Disturbance attenuation observer design for fuzzy time-delay system with fault 1

Abstract

The T-S fuzzy system with disturbance and fault is considered in the paper, and an observer with disturbance attenuation and fault detection is designed. With the linear matrix inequalities method, a sufficient condition on the existence of the observer with disturbance attenuation is provided, and a sufficient condition on the existence of the observer with fault detection is given, the two performance indices are then combined, and an optimization problem is formulated to solve the two problems simultaneously. A numerical example is given to show the effectiveness of the proposed method.

Keywords

T-S fuzzy system disturbance attenuation fault detection

1 Introduction

Almost all actual systems are nonlinear, and the nonlinearity makes the control and design problem more difficult to solve [1]. Therefore the control theory of nonlinear systems developed more slowly than the of linear systems. Since the T-S fuzzy model was proposed by Takagi and Sugeno [2], the theory of nonlinear systems have attracted more attention, and many control problems have been re-visited, including stability analysis [3, 4], model approximation design [5], filter design [6, 7], output feedback control [8], and H_∞ control.

The purpose of disturbance attenuation is to ensure the effect of the disturbance is reduced to an acceptable level. H_∞ and H₂ control are the specific versions of this problem. They are applied in almost all the control fields, for example satellite control in [9]. The abrupt variations of system parameters and structure are common phenomena in practise, such as network interconnection failure, parameter shifting, sudden environmental disturbances. It is necessary to detect the fault so as to overcome the effect. So disturbance attenuation and fault detection are hot research topics in control theory, see [10 –18] and the references therein.

In [10] a discrete-time T-S fuzzy model was studied with actuator fault and disturbance, and based on state transformation, a reduced-order fault estimation observer was designed with H_∞ performance index. The note in [11] dealt with a T-S fuzzy model with sensor faults and unknown bounded disturbance, a descriptor system approach was used, then an observer with best robustness to disturbance and sensitivity to faults was obtained. The work in [12] investigated T-S fuzzy systems with uncertainty and output disturbance, also based on a descriptor system approach, so an observer can estimate the system state and the output disturbance was designed. The paper [13] also solves the observer design for a T-S fuzzy system with output disturbance. Based on an augmented fuzzy descriptor model, a fuzzy disturbance observer was designed for detecting sensors.

A number of studies including [10], [11] and [13] have considered T-S fuzzy systems with disturbance and fault simultaneously, with solutions based on the descriptor model theory, or on the model transformation. However, all the work mentioned above did not consider the time delay, which is normal in actual control, also a factor that leads to system performance degradation. Without introducing descriptor model and model transformation, we investigate the T-S fuzzy time-delay system that is subject to disturbance and sensor fault. Based on Lyapunov stability theory, we derive the existing condition for the observer with disturbance attenuation, then the observer design with fault detection, an optimal algorithm will be proposed to balance the twoperformances.

For attenuating disturbance, the smaller the effect of disturbance to system output, the better. A lowest bound on the effect index is to be reached, and so is the highest detected index. But the two indices can not be reached simultaneously always, it is necessary for us to balance between the two indices when the disturbance and the fault appear in a system simultaneously. And a controller with disturbance attenuation and fault attenuation simultaneously is not the best one for disturbance attenuation and fault attenuation separately, so it is a tradeoff between the twobounds.

The rest of the paper is organized as follows. The problem of disturbance attenuation and fault detection for a T-S fuzzy system is formulated in Section 2. The main results of the paper are presented in Section 3. A numerical example used to verify the proposed theory is given in Section 4, and the last section concludes the paper.

2 Problem formulation

Consider a nonlinear multi-input multi-output system subject to external disturbances and sensor faults, the nonlinear dynamical behaviour can be represented by T-S fuzzy model that incorporates both fuzzy inference rules and local linearized dynamical models. And the i subsystem has the following form:

Rule i

IF θ₁ (t) is $F_{i}^{1},$ θ₂ (t) is $F_{i}^{2},$ …, θ_g (t) is $F_{i}^{g},$ THEN:

${\begin{matrix} \dot{x} (t) = & (A_{i} + Δ A_{i}) x (t) \\ + (A_{τ i} + Δ A_{τ i}) x (t - τ (t)) \\ + B_{i} u (t) + B_{di} d (t) + B_{fi} f (t), \\ y (t) = & C_{i} x (t) + C_{τ i} x (t - τ (t)) \\ + D_{di} d (t) + D_{fi} f (t), \\ x (t) = & α (t), t \in [- τ_{b}, 0], i = 1, 2, \dots, r, \end{matrix}$ (1)where $x (t) \in ℝ^{n}$ is the state, $y (t) \in ℝ^{m}$ is the output, $u (t) \in ℝ^{p}$ is the controlled input, $d (t) \in 𝕃^{n 1} [0, \infty)$ is the unknown disturbance, and it is assumed that the premise variables do not depend on the disturbance, $f (t) \in 𝕃^{n 2} [0, \infty)$ is the fault to be detected. $0 \leq τ_{a} \leq τ (t) \leq τ_{b} < \infty, \dot{τ} (t) \leq$ τ_m < 1, α (t) is the continuous function defined on the interval [- τ_b, 0], θ₁ (t) , θ₂ (t) , …, θ_g (t) are the premise variables, and g is the number of premise variable. $F_{i}^{j}, i = 1, 2, \dots, r$ ;j = 1, 2, …, g are the fuzzy sets, r is the number of IF-THEN rules. A_i, A_τi, B_i, B_di, B_fi, C_i, C_τi, D_diand D_fi are constant matrices of appropriate dimensions.

