Mixed H∞ and passive control for a class of nonlinear switched systems with average dwell time via hybrid control approach
Introduction
During recent decades, the switched systems have been extensively investigated because many systems encountered in practice possess switching features [1]. Generally speaking, a typical switched system comprises some discrete-time or continuous-time subsystems and a switching signal. The switching signal orchestrates the switching among these subsystems. So far, the problems of stability analysis and control synthesis for switched systems have received considerable attention. Some useful methodologies have been developed to solve the above problems [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. For the stability analysis of switched systems under arbitrary switching, the main method is constructing a common quadratic Lyapunov function for all subsystems [1]. For switched systems under constrained switching, together the multiple Lyapunov functions approach [2] with the average dwell time switching [3] may lead to well analysis results.
Most of the above work are focused on the study about linear switched systems. As we know, the nonlinearity is inevitably encountered in the real world. Therefore, it is theoretical significance and practical importance for us to study the nonlinear switched systems. However, the existence of nonlinearity makes it difficult to analyze nonlinear switched systems directly. For nonlinear control systems, it has been proven that the T–S fuzzy model is an effective approach to deal with the nonlinearity [15]. The T–S fuzzy model utilizes the local linear system description for each rule, then connects these local models using if-then rules. Since the linear system theory can be used to investigate the stability analysis and controller synthesis problems of nonlinear systems by the T–S fuzzy model, many relevant results have be reported in the literature [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Furthermore, using the T–S fuzzy model to represent every nonlinear subsystem of the nonlinear switched systems, the nonlinear switched systems can be modeled as the switched T–S fuzzy systems. Recently, some efforts have been made to study the nonlinear switched systems through the T–S fuzzy model [30], [31], [32].
During the past few decades, the H∞ control theory has attracted more and more attention due to its significant advances [33], [34]. Many valuable results about the H∞ controller design for switched systems have been reported in the recent literature [7], [8], [9]. On the other hand, the passivity theory is another effective tool to study the dynamical systems [35]. Furthermore, the passivity concept has been extended to switched systems [36], [37]. Then, a natural and challenging problem arises: Can the H∞ control problem and the passive control problem for switched systems be solved in a unified framework?
Very recently, the mixed H∞ and passive control/filtering problems are studied for a variety of dynamical systems [38], [39], [40]. However, as far as we know, less attention has been paid to the mixed H∞ and passive control problem for switched systems. The study of the mixed H∞ and passive control problem for switched systems is different from that for the above systems [38], [39], [40]. The difficulties of studying the mixed H∞ and passive control problem for switched systems exist in the following two aspects: On one hand, it has been pointed in [41] that the switched systems with average dwell time can only achieve a weighted H∞ performance index; On the other hand, the existence of switching among the subsystems makes the mixed H∞ and passive control problem of switched systems much more complicated.
To overcome the above two difficulties, the following two steps are adopted in our work. Firstly, a new performance index is proposed for switched systems. This new performance index can be viewed as the mixed weighted H∞ and passivity performance index. Secondly, the hybrid control strategy is used to stabilize the switched systems. The hybrid control strategy can make full use of the switching behaviors. Under the hybrid control strategy, the boundary conditions of switched systems can be incorporated into the synthesis problem in a convex formulation.
In practice, state variables are not always available in many control systems. To deal with this problem, an effective method is to design output feedback controllers using the measurable output signals. Recently, the design of dynamic output feedback controllers has become a hotspot [7], [8], [16], [42], [43]. On one hand, additional dynamics of the controllers make the dynamic output feedback controllers more flexible than static output feedback controllers. On the other hand, the problem of designing dynamic output feedback controllers can be formulated as a convex optimisation over LMIs. Based on the above considerations, the dynamic output feedback controllers are constructed for every subsystem. The hybrid controllers of our work also include the state updating controllers at the switching instant. Some existing work have studied the hybrid control problem for switched systems. [7], [8]. The main differences between our work with [7], [8] lie in: The systems studied in [7], [8] were linear switched systems. The systems considered in our work are nonlinear switched systems; The mixed weighted H∞ and passivity performance index is used in our work, which can include the weighted H∞ performance index used in [7], [8] as a special case.
