Fuzzy affine model-based output feedback controller design for nonlinear impulsive systems

doi:10.1016/j.cnsns.2019.104894

Communications in Nonlinear Science and Numerical Simulation

Volume 79, December 2019, 104894

https://doi.org/10.1016/j.cnsns.2019.104894 Get rights and content

Highlights

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This paper studies the robust output feedback control problem for a class of nonlinear systems with impulsive effects via T-S fuzzy affine models. By employing a state-input augmentation scheme, some new results on the piecewise affine (PWA) output feedback controller synthesis for T-S fuzzy affine impulsive systems are obtained through piecewise quadratic Lyapunov functions. Sufficient criteria for exponential stability analysis of the resulting closed-loop system are given within a convex optimization setup. A simulation example is shown to verify the effectiveness of the proposed approach.

Abstract

This paper studies the robust output feedback control problem for a class of nonlinear systems with impulsive effects via T-S fuzzy affine models. By employing a state-input augmentation scheme, some new results on the piecewise affine (PWA) output feedback controller synthesis for T-S fuzzy affine impulsive systems are obtained through piecewise quadratic Lyapunov functions. Sufficient criteria for exponential stability analysis of the resulting closed-loop system are given within a convex optimization setup. A simulation example is shown to verify the effectiveness of the proposed approach.

Introduction

In modern engineering and science, there usually exist some natural phenomena that the system state variables experience an abrupt change at certain instants for many real world situations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. As a consequence, it is natural to characterize the above-mentioned instantaneous perturbed actions as impulsive effects. The systems with impulsive effects exist in a large amount of practical evolutionary processes, for instance, some biological systems involving pathology bursting rhythm models, biological neural networks and thresholds, economics, telecommunications, robotics and flying object motions [12], [13], [14]. Consequently, numerous efforts have been devoted to impulsive systems [15], [16], [17], [18], [19], [20], [21], [22]. For instance, the robust $H_{\infty}$ control of uncertain singular impulsive systems was studied in [16]. In [17], a model predictive controller was synthesized for linear systems with impulses. The authors in [20] presented some new input-to-state stability analysis results on stochastic impulsive systems.

In past few decades, the methods based on Takagi-Sugeno (T-S) fuzzy models have been extensively adopted to control complex nonlinear systems [23], [24], [25]. The T-S fuzzy model consists of a set of local affine or linear models, which are smoothly connected through fuzzy membership functions (MFs) [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. Recently, many significant research results on analysis and synthesis for T-S fuzzy impulsive systems have been reported [37], [38], [39], [40]. The authors in [37] synthesized a state feedback controller for fuzzy impulsive time-delay systems such that the resulting closed-loop system was exponentially stable. [38] investigated the robust/exponential stability analysis problem for uncertain T-S fuzzy impulsive systems. [40] proposed an indirect fuzzy adaptive tracking control method for interval type-2 fuzzy impulsive systems.

Note that the results in [37], [38], [39], [40] were derived with an assumption that all the system states are available. Unfortunately, the system states are usually not fully measurable within various practical situations. To relax the above-mentioned restrictive assumption, some researchers have devoted efforts to output feedback control (OFC) for T-S fuzzy impulsive systems [41], [42]. The authors in [41] proposed an observer-based OFC scheme for T-S fuzzy time-delay impulsive systems for both delay-dependent and delay-independent cases. [42] addressed the OFC problem for T-S fuzzy impulsive networked systems by adopting quantized measurements. Nevertheless, the results in [41], [42] were mainly attained relying on a common quadratic Lyapunov function (CQLF), which tends to be conservative. Additionally, the existing controller synthesis results for nonlinear impulsive systems were obtained merely through T-S fuzzy systems with local linear models, and it has been well recognized that the T-S fuzzy affine models possess much enhanced function approximation capability [28]. Nevertheless, there have been few results on OFC problem for nonlinear impulsive systems through T-S fuzzy affine models and piecewise quadratic Lyapunov functions (PQLFs), which motivates this study.

This work attempts to propose a piecewise affine (PWA) output feedback controller synthesis approach for nonlinear systems subject to impulsive effects via T-S fuzzy affine models. By a state-input augmentation scheme, the conventional closed-loop system will be firstly formulated into a descriptor system. Based on PQLFs, some new sufficient criteria for robust exponential stability analysis of the resulting closed-loop system are given under a convex optimization framework. The main contributions of this paper are given as follows. (1) The exponential stability analysis of the closed-loop system is derived based on PQLFs, and the conservatism can be reduced. (2) The uncertain nonlinear impulsive systems are described by T-S fuzzy affine models with more approximation accuracy. (3) A new robust piecewise affine output feedback controller is designed for the T-S fuzzy affine impulsive systems. (4) It is also noted that parameter uncertainties are allowed to exist in both control input matrices and output matrices.

