Stochastic stability and robust control for sampled-data systems with Markovian jump parameters

In this paper, the problems of stochastic stability and robust control for a class of uncertain sampled-data systems are studied. The systems consist of random jumping parameters described by finite-state semi-Markov process. Sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented. The conditions for the existence of a sampled-data feedback control and a multirate sampled-data optimal control for the continuous-time uncertain Markovian jump systems are also obtained. The design procedure for robust multirate sampled-data control is formulated as linear matrix inequalities (LMIs), which can be solved efficiently by available software toolboxes. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques.     

Stochastic stability of positive Markov jump linear systems with fixed dwell time

This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.     

受控马尔可夫跳线性系统的随机鲁棒稳定性

本文讨论了受控连续和离散时间马尔可夫跳线性系统的随机鲁棒稳定性,并且给出了此时该系统发生马尔可夫跳的转移速率的一个界.     

State feedback controller design for singular positive Markovian jump systems with partly known transition rates

The paper deals with the problem of state feedback controller design for singular positive Markovian jump systems with partly known transition rates. First, by applying an appropriate linear co-positive type Lyapunov–Krasovskii function, stochastic stability of the underlying systems is discussed. Based on the results obtained, a state feedback controller is constructed such that the closed-loop singular Markovian jump system is regular, impulse-free, positive and stochastically stable. All the provided conditions are based on a reliable computational approach in linear programming. Finally, an example is given to demonstrate the validity of the main results.     

Improved results for stochastic stabilization of a class of discrete-time singular Markovian jump systems with time-varying delay

In this paper, the problem of stochastic stabilization for a class of discrete-time singular Markovian jump systems with time-varying delay is investigated. By using the Lyapunov functional method and delay decomposition approach, improved delay-dependent sufficient conditions are presented, which guarantee the considered systems to be regular, causal and stochastically stabilizable. Finally, some numerical examples are provided to illustrate the effectiveness of the obtained methods.     

Robust stabilization of a class of state-dependent jump linear systems

In this article, we consider a continuous-time state-dependent jump linear system (SDJLS), a kind of stochastic hybrid system, with the presence of uncertainties in system parameters. In SDJLS, we consider that the transition rates of the underlying random jump process depend on the state variable. In particular, we assume the transition rates to have different values across suitably defined sets to which the state of the system belongs, and address a problem of robust stability and stabilization analysis. We obtain sufficient conditions for robust stability and state-feedback stabilization in terms of linear matrix inequalities (LMIs). We validate the obtained sufficient robust stability and stabilization conditions with numerical examples.     

Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters

In this paper, the problem of stochastic stability for a class of time-delay Hopfield neural networks with Markovian jump parameters is investigated. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. Without assuming the boundedness, monotonicity and differentiability of the activation functions, some results for delay-dependent stochastic stability criteria for the Markovian jumping Hopfield neural networks (MJDHNNs) with time-delay are developed. We establish that the sufficient conditions can be essentially solved in terms of linear matrix inequalities.     

Lyapunov coupled equations for continuous-time infinite Markov jump linear systems

This paper deals with Lyapunov equations for continuous-time Markov jump linear systems (MJLS). Out of the bent which wends most of the literature on MJLS, we focus here on the case in which the Markov chain has a countably infinite state space. It is shown that the infinite MJLS is stochastically stabilizable (SS) if and only if the associated countably infinite coupled Lyapunov equations have a unique norm bounded strictly positive solution. It is worth mentioning here that this result do not hold for mean square stabilizability (MSS), since SS and MSS are no longer equivalent in our set up (see, e.g., [J. Baczynski, Optimal control for continuous time LQ-problems with infinite Markov jump parameters, Ph.D. Thesis, Federal University of Rio de Janeiro, UFRJ/COPPE, 2000]). To some extent, a decomplexification technique and tools from operator theory in Banach space and, in particular, from semigroup theory are the very technical underpinning of the paper.     

Stability of a class of networked control systems with Markovian characterization

This paper studies the stability problem for a class of networked control systems (NCSs) with the plant being a Markovian jump system. The random delays from the sensor to the controller and from the controller to the actuator are modeled as two Markov chains. The necessary and sufficient conditions for the stochastic stability are established. The state-feedback controller gain that depends on not only the delay modes but also the system mode is obtained through the iterative linear matrix inequality approach. An illustrative example is presented to demonstrate the effectiveness of the proposed method.     

Robust H∞ control for linear Markovian jump systems with unknown nonlinearities

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H_∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique.     

