is the time constant of the air-condition room (min). k₁ is the amplificatory coefficient of the constant temperature room (^oC/^oC). T₂ is the time constant of the steam heater (min). k₂ is the amplificatory coefficient of the electric executor (^oC/^oC). u is the control variable. When the temperature is lower,T₁ = 20.30 min, T₂ = 1 min. When the temperature is higher, T₁ = 30.40 min, T₂ = 2.5 min.

To illustrate the stability of this system when the temperature is 20^oC, having coordinates transform and choosing x₁ = T_n - 20, ${\dot{x}}_{1} = x_{2}$ , transform it into zero stability problem. Considering redundant circuit problem, the common fuzzy model is translated into the following switched fuzzy model to improve the speed of the air-condition to the predetermined temperature. $\begin{matrix} R_{1}^{1} : if x_{1} (t) is P_{11}^{1}, Close to the positive, then \\ {\begin{matrix} \dot{x} (t) = A_{11} x (t) + D_{11} x (t - τ) + B_{11} u_{1} (t) + ω (t) \\ y (t) = C_{11} x (t) \end{matrix} \\ R_{1}^{2} : if x_{1} (t) is N_{11}^{2}, Close to the negative, then \\ {\begin{matrix} \dot{x} (t) = A_{12} x (t) + D_{12} x (t - τ) + B_{12} u_{1} (t) + ω (t) \\ y (t) = C_{12} x (t) \end{matrix} \\ R_{2}^{1} : if x_{1} (t) is P_{21}^{1}, Close to the positive, then \\ {\begin{matrix} \dot{x} (t) = A_{21} x (t) + D_{21} x (t - τ) + B_{21} u_{2} (t) + ω (t) \\ y (t) = C_{21} x (t) \end{matrix} \\ R_{2}^{2} : if x_{1} (t) is N_{21}^{2}, Close to the negative, then \\ {\begin{matrix} \dot{x} (t) = A_{22} x (t) + D_{22} x (t - τ) + B_{22} u_{2} (t) + ω (t) \\ y (t) = C_{22} x (t) \end{matrix} \end{matrix}$

Accordingly, the switching law (40) is introduced in order to clearly express time-delay switched fuzzy system. And to illustrate and compare the effectiveness between our time-delay switched fuzzy system and other approach, an R/C hovercraft for Example 1 in [15] is proposed as follows. $\begin{matrix} Region Rule 1 : if x_{2} (t) \geq 1 then \\ Local Region Rule 1 : if x_{2} (t) is h_{11} (x_{2} (t)) \\ then {\begin{matrix} \dot{x} (t) = A_{11} x (t) + D_{11} x (t - τ) + B_{11} u (t) + ω (t) \\ y (t) = C_{11} x (t) \end{matrix} \\ Local Region Rule 2 : if x_{2} (t) is h_{12} (x_{2} (t)) \\ then {\begin{matrix} \dot{x} (t) = A_{12} x (t) + D_{12} x (t - τ) + B_{12} u (t) + ω (t) \\ y (t) = C_{12} x (t) \end{matrix} \end{matrix}$ $\begin{matrix} Region Rule 2 : if x_{2} (t) < 1 then \\ Local Region Rule 1 : if x_{2} (t) is h_{21} (x_{2} (t)) \\ then {\begin{matrix} \dot{x} (t) = A_{21} x (t) + D_{21} x (t - τ) + B_{21} u (t) + ω (t) \\ y (t) = C_{21} x (t) \end{matrix} \\ Local Region Rule 2 : if x_{2} (t) is h_{22} (x_{2} (t)) \\ then {\begin{matrix} \dot{x} (t) = A_{22} x (t) + D_{22} x (t - τ) + B_{22} u (t) + ω (t) \\ y (t) = C_{22} x (t) \end{matrix} \end{matrix}$

Where $\begin{matrix} A_{11} & = & [\begin{matrix} - 0.5 & 4 \\ - 0.943 & - 1.0493 \end{matrix}], B_{11} = [\begin{matrix} 0 \\ 0.4926 \end{matrix}], \\ A_{12} & = & [\begin{matrix} - 10.5 & 3 \\ - 10.132 & - 0.4529 \end{matrix}], B_{12} = [\begin{matrix} 0 \\ 0.1316 \end{matrix}], \\ A_{21} & = & [\begin{matrix} - 8 & 2 \\ - 10.2941 & - 1.4321 \end{matrix}], B_{21} = [\begin{matrix} 0 \\ 0.5765 \end{matrix}], \\ A_{22} & = & [\begin{matrix} - 19.2 & 2 \\ - 0.4706 & - 0.7535 \end{matrix}], B_{22} = [\begin{matrix} 0 \\ 0.1765 \end{matrix}], \\ A_{r} & = & [\begin{matrix} - 12.9 & 1.4 \\ - 9 & - 0.6 \end{matrix}], D_{11} = [\begin{matrix} 2 & - 1 \\ - 2 & 3 \end{matrix}], \\ D_{12} & = & [\begin{matrix} 2 & - 1 \\ 0 & - 3 \end{matrix}], D_{21} = D_{22} = [\begin{matrix} 1 & 1 \\ 2 & 1 \end{matrix}], \\ C_{11} & = & C_{12} = C_{21} = C_{22} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] . \end{matrix}$

The membership functions are as follows: $\begin{matrix} μ_{P_{11}^{1}} (x_{1} (t)) & = & μ_{P_{21}^{1}} (x_{1} (t)) = 1 - \frac{1}{1 + e^{- 2 x_{1} (t)}}, \\ μ_{N_{11}^{2}} (x_{1} (t)) & = & μ_{N_{21}^{2}} (x_{1} (t)) = \frac{1}{1 + e^{- 2 x_{1} (t)}} . \end{matrix}$