The rest of this paper is constructed as follows. Section 2 is devoted to preliminaries. The main results for the PWA output feedback controller synthesis are shown in Section 3. Simulation studies are given to show the effectiveness of the proposed method in Section 4. Conclusions are presented in Section 5.

Notations. $Sym {S}$ is short for $S + S^{T}$ . ℜ^m represents the m-dimensional Euclidean space and $ℜ_{+} = [0, \infty)$ . λ_max(S) (λ_min(S)) refers to the largest (smallest) eigenvalue of matrix S.

Section snippets

Uncertain T-S fuzzy affine impulsive models

Consider a T-S fuzzy affine impulsive system with r fuzzy IF-THEN rules as follows.

Plant Rule $R^{l}$ : IF θ₁(x(t)) is $F_{1}^{l}$ and ⋅⋅⋅ and θ_φ(x(t)) is $F_{φ}^{l},$ THEN ${\begin{matrix} \dot{x} (t) = (A_{l} + Δ A_{l}) x (t) + a_{l} + Δ a_{l} + (B_{l} + Δ B_{l}) u (t), t \in (t_{j}, t_{j + 1}] \\ x (t_{j}^{+}) = (I + D_{j l}) x (t_{j}) + d_{j l}, t = t_{j} \\ y (t) = (C_{l} + Δ C_{l}) x (t), l \in L : = {1, \dots, r}, j = 0, 1, 2, \dots \end{matrix}$ where $R^{l}$ represents the l-th fuzzy inference rule; $F_{ϕ}^{l} (ϕ = 1, \dots, φ)$ stands for fuzzy sets; θ(x(t)) ≔ [θ₁(x(t)), ⋅⋅⋅, θ_φ(x(t))] represents the measurable premise variables; r stands for the number of inference rules; $x (t) \in ℜ^{n_{x}}$

Main results

New sufficient criteria for robust exponential stability analysis of the closed-loop system will be derived through PQLFs in this section. To ensure the PQLFs continuous among the subspace boundaries, along the procedures in [28], matrices ${\tilde{F}}_{i} = [\begin{matrix} f_{i} & F_{i} \end{matrix}],$ $i \in I,$ with $f_{i} = 0$ for $i \in I_{0}$ are constructed to characterize the boundary across the subspaces, ${\tilde{F}}_{i} [\begin{matrix} 1 \\ x (t) \end{matrix}] = {\tilde{F}}_{s} [\begin{matrix} 1 \\ x (t) \end{matrix}], x (t) \in S_{i} ⋂ S_{s}, i, s \in I .$

The S-procedure will be also utilized to further reduce the conservatism via constructing matrices ${\tilde{W}}_{i} = [\begin{matrix} w_{i} & W_{i} \end{matrix}],$ $i \in I,$

Simulation studies

Consider a nonlinear continuous stirred tank reactor (CSTR) system [28], which is characterized by the subsequent equation ${\begin{matrix} \dot{T} = M_{0} (T_{s} - T) + M_{1} M_{0} e^{- \frac{M_{2}}{T}} C_{A} - M_{3} T + M_{3} u \\ {\dot{C}}_{A} = M_{0} (C_{0} - C_{A}) - M_{0} e^{- \frac{M_{2}}{T}} C_{A} \end{matrix}$ where C_A and T represent the concentration and temperature of the reactor, respectively. The control input u represents the temperature of the coolant stream. In this example, we choose $x_{1} = T$ and $x_{2} = C_{A}$ as the system states. The nominal parameter values are shown in Table 1 with $M_{0} = \frac{F}{V},$ $M_{1} = \frac{- Δ H}{ρ C_{p}},$ $M_{2} = \frac{E}{R},$ and $M_{3} = \frac{U A}{V ρ C_{p}}$ .

