Vasile Dragan

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.

Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise

In this paper we consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping. Our aim is to provide a mathematical background in order to give unified approach for a large class of problems associated to linear controlled systems subjected both to multiplicative white noise perturbations and Markovian jumping. First we prove an Itô type formula. Our result extends the result of Ref. [24], to the case when the stochastic process x(t) has not all moments bounded. Necessary and sufficient conditions assuring the exponential stability in mean square for the zero solution of a linear stochastic system with multiplicative white noise and Markovian jumping are provided. Some estimates for solutions of affine stochastic systems are derived, and necessary and sufficient conditions assuring the stochastic stabilizability and stochastic detectability are given. A stochastic version of Bounded Real Lemma is proved and several aspects of the problem of robust stabilization by state feedback for a class of linear systems with multiplicative white noise and Markovian jumping are investigated.

The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping

In this paper, the linear quadratic optimization problem for a class of linear stochastic systems subject both to multiplicative white noise and Markovian jumping is investigated. Two classes of admissible controls are considered. One of these classes contains controls with additional property that corresponding trajectories tend to zero (in mean square) when tends to /spl infin/, while concerning the controls contained in the second class of admissible controls there is not any stability assumption. In the optimization problem over the first class of admissible controls, the cost functional could have indefinite sign of weights matrices. An iterative procedure to compute the maximal solution of the systems of generalized Riccati equations is provided. A numerical example to illustrate the applicability of the iterative procedure is given.

A small gain theorem for linear stochastic systems

Abstract A well-known result in linear control theory is the so-called “small gain” theorem stating that if given two plants with transfer matrix functions T 1 and T 2 in H ∞ such that ‖ T 1 ‖ γ and ‖ T 2 ‖ γ , when coupling T 2 to T 1 such that u 2 = y 1 and u 1 = y 2 one obtains an internally stable closed system. The aim of the present paper is to describe a corresponding result for stochastic systems with state-dependent white noise.

H2 Optimal control for linear stochastic systems

The aim of the present paper is to provide an optimal solution to the H^2 state-feedback and output-feedback control problems for stochastic linear systems subjected both to Markov jumps and to multiplicative white noise. It is proved that in the state-feedback case the optimal solution is a static gain which is also optimal in the class of all higher-order controllers. In the output-feedback case the optimal H^2 controller has the same order as the given stochastic system. The realization of the optimal controllers depend on the stabilizing solutions of some appropriate systems of Riccati-type coupled equations. An effective iterative convergent algorithm to compute these stabilizing solutions is also presented. The paper gives some illustrative numerical example allowing to compare the results obtained by the proposed design approach with the ones presented in the recent control literature.

A /spl gamma/-attenuation problem for discrete-time time-varying stochastic systems with multiplicative noise

The aim of this paper is to develop an H/sub /spl infin//-type theory for discrete time time-varying systems with multiplicative noise. Based on a version of the bounded real lemma corresponding to this class of systems, necessary and sufficient conditions for the existence of a stabilizing deterministic controller ensuring an imposed level of attenuation are derived in terms of the solutions of two linear matrix inequalities (LMI) satisfying a complementary rank condition. Moreover, explicit formulae for such controller are obtained in the case when some additional conditions are assumed. As a particular case, we consider the same problem for discrete-time time-varying periodic systems with multiplicative noise, for which it is shown that a periodic /spl gamma/-attenuating controller may be computed as function of a certain extended LMI system.

Mean Square Exponential Stability for some Stochastic Linear Discrete Time Systems

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. Four different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other three definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain. The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions. Unlike the continuous time framework, for the discrete time linear stochastic systems with Markovian jumping two types of Lyapunov operators are introduced. Therefore in the case of discrete time linear stochastic systems subject to Markovian perturbations one obtains characterizations of the mean square exponential stability which do not have an analogous in the continuous time. One of the aim of this paper is to show that in the general case of discretetime time-varying linear stochastic systems subject to an homogeneous or an inhomogeneous Markov chain, exponential stability mean square defined in terms of state space trajectories of the systems cannot be always characterized via quadratic Lyapunov functions. The results developed in this paper may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems.

The Linear Quadratic Regulator Problem for a Class of Controlled Systems Modeled by Singularly Perturbed Itô Differential Equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of

Differential equations with positive evolutions and some applications

O(\sqrt{\varepsilon})

INFINITE DIMENSIONAL TIME VARYING SYSTEMS WITH NONLINEAR OUTPUT FEEDBACK

approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an