In this paper, the mixed H∞ and passive control problem for switched T–S fuzzy systems is studied. The main contributions of this work are listed as follows: (I) A new performance index is proposed for switched systems. (II) Using this new performance index, the weighted H∞ control problem and the passive control problem for switched T–S fuzzy systems via the hybrid control strategy are solved in a unified framework. (III) New sufficient conditions for the existence of hybrid controllers are obtained. (IV) The obtained conditions are more general because of the use of this new performance index.
The organization of this paper is given as follows. The problem formulation and preliminaries are presented in Section 2. The main results are given in Section 3. In Section 4, a numerical example is presented. Finally, some conclusions are drawn in Section 5.
Notations: The notations used in this paper are fairly standard. Rn represents the n-dimensional Euclidean space. In a matrix, the symbol “*” stands for the transposed elements in the symmetric positions. MT denotes the transpose of the matrix M. For a matrix M, He{M} denotes . I and 0 represent the identity matrix and zero matrix in the block matrix, respectively. For a vector, ‖ · ‖ denotes its Euclidean norm. The space of square-integrable functions is denoted by L2[0, ∞). For v(t) ∈ L2[0, ∞), represents its norm. The space of continuously differentiable function is represented by C1. The symbol M > 0 ( ≥ 0, < 0, ≤ 0) is used to denote a positive definite (semi-positive definite, negative definite, semi-negative definite) matrix M. If not explicitly stated, matrices are assumed to have compatible dimensions.
Section snippets
Problem formulation and preliminaries
Let us consider the following continuous-time nonlinear switched systems where is the state vector, is the control input, is the measurement output, is the controlled output, and is the disturbance input that belongs to L2[0, ∞). fσ(t)( · ), sσ(t)( · ) and gσ(t)( · ) are nonlinear functions. A piecewise constant, right-continuous function of time is the
Main results
Lemma 2 For the given nonlinear switched systems (1) and constants α > 0, γ > 0, μ ≥ 1, 0 ≤ θ ≤ 1, if there exist positive definite C1 functions : Rn → R, with satisfying
and
where then the nonlinear switched systems (1) are GUAS with a mixed weighted H∞ and passivity performance index γ for any switching signal satisfying
Proof
Numerical example
To show the effectiveness of the obtained results, a numerical example is given. Consider the following switched T–S fuzzy system, which contains two subsystems (), and each subsystem has two fuzzy rules ()
Conclusion
This paper, based on the hybrid control strategy, investigates the mixed H∞ and passive control problem for a class of nonlinear switched systems. Using the T–S fuzzy model to approximate every subsystem, the nonlinear switched systems are modeled as the switched T–S fuzzy systems. Then, the mixed weighted H∞ and passivity performance index is proposed for switched systems. Combining the ADT technique with the MLFs approach, new sufficient conditions for the hybrid controller design are
Acknowledgment
This work was supported by the Open Research Fund of AnHui Key Laboratory of Detection Technology and Energy Saving Devices, AnHui Polytechnic University (Grant no. 2017070503B026-A07), the Natural Science Research of Anhui higher education promotion program (Grant no. TSKJ2017B25), the Priming Scientific Research Foundation for the Introduction Talent in Anhui Polytechnic University (Grant no. 2017YQQ002), the Anhui Natural Science Foundation (Grant no. 1408085ME105), and the National Natural
References (46)
- et al.
Asynchronously switched control of switched linear systems with average dwell time
Automatica
(2010) - et al.
Disturbance tolerance and rejection of discrete switched systems with time-varying delay and saturating actuator
Nonlinear Anal. Hybrid Syst.
(2015) - et al.
Output feedback stabilization of switching discrete-time linear systems with parameter uncertainties
J. Frankl. Inst.
(2017) - et al.
Finite-time H∞ output tracking control for a class of switched neutral systems with mode-dependent average dwell time method
Int. J. Innov. Comput. Inf. Control
(2017) - et al.
A novel approach to output feedback control of fuzzy stochastic systems
Automatica
(2014) - et al.
Robust static output feedback H∞ control for uncertain fuzzy systems
Fuzzy Sets Syst.
(2015) - et al.
Robust state/fault estimation and fault tolerant control for T–S fuzzy systems with sensor and actuator faults
J. Frankl. Inst.
(2016) - et al.
Adaptive fuzzy tracking control for a class of pure-feedback nonlinear systems with time-varying delay and unknown dead zone
Fuzzy Sets Syst.
(2017) - et al.
Asynchronous H∞ fuzzy control for a class of switched nonlinear systems via switching fuzzy Lyapunov function approach
Neurocomputing
(2016) - et al.