Conclusions

This paper investigates the robust output feedback control problem for T-S fuzzy affine impulsive systems. Through applying a state-input augmentation approach, sufficient creteria for the exponential stability analysis of the closed-loop system are attained based on PQLFs. A simulation example is presented to show the effectiveness of the proposed method. One of our future research works is the study on reliable output feedback control for T-S fuzzy affine impulsive systems. For practical

References (42)

W.H. Chen et al.
Robust stability and $H_{\infty}$ control of uncertain impulsive systems with time-delay
Automatica
(2009)
X.D. Li et al.
Impulsive differential equations: periodic solutions and applications
Automatica
(2015)
Y. Yang et al.
Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method
Commun Nonlinear Sci NumerSimul
(2017)
I. Stamova et al.
Stability analysis of impulsive functional systems of fractional order
Commun Nonlinear Sci NumerSimul
(2014)
H. Zhang et al.
Reliable dissipative control for stochastic impulsive systems
Automatica
(2008)
C. Li et al.
Impulsive effects on stability of high-order BAM neural networks with time delays
Neurocomputing
(2011)
C. Briat
Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints
Automatica
(2016)
H.K. Lam
A review on stability analysis of continuous-time fuzzy-model-based control systems: from membership-function-independent to membership-function-dependent analysis
Eng Appl ArtifIntell
(2018)
H. Yan et al.
Input-output finite-time mean square stabilization of nonlinear semi-markovian jump systems
Automatica
(2019)
X. Zhang et al.
Robust stability of impulsive takagi-sugeno fuzzy systems with parametric uncertainties
Inf Sci
(2011)

V. Lakshmikantham et al.

Theory of impulsive differential Equations

(1989)

D.D. Bainov et al.

Systems with impulse effect: stability, theory and applications

(1989)

X.D. Li et al.

Stabilization of delay systems: delay-dependent impulsive control

IEEE Trans Autom Control

(2017)

X.D. Li et al.

An impulsive delay inequality involving unbounded time-varying delay and applications

IEEE Trans Autom Control

(2017)

W.H. Chen et al.

Global exponential synchronization of nonlinear time-delay lur’e systems via delayed impulsive control

Commun Nonlinear Sci NumerSimul

(2011)

I. Stamova

Global mittag-leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays

Nonlinear Dyn

(2014)

I. Stamova et al.

Global exponential stability of a class of impulsive cellular neural networks with supremums

Int J Adapt ControlSignal Process

(2014)

Z.H. Guan et al.

Decentralized stabilization of singular and time-delay large scale control systems with impulsive solutions

IEEE Trans Autom Control

(1995)

B. Liu et al.

Robust stability of uncertain discrete impulsive systems

IEEE Trans Circuit Syst II

(2007)

Z.H. Guan et al.

Robust decentralized stabilization for a class of large-scale time-delay uncertain impulsive dynamical systems

Automatica

(2002)

Z.H. Guan et al.

Robust $H_{\infty}$ control of singular impulsive systems with uncertain perturbations

IEEE Trans Circuit Syst II

(2005)

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^☆: This work was supported in part by the Self-Planned Task of State Key Laboratory of Robotics and Systems of Harbin Institute of Technology (SKLRS201801A03), the National Natural Science Foundation of China (61873311).

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Research paperFuzzy affine model-based output feedback controller design for nonlinear impulsive systems☆

Highlights

Abstract

Introduction

Section snippets

Uncertain T-S fuzzy affine impulsive models

Main results

Simulation studies

Conclusions

Automatica

Automatica

Commun Nonlinear Sci NumerSimul

Commun Nonlinear Sci NumerSimul

Automatica

Neurocomputing

Automatica

Eng Appl ArtifIntell

Automatica

Inf Sci

Theory of impulsive differential Equations

Systems with impulse effect: stability, theory and applications

Stabilization of delay systems: delay-dependent impulsive control

IEEE Trans Autom Control

An impulsive delay inequality involving unbounded time-varying delay and applications

IEEE Trans Autom Control

Global exponential synchronization of nonlinear time-delay lur’e systems via delayed impulsive control

Commun Nonlinear Sci NumerSimul

Global mittag-leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays

Nonlinear Dyn

Global exponential stability of a class of impulsive cellular neural networks with supremums

Int J Adapt ControlSignal Process

Decentralized stabilization of singular and time-delay large scale control systems with impulsive solutions

IEEE Trans Autom Control

Robust stability of uncertain discrete impulsive systems

IEEE Trans Circuit Syst II

Robust decentralized stabilization for a class of large-scale time-delay uncertain impulsive dynamical systems

Automatica

Robust H∞ control of singular impulsive systems with uncertain perturbations

IEEE Trans Circuit Syst II

Research paper
Fuzzy affine model-based output feedback controller design for nonlinear impulsive systems☆

Robust $H_{\infty}$ control of singular impulsive systems with uncertain perturbations