A synthesis of observer-based controller for stabilizing uncertain T-S fuzzy systems

Abstract

This study considers the robust stabilization problem of uncertain Takagi-Sugeno (T-S) fuzzy systems in which some or all state variables are unavailable. The considered uncertainties of the system are bounded and satisfy a certain condition. A sufficient condition for the state estimation and robust stabilization of the closed system is proposed. Based on the condition, the observer is designed and observer-based controller is synthesized. In the observer and controller synthesis process, with the aid of some special derivations, Bilinear Matrix Inequalities (BMIs) are converted into Linear Matrix Inequalities (LMIs) in one step such that LMI tools can be used to solve the problem. Finally, both numerical and practical examples are given to verify the effectiveness of the proposed approach.

Keywords

Nonlinear system fuzzy control system observer-based controller uncertainties LMIs

1 Introduction

Solving the problems of nonlinear systems is still a challenge and it receives a lot of attraction from researchers. For example, Markovian jumping linear systems which represent a nonlinear system in term of linear subsystems have been studied in [1, 2]. Besides, a combination of neural network and nonlinear model predictive control was studied to control the multirate networked industrial process [3]. In addition, the Takagi-Sugeno (T-S) fuzzy dynamic model was introduced by Takagi and Sugeno in 1985 and proposes a set of local linear input/output relations to represent a nonlinear system. In other words, this model uses a number of fuzzy rules to describe a global nonlinear system in terms of a set of local linear models which are smoothly connected by fuzzy membership functions [4].The T-S fuzzy model has become a distinguished landmark in the history of fuzzy control theory evolution. Numerous fuzzy control problems, such as stability analysis, robust stability and optimality, can be addressed within the framework of the T-S model [5]. In recent decades, many studies have put lots of effort in the control design of T-S fuzzy systems [6 –18]. The studies on fuzzy control systems can be classified into three approaches, namely, model-free, mathematical model, and fuzzy model approaches. The model-free approach, as its name suggests, does not require a certain model for the plant. Engineers design fuzzy controllers through some linguistic rules for the complex plants by incorporating human experience or expert knowledge [20 –22]. The whole process is not only intuitive but also easily understandable. For the mathematical model approach, the authors of [23, 24] investigated closed-loop system stabilization with fuzzy proportional-derivative (PD), proportional-integral (PI), and proportional-integral-derivative (PID) controllers based on the small gain theorem. In terms of the fuzzy model approach, one can use the concepts of sector nonlinearity and local approximation [5] to transform a nonlinear dynamic system into a set of linear system models. Moreover, the parallel distributed compensation (PDC) [25, 26] is a specific fuzzy control form for controlling a given T-S fuzzy model [19]. Based on the Lyapunov theory, stability and performance specifications for T-S fuzzy models can be derived into a Linear Matrix Inequality (LMI) form which can be solved efficiently by means of some powerful tools.

In many practical control problems, some state variables may not be measurable directly but output information is available. Hence, there are several methods reported in [6 , 28] studying the output feedback control or observer-based controller design for fuzzy systems. For instance, the suboptimal H₂ control performance with a feasible H_∞ disturbance rejection constraint was achieved by using the fuzzy observer-based mixed H₂/H_∞ controller [6]. In addition, a new fuzzy modeling based on a fuzzy linear fractional transformations model was investigated to provide a flexible tool for handling intricate nonlinear systems and deal efficiently with the output feedback parallel distributed compensation problem [27]. A method based on output feedback to deal with the output tracking control problem for nonlinear systems was proposed in [28]. Moreover, a fuzzy H_∞ model reference tracking control scheme was developed without feedback linearization technique and complicated adaptive method [29]. Regarding the nonlinear interconnected systems, an observer-based state feedback decentralized fuzzy controller was proposed in [30] to resolve the H_∞ tracking control problem. A method was developed in [31] to design an observer-based controller for a T-S fuzzy system in which the membership functions of the T-S fuzzy model plant, observer and observer-based controller do not need to be the same. Additionally, in [31], the stability conditions in terms of Bilinear Matrix Inequalities (BMIs) were resolved by using the global searching algorithm and convex programming techniques.

Since uncertainties often deteriorate a system’s stability and its performance. Owing to this reason, the issue of the robust fuzzy controller design for uncertain fuzzy systems is regarded as an essential topic. For instance, a robust single-grid-point (SGP) fuzzy controller was synthesized to deal with the presence of uncertainties in T-S fuzzy systems [19]. In [43], a state-feedback fuzzy controller for the uncertain T–S fuzzy time-delay system was designed to make the closed-loop system exponentially stable with a prescribed decay rate. Additionally, an approach based on pole placement and H_∞ technique to synthesize a dynamic parallel distributed compensation (DPDC) for the T-S fuzzy system with uncertain parts was presented in [44]. However, if some states of the plant are un-measurable, designing a robust observer-based controller is necessary. There have been many papers such as [6 , 38–40], which study this problem. The observer-based controller was designed to eliminate the effects of external disturbance and the uncertainties due to the difference between the nonlinear system and the transformed T-S fuzzy system [32]. Moreover, a fuzzy robust output tracking control for uncertain T-S fuzzy systems was presented in [33] where the robust observer-based controller was designed for the existence of unknown state variables. In [34], the observer-based fuzzy controller was constructed to solve the H_∞ problem for T-S fuzzy systems in the presence ofuncertainties.

In general, we will encounter BMI problems in fuzzy controller design work. However, BMI problems are very difficult to solve since there exist cross coupled unknown parameters in the inequalities. The method used in [6 , 39] to convert BMIs into LMIs is called a “two-stage procedure”. The main results proposed by those papers contain a big matrix which must be negative definite, but it is a BMI problem unfortunately. In the first step of the two-stage procedure, we choose one element of the big matrix, which is a BMI, and convert it into an LMI, and then solve it using LMI tools. Secondly, the solution from the first step is substituted back into the big matrix so that the big matrix becomes an LMI problem. Then we can easily solve it using LMI tools. In the paper [40], the sufficient conditions for synthesizing the observer-based controller were expressed in term of a lot of LMIs which cause a heavy computation load. Furthermore, the sufficient conditions expressed in the BMIs were solved by applying the Particle Swarm Optimization (PSO) to find the feasible solution [31]. However, the method is too time-consuming and has a heavy computation load. In order to overcome the above drawbacks, this study proposes a novel skill to transform BMIs into LMIs in the main theorem, such that the objective can be achieved. The main contribution of this paper is three-fold. (i) The robust stabilization design problem of uncertain nonlinear systems is solved via a fuzzy observer-based controller; (ii) the stability of the fuzzy observer-based control system and performance of the fuzzy observer are guaranteed; and (iii) a new method which transforms BMIs into LMIs is proposed.