ΔA_i, ΔA_τi are uncertainties in system matrices A_i, A_τi, respectively, which are assumed to be of the following forms

$[\begin{matrix} Δ A_{i} & Δ A_{τ i} \end{matrix}] = DF (t) [\begin{matrix} E_{ai} & E_{τ i} \end{matrix}]$ (2)where D, E_ai, E_τi are constant matrices with corresponding dimensions, F (t) is an unknown, real and time-varying matrix with Lebesgue measurable elements satisfying $\begin{matrix} F^{T} (t) F (t) \leq I, \forall t \end{matrix}$

Let θ (t) = [θ₁ (t) , θ₂ (t) , …, θ_g (t)]. If a standard fuzzifier and weighted center-average defuzzifier are used, then the T-S fuzzy dynamic systems can be obtained as follows:

${\begin{matrix} \dot{x} (t) = & \sum_{i = 1}^{r} h_{i} (θ (t)) [(A_{i} + Δ A_{i}) x (t) \\ + (A_{τ i} + Δ A_{τ i}) x (t - τ (t)) \\ + B_{i} u (t) + B_{di} d (t) + B_{fi} f (t)], \\ y (t) = & \sum_{i = 1}^{r} h_{i} (θ (t)) [C_{i} x (t) + C_{τ i} x (t - τ (t)) \\ + D_{di} d (t) + D_{fi} f (t)], \\ x (t) = & α (t), t \in [- τ_{b}, 0], i = 1, 2, \dots, r, \end{matrix}$ (3)where $\begin{matrix} {\begin{matrix} h_{i} (θ (t)) & = & \frac{ω_{i} (θ (t))}{\sum_{i = 1}^{r} ω_{i} (θ (t))}, \\ ω_{i} (θ (t)) & = & \prod_{j = 1}^{g} F_{i}^{j} (θ_{j} (t)), \end{matrix} \end{matrix}$ in which $F_{i}^{j} (θ_{j} (t))$ is the grade of membership of θ_j (t) in the fuzzy set $F_{i}^{j}$ , in general, $\begin{matrix} {\begin{matrix} ω_{i} (θ (t)) \geq 0, \\ \sum_{i = 1}^{r} h_{i} (θ (t)) = 1, \\ 0 \leq h_{i} (θ (t)) \leq 1, i = 1, 2, \dots, r . \end{matrix} \end{matrix}$

For system (3), we will design a fuzzy observer with the same premise variables as system (1):

Observer Rule i

IF θ₁ (t) is $F_{i}^{1}$ , θ₂ (t) is $F_{i}^{2}$ , …, θ_g (t) is $F_{i}^{g}$ , THEN:

${\begin{matrix} \dot{\hat{x}} (t) = & (A_{i} + Δ A_{i}) \hat{x} (t) + (A_{τ i} \\ + Δ A_{τ i}) x (t - τ (t)) + B_{i} u (t) \\ + H_{i} (y (t) - \hat{y} (t)), \\ \hat{y} (t) = & C_{i} \hat{x} (t) + C_{τ i} \hat{x} (t - τ (t)), \\ \hat{x} (t) = & β (t), t \in [- τ_{b}, 0], i = 1, 2, \dots, r, \end{matrix}$ (4)where $\hat{x} (t) \in ℝ^{n}$ is the observer state, $\hat{y} (t) \in ℝ^{m}$ is the observer output, β (t) are the initial estimation states, H_i, i = 1, 2, …, r are the observer gains to be designed.

Then the fuzzy observer can be obtained as follows:

${\begin{matrix} \dot{\hat{x}} (t) = & \sum_{i = 1}^{r} h_{i} (θ (t)) [(A_{i} + Δ A_{i}) \hat{x} (t) \\ + (A_{τ i} + Δ A_{τ i}) \hat{x} (t - τ (t)) + B_{i} u (t) \\ + H_{i} (y (t) - \hat{y} (t))], \\ \hat{y} (t) = & \sum_{i = 1}^{r} h_{i} (θ (t)) [C_{i} \hat{x} (t) + C_{τ i} \hat{x} (t - τ (t))], \\ \hat{x} (t) = & β (t), t \in [- τ_{b}, 0], i = 1, 2, \dots, r . \end{matrix}$ (5)

