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Journal Article•DOI•

Stochastic stability properties of jump linear systems

TL;DR: In this paper, the authors studied stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties, and showed that all second moment stability properties are equivalent and are sufficient for almost sure sample path stabilisation.
Abstract: Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented. Finally, for one-dimensional jump linear system, it is proved that the region for delta -moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0/sup +/. >
Citations
More filters
Dissertation•
01 Jan 1998
TL;DR: In the thesis it is shown how to analyze stability and expected performance of linear controllers where the network delays are described by one of the two network models above.
Abstract: Control loops that are closed over a communication network get more and more common. A problem with such systems is that the transfer delays will be varying with different characteristics depending on the network hardware and software. The network delays are typically varying due to varying network load, scheduling policies in the network and the nodes, and due to network failures. Two network models of different complexity are studied: Random delays that are independent from transfer to transfer, Random delays with probability distribution functions governed by an underlying Markov chain. The delay models are verified by experimental measurements of network delays. In the thesis it is shown how to analyze stability and expected performance of linear controllers where the network delays are described by one of the two network models above. Methods to evaluate quadratic cost functions are developed. Through the same analysis we find criteria for mean square stability of the closed loop for the different network models. The Linear Quadratic Gaussian (LQG) optimal controller is developed for the two delay models. The derived controller uses knowledge of old time delays. These can be calculated using ``timestamping'' of messages in the network. ``Timestamping'' means that every transfered signal is marked with the time of generation. The receiving node can then calculate how long the transfer delay was by comparing the timestamp with the node's internal clock. (Less)

1,202 citations

Journal Article•DOI•
TL;DR: Using Linear matrix inequalities (LMIs) approach, sufficient conditions are proposed to guarantee the stochastic stability of the underlying system and a reaching motion controller is designed such that the resulting closed-loop system can be driven onto the desired sliding surface in a limited time.
Abstract: In this note, we consider the problems of stochastic stability and sliding-mode control for a class of linear continuous-time systems with stochastic jumps, in which the jumping parameters are modeled as a continuous-time, discrete-state homogeneous Markov process with right continuous trajectories taking values in a finite set. By using Linear matrix inequalities (LMIs) approach, sufficient conditions are proposed to guarantee the stochastic stability of the underlying system. Then, a reaching motion controller is designed such that the resulting closed-loop system can be driven onto the desired sliding surface in a limited time. It has been shown that the sliding mode control problem for the Markovian jump systems is solvable if a set of coupled LMIs have solutions. A numerical example is given to show the potential of the proposed techniques.

613 citations

Journal Article•DOI•
TL;DR: In this paper, the authors considered the problem of robust H/sup /spl infin/ filtering for uncertain Markovian jump linear systems with time-delays which are time-varying and depend on the system mode.
Abstract: This paper considers the problem of robust H/sup /spl infin// filtering for uncertain Markovian jump linear systems with time-delays which are time-varying and depend on the system mode. The parameter uncertainties are time-varying norm-bounded. The aim of this problem is to design a Markovian jump linear filter that ensures robust exponential mean-square stability of the filtering error system and a prescribed L/sub 2/- induced gain from the noise signals to the estimation error, for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained. The desired filter can be constructed by solving a set of linear matrix inequalities. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.

536 citations

Journal Article•DOI•
TL;DR: A new criterion for testing the robust stability of Markovian jump linear systems with uncertain switching probabilities is established in terms of linear matrix inequalities, and a sufficient condition is proposed for the design of robust state-feedback controllers.

524 citations

Journal Article•DOI•
TL;DR: This paper addresses two strategies for the stabilization of continuous-time, switched linear systems and is designed from the solution of what the authors call Lyapunov-Metzler inequalities from which the stability condition (including chattering) is expressed.
Abstract: This paper addresses two strategies for the stabilization of continuous-time, switched linear systems. The first one is of open loop nature (trajectory independent) and is based on the determination of a minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function to be uniformly decreasing at every switching time. The second one is of closed loop nature (trajectory dependent) and is designed from the solution of what we call Lyapunov-Metzler inequalities from which the stability condition (including chattering) is expressed. Being nonconvex, a more conservative but simpler-to-solve version of the Lyapunov-Metzler inequalities is provided. The theoretical results are illustrated by means of examples.

481 citations

References
More filters
Book•
17 Jan 2012
TL;DR: In this article, a book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes is presented, which is based on the work of this article.
Abstract: Book on stochastic stability and control dealing with Liapunov function approach to study of Markov processes

1,293 citations

Journal Article•DOI•
TL;DR: In this paper, necessary and sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite-time Markovian jump linear quadratic (JLQ) problem are established.
Abstract: Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem. >

955 citations

01 Jan 1970
TL;DR: In this article, the authors discuss control processes and optimization problems solutions by stochastic differential equations, discussing dynamic models and programming, linear filtering and optimal feedback, and discuss linear filtering with optimal feedback.
Abstract: Control processes and optimization problems solutions by stochastic differential equations, discussing dynamic models and programming, linear filtering and optimal feedback

458 citations

Journal Article•DOI•
TL;DR: The main purpose of this manuscript is to give some understanding of the subject of stability of stochastic systems by presenting some of the basic ideas as well as a survey of results that have appeared in the literature.

434 citations

Journal Article•DOI•
TL;DR: In this paper, a class of linear systems are studied which are subject to sudden changes in parameter values and an algorithm similar in form to Kushner's stochastic maximum principle is derived.
Abstract: A class of linear systems are studied which are subject to sudden changes in parameter values An algorithm similar in form to Kushner's stochastic maximum principle is derived and the relationship between these algorithms discussed Systems in which the performance measure is quadratic are investigated in detail and a differential equation is derived which yields the optimal feedback gains

347 citations