Abstract
In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed Itô differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution (satisfying a prescribed sign condition) to the corresponding stochastic algebraic Riccati equation is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed.
Highlights
In the last 40 years, special attention was paid to the singular perturbation techniques applied in both analysis and synthesis of control laws with prescribed performance specifications for the regulation of systems whose mathematical models are described by a system of differential equations of high dimension, and contain a number of small parameters multiplying derivatives of a part of the state variables of the physical phenomenon under discussion.The large number of differential equations of the mathematical model of a physical process may be caused by the presence of some “parasitic” parameters such as small time constants, resistances, inductances, capacitances, moments of inertia, small masses, etc.The presence of such small parameters is often a source of stiffness due to the simultaneous occurrence of slow and fast phenomena
Unlike the deterministic case, where the reduced model is obtained by removing the small parameters, in the case of stochastic optimal control problems driven by systems of singularly perturbed Itô differential equations, the definition of the reduced model is not always intuitive and it is strongly dependent upon the intensity of the white noise type perturbations affecting the diffusion part of the fast equations of the mathematical model
In the stochastic framework, when the controlled systems are modeled by singularly perturbed Itô differential equations, the definition of the system of reduced algebraic Riccati equations cannot be done by a simple neglection of the small parameters
Summary
In the last 40 years, special attention was paid to the singular perturbation techniques applied in both analysis and synthesis of control laws with prescribed performance specifications for the regulation of systems whose mathematical models are described by a system of differential equations of high dimension, and contain a number of small parameters multiplying derivatives of a part of the state variables of the physical phenomenon under discussion. Unlike the deterministic case, where the reduced model is obtained by removing the small parameters, in the case of stochastic optimal control problems driven by systems of singularly perturbed Itô differential equations, the definition of the reduced model is not always intuitive and it is strongly dependent upon the intensity of the white noise type perturbations affecting the diffusion part of the fast equations of the mathematical model. Employing the stabilizing solution of the system of the reduced equations and the corresponding stabilizing gain matrices we show that one may apply the implicit functions theorem to obtain the existence, as well as the asymptotic structure of, the stabilizing solution of the algebraic Riccati equation associated with the optimal control problem under consideration. We show how the asymptotic structure of the stabilizing feedback gain can be used to construct a near optimal control
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