For simplicity, the controlled T-S fuzzy system (3) will be translated into the following form: $\begin{matrix} {\begin{matrix} \dot{x} (t) = & (A_{h} + Δ A_{h}) x (t) + (A_{τ h} \\ + Δ A_{τ h}) x (t - τ (t)) + B_{h} u (t) \\ + B_{dh} d (t) + B_{fh} f (t), \\ y (t) = & C_{h} x (t) + C_{τ h} x (t - τ (t)) \\ + D_{dh} d (t) + D_{fh} f (t), \\ x (t) = & α (t), t \in [- τ_{b}, 0], i = 1, 2, \dots, r, \end{matrix} \end{matrix}$ (6)where

$\begin{matrix} A_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) A_{i}, & Δ A_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) Δ A_{i}, \\ A_{τ h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) A_{τ i}, & Δ A_{τ h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) Δ A_{τ i}, \\ B_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) B_{i}, & B_{dh} & = \sum_{i = 1}^{r} h_{i} (θ (t)) B_{di}, \\ B_{fh} & = \sum_{i = 1}^{r} h_{i} (θ (t)) B_{fi}, & C_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) C_{i}, \\ C_{τ h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) C_{τ i}, & D_{dh} & = \sum_{i = 1}^{r} h_{i} (θ (t)) D_{di}, \\ D_{fh} & = \sum_{i = 1}^{r} h_{i} (θ (t)) D_{fi} . \end{matrix}$ (7)

Similarly the designed fuzzy observer (5) will be translated into the following simple form:

${\begin{cases} \dot{\hat{x}} (t) = (A_{h} + Δ A_{h}) \hat{x} (t) + (A_{τ h} \\ + Δ A_{τ h}) \hat{x} (t - τ (t)) + B_{h} u (t) \\ + H_{h} (y (t) - \hat{y} (t)), \\ \hat{y} (t) = C_{h} \hat{x} (t) + C_{τ h} \hat{x} (t - τ (t)), i = 1, 2, \dots, r, \end{cases}$ (8) $where H_{h} = \sum_{i = 1}^{r} h_{i} (θ (t)) H_{i} .$

For meeting the design requirements and without loss of generality, the following assumption is introduced:

Assumption 1. The T-S fuzzy system (6) is stabilizing and [C_i, A_i] , i = 1, 2, …, r is observable.

Let the estimation error e (t) be $x (t) - \hat{x} (t)$ , and $r (t) = y (t) - \hat{y} (t)$ be the residual signal. The global T-S fuzzy dynamical model can be given by:

${\begin{matrix} \dot{\tilde{x}} (t) = & {\hat{A}}_{h} \tilde{x} (t) + {\hat{A}}_{τ h} \tilde{x} (t - τ (t)) \\ + {\hat{B}}_{dh} d (t) + {\hat{B}}_{fh} f (t), \\ r (t) = & {\hat{C}}_{h} \tilde{x} (t) + {\hat{C}}_{τ h} \tilde{x} (t - τ (t)) \\ + {\hat{D}}_{dh} d (t) + {\hat{D}}_{fh} f (t), \end{matrix}$ (9)where

$\begin{matrix} \tilde{x} (t) & = [\begin{matrix} x (t) \\ e (t) \end{matrix}], \\ {\hat{A}}_{h} & = [\begin{matrix} A_{h} + Δ A_{h} & 0 \\ 0 & A_{h} + Δ A_{h} - H_{h} C_{h} \end{matrix}], \\ {\hat{A}}_{τ h} & = [\begin{matrix} A_{τ h} + Δ A_{τ h} & 0 \\ 0 & A_{τ h} + Δ A_{τ h} - H_{h} C_{τ h} \end{matrix}], \\ {\hat{B}}_{dh} & = [\begin{matrix} B_{dh} \\ B_{dh} - H_{h} D_{dh} \end{matrix}], {\hat{B}}_{fh} = [\begin{matrix} B_{fh} \\ B_{fh} - H_{h} D_{fh} \end{matrix}], \\ {\hat{C}}_{h} & = [\begin{matrix} 0 & C_{h} \end{matrix}], {\hat{C}}_{τ h} = [\begin{matrix} 0 & C_{τ h} \end{matrix}], {\hat{D}}_{dh} = D_{dh}, \\ {\hat{D}}_{fh} & = D_{fh} . \end{matrix}$ (10)

With the above discussion, the problem can be described.

Given the T-S fuzzy time-delay uncertainty system (3) subject to disturbance and fault, design a fuzzy observer in the form of (5), such that the follow requirements are satisfied.

(R1) The global dynamic system (9)-(10) is asymptotically stable;

(R2) The designed fuzzy observer (8) has better attenuation to the external disturbance and higher sensitivity to the fault.

3 Main results

For developing the main results of the paper, let us recall the following lemma.

