H∞ reduced-order observer-based controller synthesis approach for T-S fuzzy systems

doi:10.1016/j.jfranklin.2019.06.004

Journal of the Franklin Institute

Volume 356, Issue 12, August 2019, Pages 6388-6400

https://doi.org/10.1016/j.jfranklin.2019.06.004 Get rights and content

Abstract

For continuous-time nonlinear systems represented by Takagi–Sugeno fuzzy models, a new H_∞ reduced-order-observer based controller synthesis structure is investigated in this paper. By the fuzzy reduced-order observer and fuzzy controller, an augmented error system composed of the estimation and control errors is obtained. The fuzzy modeling residual terms are seen as part of the external disturbance, and an extra design matrix is added to facilitate the design process. The robustness and stability conditions are given based on Lyapunov function approach, then the conditions are transformed into convex form to facilitate the numerical solving process. Finally, by the comparison with existing methods in simulation section, the control performance and conservativeness reduction effects of the proposed methods are verified.

Introduction

In recent years, with the great demands of advanced control performance for engineering systems, the control problems of nonlinear systems have drawn increasing attentions [1], [2]. However, controller design tasks for the nonlinear system are still difficult to tackle due to the inherent complexity of the system. The T-S fuzzy system, introduced in [3], provides a simplified approach to represent nonlinear systems thanks to its universal approximation ability. With the T-S fuzzy system, tremendous mature linear control system theories can be used on the complicated nonlinear system [4], [5], [6], [7], [8], [9], [10], [11].

Due to the economic and physical limitations, system state may be not measurable under certain circumstance. For this issue, various approaches for designing output feedback control structures are brought up, which can be categorised into three aspects: static output feedback control [12], [13], dynamic output feedback control [14], [15], [16], and observer-based control [17], [18]. Under certain cases, system estimation function is very important. For example, some unmeasurable plane parameters must be estimated to show the pilot for safety purpose during the flight phrase. For the system state estimation advantages, the observer-controller structure receives widely attentions: for a T-S fuzzy descriptor system, robust method [19] and relaxed stability synthesis approach [20] are investigated. For a continuous T-S fuzzy system, conservatism relaxed robust approaches are studied in [21], [22], [23], [24]. And the discrete case can be found in [25], the results in linear system aspects could be also refereed as in [26]. For switched fuzzy systems, the observer-based output-feedback control schemes are developed both for the measurable and unmeasurable premise variable cases in [27], [28]. To reduce the conservatism in stability analysis, separation principle for T-S fuzzy systems observer-controller structure is studied in [29] and references therein, further results are given under network environment as in [30] and under external disturbance in [31]. Aimed at further reducing the conservatism caused by the full frequency analysis methods, the problem of observer-controller based H_∞ control in finite frequency domain case is investigated in [32]. For a real world engineering system, the state may be partly measurable, but for the full-order observer, the measurable state will also be estimated, this redundancy structure will induce certain conservatism in stability analysis. Reduced-order observer will avoid the estimation process for measurable system state, then the conservatism will certainly be reduced in stability analysis.

For T-S fuzzy systems, a separation principle structure for full/reduced-order observer-controller is proposed in [33], [34]. Faults and disturbance reconstruction methods by reduced-order observer can be found in [35]. For a discrete time T-S fuzzy system, a recursive reduced-order observer design method is proposed in [36]. In [37], a robust state/fault estimation reduced-order observer and fault tolerant controller are given, and the separation principle still holds under the fault case. A piecewise Lyapunov function based full/reduced-order observer approach is investigated in [38], [39]. But till now, for reduced-order observer-controller method, how to reduce the conservatism during robust performance analysis remains to be an open problem.

In this paper, a novel reduced-order-observer based controller synthesis method is proposed to reduce the conservatism of T-S fuzzy systems. A series of sufficient conditions for system stability and robustness are obtained, the conditions are in convex form and can be solved numerically. Finally, a suspended floater following system braking control task is carried out to show the effectiveness of the proposed method. The contributions of this paper can be summarized as follows: 1) To the best of the authors’ knowledge, the convex form conditions based H_∞ reduced-observer-controller synthesis approach is proposed for the first time. 2) Compared with existing results, the conservatism is further reduced by the obtained stability and robustness conditions.