The paper is organized as follows. In Section 2,the considered system formulation and problem description are presented. The main theorems and the systematic design procedure for the fuzzy observer and observer-based controller are provided inSection 3. In Section 4, two examples are offered to demonstrate the effectiveness of the synthesized fuzzy observer and observer-based controller. Finally, the conclusion is given in Section 5.

Notations: In this paper, A > 0 denotes the positive definite matrix A. A^T denotes the transpose of matrix A; A^-1 denotes the inverse of A; I denotes an identity matrix. The symbol $R^{n \times m}$ denotes the set of n × m matrices and the symbol (*) denotes the transpose elements in the symmetric positions.

2 System and problem description

Let us consider an uncertain nonlinear system as (1). ${\begin{matrix} \dot{x} (t) = (A (x) + Δ A (x)) x (t) + (B (x) + Δ B (x)) u (t) \\ y (t) = Cx (t) \end{matrix}$ (1) where A (x) and B (x) are nonlinear matrices; ΔA (x) and ΔB (x) are uncertainties; $y (t) \in R^{p}$ is the output vector and $C \in R^{p \times n}$ is the constant output matrix. Furthermore, $x (t) \in R^{n}$ is an unavailable state vector, and $u (t) \in R^{m}$ is the control input vector. It is seen from (1) that the output equation is linear, but the dynamic state equation is nonlinear. Using the sector nonlinearity method, the system (1) can be transformed into a T-S fuzzy system as (2).

Plant Rule i:

$\begin{matrix} IF θ_{1} (t) is Q_{i 1}, \dots, and θ_{s} (t) is Q_{is}, THEN \\ \dot{x} (t) = (A_{i} + Δ A (x)) x (t) + (B_{i} + Δ B (x)) u (t), \\ i = 1, 2, \dots, r . \end{matrix}$ (2) where θ₁ (t) , …, θ_s (t) are the premise variables which are the functions of system states, Q_ij (i = 1, 2, …, r ; j = 1, 2, …, s) denotes the fuzzy set of θ_j (t), r is the number of rules, and s is the number of premise variables. The matrices $A_{i} \in R^{n \times n}$ and $B_{i} \in R^{n \times m}$ represent the system matrix and control matrix, respectively. Here, we consider that ΔA (x) and ΔB (x) are nonlinear uncertainties and are not handled by local partitions. The overall uncertain fuzzy system inferred from plant rules (2) is as follows

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{r} β_{i} (θ) [(A_{i} + Δ A (x)) x (t) \\ + (B_{i} + Δ B (x)) u (t)], \end{matrix}$ (3) where i = 1, 2, … r, $β_{i} (θ) = \frac{w_{i} (θ)}{\sum_{i = 1}^{r} w_{i} (θ)}, w_{i} (θ) = \prod_{j = 1}^{s} Q_{ij} (θ),$ and θ (t) = [θ₁ (t) , …, θ_s (t)] is a vector. Here, Q_ij (θ (t)) is the membership function of θ_j (t). In addition, β_i (θ) is regarded as the normalized weight of each IF-THEN rule satisfying β_i (θ) ≥0 and $\sum_{i = 1}^{r} β_{i} (θ) = 1$ . Then, the entire form of the T-S fuzzy system with output y (t) is ${\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{r} β_{i} (θ) [(A_{i} + Δ A (x)) x (t) + (B_{i} + Δ B (x)) u (t)] \\ y (t) = Cx (t), \end{matrix}$ (4)

Assumption 1: Assume that the uncertainties ΔA (x) and ΔB (x) can be written in the following forms, $\begin{matrix} Δ A (x) = D_{a} Δ_{a} (x) E_{a} \end{matrix}$ (5a) $\begin{matrix} Δ B (x) = D_{b} Δ_{b} (x) E_{b} \end{matrix}$ (5b) where $D_{a} \in R^{n \times h}, E_{a} \in R^{s \times n}, D_{b} \in R^{n \times l}$ and $E_{b} \in R^{q \times m}$ are known real matrices with appropriate dimensions, $Δ_{a} (x) \in R^{h \times s}$ and $Δ_{b} (x) \in R^{l \times q}$ are unknown nonlinear matrices with bounds $Δ_{a} (x) Δ_{a}^{T} (x) \leq γ_{a} I$ and $Δ_{b} (x) Δ_{b}^{T} (x) \leq γ_{b} I$ ; γ_a, γ_b are two positive scalars and I is identify matrix with appropriate dimensions.

It is noted that although Assumption 1 is a constraint, the existence of D_a, E_a, D_b and E_b may cause the upper bounds of the uncertainties to be small. Since the states of (3) may be unavailable, under the concept of parallel distributed compensation (PDC), the rules of the fuzzy observer-based controllers in the following are employed to stabilize the T-S fuzzy model (3).

Observer-based Controller Rule i:

$\begin{matrix} IF θ_{1} (t) is Q_{i 1}, \dots, and θ_{s} (t) is Q_{ij}, THEN \\ u (t) = G_{j} \hat{x} (t) \end{matrix}$ (6) where $G_{j} \in R^{m \times n}$ is the controller gain. Hence, the fuzzy observer-based controller is as follows $u (t) = \sum_{j = 1}^{r} β_{j} (θ) G_{j} \hat{x} (t) .$ (7)

Here, $\hat{x} (t)$ is the estimated state vector which should be obtained from a fuzzy observer. Hence, the purpose of this study is to design a fuzzy observer to estimate the system states of (4) and synthesize a fuzzy observer-based controller for stabilizing the whole closed-loop system.

3 Fuzzy observer-based controller synthesis

Let the fuzzy observer for the fuzzy system (4) be as follows ${\begin{matrix} \dot{\hat{x}} (t) = \sum_{i = 1}^{r} β_{i} (θ) [A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} (y (t) - \hat{y} (t))] \\ \hat{y} (t) = C \hat{x} (t) \end{matrix}$ (8) where $\hat{x} (t) \in R^{n}$ is the estimated state vector of x (t), and $L_{i} \in R^{n \times p}$ is the observer gain matrix. Define the error between the real system and the estimated state variables as $e (t) = x (t) - \hat{x} (t)$ . Substituting (7) into (3) yields the closed-loop system (9).

