Robust and non-fragile finite-time H∞ control for uncertain Markovian jump nonlinear systems
Introduction
In the past decades, Markovian jump systems (MJSs), as a special kind of hybrid systems, have received increasing attention due to their extensive applications in a variety of areas, including economics, biomedicine, fault tolerant systems, communication networks, and so on. It has been recognized that a number of dynamic systems with stochastic abrupt changes can be represented by MJSs, such as sudden environmental changes, changing subsystem interconnections and component failures [1], [2], [3], [4]. Therefore, many appealing results related to MJSs have been investigated by applying linear matrix inequality (LMI) approach and stochastic analysis techniques. For example, the robust stability and stabilization problems were considered in [5], [6], the analysis and design of the H∞ control and filtering were discussed in [7], [8], [9], [10], the observer design and fault-tolerant control were presented in [11], [12], [13], the non-fragile and robust control problems were investigated in [14], [15]. It should be mentioned that the study of Markovian jump nonlinear systems (MJNSs) is rarely investigated and at present mainly concentrates on the specially structured MJNSs. Generally speaking, nonlinear subsystems of such MJNSs should be satisfying sector-bounded condition, tridiagonal structure, Lipschitz condition, fuzzy Markovian jump form or other special structures [16], [17], [18], [19], [20], [21], [22].
On the other hand, in practical applications, the main concern is the transient behavior of the system which described system state does not exceed a certain threshold over a given finite time interval. For example, there exists some cases where large values of the system states are not acceptable in the presence of saturations [23]. In order to deal with the transient behavior, finite-time stability was introduced in [24]. Then, finite-time stability, finite-time boundedness and finite-time stabilization in [25], [26] were investigated for continuous- or discrete-time systems by using Lyapunov function approach and LMI techniques. Recently, the analysis and design of finite-time state estimation, finite-time H∞ control and finite-time H∞ filtering were also tackled for varieties of systems in [27], [28], [29], [30], [31], [32], [33], [34], [35]. In this work, a class of MJNSs are considered that every nonlinear subsystem lies in a hypersphere and whose center is a linear subsystem with uncertainties and whose radius is bounded by the norm of another linear subsystem. The non-fragile controller design method is an attractive topic in that it can minimize the cost of implementation, and allow for the adjustment online of control parameters in theory analysis and practical applications. The main objective of non-fragile control is design a feedback control such that it can tolerate some level of controller gain variations in feedback control gains in [36], [37], [38], [39]. It is noted that the study of finite-time control is rarely tackled for nonlinear systems in the literature [20], [21], [40], [41], [42], [43]. To the best of our knowledge, the problem of robust and non-fragile finite-time H∞ control has not been investigated for the class of uncertain MJNSs, which still keeps open and motivates our study.
In this paper, the problem of non-fragile and robust finite-time H∞ control is studied for a class of uncertain MJNSs with bounded parametric uncertainties and norm-bounded disturbance. Firstly, sufficient criteria of stochastic finite-time boundedness and stochastic H∞ finite-time boundedness are provided for the class of stochastic jump systems by utilizing stochastic analysis techniques and LMI methods. Then, a controller is designed such that the class of stochastic nonlinear dynamics are stochastically finite-time bounded and have an H∞ attention performance level. Furthermore, the analysis and design of non-fragile and robust finite-time controller are provided to guarantee that the class of uncertain MJNSs are stochastically finite-time bounded with a prescribed attention index by employing non-fragile control techniques. As a special case, we also deal with the analysis and design of stochastic finite-time stability and stochastic finite-time stabilization. All the criterions can be formulated in terms of LMIs. Finally, two examples are also given to illustrate the effectiveness of obtained results. The contributions of this paper lies in the following three aspects: (i) by applying the existing Matlab LMI toolbox and matrix decomposition techniques, stochastic H∞ finite-time boundedness analysis is provided for the class of stochastic jump nonlinear systems; (ii) when additive bounded control gains appear in closed-loop MJNSs, a sufficient criteria to design a non-fragile finite-time controller is presented in terms of LMIs by applying non-fragile control methods; and (iii) as affiliated results, conditions on stochastic finite-time stability and stochastic finite-time boundedness are also established for uncertain MJNSs with or without the presence of external disturbance. Therefore, the main purpose of this paper is to make the first attempt to tackle the aforementioned contributions.
The rest of the paper is organized as follows. The problem formulation is given in Sections 2 and 3 presents our main results. Numerical examples are provided in Section 4 to illustrate the effectiveness of the developed approach. Finally, Section 5 concludes the paper.
Notations. and represent the sets of n component real vectors and n × m real matrices, respectively. E{.} denotes the expectation operator with respective to some probability measure . X > 0 means that X is a real symmetric and positive matrix. ‖x‖ is the Euclidean norm of vector x. I is used to stand for an identity matrix of appropriate dimensions. The superscript T denotes for matrix transposition. The symbol * stands for the transposed elements in the symmetric positions of a matrix, and diag{⋅⋅⋅} stands for a block-diagonal matrix. In addition, He(B) denotes . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
Section snippets
Problem formulation
In this paper, {rt, t ≥ 0} is right-continuous Markov stochastic process taking values in a finite space with transition matrix and the transition probabilities are described as follows: where πij satisfies πij ≥ 0(i ≠ j) and for all i ∈ Λ. To simplify the presentation of this paper, in the sequel, for each possible matrix M(rk) will be denoted by Mi, will be denoted by
Main results
In this section, our main objective is to design a robust and non-fragile finite-time H∞ controller (8) to guarantee that the MJNS (9a) and (9b) is SH∞FTB. We first give the following conditions which can ensure the stochastic finite-time boundedness of the MJNS (9a), which will derive our main results.
Theorem 1 Given that T > 0, c1 > 0, d > 0 and α ≥ 0. The closed-loop MJNS (9a) is SFTB with respect to (c1, c2, T, Ri, d), if there exist positive scalars η1, η2, c2, a set of scalars {ϵi > 0, i ∈ Λ},
Numerical examples
In this section, we provide two numerical examples to show the proposed results.
Example 1 Consider a two-mode MJNS (9a) and (9b) with
The controlled output is chosen as where
In addition, the nonlinear functions are chosen by
Thus, in order to satisfy the constraint (3), we can choose
Conclusions
This paper is concerned with non-fragile and robust finite-time H∞ control for uncertain MJNSs with bounded parametric uncertainties and norm-bounded disturbance. Sufficient criteria of stochastic finite-time stability, stochastic finite-time boundedness and stochastic H∞ finite-time boundedness are provided for the class of stochastic jump systems by utilizing stochastic analysis and LMI techniques. The main contribute of this paper is to design a non-fragile and robust finite-time controller
Acknowledgements
This work was partially supported by the Natural Science Foundation of Henan Province of China (132300410013), the Plan of Nature Science Fundamental Research in Henan University of Technology (2012JCYJ13), the Australian Research Council (DP140102180, LP140100471), the 111 Project (B12018), the National Natural Science Foundation of China (61174058, 61573112), and the Joint Key Grant of National Natural Science Foundation of China and Zhejiang Province (U1509217). The authors would like to
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