Lemma 1. [9] Assume that X and Y are vectors or matrices with appropriate dimensions. The following inequality $\begin{matrix} X^{T} Y + Y^{T} X \leq γ X^{T} X + γ^{- 1} Y^{T} Y \end{matrix}$ holds for any constant γ > 0.

The stability condition of the original controlled system under the external disturbance and sensor fault are all zeros is established, which is the base of the following theory of disturbance attenuation and fault detection.

Theorem 1. The T-S fuzzy system (9)-(10) (with both d (t) and f (t) are 0) is asymptotically stable if there exist symmetric positive-definite matrices P_11j, P_22j, Q_11j, Q_22j, j = 1, 2, … , r ; constants α₁, α₂, α₃, α₄ and matrices H_i, i = 1, 2, …, r such that the following inequalities hold: $\begin{matrix} [\begin{matrix} Φ_{11} & 0 & Φ_{13} & 0 & P_{11 j} D & 0 & P_{11 j} D & 0 \\ * & Φ_{22} & 0 & Φ_{24} & 0 & P_{22 j} D & 0 & P_{22 j} D \\ * & * & Φ_{33} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & Φ_{44} & 0 & 0 & 0 & 0 \\ * & * & * & * & - α_{1} & 0 & 0 & 0 \\ * & * & * & * & * & - α_{2} & 0 & 0 \\ * & * & * & * & * & * & - α_{3} & 0 \\ * & * & * & * & * & * & * & - α_{4} \end{matrix}] < 0 \\ (i = 1, 2, \dots, r; j = 1, 2, \dots, r) \end{matrix}$ (11)where $\begin{matrix} Φ_{11} = & P_{11 j} A_{i} + A_{i}^{T} P_{11 j} + Q_{11 j} + α_{1} E_{ai}^{T} E_{ai}, \\ Φ_{13} = & P_{11 j} A_{τ i}, \\ Φ_{22} = & P_{22 j} A_{i} + A_{i}^{T} P_{22 j} + Q_{22 j} + α_{2} E_{ai}^{T} E_{ai} \\ - P_{22 j} H_{i} C_{i} - C_{i}^{T} H_{i}^{T} P_{22 j}, \\ Φ_{24} = & P_{22 j} A_{τ i} - P_{22 j} H_{i} C_{τ i}, \\ Φ_{33} = & - (1 - τ_{m}) Q_{11 j} + α_{3} E_{τ i}^{T} E_{τ i}, \\ Φ_{44} = & - (1 - τ_{m}) Q_{22 j} + α_{4} E_{τ i}^{T} E_{τ i} . \end{matrix}$

Proof. The Lyapunov functional candidate is chosen as $\begin{matrix} V (x (t), τ) = & {\tilde{x}}^{T} (t) P_{h} \tilde{x} (t) \\ + \int_{t - τ (t)}^{t} {\tilde{x}}^{T} (s) Q_{h} \tilde{x} (s) ds, \end{matrix}$ (12)where $P_{h} = \sum_{i = 1}^{r} h_{i} (θ (t)) P_{i}$ , $Q_{h} = \sum_{i = 1}^{r} h_{i} (θ (t)) Q_{i}$ .

The time derivative of V with respect to t, along the trajectories of system (9), is given by

$\begin{matrix} \dot{V} (x (t), τ) = & 2 {\tilde{x}}^{T} (t) P_{h} \dot{\tilde{x}} (t) + {\tilde{x}}^{T} (t) Q_{h} \tilde{x} (t) \\ - (1 - \dot{τ} (t)) {\tilde{x}}^{T} (t - τ (t)) Q_{h} \tilde{x} (t - τ (t)) \\ \leq & {\tilde{x}}^{T} (t) (P_{h} {\hat{A}}_{h} + {\hat{A}}_{h}^{T} P_{h} + Q_{h}) \tilde{x} (t) \\ + 2 {\tilde{x}}^{T} (t) P_{h} {\hat{A}}_{τ h} \tilde{x} (t - τ (t)) - (1 - τ_{m}) \\ \times {\tilde{x}}^{T} (t - τ (t)) Q_{h} \tilde{x} (t - τ (t)) . \end{matrix}$ (13)

Considering the structure of parameters in (2), (7) and (10), and defining $\begin{matrix} P_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) [\begin{matrix} P_{11 i} & 0 \\ 0 & P_{22 i} \end{matrix}], \\ Q_{h} & = \sum_{i = 1}^{r} h_{i} (θ (t)) [\begin{matrix} Q_{11 i} & 0 \\ 0 & Q_{22 i} \end{matrix}] . \end{matrix}$