The rest of the paper is organized as follows. Section 2 introduces the fuzzy system and concerned problems. Section 4 describes the reduced-order observer-based controller design process. Section 5 presents the stability analysis. Section 6 illustrates the simulation process. Finally, Some closing remarks are presented.

Notations: Throughout the paper, Rⁿ denotes the n-dimensional Euclidean space. X^T and $X^{- 1}$ denote the transpose and the inverse of a matrix X, respectively. 2-norm of a vector * is given as ${∥ * ∥}_{2}^{2} = \int_{o}^{\infty} *^{T} * d t$ . ⋆ denotes the transpose of the corresponding sub-matrix. I is an identity matrix, and 0 denotes a zero matrix or a zero vector. The Hermitian section of X is denoted by $He {X} = X + X^{T}$ . diag{} denotes a diagonal matrix.

Section snippets

Problem statement

Consider a class of T-S fuzzy system: $Rule i : IF β_{1} (t) is M_{i 1} . . . and β_{p} (t) is M_{i p} THEN$ $\begin{matrix} \dot{x} (t) & = & A_{i} x (t) + B_{i} u (t) + a_{i} + E_{i} w (t) \\ z (t) & = & C_{1 i} x (t) \\ y (t) & = & C_{2 i} x (t) \end{matrix}$ where M_ij are fuzzy sets; $i = 1, 2, \dots, l$ with l denoting the number of fuzzy rules; $j = 1, 2, \dots, p$ with p denoting the number of premise variables; $β (t) = [β_{1} (t) . . .$ β_p(t)]^T ∈ R^p is the premise variable; x(t) ∈ Rⁿ is the system state; y(t) ∈ R^q is the measured output; u(t) ∈ R^m is the control input; z(t) ∈ Rⁿ is the output state of interests; w(t) ∈ R^o is the unknown external

Reduced-order observer

Assumption 1

The pairs (A_i, C_2i) are observable, and rank $C_{2 i} = q,$ where $i = 1, 2, \dots, l$ .

Define matrix $P_{i} = {[{C_{2 i}}^{T} {R_{i}}^{T}]}^{T},$ where $R_{i} \in R^{(n - q) \times n}$ . Consider that P_i is nonsingular, so R_i is not unique and is arbitrary. Given $F_{i} = {P_{i}}^{- 1} = [F_{i 1} F_{i 2}],$ and it is obvious that: $I = P_{i} F_{i} = [\begin{matrix} C_{2 i} \\ R_{i} \end{matrix}] [F_{i 1} F_{i 2}] = [\begin{matrix} C_{2 i} F_{i 1} & C_{2 i} F_{i 2} \\ R_{i} F_{i 1} & R_{i} F_{i 2} \end{matrix}]$ where F_i1 ∈ R^n × q, $F_{i 2} \in R^{n \times (n - q)},$ and $C_{2 i} F_{i 1} = I_{q},$ $C_{2 i} F_{i 2} = 0$ . Consider a liner nonsingular transformation $\bar{x} = \sum_{i = 1}^{l} h_{i} P_{i} x,$ ${\bar{x}}_{1 i} = C_{2 i} x$ and ${\bar{x}}_{2 i} = R_{i} x,$ a new T-S fuzzy system can be obtained as: $\begin{matrix} \dot{\bar{x}} & = \sum_{i = 1}^{l} h_{i} {[\begin{matrix} {\bar{x}}_{1 i}^{T} & {\bar{x}}_{2 i}^{T} \end{matrix}]}^{T} = \sum_{i = 1}^{l} h_{i} (A_{i} \end{matrix}$

System stability analysis

Lemma 1

[20] For real matrices Ω < 0 and X with appropriate dimension such that X^TΩX ≤ 0, and a scalar α, the following inequality holds: $X^{T} Ω X \leq - α (X^{T} + X) - α^{2} Ω^{- 1}$

Definition 1

[31] For the T-S fuzzy system (15), the following fuzzy H_∞ index is presented to describe the external disturbance attenuation performance: ${∥ z ∥}_{2}^{2} < γ^{2} \sum_{i = 1}^{l} h_{i} {∥ d_{i} ∥}_{2}^{2}$ where γ is a positive scalar.

