Marcos G. Todorov

Continuous-Time Markov Jump Linear Systems

1.Introduction.- 2.A Few Tools and Notations.- 3.Mean Square Stability.- 4.Quadratic Optimal Control with Complete Observations.- 5.H2 Optimal Control With Complete Observations.- 6.Quadratic and H2 Optimal Control with Partial Observations.- 7.Best Linear Filter with Unknown (x(t), theta(t)).- 8.H_

A Detector-Based Approach for the

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H_{2}

Control.- 9.Design Techniques.- 10.Some Numerical Examples.- A.Coupled Differential and Algebraic Riccati Equations.- B.The Adjoint Operator and Some Auxiliary Results.- References.- Notation and Conventions.- Index.

Control of Markov Jump Linear Systems With Partial Information

In this paper, we study the H2-control for discrete-time Markov Jump Linear Systems (MJLS) with partial information. We consider the case in which we do not have access to the Markov jump parameter but, instead, there is a detector that emits signals which provides information on this parameter. A salient feature of our formulation is that it encompasses, for instance, the cases with perfect information, no information and cluster observations of the Markov parameter, which were previously analyzed in the Markov jump control literature. The goal is to derive a feedback linear control using the information provided by the detector in order to stochastically stabilize the closed loop system. We present two Lyapunov like equations for the stochastic stability of the system. In addition, we show that a Linear Matrix Inequalities (LMI) formulation can be obtained in order to design a stochastically stabilizing feedback control. In the sequel we deal with the H2 control problem and we show that, again, an LMI optimization problem can be formulated in order to design a stochastically stabilizing feedback control with guaranteed H2-cost. We also present two special cases, one of them always satisfied for the limit case in which the detector provides perfect information on the Markov parameter, and the Bernoulli jump case, under which LMI conditions become necessary and sufficient for the stochastic stabilizability of the system and the LMI optimization problems provide the optimal H2 cost. For the Bernoulli jump case we show that our formulation generalizes previous ones. The case with convex polytopic uncertainty on the parameters of the system and on the transition probability matrix is also considered. The paper is concluded with some numerical examples.

Output Feedback

The output feedback H∞ control is addressed for a class of continuous-time Markov jump linear systems with the Markov process taking values in an infinite countable set S. We consider that only an output and the jump parameters are available to the controller. Via a certain Bounded Real Lemma, together with some extensions of Schur complements and of the Projection Lemma, a theorem which characterizes whether there exist or not a full-order solution to the disturbance attenuation problem is devised in terms of two different Linear Matrix Inequality (LMI) feasibility problems. This result connects the so-called projective approach to an LMI problem which is more amenable to computer solution, and hence for design. We conclude the paper with two design algorithms for the construction of such controllers.

H_\infty

A bounded real lemma is established, providing an equivalent condition to stochastic stability (SS) of a continuous- time infinite Markovian jump linear system (MJLS) with a prescribed L2-stochastic disturbance attenuation level gamma in terms of existence of solutions to an infinite set of coupled linear matrix inequalities (LMIs). Besides the interest in its own right, the main result provides a fundamental tool for the development of an Hinfin-like theory devoted to this class of systems.

Control of Continuous-Time Infinite Markovian Jump Linear Systems via LMI Methods

This paper is concerned with the robust analysis and control of continuous-time Markov jump linear systems. The centerpiece of the paper is an alternative to the small-gain theorem, which hinges on the robustification of a certain adjoint Lyapunov operator. By means of this technique, it is proven that the small-gain theorem of Markov jump linear systems may sometimes be arbitrarily conservative, even when nonlinear Lipschitz disturbances are taken into account. The adjoint approach, on the other hand, provides the maximal degree of robustness for this particular setup. In addition, we prove that the adjoint design of controllers is solved more efficiently than the design based on small-gain analysis. Bearing these new facts in mind, we derive an iterative algorithm for the design of robust controllers, based on linear matrix inequalities. By means of numerical examples, regarding the robust control of an underactuated manipulator arm and of a simplified power systems model, it is shown that the adjoint design methodology can be much more advantageous than its small-gain counterpart.

Infinite Markov jump-bounded real lemma

In this paper we introduce the subject of stability radii for continuous-time infinite Markov jump linear systems (MJLS) with respect to unstructured perturbations. By means of the small-gain approach, a lower bound for the complex radius is derived along with a linear matrix inequality (LMI) optimization method which is new in this context. In this regard, we propose an algorithm to solve the optimization problem, based on a bisectional procedure, which is tailored in such a way that avoids the issue of scaling optimization. In addition, an easily computable upper bound for the real and complex stability radii is devised, with the aid of a spectral characterization of the problem. This seems to be a novel approach to the problem of robust stability, even when restricted to the finite case, which in turn allows us to obtain explicit formulas for the stability radii of two-mode scalar MJLS. We also introduce a connection between stability radii and a certain margin of stability with respect to perturbations on the transition rates of the Markov process. The applicability of the main results is illustrated with some numerical examples.

A new perspective on the robustness of Markov jump linear systems

This paper addresses the robust stochastic stability and stabilization of continuous-time Markov jump linear systems (MJLS), with the Markov jump parameters taking values in a countably infinite set. It is assumed that the state and input matrices are subjected to norm-bounded uncertainty with a prespecified structure, which encompasses the block-diagonal setting. We introduce new robust analysis and synthesis characterizations such that, unlike previous approaches in the MJLS literature, the scaling parameters are treated as decision variables in linear matrix inequalities. As a by-product, new contributions to the theory of stability radii of MJLS are provided. When restricted to the finite case, we further introduce new adjoint linear matrix inequality (LMI) characterizations for each of the robust analysis and synthesis problems, as well as for stability radii. Besides the interest in its own right, the adjoint approach allows us to verify that, in the general MJLS case, there is a gap between the complex stability radius and what can be assessed with scaled versions of the small-gain theorem. This suggests a fundamental limitation of the robustness against linear perturbations that the H

On the stability radii of continuous-time infinite Markov jump linear systems

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