$\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{r} β_{i} (θ) {(A_{i} + Δ A (x)) x (t) + \\ (B_{i} + Δ B (x)) \sum_{j = 1}^{r} β_{j} (θ) G_{j} \hat{x} (t)} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {A_{i} x (t) + Δ A (x) x (t) \\ + B_{i} G_{j} \hat{x} (t) + Δ B (x) G_{j} \hat{x} (t) - B_{i} G_{j} x (t) \\ + B_{i} G_{j} x (t) + Δ B (x) G_{j} x (t) - Δ B (x) G_{j} x (t)} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {(A_{i} + B_{i} G_{j}) x (t) \\ + (Δ A (x) + Δ B (x) G_{j}) x (t) - B_{i} G_{j} e (t) \\ - Δ B (x) G_{j} e (t)} \end{matrix}$ (9) $\begin{matrix} \dot{e} (t) = \dot{x} (t) - \dot{\hat{x}} (t) \\ = \sum_{i = 1}^{r} β_{i} (θ) {(A_{i} + Δ A (x)) x (t) \\ + (B_{i} + Δ B (x)) u (t) - A_{i} \hat{x} (t) - B_{i} u (t) \\ - L_{i} (y (t) - \hat{y} (t))} \\ = \sum_{i = 1}^{r} β_{i} (θ) {(A_{i} + Δ A (x)) x (t) + (B_{i} + Δ B (x)) \\ \sum_{j = 1}^{r} β_{j} (θ) G_{j} \hat{x} (t) - A_{i} \hat{x} (t) - B_{i} \\ \sum_{j = 1}^{r} β_{j} (θ) G_{j} \hat{x} (t) - L_{i} (y (t) - \hat{y} (t))} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {A_{i} e (t) \\ + Δ A (x) x (t) + Δ B (x) G_{j} \hat{x} (t) - L_{i} Ce (t)} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {(A_{i} - L_{i} C) e (t) \\ + Δ A (x) x (t) + Δ B (x) G_{j} \hat{x} (t) - Δ B (x) G_{j} x (t) \\ + Δ B (x) G_{j} x (t)} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {(A_{i} - L_{i} C) e (t) + (Δ A (x) \\ + Δ B (x) G_{j}) x (t) - Δ B (x) G_{j} e (t)} \end{matrix}$ (10)

From Equations (3), (7) and (8), the error dynamics can be written in (10). Then, the rest of the problem is to find the proper matrices L_i and G_j such that the error dynamics (10) and the closed-loop system (9) are robustly stabilized. In order to achieve the above objective, the following Lemma 1 isneeded.

Lemma 1.For any two real matricesXandYwith appropriate dimensions, there exists a positive constantɛ > 0 such that the following inequality is satisfied. $X^{T} Y + Y^{T} X \leq ɛ X^{T} X + ɛ^{- 1} Y^{T} Y .$ (11)

Based on Assumption 1 and Lemma 1, we have Theorem 1 below.

Theorem 1.The uncertain fuzzy closed-loop system (9) is stable and the estimation errore (t) converges to zero asymptotically, if there exist matricesL_i, G_jand symmetric positive definite matricesPandRsuch that $[\begin{matrix} H_{ij}^{(11)} & 0 \\ 0 H_{ij}^{(22)} \end{matrix}] < 0$ (12) where $\begin{matrix} H_{ij}^{(11)} = A_{i} P + {PA}_{i}^{T} + G_{j}^{T} B_{i}^{T} P + {PB}_{i} G_{j} \\ + 2 E_{a}^{T} E_{a} + γ_{a} {PD}_{a} D_{a}^{T} P \\ + 2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} + 2 γ_{b} {PD}_{b} D_{b}^{T} P + {PB}_{i} B_{i}^{T} P, \\ H_{ij}^{(22)} = 2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} + A_{i}^{T} R + {RA}_{i} \\ - C^{T} L_{i}^{T} R - {RL}_{i} C \\ + γ_{a} {RD}_{a} D_{a}^{T} R + 2 γ_{b} {RD}_{b} D_{b}^{T} R + G_{j}^{T} G_{j}, \\ and i, j = 1, 2, \dots, r . \end{matrix}$ Proof: Let us define a Lyapunov function for the system (9) and estimation error e (t) as $V (t) = x^{T} (t) Px (t) + e^{T} (t) Re (t) .$ (13)

The time derivative of V (t) is $\dot{V} (t) = 2 {\dot{x}}^{T} (t) Px (t) + 2 {\dot{e}}^{T} (t) Re (t) .$ (14)

