Stability and L1-gain analysis for switched positive T–S fuzzy systems under asynchronous switching
Introduction
Positive systems have received increasing attention because of their wide applications in the areas of economics, biological network, epidemiological network [1], [2], [3]. A dynamical system is called a positive system if its initial condition is nonnegative then its state is always confined to the positive orthant. In recent years, huge efforts have been devoted to the study of positive systems [4], [5], [6], [7], [8], [9], [10]. Switched positive systems, which are comprised of a group of positive subsystems and a switching signal governing the switching among them [11], [12], have also received considerable attentions [13], [14], [15], [16]. The applications of such systems can be found in many engineering areas, such as congestion control [17], wireless power control [18], synchronization problem [19].
Time delays and disturbances are often encountered in many practical systems, such as biological and engineering systems [20], [21], [22], [23]. They are the source of instability and even deteriorate the system performance. Until now, many efforts have been paid to systems with these issues [24], [25], [26], [27], [28], [29]. In [27], the stability problem of positive switched linear systems was studied. A weak excitation condition was proposed such that the studied system is exponentially stable. In [28], stability analysis and controller synthesis problems of a class of positive systems were investigated. Sufficient conditions guaranteeing the exponential stability were developed by using the average dwell time scheme. It is worth noting that most of the existing results on positive systems or switched positive systems mainly focused on linear dynamical systems. In fact, nonlinearities commonly exist in many physical systems. Due to the existence of nonlinearities, the methods developed for linear positive systems cannot be applied to nonlinear positive systems directly. It has been proven that Takagi–Sugeno (T–S) fuzzy model can provide an effective representation of a certain class of complex nonlinear systems under some mild conditions. Up to now, many numerous excellent results for T–S fuzzy systems have been reported (see, for example, [30], [31], [32], just to name a few). And in recent years, more increasing attentions have been paid to the stability analysis of positive T–S fuzzy systems [33], [34]. However, until now, there are few results on the stability analysis and controller synthesis problem of switched positive T–S fuzzy systems [35], [36], [37]. In [35], the stability problem of continuous-time switched positive T–S fuzzy systems were investigated. Some simple sufficient conditions guaranteeing the exponential stability were derived. Both the stability analysis and controller synthesis problems of switched positive T–S fuzzy systems were studied in [36]. In [37], the exponential stability and L2-gain analysis problem of T–S fuzzy systems was investigated by using the average dwell time technique.
On the other hand, it should be noted that most of existing results on switched positive T–S fuzzy systems were based on an ideal assumption that the controller has instant access to both the system state and switching signal. As pointed out by Wang et al. [38], the switching delays often exist in practical systems, which will make the dynamics of the closed-loop systems more complex. Therefore, proposing a more practical controller for switched systems under asynchronous switching is very necessary. Though many results for switched systems under asynchronous switching have been reported [38], [39], [40], [41], [42], [43], only few results have been obtained for switched positive systems with asynchronous switching [44]. To the best of our knowledge, the asynchronous control problem of switched positive T–S fuzzy systems has not been investigated yet. This is the main motivation of the study. In addition, as aforementioned, time delays and disturbance are often encountered in practical systems, how to analyze the asynchronous L1-gain performance of T–S fuzzy systems under both time-varying delays and disturbance is another motivation of the paper.
In the paper, the stability and L1-gain analysis problem of switched positive T–S fuzzy systems under asynchronous switching is investigated. The contributions of the paper can be summarized as follows. First, novel linear copositive Lyapunov functionals, which will increase during the running time of activated systems are designed. Second, a switching signal satisfying a mode-dependent dwell time (MDADT) is presented. Based the designed Lyapunov functionals and such a switching signal, solvable conditions of the exponential stability and L1-gain are developed for the studied system. Third, the desired controllers are also derived and an algorithm for solving such controllers are provided with the help of such solvable conditions. A numerical example is provided to illustrate the validity of the proposed method.
The rest of the paper is organized as follows. Some preliminary knowledges and problem formulation are provided in Section 2. Section 3 is devoted to the stability and L1-gain performance analysis, while Section 4 addresses the controller synthesis problem. In Section 5, a numerical example is provided to show the effectiveness of the proposed method. Section 6 draws the conclusion.
Notation. Throughout the paper, A ≫ 0(A ≪ 0) means all the entries of matrix A are nonnegative (nonpositive), and matrix A is called a nonnegative matrix (nonpositive matrix). denote the sets of nonnegative real numbers, positive real numbers, and nonnegative integer, respectively. denotes the n-dimensional Euclidean space. denotes a vector whose entries are all one. denotes a diagonal matrix. with xi being the ith element of .