The matrices are obtained to solve (39) with l, h, j, θ, n = 1, 2, i = 1, 2, δ = 1, τ = 1, $\tilde{γ} = 1$ , β_ij = 1: ${\bar{X}}_{1} = [\begin{matrix} 0.1613 & - 0.0093 \\ - 0.0093 & 0.4480 \end{matrix}], {\bar{X}}_{2} = [\begin{matrix} 0.8827 & 0.6578 \\ 0.6578 & 5.4212 \end{matrix}],$

$\begin{matrix} γ & = & \frac{1}{2}, L_{11} = 0.7049, L_{12} = 0.7379, \\ L_{21} & = & 1.3780, L_{22} = 0.9152, \\ S_{11} & = & 10^{4} \cdot [\begin{matrix} 1.7341 & - 0.0081 \\ - 0.0081 & 1.6155 \end{matrix}], \\ S_{12} & = & 10^{4} \cdot [\begin{matrix} 1.7342 & - 0.0081 \\ - 0.0081 & 1.6156 \end{matrix}], \\ S_{13} & = & 10^{4} \cdot [\begin{matrix} 1.7601 & 0.2646 \\ 0.2646 & 5.9503 \end{matrix}], \\ S_{14} & = & 10^{4} \cdot [\begin{matrix} 1.7463 & 0.1149 \\ 0.1149 & 2.8850 \end{matrix}], \\ S_{15} & = & 10^{4} \cdot [\begin{matrix} 1.7343 & - 0.0082 \\ - 0.0082 & 1.6157 \end{matrix}], \\ S_{21} & = & 10^{4} \cdot [\begin{matrix} 1.7345 & - 0.0040 \\ - 0.0040 & 1.7068 \end{matrix}], \\ S_{22} & = & 10^{4} \cdot [\begin{matrix} 1.7346 & - 0.0040 \\ - 0.0040 & 1.7069 \end{matrix}], \\ S_{23} & = & 10^{4} \cdot [\begin{matrix} 1.7348 & - 0.0024 \\ - 0.0024 & 1.7190 \end{matrix}], \\ S_{24} & = & 10^{4} \cdot [\begin{matrix} 1.7348 & - 0.0024 \\ - 0.0024 & 1.7190 \end{matrix}], \\ S_{25} & = & 10^{4} \cdot [\begin{matrix} 1.7529 & 0.1245 \\ 0.1245 & 2.6131 \end{matrix}], \\ K_{11} & = & [\begin{matrix} - 0.0268 & 0.0077 \end{matrix}], \\ K_{21} & = & [\begin{matrix} - 3.5305 & - 25.2099 \end{matrix}], \\ K_{12} & = & [\begin{matrix} 0.2084 & - 0.0541 \end{matrix}], \\ K_{22} & = & [\begin{matrix} - 2.1393 & - 33.7188 \end{matrix}] . \end{matrix}$

Fig.1

Pulse wave signal of r (t) and ω (t).

Fig.2

(a) Switched fuzzy system state x₁ (t) tracking the reference state x_r1 (t) with Pulse wave. (b) Switched fuzzy system state x₂ (t) tracking the reference state x_r2 (t) with Pulse wave. (c) The control signal of Switched fuzzy system with Pulse wave. (d) The error responses of Switched fuzzy system with Pulse wave. (e) The mean square error of Switched fuzzy system with Pulse wave.

Fig.3

Gaussian noise signal of ω (t).

Fig.4

Switched fuzzy system state x₁ (t) tracking the reference state x_r1 (t) with Gaussian noise. (b) Switched fuzzy system state x₂ (t) tracking the reference state x_r2 (t) with Gaussian noise. (c) The control signal of Switched fuzzy system with Gaussian noise. (d) The error responses of Switched fuzzy system with Gaussian noise. (e) The mean square error of Switched fuzzy system with Gaussian noise.

Fig.5

(a) Switching fuzzy system state x₁ (t) tracking the reference state x_r1 (t). (b) Switching fuzzy system state x₂ (t) tracking the reference state x_r2 (t). (c) The control signal of Switching fuzzy system. (d) The error responses of Switching fuzzy system.

Taking the initial condition as x (0) = [5, - 10] ^T, with r (t) and ω (t) are generated by pulse wave form in Fig. 1. The simulation result of SF time-delay system is depicted in Fig. 2.

Furthermore, we consider ω (t) as a Gaussian noise, the simulation result of SF time-delay system is depicted in Fig. 3 for the initial condition x (0) = [5, - 10] ^T.

For the same data of Example 1 in [15] and the same initial condition x (0) = [5, - 10] ^T, the simulation result is depicted in Fig. 5. It should be noted that time-delay switched fuzzy system can improve the transient characteristic. Figures 2, 4 and 5 indicate that the proposed method gives better results in convergence.

5 Conclusion

In this paper, observer-based tracking controller design approaches have been considered for a class of time-delay switched fuzzy system as unmeasurable premise variable. The possibility of designing switching control law based on measured output instead of the state information is considered. Sufficient conditions for the solvability are then developed in order to ensure from one side the tracking between the time-delay switched fuzzy model and one reference model, and the stability convergence of the closed loop system from the other side. By using linear matrix inequalities, the controller design problem can be solved efficiently. The numerical example shows the feasibility and validity of the proposed design methods.

Footnotes

Acknowledgments

This work is supported by National Nature Science Foundation under Grant 61004039, Program for Liaoning Excellent Talents in University under Grant LR2015043, and Project of Natural Science Foundation of Liaoning Province under Grant 201602529.

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