A flexible terminal approach to stochastic stability and stabilization of continuous-time semi-Markovian jump systems with time-varying delay
Introduction
To date, Markovian jump systems (MJSs) have received much attention due to their merits in describing complex systems with random abrupt variations. MJSs have wide applications in many fields, such as chemical processing, converter applications, aircraft control, robotics, etc. [1], [2], [3], [4], [5], [6], [7], [8], [9]. Over the past decades, the research of MJSs has found the framework of stability analysis [10], [11], [12], [13], [14], H∞ control, and filtering [15], [16], [17], [18]. It is noted that in [19], [20], [21], [22], [23], [24], the transition probabilities (TPs) are time-invariant. However, this assumption is not realistic in many real applications because the TPs in applications are not fixed when the environment suddenly varies. To overcome this shortcoming, nonhomogenous MJSs are introduced in [25], [26], where the TPs are allowed to be time-varying. In the reported literature [27], [28], the sojourn-time (ST) in nonhomogeneous MJSs obey the exponential probability distribution, which plays a limited role with jump time. To deal with this, SMJSs are proposed, where the TPs are time-varying. Note that SMJSs are more general than MJSs in real situations, because the ST obeys the Weibull distribution instead of an exponential distribution. Thereby, SMJNNs can be employed to model dynamic complex systems, which cover the conventional MJSs as special cases. Results for the stochastic stability analysis and control synthesis of SMJSs are available in [29], [30], [31], [32], [33], [34], [35].
Time delay is commonly encountered in various dynamic systems, for instance, networked control systems [36], [37], [38], [39], [40], [41], neural networks [42], [43], and chaotic systems [44], [45], [46], [47], [48]. It is a source of poor performance or instability of dynamic systems. To overcome the shortcomings of the time delay, widespread techniques have been employed, and numerous investigations on delay-dependent stability conditions of MJSs have been reported in the literature [49], [50]. Note that the stability analysis for MJSs subject to time delay are divided into two types: delay-dependent stability and delay-independent stability. When a time interval is utilized in delay-dependent stability conditions, this may achieve less conservative criteria than with delay-independent stability conditions. Recently, to achieve less conservative delay-dependent stability conditions, efficient augmented LKFs and free-weighting matrix techniques have been used. It can be seen from the existing literature that MJSs with time delays have attracted considerable research [51], [52], [53], [54]. However, to our knowledge, little effort has been devoted to the stabilization problem of SMJSs with time-varying delays. The aforementioned work of SMJSs can be improved by reducing conservatism, which is theoretically challenging and open in the research community. This motivates the current study.
The purpose of this study is to propose improved stability and stabilization criteria for SMJSs with time-varying delays for realistic circumstances. By using a flexible terminal approach and a novel SMLKF, improved stochastic stabilization conditions are developed. Finally, two examples are illustrated to depict the effectiveness and less conservatism of the proposed criteria.
Notation: In this work, all matrices are assumed to have proper dimensions. represents the n dimensional Euclidean space, the symbol * is used as an ellipsis for terms for symmetry. . In Ω, and respectively, stand for the sample space, subsets of sample space, and the probability measure. stands for the expectation operator.
Section snippets
Preliminaries
Let us consider the following SMJSs, defined on a fixed probability space:where is the state vector, is the input control, and and are constant matrices with proper dimensions.
The continuous-time semi-Markov chain rt takes values of subject to the transition probability matrix (TPM) as follows:where πpq(h) denotes the transition rate (TR)
Stochastic stability analysis
First of all, we consider the stability analysis of SMJS (1) with u(t) ≡ 0: Theorem 3.1 For given scalars τm > 0, τM > 0, μm > 0, and μM > 0, the SMJS (4)is stochastically stable, if there exist symmetric matrices Qip > 0, Si Pp > 0, R1 > 0, R2 > 0, H1 > 0, H2 > 0, U1 > 0, U2 > 0, W1 > 0, W2 > 0, and matrices Xj Jlp with proper dimensions, such that
Numerical examples and simulation
Example 1 Consider that the SMJS (4) with two subsystems has the following parameters [48]:
It is assumed that and and the TPM is given by . The largest value of the allowable upper bound τM can be obtained by using the results in Theorem 3.1, which is illustrated in Table 1. Compared with the existing results in [32], [57], [58], [60], one can see that a less conservative result has been achieved, see the
Conclusions
The stability and stabilization problems for SMJSs with time-varying delay have been investigated by using a flexible terminal approach. With the help of SMLKF and improved RCI techniques, some delay-dependent less conservative stability and stabilization conditions are proposed for SMJSs with time-varying delay. Note that to further reduce the synthesis of conservatism of developed criteria will be considered in the future work. Furthermore, the proposed results may be applied and extended to
Acknowledgment
This work was supported by the National Natural Science Foundation of China (61703150, 11701163), the Natural Science Foundation of Shandong Provinces of China (ZR2018LC010), the Program for Innovative Research Team of the Higher Education Institutions of Hubei Province (T201812).
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Stability and stabilization analysis of Markovian jump systems with generally bounded transition probabilities
2020, Journal of the Franklin InstituteCitation Excerpt :In recent years, Markovian jump systems (MJSs) have been intensively studied, in consequence of their excellent ability to model practical systems subjected to random abrupt variations. So far, great efforts have been made for either continuous-time or discrete-time MJSs, concerning various issues such as stability and stabilization analyses [1–13], robust controller design [14–24], and observer or filter design [22,25–33]. It is worthy to point out that almost all those methods on the design of controller, observer, or filter are developed on the basis of the stability and stabilization theories, which can facilitate the stability proofs of the closed-loop systems and observation error systems.