then (13) can be converted into the following form: $\begin{matrix} \dot{V} (x (t), τ) = \sum_{i = 1}^{r} \sum_{j = 1}^{r} h_{i} (θ (t)) h_{j} (θ (t)) {x^{T} (t) [P_{11 j} A_{i} \\ + A_{i}^{T} P_{11 j} + Q_{11 j} + P_{11 j} DF (t) E_{ai} + E_{ai}^{T} F^{T} (t) D^{T} \\ \times P_{11 j}] x (t) + e^{T} (t) [P_{22 j} A_{i} + A_{i}^{T} P_{22 j} - P_{22 j} H_{i} C_{i} \\ - C_{i}^{T} H_{i}^{T} P_{22 j} + P_{22 j} DF (t) E_{ai} + E_{ai}^{T} F^{T} (t) D^{T} P_{22 j} \\ + Q_{22 j}] e (t) + x^{T} (t) [P_{11 j} A_{τ i} + P_{11 j} DF (t) E_{τ i}] \\ \times x (t - τ (t)) + x^{T} (t - τ (t)) [E_{τ i}^{T} F^{T} (t) D^{T} P_{11 j} \\ + A_{τ i}^{T} P_{11 j}] x (t) + e^{T} (t) [P_{22 j} A_{τ i} + P_{22 j} DF (t) E_{τ i} \\ - P_{22 j} H_{i} C_{τ i}] e (t - τ (t)) + e^{T} (t - τ (t)) [A_{τ i}^{T} P_{22 j} \\ + E_{τ i}^{T} F^{T} (t) D^{T} P_{22 j} - C_{τ i}^{T} H_{i}^{T} P_{22 j}] e (t) \\ - x^{T} (t - τ (t)) (1 - τ_{m}) Q_{11 j} x (t - τ (t)) \\ - e^{T} (t - τ (t)) (1 - τ_{m}) Q_{22 j} e (t - τ (t))} . \end{matrix}$ (14) Using Lemma 1, one can obtain $\begin{matrix} P_{11 j} DF (t) E_{ai} + E_{ai}^{T} F^{T} (t) D^{T} P_{11 j} \\ \leq & α_{1} E_{ai}^{T} E_{ai} + α_{1}^{- 1} P_{11 j} {DD}^{T} P_{11 j}, \\ P_{22 j} DF (t) E_{ai} + E_{ai}^{T} F^{T} (t) D^{T} P_{22 j} \\ \leq & α_{2} E_{ai}^{T} E_{ai} + α_{2}^{- 1} P_{22 j} {DD}^{T} P_{22 j} . \end{matrix}$ (15)

$\begin{matrix} x^{T} (t) P_{11 j} DF (t) E_{τ i} x (t - τ (t)) \\ + x^{T} (t - τ (t)) E_{τ i}^{T} F^{T} (t) D^{T} P_{11 j} x (t) \\ \leq α_{3} x^{T} (t - τ (t)) E_{τ i}^{T} E_{τ i} x (t - τ (t)) \\ + α_{3}^{- 1} x^{T} (t) P_{11 j} {DD}^{T} P_{11 j} x (t), \\ e^{T} (t) P_{22 j} DF (t) E_{τ i} e (t - τ (t)) \\ + e^{T} (t - τ (t)) E_{τ i}^{T} F^{T} (t) D^{T} P_{22 j} e (t) \\ \leq α_{4} e^{T} (t - τ (t)) E_{τ i}^{T} E_{τ i} e (t - τ (t)) \\ + α_{4}^{- 1} e^{T} (t) P_{22 j} {DD}^{T} P_{22 j} e (t) . \end{matrix}$ (16) Substituting (15-16) into the equality (14), also noting that h_i (θ (t)) h_j (θ (t)) , i = 1, 2, …, r ; j = 1, 2, …, r are positive, and defining $ξ (t) = {[\begin{matrix} x^{T} (t) & e^{T} (t) & x^{T} (t - τ (t)) & e^{T} (t - τ (t)) \end{matrix}]}^{T},$ it yields: $\dot{V} (x (t), τ) = ξ^{T} (t) [\begin{matrix} Φ_{11} & 0 & Φ_{13} & 0 \\ * & Φ_{22} & 0 & Φ_{24} \\ * & * & Φ_{33} & 0 \\ * & * & * & Φ_{44} \end{matrix}] ξ (t) + 𝔼 𝔻 𝔼^{T},$ where $\begin{matrix} 𝔼 & = [\begin{matrix} P_{11 j} D & 0 & P_{11 j} D & 0 \\ 0 & P_{22 j} D & 0 & P_{22 j} D \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], \\ 𝔻 & = [\begin{matrix} α_{1}^{- 1} & 0 & 0 & 0 \\ * & α_{2}^{- 1} & 0 & 0 \\ * & * & α_{3}^{- 1} & 0 \\ * & * & * & α_{4}^{- 1} \end{matrix}] . \end{matrix}$

In view of Schur Complement Lemma, it is easy to obtain the result (11), thus it completes the proof.□

Remark 1. Theorem 1 solved robust stability problem of T-S fuzzy system with uncertainty by using a fuzzy observer, and the result in which is the foundation of the further problem solution.

