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In this study, we investigate the optimal control of a class of singularly perturbed linear stochastic systems with Markovian jumping parameters. After establishing an asymptotic structure for the stabilizing solution of the coupled stochastic algebraic Riccati equations, a parameter-independent composite controller is derived. Furthermore, the cost degradation in a reduced-order controller is discussed. Thus, the exactness of the proposed approximate control is discussed for the first time. As an additional important contribution, a numerical algorithm for solving the coupled stochastic algebraic Riccati equations is proposed, and the feature of the resulting higher-order controller is shown. Finally, a simple example is presented to demonstrate the validity of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper a linear quadratic optimization problem is addressed under the assumption that the state space representation of the controlled system is described by a system of Itoô differential equations with two fast time scales. It is well known that under some standard assumptions this optimization problem has a unique optimal control which is in a state feedback form. The gain matrix of the optimal control is computed based on the stabilizing solution of a suitable algebraic Riccati equation. The presence of the small parameters associated to the fast time scales may produce some ill conditioning of the numerical computations of the stabilizing solution. Our goal is to perform a detailed study of the dependence of the stabilizing solution of the Riccati equation with respect to the small parameters which are describing the fast time scales. In this way, we are able to obtain a near optimal control whose gain matrix does not depend upon the small parameters (which may be unknown). An estimation of the loss of the performance of the near optimal control is also done. [ABSTRACT FROM AUTHOR]

The aim of the paper is to present a design procedure of the optimal controller minimizing the H2-type norm of discrete-time stochastic linear systems with periodic coefficients simultaneously affected by a nonhomogeneous but periodic Markov chain and state and control multiplicative white noise perturbations. Firstly, two H2-type norms for the linear stochastic systems under consideration were introduced. These H2-type norms may be viewed as measures of the effect of the additive white noise perturbations on the regulated output of the considered system. Before deriving of the state space representation of the optimal controller, some useful formulae of the two H2-type norms were obtained. These formulae are expressed in terms of periodic solutions of some suitable linear equations and are derived in the absence of some additional assumptions regarding the Markov chain other than the periodicity of the sequence of the transition probability matrices. Further, it is shown that the optimal H2 controller depends on the stabilizing solutions of some specific systems of coupled Riccati equations, which generalize the well-known control and filtering equations from linear time invariant case. For the readers convenience, the paper presents iterative numerical algorithms for the computations of the stabilizing solutions of these Riccati type systems. The theoretical developments are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We consider a class, denoted by Q, of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system, namely a quadratic target. We say that a system in Q undergoes a globally exact quadratization. Here "globally" adds up to a slight extension of the notion of C8 immersion (of systems), namely a dense immersion, which amounts to saying that it is defined on the whole manifold of the system states, except possibly a zero-measure set. It is proven that the class Q includes all systems characterized by vector fields whose components are analytic integral closed-form functions (ICFFs). The result is first proven for algebraic system functions, by means of a constructive proof, and next extended up to analytic ICFFs. For nonanalytic ICFFs a weaker result is proven as well. Also the case of a partially observed system is considered, as well as the internal structure of every quadratic representation, which is proven to be always a feedback interconnection of bilinear systems. Finally, examples are presented for which the constructive proof given earlier is turned into a quadratization algorithm, which can be carried out by hand, and the resulting differential equations of the quadratic representation are presented. [ABSTRACT FROM AUTHOR]