By substituting (5a), (5b), (9), and (10) into (14), then using Assumption 1 and Lemma 1, it yields in (15). Since the assumption that $Δ_{a} (x) Δ_{a}^{T} (x) \leq γ_{a} I$ and $Δ_{b} (x) Δ_{b}^{T} (x) \leq γ_{b} I$ , hence the Equation (15) can be rewritten in Equation (16) $\begin{matrix} \dot{V} (t) = \sum_{i = 1}^{r} β_{i} (θ) \sum_{j = 1}^{r} β_{j} (θ) 2 {(A_{i} + B_{i} G_{j}) x (t) \\ + (Δ A (x) + Δ B (x) G_{j}) x (t) - B_{i} G_{j} e (t) \\ - Δ B (x) G_{j} e (t)}^{T} Px (t) \\ + \sum_{i = 1}^{r} β_{i} (θ) \sum_{j = 1}^{r} β_{j} (θ) 2 {(A_{i} - L_{i} C) e (t) \\ + (Δ A (x) + Δ B (x) G_{j}) x (t) \\ - Δ B (x) G_{j} e (t)}^{T} Re (t) \\ = \sum_{i = 1}^{r} β_{i} (θ) \sum_{j = 1}^{r} β_{j} (θ) {2 A_{i} x (t) \\ + 2 B_{i} G_{j} x (t) + 2 D_{a} Δ_{a} (x) E_{a} x (t) \\ + 2 D_{b} Δ_{b} (x) E_{b} G_{j} x (t) \\ - 2 B_{i} G_{j} e (t) - 2 D_{b} Δ_{b} (x) E_{b} G_{j} e (t)}^{T} Px (t) \\ + \sum_{i = 1}^{r} β_{i} (θ) \sum_{j = 1}^{r} β_{j} (θ) {2 A_{i} x (t) - 2 L_{i} Ce (t) \\ + 2 D_{a} Δ_{a} (x) E_{a} x (t) + 2 D_{b} Δ_{b} (x) E_{b} G_{j} x (t) \\ - 2 D_{b} Δ_{b} (x) E_{b} G_{j} e (t)}^{T} Re (t) \\ \leq & \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {x^{T} (t) A_{i}^{T} Px (t) + x^{T} (t) \\ {PA}_{i} x (t) + x^{T} (t) G_{j}^{T} B_{i}^{T} Px (t) \\ + x^{T} (t) {PB}_{i} G_{j} x (t) \\ + x^{T} (t) E_{a}^{T} E_{a} x (t) + x^{T} (t) {PD}_{a} Δ_{a} (x) Δ_{a}^{T} (x) \\ D_{a}^{T} Px (t) + x^{T} (t) G_{j}^{T} E_{b}^{T} E_{b} G_{j} x (t) + x^{T} (t) \\ {PD}_{b} Δ_{b} (x) Δ_{b}^{T} (x) D_{b}^{T} Px (t) \\ + e^{T} (t) G_{j}^{T} G_{j} e (t) + x^{T} (t) {PB}_{i} B_{i}^{T} Px (t) \\ + e^{T} (t) G_{j}^{T} E_{b}^{T} E_{b} G_{j} e (t) \\ + x^{T} (t) {PD}_{b} Δ_{b} (x) Δ_{b}^{T} (x) D_{b}^{T} Px (t) \\ + e^{T} (t) A_{i}^{T} Re (t) + e^{T} (t) {RA}_{i} e (t) \\ - e^{T} (t) C^{T} L_{i}^{T} Re (t) - e^{T} (t) {RL}_{i} Ce (t) \\ + x^{T} (t) E_{a}^{T} E_{a} x (t) \\ + e^{T} (t) {RD}_{a} Δ_{a} (x) Δ_{a}^{T} (x) D_{a}^{T} Re (t) \\ + x^{T} (t) G_{j}^{T} E_{b}^{T} E_{b} G_{j} x (t) \\ + e^{T} (t) {RD}_{b} Δ_{b} (x) Δ_{b}^{T} (x) D_{b}^{T} Re (t) \\ + e^{T} (t) G_{j}^{T} E_{b}^{T} E_{b} G_{j} e (t) \\ + e^{T} (t) {RD}_{b} Δ_{b} (x) Δ_{b}^{T} (x) D_{b}^{T} Re (t)} \end{matrix}$ (15)

$\begin{matrix} \dot{V} (t) \leq \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {x^{T} (t) \\ [A_{i}^{T} P + {PA}_{i} + G_{j}^{T} B_{i}^{T} P \\ + {PB}_{i} G_{j} + 2 E_{a}^{T} E_{a} + γ_{a} {PD}_{a} D_{a}^{T} P \\ + 2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} \\ + 2 γ_{b} {PD}_{b} D_{b}^{T} P + {PB}_{i} B_{i}^{T} P] x (t) + e^{T} (t) \\ [2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} \\ + A_{i}^{T} R + {RA}_{i} - C^{T} L_{i}^{T} R - {RL}_{i} C \\ + γ_{a} {RD}_{a} D_{a}^{T} R \\ + 2 γ_{b} {RD}_{b} D_{b}^{T} R + G_{j}^{T} G_{j}] e (t)} . \end{matrix}$ (16)

The Equation (16) can be rewritten as (17).

\begin{matrix} \dot{V} (t) = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {[\begin{matrix} x^{T} (t) & e^{T} (t) \end{matrix}] \\ \times [\begin{matrix} H_{i j}^{(11)} & 0 \\ 0 & H_{i j}^{(22)} \end{matrix}] [\begin{matrix} x (t) \\ e (t) \end{matrix}]} \end{matrix}

(17)

where

\begin{matrix} H_{i j}^{(11)} = A_{i} P + P A_{i}^{T} + G_{j}^{T} B_{i}^{T} P + P B_{i} G_{j} + 2 E_{a}^{T} E_{a} \\ + γ_{a} P D_{a} D_{a}^{T} P + 2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} \\ + 2 γ_{b} P D_{b} D_{b}^{T} P + P B_{i} B_{i}^{T} P \\ H_{i j}^{(22)} = 2 G_{j}^{T} E_{b}^{T} E_{b} G_{j} + A_{i}^{T} R + R A_{i} - C^{T} L_{i}^{T} R - R L_{i} C + γ_{a} R D_{a} D_{a}^{T} R + 2 γ_{b} R D_{b} D_{b}^{T} R + G_{j}^{T} G_{j} \end{matrix}

if 12 holds,

\begin{matrix} \dot{V} (t) = \sum_{i = 1}^{r} \sum_{j = 1}^{r} β_{i} (θ) β_{j} (θ) {[\begin{matrix} x^{T} (t) & e^{T} (t) \end{matrix}] \\ \times [\begin{matrix} H_{i j}^{(11)} & 0 \\ 0 & H_{i j}^{(22)} \end{matrix}] [\begin{matrix} x (t) \\ e (t) \end{matrix}]} < 0. \end{matrix}

(18)

From (18), it can be inferred that the uncertain closed-loop fuzzy system (9) is stable and the estimation error e (t) converges to zero asymptotically. The proof is completed successfully.

It is noted that the condition (12) of Theorem 1 is a Bilinear Matrix Inequality (BMI) which cannot be solved easily. Hence, in order to overcome this difficulty, the following Lemma 2 will be used for the transformation process from BMI toLMI.