3.1 Disturbance attenuation

To measure the effect of the disturbance on the system output, firstly we set f (t) =0. In this case, the system is:

${\begin{matrix} \dot{\tilde{x}} (t) & = & {\hat{A}}_{h} \tilde{x} (t) + {\hat{A}}_{τ h} \tilde{x} (t - τ (t)) + {\hat{B}}_{dh} d (t), \\ r (t) & = & {\hat{C}}_{h} \tilde{x} (t) + {\hat{C}}_{τ h} \tilde{x} (t - τ (t)) + {\hat{D}}_{dh} d (t) . \end{matrix}$ (17)then we use a performance index of the following form $η_{1} = \frac{∥ r (t) ∥}{∥ d (t) ∥}$ (18)where $∥ x (t) ∥ = \int_{0}^{𝕋} (x^{T} (t) x (t)) dt$ , to system (9), where $𝕋$ is the period to evaluate the disturbance attenuation performance. It follows that when attenuating the disturbance, the smaller the index, the better the performance.

The main objective of the subsection is to find an observer in the form of (8), such that the overall T-S fuzzy system (9) is stable and under zero-initial condition, ∥r (t)∥ < η₁ ∥ d (t) ∥ is satisfied for all non-zero d(t).

Theorem 2. The T-S fuzzy system (9) (f (t) =0) is asymptotically stable, and also has a prescribed performance level η₁ > 0 if there exist symmetric positive-definite matrices P_11j, P_22j, Q_11j, Q_22j, j = 1, 2, … , r ; constants α₁, α₂, α₃, α₄ and matrices H_i, i = 1, 2, …, r such that the following inequalities holds: $Ψ = [\begin{matrix} ψ_{11} & ψ_{12} \\ * & ψ_{22} \end{matrix}] < 0,$ (19) $(i = 1, 2, \dots, r; j = 1, 2, \dots, r)$

with $\begin{matrix} ψ_{11} = [\begin{matrix} Φ_{11} & 0 & Φ_{13} & 0 & P_{11 j} B_{di} \\ * & {\tilde{ψ}}_{22} & 0 & {\tilde{ψ}}_{24} & {\tilde{ψ}}_{25} \\ * & * & Φ_{33} & 0 & 0 \\ * & * & * & {\tilde{ψ}}_{44} & C_{τ i} D_{di} \\ * & * & * & * & - η_{1} + D_{di}^{T} D_{di} \end{matrix}], \\ ψ_{12} = [\begin{matrix} P_{11 j} D & 0 & P_{11 j} D & 0 \\ 0 & P_{22 j} D & 0 & P_{22 j} D \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], \\ ψ_{22} = [\begin{matrix} - α_{1} & 0 & 0 & 0 \\ * & - α_{2} & 0 & 0 \\ * & * & - α_{3} & 0 \\ * & * & * & - α_{4} \end{matrix}], \end{matrix}$ (20)where $\begin{matrix} {\tilde{ψ}}_{22} = Φ_{22} + C_{i}^{T} C_{i}, \\ {\tilde{ψ}}_{24} = Φ_{24} + C_{i}^{T} C_{τ i}, \\ {\tilde{ψ}}_{25} = P_{22 j} B_{di} - P_{22 j} H_{i} D_{di} + C_{i}^{T} D_{di}, \\ {\tilde{ψ}}_{44} = Φ_{44} + C_{τ i}^{T} C_{τ i} . \end{matrix}$

Proof. By using the same Lyapunov functional candidate in Theorem 1, and introducing the following cost function

$J_{1} = \int_{0}^{𝕋} r^{T} (t) r (t) dt - η_{1} \int_{0}^{𝕋} d^{T} (t) d (t) dt .$ (21)

Under zero-initial condition, we introduce another index: $\begin{matrix} \tilde{J_{1}} & = \int_{0}^{𝕋} r^{T} (t) r (t) dt - η_{1} \int_{0}^{𝕋} d^{T} (t) d (t) dt + V (x (t), τ) \\ = \int_{0}^{𝕋} (r^{T} (t) r (t) - η_{1} d^{T} (t) d (t) + \dot{V} (x (t), τ)) dt . \end{matrix}$ (22) Similar to the derivation in Theorem 1, and by defining $ξ_{1} (t) = {[\begin{matrix} x^{T} (t) & e^{T} (t) & x^{T} (t - τ (t)) & e^{T} (t - τ (t)) & d^{T} (t) \end{matrix}]}^{T},$ the index $\tilde{J_{1}}$ can be changed into $\begin{matrix} \tilde{J_{1}} = ξ_{1}^{T} (t) Ψ ξ_{1} (t) . \end{matrix}$

Considering the inequality (19), then $\tilde{J_{1}} < 0$ . Noting that Lyapunov functional is positive, so we can conclude that J₁ < 0. Thus the asymptotically stability of the system (17) and the performance index J₁ < 0 can be guaranteed. This completes the proof.□

3.2 Fault detection

For system (9) when d (t) =0, it becomes into

${\begin{matrix} \dot{\tilde{x}} (t) & = & {\hat{A}}_{h} \tilde{x} (t) + {\hat{A}}_{τ h} \tilde{x} (t - τ (t)) + {\hat{B}}_{fh} f (t), \\ r (t) & = & {\hat{C}}_{h} \tilde{x} (t) + {\hat{C}}_{τ h} \tilde{x} (t - τ (t)) + {\hat{D}}_{fh} f (t) . \end{matrix}$ (23)

To detect the fault, a performance index of the following form $η_{2} = \frac{∥ r (t) ∥}{∥ f (t) ∥}$ (24)is introduced, then in order to make the detection more sensitive, the bigger the index, the better the performance.