Stability analysis for stochastic jump systems with time-varying delay

This paper deals with the mean-square asymptotic stability of stochastic Markovian jump systems with time-varying delay. Based on a new stochastic inequality and convex analysis property, some novel stability conditions are presented. In the derivation, the information of the time-varying delay is retained and the estimation of it by the worst-case enlargement is not involved. Some special cases of the systems under consideration are also investigated. Illustrative examples are given to show the effectiveness of the proposed approach.     

Stability,stabilizability and detectability for Markov jump discrete-time linear systems with multiplicative noise in Hilbert spaces

In this article we discuss stability, stabilizability and detectability problems for Markov-jump discrete-time linear systems (MJDLSs) with multiplicative noise (MN) and countably infinite state space of the Markov chain. On the basis of a new solution representation formula, we give new deterministic characterizations of the stability and the detectability properties of MJDLSs with MN. These results are obtained using an operatorial approach and the properties of certain positive evolution operators defined on ordered Banach spaces of sequences of nuclear operators. Assuming detectability conditions and avoiding stochastic proofs, we prove that any global, nonnegative and bounded solution of the Riccati equation of control is stabilizing for the MJDLSs with MN and control. Finally, we apply our results to solve a linear quadratic optimal control problem. The theory is illustrated by an example.     

Stochastic versus mean square stability in continuous time linear infinite Markov jump parameter systems

We deal with linear systems with Markovian Jump Parameters (LSMJP). Most of the literature on this matter adopts a finite state space for the Markov chain. In this paper we focus on the countably infinite state space case showing that, unlike the finite state space case, two important concepts in optimal control theory, namely, stochastic stability (SS) and mean square stability (MSS) are no longer equivalent in this setting.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Optimal Guaranteed Cost Control of Stochastic Discrete-Time Systems with States and Input Dependent Noise Under Markovian Switching

A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples are given to demonstrate the potential of the proposed techniques and obtained results.     

HMM-based quantized dissipative control for 2-D Markov jump systems

In this paper, the dissipative quantized control problem is addressed for Markov jump two-dimensional systems based on Roesser model, in which both asynchronous phenomenon and signal quantization between system modes and controller modes are taken into consideration simultaneously. Moreover, the hidden Markov model (HMM) is adopted to tackle such an asynchronous phenomenon. The principal goal is to devise a state feedback controller, which guarantees that the established closed-loop system achieves asymptotic mean square stability as well as satisfies a prescribed extended dissipative property. Drawing support from Lyapunov function approach and inequality technique, some less conservative criteria ensuring the implementability of the desired controller are derived. Ultimately, the availability and practicability of the developed results are certified through a simulation example.     

An efficient model predictive control for Markovian jump systems with all unstable modes

This paper is concerned with the efficient model predictive control (EMPC) problem for a class of Markovian jump systems (MJSs) with unstable modes under polytopic uncertainties and hard constraints. The transition probability matrix and a dual-mode control strategy in the framework of EMPC are co-designed. To achieve a nice tradeoff among the computation burden, the initial feasible region, and the control performance, the EMPC is proposed, whose main idea is two-fold: (1) the terminal constraint set, the corresponding feedback gain, and proper switching rules (i.e. the transition probability) are designed simultaneously by solving an off-line “min–max” problem related to subsystem modes; and (2) a fairly large initial feasible region is obtained off-line by adjusting the dimension of the control perturbation sequence, meanwhile such a perturbation sequence is designed online to steer the system state belonging to initial feasible region into the terminal constraint set within the pre-determined steps. Furthermore, sufficient conditions are presented to rigidly guarantee the feasibility of the proposed EMPC algorithm and the mean-square stability of the underlying MJS. Finally, an illustrative example regarding the economic system is provided to verify the feasibility and effectiveness of the developed algorithm.     

H∞ robust control based on event‐triggered sampling for hybrid systems with singular Markovian jump

The problem of H_∞ robust control based on event‐triggered sampling for a class of singular hybrid systems with Markovian jump is considered in this paper. The primary object of this paper here is to design the event‐triggered sampling controller for a class of uncertain singular Markovian systems, and two fundamental issues on mean square exponential admissibility and H_∞ robust performance are fully addressed. By making use of a suitable Lyapunov functional, in combination with both infinitesimal operator and linear matrices inequalities(LMIs), the sufficient criteria are derived to guarantee the controlled singular hybrid system with Markovian jump is robustly exponentially mean‐square admissible and has a prescribed H_∞ performance γ. Finally, a typical RLC circuit system is given to show the effectiveness of the proposed control method.