Lemma 2. [35] There is an inequalityX^TΦX ≤ 0, where the matrixΦis negative definite, andXis a matrix with appropriate dimensions. Then, there exists a real constantδsatisfying the following inequality. $X^{T} Φ X \leq - δ (X^{T} + X) - δ^{2} Φ^{- 1}$ (19)

Theorem 2.The uncertain fuzzy system (9) is stable and the estimated error (10) converges to zero asymptotically, if there exists matricesZ_j = G_jW, Y_i = RL_i, a constantδ, positive definiteW = P^-1, andRsuch that

$\begin{matrix} [\begin{matrix} Γ & {RD}_{a} & {RD}_{b} \\ (*) & - γ_{a}^{- 1} I & 0 \\ (*) & 0 & - 0.5 γ_{b}^{- 1} I \end{matrix}] < 0 \end{matrix}$ (20a)

$\begin{matrix} [\begin{matrix} \sum_{ij}^{(11)} & 0 \\ 0 & \sum_{ij}^{(22)} \end{matrix}] < 0, \end{matrix}$ (20b)

\sum_{i j}^{(11)} = [\begin{matrix} W A_{i}^{T} + A_{i} W + B_{i} Z_{j} + Z_{j}^{T} B_{i}^{T} + γ_{a} D_{a} D_{a}^{T} + 2 γ_{b} D_{b} D_{b}^{T} + B_{i} B_{i}^{T} & (*) & (*) \\ E_{a} W & - 0.5 I & 0 \\ E_{b} Z_{j} & 0 & - 0.5 I \end{matrix}]

(21)

$\begin{matrix} \sum_{ij}^{(22)} = [\begin{matrix} - 2 δ W & (*) & (*) & (*) & 0 & 0 \\ E_{b} Z_{j} & - 0.5 I & 0 & 0 & 0 & 0 \\ Z_{j} & 0 & - I & 0 & 0 & 0 \\ δ I & 0 & 0 & Γ & {RD}_{a} & {RD}_{b} \\ 0 & 0 & 0 & (*) & - γ_{a}^{- 1} I & 0 \\ 0 & 0 & 0 & (*) & 0 & - 0.5 γ_{b}^{- 1} I \end{matrix}] \\ where \\ Γ = A_{i}^{T} R + {RA}_{i} - C^{T} Y_{i}^{T} - Y_{i} C \end{matrix}$ (22) where $\sum_{ij}^{(11)}$ and $\sum_{ij}^{(22)}$ are defined as (21), (22), $Γ = A_{i}^{T} R + {RA}_{i} - C^{T} Y_{i}^{T} - Y_{i} C$ , and i, j = 1, 2, …, r ..

Proof: In order to simplify the design process, a new matrix is introduced $\tilde{W} = [\begin{matrix} W & 0 \\ 0 & W \end{matrix}]$ (23)

$[\begin{matrix} \begin{matrix} {WA}_{i}^{T} + A_{i} W + {WG}_{j}^{T} B_{i}^{T} + B_{i} G_{j} W \\ + 2 {WE}_{a}^{T} E_{a} W + γ_{a} D_{a} D_{a}^{T} + 2 {WG}_{j}^{T} E_{b}^{T} E_{b} G_{j} W \\ + 2 γ_{b} D_{b} D_{b}^{T} + B_{i} B_{i}^{T} \end{matrix} & 0 \\ 0 \begin{matrix} 2 {WG}_{j}^{T} E_{b}^{T} E_{b} G_{j} W + {WA}_{i}^{T} RW + {WRA}_{i} W \\ - {WC}^{T} L_{i}^{T} RW - {WRL}_{i} CW + γ_{a} {WRD}_{a} D_{a}^{T} RW \\ + 2 γ_{b} {WRD}_{b} D_{b}^{T} RW + {WG}_{j}^{T} G_{j} W \end{matrix} \end{matrix}] < 0$ (24) where W = P^-1. Let the right-hand side and left-hand side of the Equation (12) be multiplied by (23), then the following inequality (24) is yielded.

Define new variables $\begin{matrix} Z_{j} & = & G_{j} W, \end{matrix}$ (25a) $\begin{matrix} N & = & A_{i}^{T} R + {RA}_{i} - C^{T} L_{i}^{T} R - {RL}_{i} C \\ + γ_{a} {RD}_{a} D_{a}^{T} R + 2 γ_{b} {RD}_{b} D_{b}^{T} R, \end{matrix}$ (25b) and substitute them into (24). The Equation (24) is equivalent to the following linear matrix inequality (27).

One can realize that there are bilinear terms in the first and second diagonal blocks of (27). Therefore, firstly, let the Schur complement be applied to the first-diagonal-block of (27) yielding as (28).

If the condition (20a) holds and applies Schur complement, (20) is equivalent to

$\begin{matrix} A_{i}^{T} R + {RA}_{i} - C^{T} L_{i}^{T} R - {RL}_{i} C + γ_{a} {RD}_{a} D_{a}^{T} R \\ + 2 γ_{b} {RD}_{b} D_{b}^{T} R < 0 . \end{matrix}$ (26)

From (25b) and (26), it implies that N < 0, therefore, Lemma 2 can be applied for the second-diagonal-block of (27) and it can be replaced by (29).

$[\begin{matrix} \begin{matrix} {WA}_{i}^{T} + A_{i} W + B_{i} Z_{j} + Z_{j}^{T} B_{i}^{T} + 2 {WE}_{a}^{T} E_{a} W \\ + γ_{a} D_{a} D_{a}^{T} + 2 Z_{j}^{T} E_{b}^{T} E_{b} Z_{j} + 2 γ_{b} D_{b} D_{b}^{T} + B_{i} B_{i}^{T} \end{matrix} & 0 \\ 0 WNW + 2 Z_{j}^{T} E_{b}^{T} E_{b} Z_{j} + Z_{j}^{T} Z_{j} \end{matrix}] < 0$ (27)

$\sum_{ij}^{(11)} = [\begin{matrix} {WA}_{i}^{T} + A_{i} W + B_{i} Z_{j} + Z_{j}^{T} B_{i}^{T} + γ_{a} D_{a} D_{a}^{T} + 2 γ_{b} D_{b} D_{b}^{T} + B_{i} B_{i}^{T} & (*) (*) \\ E_{a} W - 0.5 I 0 \\ E_{b} Z_{j} 0 - 0.5 I \end{matrix}]$ (28)

$\begin{matrix} WNW + 2 Z_{j}^{T} E_{b}^{T} E_{b} Z_{j} + Z_{j}^{T} Z_{j} \leq - 2 δ W - δ^{2} N^{- 1} \\ + 2 Z_{j}^{T} E_{b}^{T} E_{b} Z_{j} + Z_{j}^{T} Z_{j} \end{matrix}$ (29)

Furthermore, using the Schur complement repeatedly on (29), we can obtain the new form without any bilinear terms as (30). Therefore, the proof is completed.