Theorem 1

For $\forall i, j = 1, 2, \dots, l,$ the fuzzy system (15) is asymptotically stable, and the H_∞ index ${∥ z ∥}_{2}^{2} < γ^{2} \sum_{i = 1}^{l} h_{i} {∥ d_{i} ∥}_{2}^{2}$ is achieved, if there exist matrices $Q = Q^{T} \in R^{2 n \times 2 n} > 0,$ ${\bar{W}}_{i j} \in R$

Performance analysis

The braking control task of suspended floater following control system is adopted here to verify the effectiveness of the proposed method. As shown in Fig. 1, the kinetics and dynamics of the system can be described as: $\begin{matrix} \dot{x} & = f_{1} (x, u) + f_{2} (x, d) \\ z & = C_{1} x \\ y & = C_{2} x \end{matrix}$ where: $\begin{matrix} C_{1} & = [\begin{matrix} 1 & 0 & 0 \end{matrix}], C_{2} = I^{3 \times 3} \end{matrix}$ $\begin{matrix} x = [\begin{matrix} θ \\ \dot{θ} \\ v \end{matrix}], f_{1} (x, u) = [\begin{matrix} \dot{θ} \\ - \frac{(M + m) C_{s}}{m M l^{2}} \dot{θ} + \frac{C_{r}}{M l} v - \frac{(m + M) g \sin θ}{M l} - \frac{1}{M l} u \\ \frac{C_{s}}{M l} \dot{θ} - \frac{C_{r}}{M} v + \frac{m g \sin θ}{M} + \frac{1}{M} u \end{matrix}], \\ f_{2} (x, d) = [\begin{matrix} 0 \\ \frac{(m + M) \cos θ}{m M l} d \\ - \frac{\cos θ}{M} d \end{matrix}] \end{matrix}$

θ is the swing angle of the suspended rope; v is the velocity of the motion platform; u is the control force of the

Conclusions

In this paper, the problem of T-S fuzzy system reduced-order observer-controller synthesis has been addressed. The observer-controller errors augmented system has been proved to be asymptotically stable, and H_∞ index is minimized when external disturbance exists. Finally, based on the braking control process of suspended floater following system, the effectiveness of the proposed method is verified. Comparison with traditional methods are also given to further illustrate the conservatism

Acknowledgments

This work is supported in part by the National Science Foundation of China (Grant nos. 61803127, 61873271, 61873335), the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, China, the Natural Science Foundation in Heilongjiang Province, China (Grant no. YQ2019F012), the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province, China (Grant no. UNPYSCT-2017093), the 111 Project (Grant no. D18003), the

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H∞ reduced-order observer-based controller synthesis approach for T-S fuzzy systems

Abstract

Introduction

Section snippets

Problem statement

Reduced-order observer

System stability analysis

Performance analysis

Conclusions

Acknowledgments

Automatica

Neurocomputing

J. Frankl. Inst.

Automatica

J. Frankl. Inst.

J. Frankl. Inst.

Neurocomputing

Neurocomputing

Fuzzy Sets Syst.

Int. J. Robust Nonlniear Control

Integral sliding mode-based attitude coordinated tracking for spacecraft formation with communication delays

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Fuzzy identification of systems and its applications to modeling and control

IEEE Trans. Syst. Man Cybern.

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IEEE Trans. Cybern.

Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation

IEEE Trans. Autom. Control

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IEEE Trans. Fuzzy Syst.

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H_∞ reduced-order observer-based controller synthesis approach for T-S fuzzy systems

H_∞ observer design for continuous-time Takagi–Sugeno fuzzy model with unknown premise variables via nonquadratic Lyapunov function