Adrian-Mihail Stoica

Mathematical methods in robust control of discrete-time linear stochastic systems

Elements of probability theory.- Discrete-time linear equations defined by positive operators.- Mean square exponential stability.- Structural properties of linear stochastic systems.- Discrete-time Riccati equations of stochastic control.- Linear quadratic optimization problems.- Discrete-time stochastic optimal control.- Robust stability and robust stabilization of discrete-time linear stochastic systems.

Mathematical methods in robust control of linear stochastic systems

Preliminaries to Probability Theory and Stochastic Differential Equations.- Exponential Stability and Lyapunov-Type Linear Equations.- Structural Properties of Linear Stochastic Systems.- The Riccati Equations of Stochastic Control.- Linear Quadratic Control Problem for Linear Stochastic Systems.- Stochastic Version of the Bounded Real Lemma and Applications.- Robust Stabilization of Linear Stochastic Systems.

Well-conditioned computation for H∞ controller near the optimum

The present paper gives a procedure for determining a H∞ optimal controller in the assumption that the game Riccati equations have stabilizing positive definite solutions at the optimum value. A specific feature of the construction is its first step consisting in balancing with respect to the positive definite stabilizing solutions of the Riccati equations. The justification is based on singular perturbations reduction.

Stochastic Version of Bounded Real Lemma and Applications

The main goal of this chapter is to investigate the robustness properties of a stable linear stochastic system with respect to various classes of uncertainties.

Preliminaries to Probability Theory and Stochastic Differential Equations

This first chapter collects for the readers convenience some definitions and fundamental results concerning the measure theory and the stochastic processes theory which are needed in the following developments of the book. Classical results concerning the measure theory, integration, stochastic processes, and stochastic integrals are presented without proofs. Appropriate references are given; thus for the measure theory we mention [33, 55, 71, 75, 118, 138]; for the probability theory we refer to [32, 71, 119, 130, 138], and for the theory of stochastic processes and stochastic differential equations we cite [6, 18, 32, 71, 72, 85, 102, 105, 120, 121, 141, 152, 153]. However several results which can be found in some references less accessible are proved.

Exponential Stability in Mean Square

In this chapter the problem of mean square exponential stability of the zero solution to the stochastic differential equations of type (1.22) is studied. The stability of a steady-state is one of the main tasks which appears in many design problems of controllers with prescribed performances.

Linear Differential Equations with Positive Evolution on Ordered Banach Spaces

In this chapter the problem of exponential stability of the zero state equilibrium of a class of linear differential equations on a real ordered Banach space is investigated.

Stochastic H 2 Optimal Control

In this chapter the problem of H 2-control of a continuous-time linear system subject to Markovian jumping and independent multiplicative and additive white noise perturbations is considered. Several kinds of H 2 type of performance criteria (often called H 2-norms) are introduced and characterized via solutions of some suitable linear equations on the spaces of symmetric matrices.

Linear Quadratic Optimization Problems for Linear Stochastic Systems

In this chapter as well as in the next chapters one shows how the mathematical results derived in the previous chapters are involved in the design of stabilizing controllers with some imposed performances for a wide class of linear stochastic systems. The design problem of some stabilizing controls minimizing quadratic performance criteria is studied. More precisely, this chapter deals with the so-called linear quadratic optimization problem (LQOP). LQOP has received much attention in control literature due to its wide area of applications.

Robust Stabilization of Linear Stochastic Systems

In the present chapter we consider the robust stabilization problem of systems subject to both multiplicative white noise and to Markovian jumps with respect to some classes of parametric uncertainty. As it is already known, a wide variety of aspects of the robust stabilization problem can be embedded in a general disturbance attenuation problem (DAP) which extends the well-known H ∞ control problem in the case of deterministic invariant linear systems. A special attention will be paid in this chapter to the attenuation problem of exogenous perturbations with a specified level of attenuation. In the same time, some particular robust stabilization problems which solutions are derived using the results in the preceding chapter will be presented. The solution of the general attenuation problem will be given in terms of some linear matrix inequalities which provides necessary and sufficient solvability conditions. Throughout this chapter we assume that (mathcal{D} ={ 1,2,ldots,d}).