Remark 1. It is clear that (20a) and (20b) are LMI forms which can be solved easily by the Matlab LMI toolbox. This theorem provides a one stage method to get the feasible solutions P, R, G_j, and L_i from the LMI conditions (20a) and (20b) in which computation load is reduced significantly compared to the previous studies [6 , 40].

Remark 2. According to Lemma 2, it is noted that there is not any constraint on the real constant δ. However, if we choose an unsuitable value for δ, it is possible to make the solutions of the LMI (20) infeasible. If a feasible solution is not found, we will arbitrarily choose another one until a feasible solution is found. In this paper, we have not found a relationship between the value of δ and the feasible solution. It may be an open problem to be solved in the future.

The procedure for synthesizing the robust observer-based controller is summarized as follows:

Step 1: Derive the T-S fuzzy model as (2) for the considered uncertain nonlinear system. (Skip this step if a T-S fuzzy plant model is already at hand or accessible in other ways).

Step 2: Determine the upper bound of the internal uncertainties from (5a) and (5b).

Step 3: Resolve conditions (20a) and (20b) to obtain P, R, G_j, and L_i.

Step 4: Construct the fuzzy observer (8) and robust fuzzy observer-based controller (7).

4 Illustrative examples

In this section, two examples are given. One is a numerical example and the other is a practical example.

4.1 Example 1

Consider the uncertain nonlinear system as (1), where $\begin{matrix} x (t) = [x_{1} (t) x_{2} (t) x_{3} (t)]^{T}, y (t) = Cx (t) \\ A = [\begin{matrix} - 3 & 1 & 0 \\ 7 & - 5 & 0 \\ 2 & 0 & 0.2 sin (y) \end{matrix}], \\ Δ A = [\begin{matrix} 0.063 sin (3 y) & 0.084 sin (3 y) & 0.042 sin (3 y) \\ 0.045 sin (3 y) & 0.06 sin (3 y) & 0.03 sin (3 y) \\ 0.0828 sin (3 y) & 0.1104 sin (3 y) & 0.0552 sin (3 y) \end{matrix}], \\ B = [\begin{matrix} 0 \\ 0.2 sin (y) \\ 1 \end{matrix}], and Δ B = [\begin{matrix} 0.064 cos (2 y) \\ 0.0768 cos (2 y) \\ 0.0544 cos (2 y) \end{matrix}] . \end{matrix}$ (31)

Let (1) be represented by a T-S plant model as follows.

Plant Rule 1.

IF θ (t) is Q₁, THEN $\dot{x} (t) = (A_{1} + Δ A (x)) x (t) + (B_{1} + Δ B (x)) u (t)$

Plant Rule 2.

IF θ (t) is Q₂, THEN $\dot{x} (t) = (A_{2} + Δ A (x)) x (t) + (B_{2} + Δ B (x)) u (t)$ where θ (t) =0.2sin (y), $Q_{1} (θ (t)) = \frac{1 + 5 θ (t)}{2} = \frac{1 + sin (y)}{2}$ and $Q_{2} (θ (t)) = \frac{1 - 5 θ (t)}{2} = \frac{1 - sin (y)}{2}$ are the membership functions of the antecedent fuzzy sets. A_i, B_i, and C are given as below $\begin{matrix} A_{1} = [\begin{matrix} - 3 & 1 & 0 \\ 7 & - 5 & 0 \\ 2 & 0 & 0.2 \end{matrix}], B_{1} = [\begin{matrix} 0 \\ 0.2 \\ 1 \end{matrix}], \\ A_{2} = [\begin{matrix} - 3 & 1 & 0 \\ 7 & - 5 & 0 \\ 2 & 0 & - 0.2 \end{matrix}], B_{2} = [\begin{matrix} 0 \\ - 0.2 \\ 1 \end{matrix}], \\ C = [\begin{matrix} 1 & 2 & 1 \end{matrix}], \end{matrix}$

Furthermore, those uncertainties satisfy the Equations (5a) and (5b) and have $\begin{matrix} D_{a} = [\begin{matrix} 0.35 \\ 0.25 \\ 0.46 \end{matrix}], D_{b} = [\begin{matrix} 0.2 \\ 0.24 \\ 0.17 \end{matrix}], \\ E_{a} = [\begin{matrix} 0.3 & 0.4 & 0.2 \end{matrix}], \\ E_{b} = 0.4, Δ_{a} (x) = 0.6 sin (3 y), Δ_{b} (x) = 0.8 cos (2 y) . \end{matrix}$

Finally, the overall uncertain fuzzy system is shown as follows ${\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{2} β_{i} (θ) {(A_{i} + Δ A (x)) x (t) + (B_{i} + Δ B (x)) u (t)} \\ y (t) = Cx (t) \end{matrix}$ (32) where ${\begin{matrix} β_{1} (θ (t)) = \frac{w_{i} (θ (t))}{\sum_{i = 1}^{r} w_{i} (θ (t))} = \frac{1 + sin (y)}{2} \\ β_{2} (θ (t)) = \frac{w_{i} (θ (t))}{\sum_{i = 1}^{r} w_{i} (θ (t))} = \frac{1 - sin (y)}{2} \end{matrix}$

Now, let us follow the design procedure to synthesize the robust fuzzy observer-based controller. Here

Step 1: This step has been done as (32).

Step 2: The upper bounds of the unknown time-varying blocks are $Δ_{a} (x) Δ_{a}^{T} (x) \leq 0.6$ and $Δ_{b} (x) Δ_{b}^{T} (x) \leq 0.8$ , respectively.