H∞ filtering for a class of nonlinear switched systems with stable and unstable subsystems
Signal Process.
(2017)
Passivity-based sliding mode control of uncertain singular time-delay systems
Automatica
Mixed H∞ and passive control for singular systems with time delay via static output feedback
Appl. Math. Comput.
Robust mixed H∞ and passive filtering for networked Markov jump systems with impulses
Signal Process.
Disturbance attenuation properties of time-controlled switched systems
J. Frankl. Inst.
Switching in Systems and Control
Multiple Lyapunov functions and other analysis tools for switched and hybrid systems
IEEE Trans. Autom. Control
Stability of switched systems with average dwell time
Proceedings of the Thirty Eighth IEEE Conference Decision and Control, Phoenix, AZ
Stability of stochastic nonlinear systems with state-dependent switching
IEEE Trans. Autom. Control
Hybrid control for switched linear systems with average dwell time
IEEE Trans. Autom. Control
Asynchronous switching output feedback control of discrete-time switched linear systems
Int. J. Control
Stability, L2-gain and asynchronous H∞ control for continuous-time switched systems
Int. J. Robust Nonlinear Control
Distributed filtering for switched linear systems with sensor networks in presence of packet dropouts and quantization
IEEE Trans. Circuits Syst. I Reg. Pap.
Adaptive NN dynamic surface controller design for nonlinear pure-feedback switched systems with time-delays and quantized input
IEEE Trans. Syst. Man Cybern. A. Syst
Cited by (34)
Asynchronous observer design for switched T–S systems with unmeasurable premises and switching mismatches
2021, Engineering Applications of Artificial IntelligenceCitation Excerpt :Note that these simulation examples have been implemented in Matlab (using the ode23 solver) and the LMI conditions of the above proposed theorems have been solved using the YALMIP Toolbox (Lofberg, 2004) with the semidefinite programming solver SeDuMi (Labit et al., 2002). The goal of this academic example is to discuss the conservatism of the LMI-based conditions proposed in Theorems 1–4, with respect to previous results (Garbouj et al., 2019; Zheng et al., 2018b; Hong et al., 2018; Belkhiat et al., 2019). Note that, from the previous literature, we failed to find suitable LMI-based conditions for switched T–S observers that exhibit both UPVs and asynchronous switched modes.
Stochastic H<inf>∞</inf> finite-time control for linear neutral semi-Markovian jumping systems under event-triggering scheme
2021, Journal of the Franklin InstituteFinite-time stabilization of switched neutral systems with time-varying delays via sampled-data control
2020, Journal of the Franklin InstituteCitation Excerpt :Switching signal with average dwell time (ADT) was introduced in [39] and has become standard since then; it includes dwell time (DT) as a special case. According to literatures [40–46], the minimum value of permissible ADT can be determined by the two mode-independent parameters. However, in terms of all mode-independent subsystems, settings of the two common parameters may result in some extent conservativeness.
Finite-time resilient fault-tolerant investment policy scheme for chaotic nonlinear finance system
2020, Chaos, Solitons and FractalsCitation Excerpt :Xin et al.[25] employed Jacobian predictor-corrector approach to design a finite-time controller to stabilize the fractional-order chaotic finance system within a precise period of time. In the existing literature, extended passivity based control has been widely investigated to eliminate unpredictable events and to handle fickle investment policy that can be regarded as external disturbances [32–36]. Recently for singular systems Chen et al.[32] developed an efficient extended passivity performance based static output feedback controller and in [36] a sufficient conditions are derived to ensure the asymptotic stability in mean-square sense for the stochastic nonlinear systems by using extended passivity control.
Indirect adaptive fuzzy control of nonlinear descriptor systems
2020, European Journal of ControlCitation Excerpt :An LMI method to design state-space H∞-controllers for linear time-invariant singular systems is proposed by Inoue et al. [16]. In [45], the mixed H∞ and passive control problem is studied for descriptor systems with time-delay in order to make the considered system regular, impulse-free and stable. Further, other design schemes for descriptor systems have been developed such as a nonlinear feedback control law based on the problem of asymptotic output tracking for a class of nonlinear descriptor systems [44] and a nonlinear model-following control for fuzzy descriptor systems [36].
Vector incremental L <inf>2</inf> -gain and incremental stability for switched nonlinear systems
2019, Journal of the Franklin Institute