Incomplete Transition Descriptions Research Articles

Improved aperiodic sampled-data control for singular Markov jump systems under incomplete transition descriptions

This paper studies the aperiodic sampled-data control problem of singular Markov jump systems under uncertain and unknown transition rates. First, taking into account the system information from x(tk) to x(tk+1), a novel two-sided looped mode-dependent Lyapunov–Krasovskii function (LKF) is constructed, which provides more freedom to test the feasibility of the LMI condition in the criterion. Based on this LKF and an improved integral inequality, a stochastic admissibility condition is given by linear matrix inequalities (LMIs), which is less conservative. Then, by using a lemma of equivalent set, a controller algorithm without equality constraint is proposed by LMIs to ensure the stability of the system. Finally, several examples are provided to verify the validity of the main results.

Nonquadratic local stabilization of nonhomogeneous Markovian jump fuzzy systems with incomplete transition descriptions

In this paper, the local stabilization problem of continuous-time nonhomogeneous Markovian jump fuzzy systems with incomplete transition descriptions is investigated using a mode-dependent nonquadratic Lyapunov function. Different from other existing studies, this paper proposes a method that can improve the feasibility of local stabilization conditions (i) by not assuming the maximum allowable bound of the time-derivative of the fuzzy basis functions, and (ii) by utilizing hidden information of the partial derivative of the fuzzy weighting functions with respect to the state variables. Furthermore, since the problem of estimating the attraction region is addressed with consideration of energy-bounded disturbances, the proposed method can be directly applied to the H∞ control problem without any other assumptions on the magnitude of disturbances.

Finite-time Stability and Stabilization of Markovian Jump Linear Systems Subject to Incomplete Transition Descriptions

Finite-time Stability and Stabilization of Markovian Jump Linear Systems Subject to Incomplete Transition Descriptions

Dissipative control of interval type-2 nonhomogeneous Markovian jump fuzzy systems with incomplete transition descriptions

This paper investigates the problem of designing a dissipative controller for interval type-2 nonhomogeneous Markovian jump fuzzy systems (MJFSs) with incomplete transition descriptions. To explore a generalized model for MJFSs under realistic situations, the coexistence of incompletely known transition rates and mismatched membership functions is considered in MJFS models, and the corresponding stabilization conditions are derived in terms of multi-parameterized linear matrix inequalities (M-PLMIs). In addition, to handle the problem of M-PLMIs, two novel relaxation techniques are additionally proposed such that the use of redundant constraints on multiple time-varying parameters can be significantly reduced. Finally, the validity of the proposed method is demonstrated via two illustrative examples.

Stability and Stabilization for Discrete-time Markovian Jump Stochastic Systems with Piecewise Homogeneous Transition Probabilities

In this paper, the stability and stabilization problems for discrete-time Markovian jump stochastic systems with time-varying transition probabilities are investigated. The time-varying character of the transition probabilities is considered to be finite piecewise homogeneous and the variations in the finite set are considered to be a stochastic variation. First, a stability criterion is derived to guarantee the stochastic stability of the considered piecewise homogeneous Markovian jump stochastic system. Further, a sufficient condition on the existence of a mode-dependent state feedback controller is proposed such that the resulting closed-loop system is stochastically stable. In addition, the stability and stabilization problems are studied for the piecewise homogeneous Markovian jump stochastic system with incomplete transition descriptions. Finally, some simulation results are given to show the validity and potential of the developed results.

Stochastic stability analysis of Markovian jump linear systems with incomplete transition descriptions

In this study, the authors focus on the stability analysis for Markovian jump linear systems with partly known transition rates in the continuous-time domain and partly known transition probabilities in the discrete-time domain. By using the properties of the transition rates and transition probabilities, two new sufficient conditions are derived for the stochastic stability of the continuous-time and discrete-time Markovian jump linear systems, respectively. The main advantage of the proposed stability conditions is that the total number of linear matrix inequalities (LMIs) in the proposed stability conditions is much less than that in some existing results. Based on the presented stability conditions, two state feedback controllers are designed for the considered systems in terms of LMIs. In addition, two kinds of stability criteria are developed for the stochastic stability of the considered systems with incomplete transition descriptions by the existence of the unique positive definite solution of the coupled Lyapunov matrix equations. Finally, two numerical examples and one practical example are provided to verify the correctness of the theoretical results.