Step 3: Solve (20a) and (20b) using the MATLAB LMI toolbox with δ = 5.1. The feasible solutions can be found as below $\begin{matrix} W = [\begin{matrix} 0.0707 & - 0.0075 & 0.0016 \\ - 0.0075 & 0.0701 & 0.0076 \\ 0.0016 & 0.0076 & 0.0509 \end{matrix}], \\ R = (1.0 e^{03}) \times [\begin{matrix} 5.8533 & - 5.0553 & - 0.9420 \\ - 5.0553 & 5.5734 & 1.1796 \\ - 0.9420 & 1.1796 & 0.3852 \end{matrix}], \\ Y_{1} = (1.0 e^{05}) \times [\begin{matrix} 0.1805 \\ 1.1569 \\ 0.5002 \end{matrix}], \\ Y_{2} = (1.0 e^{06}) \times [\begin{matrix} 0.6927 \\ 1.3947 \\ 0.6964 \end{matrix}], \\ Z_{1} = [\begin{matrix} - 0.1672 & - 0.1393 & - 0.8286 \end{matrix}], and \\ Z_{2} = [\begin{matrix} - 0.2285 & - 0.3499 & - 0.1856 \end{matrix}] . \end{matrix}$ Based on the above matrices, we can obtain P, G_j, and L_i. These matrices are as follows $\begin{matrix} P = [\begin{matrix} 14.3291 & 1.6154 & - 0.6865 \\ 1.6154 & 14.6707 & - 2.2340 \\ - 0.6865 & - 2.2340 & 20.0148 \end{matrix}], \\ L_{1} = [\begin{matrix} 82.2001 \\ 76.3064 \\ 106.9611 \end{matrix}], \\ L_{2} = (1.0 e^{03}) [\begin{matrix} 1.3765 \\ 1.1475 \\ 1.6598 \end{matrix}], \\ G_{1} = [\begin{matrix} - 2.0526 & - 0.4623 & - 16.1574 \end{matrix}], and \\ G_{2} = [\begin{matrix} - 3.7120 & - 5.0882 & - 2.7760 \end{matrix}] . \end{matrix}$

Step 4: Then, the observer and observer-based controller are synthesized.

The Figs. 1–3 show the simulation results which are carried out with the different initial conditions $x (0) = [\begin{matrix} 0.04 & 0.05 & 0.05 \end{matrix}]^{T}$ and $\hat{x} (0) = [\begin{matrix} 0.1 & 0.1 & - 0.1 \end{matrix}]^{T}$ for real state variables and estimated state variables, respectively.

These simulation result in Figs. 1–3 show that all state variables and their estimation errors between real states and estimated states converge to zero asymptotically. It is seen that the synthesized observer-based controller works well for a T-S fuzzy system in the presence of uncertainties.

The second example is a backing up control problem for a practical Truck-Trailer system.

4.2 Example 2

Consider the continuous-time nonlinear dynamic system model of a Truck-Trailer moving as follows [41], $\begin{matrix} {\dot{x}}_{1} (t) = - \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t) + \frac{v \bar{t}}{{lt}_{0}} u (t), \\ {\dot{x}}_{2} (t) = \frac{v \bar{t}}{{Lt}_{0}} x_{1} (t) \\ {\dot{x}}_{3} (t) = \frac{v \bar{t}}{t_{0}} sin [x_{2} (t) + \frac{v \bar{t}}{2 L} x_{1} (t)] \\ {\dot{x}}_{4} (t) = \frac{v \bar{t}}{t_{0}} cos [x_{2} (t) + \frac{v \bar{t}}{2 L} x_{1} (t)] \end{matrix}$ (33) where x₁ (t) is the angle difference between truck and trailer, x₂ (t) is the angle of trailer corresponding to the X-axis, x₃ (t) is the vertical position of rear end of trailer on the Y-axis and x₄ (t) is the horizontal position of rear end of trailer on the X-axis. All states are shown in Fig. 4 below. u is the steering angle of the truck.

The parameters of the Truck-Trailer system are given as l = 2.8, L = 5.5, v = -1, $\bar{t} = 2$ , t₀ = 0.5, d = 10t₀/ - π . Assume there are nonlinear uncertainties ΔA (x) and ΔB (x) being added in the system (33). After converting (33) into T-S fuzzy systems based on the method in [42], the T-S fuzzy model with the presence of uncertainties is described as (34).

It is known that lining up the truck and trailer and backing the truck-trailer up to a desired location are very difficult, only highly skilled and experienced drivers can do it. This task becomes much harder when the truck-trailer system is affected by some uncertainties. Here, suppose we have sensors for the truck and trailer, x₁ (t) and x₂ (t) are measurable. But x₃ (t) is un-measurable. In this study, we would like to synthesize an observer-based controller for the system such that the lining up and backing up of the truck-trailer to the desired location are achieved simultaneously. In other words, the control objective is to back the truck-trailer up such that x₁ (t), x₂ (t) and x₃ (t) are approaching zero finally. But the position x₄ (t) is not constrained. Due to this reason, the state x₄ (t) will not be considered here. Therefore the state vector $x (t) = [\begin{matrix} x_{1} (t) & x_{2} (t) & x_{3} (t) \end{matrix}]^{T}$ contains three states only.

The fuzzy model with existing uncertainties is expressed as follows [41]:

Rule 1: IF φ (t) is about 0, THEN $\dot{x} (t) = (A_{1} + Δ A (x)) x (t) + (B_{1} + Δ B (x)) u (t)$

Rule 2: IF φ (t) is about π or -π, THEN $\dot{x} (t) = (A_{2} + Δ A (x)) x (t) + (B_{2} + Δ B (x)) u (t)$

where $φ (t) = x_{2} (t) + (v \bar{t} / - 2 L) x_{1} (t) .$

Overall uncertain fuzzy system is shown as follows

$\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{2} β_{i} (φ (t)) {(A_{i} + Δ A (x)) x (t) \\ + (B_{i} + Δ B (x)) u (t)} \\ y (t) = Cx (t), \end{matrix}$ (34) where $\begin{matrix} A_{1} = [\begin{matrix} - v \bar{t} / - {Lt}_{0} & 0 & 0 \\ v \bar{t} / - {Lt}_{0} & 0 & 0 \\ v^{2} {\bar{t}}^{2} / - 2 {Lt}_{0} & v \bar{t} / - t_{0} & 0 \end{matrix}], \\ A_{2} = [\begin{matrix} - v \bar{t} / - {Lt}_{0} & 0 & 0 \\ v \bar{t} / - {Lt}_{0} & 0 & 0 \\ {dv}^{2} {\bar{t}}^{2} / - 2 {Lt}_{0} & dv \bar{t} / - t_{0} & 0 \end{matrix}], \\ B_{1} = B_{2} = {[\begin{matrix} v \bar{t} / - {lt}_{0} & 0 & 0 \end{matrix}]}^{T}, C = [\begin{matrix} 1 & 0.5 & 1 \end{matrix}], \end{matrix}$ and the uncertainties are $\begin{matrix} Δ A (x) \\ = [\begin{matrix} 0.063 sin (3 x_{1}) & 0.084 sin (3 x_{1}) & 0.042 sin (3 x_{1}) \\ 0.045 sin (3 x_{1}) & 0.06 sin (3 x_{1}) & 0.03 sin (3 x_{1}) \\ 0.0828 sin (3 x_{1}) & 0.1104 sin (3 x_{1}) & 0.0552 sin (3 x_{1}) \end{matrix}], \\ Δ B (x) = [\begin{matrix} 0.064 sin (3 x_{1}) \\ 0.0768 sin (3 x_{1}) \\ 0.0544 sin (3 x_{1}) \end{matrix}], \end{matrix}$ where $\begin{matrix} β_{1} (φ (t)) = (1 - \frac{1}{1 + exp (- 3 (φ (t) - 0.5 π))}) \\ \times (\frac{1}{1 + exp (- 3 (φ (t) + 0.5 π))}), \\ β_{2} (φ (t)) = 1 - β_{1} (φ (t)) . \end{matrix}$