The main objective of the subsection is to find an observer in the form of (8), such that the overall T-S fuzzy system (9)-(10) is stable and under zero-initial condition, ∥r (t)∥ > η₂ ∥ f (t) ∥ is satisfied for all non-zero f(t).

Theorem 3. The T-S fuzzy system (9) (d (t) =0) is asymptotically stable, and also has a prescribed performance level η₂ > 0 if there exist symmetric positive-definite matrices P_11j, P_22j, Q_11j, Q_22j, j = 1, 2, …, r; constants α₁, α₂, α₃, α₄ and matrices H_i, i = 1, 2, …, r such that the following inequalities hold:

$Ω = [\begin{matrix} ω_{11} & ω_{12} \\ * & ω_{22} \end{matrix}] < 0,$ (25) $(i = 1, 2, \dots, r; j = 1, 2, \dots, r)$

with $\begin{matrix} ω_{11} & = & [\begin{matrix} Φ_{11} & 0 & Φ_{13} & 0 & P_{11 j} B_{fi} \\ * & Φ_{22} - C_{i}^{T} C_{i} & 0 & {\tilde{ω}}_{24} & {\tilde{ω}}_{25} \\ * & * & Φ_{33} & 0 & 0 \\ * & * & * & {\tilde{ω}}_{44} & - C_{τ i}^{T} D_{fi} \\ * & * & * & * & η_{2} - D_{fi}^{T} D_{fi} \end{matrix}], \\ ω_{12} & = & ψ_{12}, ω_{22} = ψ_{22}, \end{matrix}$ where $\begin{matrix} {\tilde{ω}}_{24} & = Φ_{24} - C_{i}^{T} C_{τ i}, \\ {\tilde{ω}}_{25} & = P_{22 j} B_{fi} - P_{22 j} H_{i} D_{fi} - C_{i}^{T} D_{fi}, \\ {\tilde{ω}}_{44} & = Φ_{44} - C_{τ i}^{T} C_{τ i} . \end{matrix}$ (26)

Proof. By using the same Lyapunov functional candidate as in Theorem 1, and introducing the following cost function $\begin{matrix} J_{2} = \int_{0}^{𝕋} r^{T} (t) r (t) dt - η_{2} \int_{0}^{𝕋} f^{T} (t) f (t) dt . \end{matrix}$

Under zero-initial condition, we introduce another performance index: $\begin{matrix} \tilde{J_{2}} & = \int_{0}^{𝕋} r^{T} (t) r (t) dt - η_{2} \int_{0}^{𝕋} f^{T} (t) f (t) dt - V (x (t), τ) \\ = \int_{0}^{𝕋} (r^{T} (t) r (t) - η_{2} f^{T} (t) f (t) - \dot{V} (x (t), τ)) dt . \end{matrix}$ (27)Similar to the derivation in Theorem 1, and by defining $ξ_{2} (t) = {[\begin{matrix} x^{T} (t) & e^{T} (t) & x^{T} (t - τ (t)) & e^{T} (t - τ (t)) & f^{T} (t) \end{matrix}]}^{T},$ the index $\tilde{J_{2}}$ can be changed into $\begin{matrix} \tilde{J_{2}} = - ξ_{2}^{T} (t) Ω ξ_{2} (t) . \end{matrix}$ Considering inequality (25), then $\tilde{J_{2}} > 0$ , also Lyapunov functional is positive, so J₂ is positive. Thus the asymptotically stability of the system (17) and the performance index (24) can be guaranteed. This completes the proof.□

Remark 2. In the above two subsections, the existence conditions for disturbance attenuation observer and fault detection observer are described by linear matrix inequalities (LMIs) (refIneqTheorem2) {and (refIneqTheorem3), which can be easily solved by using Matlab LMI Toolbox.

3.3 The trade-off performance

In the last two subsections, we designed the disturbance attenuation observer (f (t) =0) and the fault detection observer (d (t) =0) for T-S fuzzy system (9). In this subsection, we will make a tradeoff between the two performances. That is, we will find a fuzzy observer (8) such that the overall T-S fuzzy dynamical system (9) is asymptotically stable, satisfying the index (18) with η₁ > 0 and the index (24) with η₂ > 0 under zero initial conditions. So the index of the tradeoff performance observer can be described as: $\begin{matrix} J & = & min_{H_{i}, i = 1, 2, \dots, r} \frac{η_{1}}{η_{2}} \\ s . t . LMIs (19) and (25) \end{matrix}$

The algorithm for designing the observer is as following:

Step1. Solve the LMIs (19) and (25) and get the initial parameters of η₁ and η₂, that is, η_1min and η_2max.