Formula omitted] control of Markovian jump LPV systems with measurement noises: Application to a DC-motor device with voltage fluctuations

This paper concerns the stabilization problem of Markovian jump LPV systems with incomplete transition descriptions and measurement noises. The attention of this paper is focused on two aspects: (1) proposing a quadratic combination method of totally incorporating the transition rate constraints into the H2 stabilization condition, and (2) developing a nonlinear control strategy of clearly eliminating the influence of measurement disturbances in the control process. Continuing this, the derived stabilization condition is applied to the speed control of DC-motor devices with voltage fluctuations, where the features of fluctuating voltage are modeled in both stochastic and deterministic senses.

Relaxed nonquadratic stabilization conditions for Markovian jump fuzzy systems with incomplete transition descriptions

This paper focuses on deriving relaxed mean-square stability and H∞ performance conditions for discrete-time MJFSs under a general description of transition probabilities. In the derivation, some available constraints are exploited on the basis of the property of transition probabilities, and then a two-stage relaxation scheme is proposed to reduce the conservatism that can arise from incomplete knowledge of probabilities as well as from fuzzy basis functions. Finally, nonquadratic stabilization conditions are derived in terms of LMIs, whose effectiveness is shown by two illustrative examples.

Delay-dependent stability analysis for singular Markovian jump systems with incomplete transition probabilities

This paper investigates the problem of stochastic stability analysis for a class of discrete-time singular Markovian jump systems with time-varying delays and incomplete (i.e., unknown and uncertain) transition probabilities. To derive an improved delay-range-dependent stability criterion under the incomplete transition descriptions, a relaxation scheme is proposed with consideration of the free-weighting matrix method compatible with the delay partitioning technique, and its purpose is accomplished (1) by making full use of the available boundary information on incomplete transition probabilities, and (2) by employing novel slack variables that allow some useful equality constraints to be exploited.

New approach to delay‐dependent H ∞ filtering for discrete‐time Markovian jump systems with time‐varying delay and incomplete transition descriptions

This study is concerned with the delay-dependent H∞ filter design for a class of discrete-time Markovian jump linear systems (MJLSs) with time-varying delay and incomplete transition descriptions. The considered systems with incomplete transition descriptions cover the MJLSs with known transition probabilities (TPs), partially unknown TPs and uncertain TPs, which are more general. A new equivalent model is proposed for the original MJLSs by employing a two-term approximation method, which formulates the filtering problem in the framework of input–output stability. Based on a Markovian Lyapunov–Krasovskii functional combined with the scaled small gain theorem, a new delay-dependent bounded real lemma for the underlying systems is established. It is shown that by using a linearisation technique, the corresponding full- and reduced-order H∞ filter design is cast into a convex optimisation problem in terms of linear matrix inequalities. Finally, simulation examples are provided to illustrate the effectiveness and less conservatism of the proposed approach.

Improved fuzzy control design for nonlinear Markovian-jump systems with incomplete transition descriptions

This paper addresses the state feedback control of nonlinear continuous-time, Markovian-jump systems. The nonlinearity is represented by Takagi–Sugeno fuzzy models and the transition probability matrix is assumed to be partly known: some elements in the matrix are known, some are unknown but with known lower and upper bounds, and some are completely unknown. By making full use of the continuous property of the transition probability matrix, new sufficient conditions for the stochastic stability of the system are obtained in terms of linear matrix inequalities. We show that the conditions given are less conservative than or at least the same as those for existing results. Moreover, using the conditions obtained, we establish a method for design of a H∞ state feedback controller. Numerical examples illustrate the effectiveness of the proposed method.

Necessary and Sufficient Conditions for Analysis and Synthesis of Markov Jump Linear Systems With Incomplete Transition Descriptions

This technical note is concerned with exploring a new approach for the analysis and synthesis for Markov jump linear systems with incomplete transition descriptions. In the study, not all the elements of the transition rate matrices (TRMs) in continuous-time domain, or transition probability matrices (TPMs) in discrete-time domain are assumed to be known. By fully considering the properties of the TRMs and TPMs, and the convexity of the uncertain domains, necessary and sufficient criteria of stability and stabilization are obtained in both continuous and discrete time. Numerical examples are used to illustrate the results.