Substituting the values of the parameters into the system (34), we obtain $\begin{matrix} A_{1} = [\begin{matrix} 0.727 & 0 & 0 \\ - 0.727 & 0 & 0 \\ 0.727 & - 4 & 0 \end{matrix}], A_{2} = [\begin{matrix} 0.727 & 0 & 0 \\ - 0.727 & 0 & 0 \\ 1.157 & - 6.366 & 0 \end{matrix}], \\ B_{1} = B_{2} = {[\begin{matrix} - 1.429 & 0 & 0 \end{matrix}]}^{T}, and C = [\begin{matrix} 1 & 0.5 & 1 \end{matrix}] . \end{matrix}$ It is noted that the uncertainties of the systems can be written in the form of (5) and we have $\begin{matrix} D_{a} = [\begin{matrix} 0.15 \\ 0.25 \\ 0.16 \end{matrix}], D_{b} = [\begin{matrix} 0.2 \\ 0.24 \\ 0.17 \end{matrix}], \\ E_{a} = [\begin{matrix} 0.3 & 0.1 & 0.1 \end{matrix}], \\ E_{b} = 0.1, Δ_{a} (x) = 0.6 sin (3 x_{1}), and \\ Δ_{b} (x) = 0.8 \sin (3 x_{1}) . \end{matrix}$

Since the upper bounds of the unknown time-varying blocks are $Δ_{a} (x) Δ_{a}^{T} (x) \leq 0.6$ and Δ_b (x) $Δ_{b}^{T} (x) \leq 0.8$ , respectively. Then resolve (20a) and (20b) using the MATLAB LMI toolbox with δ = 2, the matrices W, R, P, G_j and L_i can be found as below. $\begin{matrix} W = [\begin{matrix} 0.0465 & 0.0215 & 0.0324 \\ 0.0215 & 0.0327 & 0.0236 \\ 0.0324 & 0.0236 & 0.0525 \end{matrix}], \\ R = (1.0 e^{03}) \times [\begin{matrix} 6.0899 & 3.0595 & 6.0957 \\ 3.0595 & 1.5571 & 3.0623 \\ 6.0957 & 3.0623 & 6.1015 \end{matrix}], \\ P = [\begin{matrix} 41.0933 & - 12.7792 & - 19.6472 \\ - 12.7792 & 49.2189 & - 14.2364 \\ - 19.6472 & - 14.2364 & 37.5987 \end{matrix}], \\ Y_{1} = (1.0 e^{06}) \times [\begin{matrix} 3.8779 \\ 1.9386 \\ 3.8775 \end{matrix}], \\ Y_{2} = (1.0 e^{06}) \times [\begin{matrix} 4.7396 \\ 2.3537 \\ 4.7366 \end{matrix}], \end{matrix}$ $\begin{matrix} L_{1} = (1.0 e^{05}) \times [\begin{matrix} 1.2110 \\ - 0.0087 \\ - 1.1991 \end{matrix}], \\ L_{2} = (1.0 e^{05}) \times [\begin{matrix} 2.2511 \\ - 0.0210 \\ - 2.2307 \end{matrix}], \\ Z_{1} = [\begin{matrix} 0.7590 & 0.0562 & 0.6590 \end{matrix}], \\ Z_{2} = [\begin{matrix} 0.6789 & 0.2174 & 0.6273 \end{matrix}], \\ G_{1} = [\begin{matrix} 17.5247 & - 16.3145 & 9.0643 \end{matrix}], and \\ G_{2} = [\begin{matrix} 12.7941 & - 6.9046 & 7.1530 \end{matrix}] . \end{matrix}$

Then the fuzzy observer is constructed as $\begin{matrix} \dot{\hat{x}} (t) = \sum_{i = 1}^{2} β_{i} (φ (t)) [A_{i} \hat{x} (t) \\ + B_{i} u (t) + L_{i} (y (t) - \hat{y} (t))] \\ \hat{y} (t) = C \hat{x} (t) \end{matrix}$

The observer-based controller is built as $u (t) = \sum_{j = 1}^{2} β_{j} (φ (t)) G_{j} \hat{x} (t)$ . The results of the simulation are depicted in Figs. 5–8 with initial conditions $x (0) = \begin{matrix} [1 & 0.1 & - 4]^{T} \end{matrix}$ , $\hat{x} (0) = \begin{matrix} [- 0.9 & - 0.8 & - 0.8]^{T} \end{matrix}$ .

Figures 5–7 present the simulation results of the real states x₁ (t), x₂ (t), x₃ (t), the estimated states ${\hat{x}}_{1} (t)$ , ${\hat{x}}_{2} (t)$ , ${\hat{x}}_{3} (t)$ and their estimation errors. As expected, the states x₁ (t), x₂ (t) and x₃ (t) approach zero asymptotically and their estimation errors converge to zero as well. Additionally, the truck-trailer has been backed up smoothly to the desired position as shown in Fig. 8.

5 Conclusion

This study has proposed a sufficient condition for the estimation and stabilization of an uncertain T-S fuzzy system where the state variables of the system may be unavailable. The uncertainties of the system must satisfy an assumption. Furthermore, the stabilization condition with BMI forms (Theorem 1) has been transformed into a new condition with LMI forms (Theorem 2) such that the observer and controller are synthesized with the aid of LMI tools. Therefore, the computation load is reduced significantly compared to the previous related studies. Finally, two examples have been illustrated to verify the effectiveness of the synthesized fuzzy observer-based controller.

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