Step2. If η_1min and η_2max are feasible for LMIs (19) and (25) simultaneously, the optimal H_i, i = 1, 2, …, r can be obtained. Otherwise, the tradeoff between the disturbance attenuation and the fault detection should be made. Let η_1n = η_1(n-1) + Δη, η_2n = η_2(n-1) - Δη, where Δη is a sufficient small positive scalar and n is the iteration step, and test the feasibility of LMIs (19) and (25), if it is feasible before n_max = floor (η_2max/Δη), then the optimal solutions of J and H_i, i = 1, 2, …, r can be obtained, otherwise there is no solution. The flowchart of the algorithm is shown in Fig. 1.

Remark 3. In Step1, the disturbance attenuation observer and fault detection observer were designed separately, so η₁ and η₂ represent the best indices for the two requirements separately. And after Step2, that is, after balance between the two performances, an optimal solution was obtained satisfying the two requirements simultaneously.

Remark 4. The two goals can be integrated is the basic property of linear system, so it is merit of T-S fuzzy system, since the consequent part is linear systems.

4 Numeral example

In this part, a T-S fuzzy system with uncertainties is considered with the following parameters: $\begin{matrix} A_{1} = [\begin{matrix} - 5 & - 3 \\ - 1 & - 10 \end{matrix}], A_{τ 1} = [\begin{matrix} 0.05 & - 0.1 \\ 0 & - 0.05 \end{matrix}], \\ A_{2} = [\begin{matrix} - 5 & - 4 \\ - 1 & - 16 \end{matrix}], A_{τ 2} = [\begin{matrix} - 0.05 & - 0.1 \\ 0 & - 0.05 \end{matrix}], \\ B_{1} = B_{2} = [\begin{matrix} 1.0 \\ 0.5 \end{matrix}], B_{d 1} = B_{d 2} = [\begin{matrix} 0.1 \\ 0 \end{matrix}], \\ B_{f 1} = B_{f 2} = [\begin{matrix} 0 \\ 0.1 \end{matrix}], C_{1} = C_{2} = [\begin{matrix} 0.6 & 0 \end{matrix}], \\ C_{τ 1} = C_{τ 2} = [\begin{matrix} 0 & 0.2 \end{matrix}], D_{d 1} = D_{d 2} = 0.1, \\ D_{f 1} = D_{f 2} = 6, E_{a 1} = E_{a 2} = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}], \\ E_{τ 1} = E_{τ 2} = [\begin{matrix} 0.3 & 0 \\ 0 & 0.3 \end{matrix}], F (t) = [\begin{matrix} \sin (t) & 0 \\ 0 & \cos (t) \end{matrix}], \\ D = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] . \end{matrix}$ the fuzzy membership functions are chosen as $h_{1} (x_{1}) = 1 - 0.25 x_{1}^{2} and h_{2} (x_{1}) = 1 - h_{1} (x_{1})$ .

Based on Theorem 2 and Theorem 3, we can obtain the lower level of η₁ and the upper level of η₂ as η₁ = 11.2233 and η₂ = 16.2922 respectively, also these are the optimal parameters, so we can get the fuzzy observer gain matrices as: $\begin{matrix} H_{1} = [\begin{matrix} - 0.2739 \\ 0.3712 \end{matrix}], H_{2} = [\begin{matrix} - 0.2784 \\ 0.2533 \end{matrix}] . \end{matrix}$

The initial state of the system is $x_{0} = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T}$ and the observer error initial value is $e_{0} = {[\begin{matrix} 1 & 3 \end{matrix}]}^{T}$ , the unknown disturbance d (t) is assumed as random number between 0 and 0.1, the fault signal f (t) is simulated as square wave of 0.02 amplitude. The unknown disturbance signal is shown in Fig. 2, the residual evaluation response is shown in Fig. 3, the observer error signal is shown in Fig. 4. It is shown that the observer errors tend to zero during 0.7s, that is to say, under the random disturbance signal and square wave fault signal, the states of the system can be observed by the observer exactly. So the theory proposed in the paper is effective.

5 Conclusion

In this paper, the disturbance attenuation and fault detection observer design for a class of T-S fuzzy uncertainty system was considered. Firstly, the sufficient existing condition for the disturbance attenuation observer was established. Then the sufficient existing condition for the fault detection observer was given by LMI. Thirdly, for realizing the two design objectives simultaneously, an optimal algorithm was proposed between the trade-off of the two performances. In the last, a numerical example has been given to demonstrate the potential of the proposed techniques.

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