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 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.     

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.     

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.     

Stability analysis of impulsive control systems

This paper studies impulsive control systems. Several stability criteria are established by employing the method of Lyapunov functions. These criteria may be used for impulsive feedback control design. As an application, impulsive control of the Lorenz chaotic system is discussed. Numerical experiments are carried out for the control of the Lorenz system. It is shown that small and frequent impulses need to be used in order to stabilize the Lorenz system.     

脉冲差分系统的Robust稳定性

运用Lyapunov函数法分析并建立了脉冲差分系统结构扰动下的R obust稳定性准则.     

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.     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Stability of linear systems with time-varying delays: A direct frequency domain approach

The stability of linear systems with uncertain bounded time-varying delays (without any constraints on the delay derivatives) is analyzed. It is assumed that the system is stable for some known constant values of the delays (but may be unstable for zero delay values). The existing (Lyapunov-based) stability methods are restricted to the case of a single non-zero constant delay value, and lead to complicated and restrictive results. In the present note for the first time a stability criterion is derived in the general multiple delay case without any constraints on the delay derivative. The simple sufficient stability condition is given in terms of the system matrices and the lengths of the delay segments. Different from the existing frequency domain methods which usually apply the small gain theorem, the suggested approach is based on the direct application of the Laplace transform to the transformed system and on the bounding technique in _L2$L_2$. A numerical example illustrates the efficiency of the method.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

Adaptive control of linear stochastic evolution systems

An adaptive control problem for some linear stochastic evolution systems in Hilbert spaces is formulated and solved in this paper. The solution includes showing the strong consistency of a family of least squares estimates of the unknown parameters and the convergence of the average quadratic costs with a control based on these estimates to the optimal average cost. The unknown parameters in the model appear affinely in the infinitesimal generator of the C ₀ semigroup that defines the evolution system. A recursive equation is given for a family of least squares estimates and the bounded linear operator solution of the stationary Riccati equation is shown to be a continuous function of the unknown parameters in the uniform operator topology     

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.     

Stability analysis and control synthesis for a class of switched neutral systems

In this paper, the problems of asymptotical stability and stabilization of a class of switched neutral control systems are investigated. A delay-dependent stability criterion is formulated in term of linear matrix inequalities (LMIs) by using quadratic Lyapunov functions and inequality analysis technique. The corresponding switching rule is obtained through dividing the state space properly. Also, the synthesis of stabilizing state-feedback controllers are done such that the close-loop system is asymptotically stable. Two numerical examples are given to show the proposed method.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.