Section snippets
Preliminaries and problem statement
T-S fuzzy systems are described by fuzzy IF-THEN rules. A switched positive fuzzy system with time-varying delay can be given by
Model Rule IF Ps1 is and Ps2 is and ⋅⋅⋅ and Psl is THEN where is the state vector, is the control input, is the measurement output, denotes external disturbance. Asj, Esi, Bsj, Fsj, Csj, Dsj are constant matrices with compatible
Exponential stability and L1-gain analysis
In this section, we will first investigate the exponential stability and the L1-gain of system (18). Then, in the next section, we will provide the controller synthesis problem for system (18).
The following theorem states one of the main results in the paper.
Theorem 1 Given for nonnegative matrices Cik, Dik, Eik, and any initial condition if there exist positive vectors such that is a Metzler matrix,
Controller synthesis
It should be noted that in previous section, Theorem 1 presents sufficient conditions for the stability and L1-gain analysis for system (18). However, due to the existence of Θ(i, i)jk and Θ(i, l)jk, inequalities (20) and (21) presented in Theorem 1 are not in the form of LMIs, which are very hard to be solved. The following theorem provides solvable conditions for the stability and L1-gain analysis for system (18).
Theorem 2 Given for nonnegative matrices Cik, Dik, Eik,
Numerical example
Consider the following switched nonlinear system where denotes the switching signal, and
Conclusion
In this paper, the stability and L1-gain analysis problem of switched positive T–S fuzzy systems with time-varying delays under asynchronous switching has been investigated. Some sufficient conditions guaranteeing the exponential stability and the expected system performance has been developed by adopting the MDADT technique. An algorithm to solve the desired controllers has also been presented. A numerical example has been provided to show the effectiveness of the proposed method. It is worth
Acknowledgment
This work was supported in part by the China Postdoctoral Science Foundation funded project under grant no. 2017M620555, National Natural Science Foundation of China under Grants 61533002, 61773086, 61722302, 61573069, and the Fundamental Research Funds for the Central Universities under Grant DUT16RC(3)033.
References (46)
- et al.
A new delay-sir model for pulse vaccination
Biomed. Signal Process. Control
(2009) - et al.
Positivity-preserving H∞ model reduction for positive systems
Automatica
(2011) - et al.
ℓ1-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach
Automatica
(2014) - et al.
Improved results on stability of continuous-time switched positive linear systems
Automatica
(2014) - et al.
Stability of switched positive linear systems with average dwell time switching
Automatica
(2012) - et al.
L1/ℓ1-gain analysis and synthesis of Markovian jump positive systems with time delay
ISA Trans.
(2016) - et al.
Guaranteed cost load frequency control for a class of uncertain power systems with large delay periods
Neurocomputing
(2015) - et al.
Input-to-state-stability and criteria for a class of hybrid dynamical systems
Appl. Math. Comput.
(2018) - et al.
Pursuing an evader through cooperative relaying in multi-agent surveillance networks
Automatica
(2017) - et al.
Stability, L1-gain and control synthesis for positive switched systems with time-varying delay
Nonlinear Anal. Hybrid Syst.
(2013)
Decentralized adaptive approximation-based fuzzy output-feedback control of uncertain switched stochastic interconnected nonlinear systems
J. Frankl. Inst.
Exponential stability analysis and L1 synthesis of positive T–S fuzzy systems with time-varying delays
Nonlinear Anal. Hybrid Syst.
Stability analysis for switched positive T–S fuzzy systems
Neurocomputing
Robust stability and L1-gain analysis of interval positive switched T— fuzzy systems with mode-dependent dwell time
Neurocomputing
Exponential stability analysis and L2-gain control synthesis for positive switched T–S fuzzy systems
Nonlinear Anal. Hybrid Syst.
Stability and L2-gain analysis of switched input delay systems with unstable modes under asynchronous switching
J. Frankl. Inst.
Lyapunov–Krasovskii functionals for switched nonlinear input delay systems under asynchronous switching
Automatica
Dynamic output feedback control for a class of switched delay systems under asynchronous switching
Inf. Sci.
Stabilization of networked switched linear systems: an asynchronous switching delay system approach
Syst. Control Lett.
Stabilization of switched delay systems with polytopic uncertainties under asynchronous switching
J. Frankl. Inst.
Asynchronous L1 control of delayed switched positive systems with mode-dependent average dwell time
Inf. Sci.
Asynchronous H∞ control of switched delay systems with average dwell time
J. Frankl. Inst.
The conservation of information, towards an axiomatized modular modeling approach to congestion control
IEEE Trans. Netw.
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