The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito 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 O(√ε) 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 O(ε) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter ε is not precisely known. © 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh

Al'brekht's Method in Infinite Dimensions

In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators

Series Solution of Discrete Time Stochastic Optimal Control Problems

In this paper we consider discrete time stochastic optimal control problems over infinite and finite time horizons. We show that for a large class of such problems the Taylor polynomials of the solutions to the associated Dynamic Programming Equations can be computed degree by degree.Comment: arXiv admin note: text overlap with arXiv:1806.0412

Stochastic HJB Equations and Regular Singular Points

IIn this paper we show that some HJB equations arising from both finite and infinite horizon stochastic optimal control problems have a regular singular point at the origin. This makes them amenable to solution by power series techniques. This extends the work of Al'brecht who showed that the HJB equations of an infinite horizon deterministic optimal control problem can have a regular singular point at the origin, Al'brekht solved the HJB equations by power series, degree by degree. In particular, we show that the infinite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a new type of algebraic Riccati equation which we call the Stochastic Algebraic Riccati Equation (SARE). If SARE can be solved then one has a complete solution to this infinite horizon stochastic optimal control problem. We also show that a finite horizon stochastic optimal control problem with linear dynamics, quadratic cost and bilinear noise leads to a Stochastic Differential Riccati Equation (SDRE) that is well known. If these problems are the linear-quadratic-bilinear part of a nonlinear finite horizon stochastic optimal control problem then we show how the higher degree terms of the solutions can be computed degree by degree. To our knowledge this computation is new

Application of optimal control theory to the design of the NASA/JPL 70-meter antenna servos

The application of Linear Quadratic Gaussian (LQG) techniques to the design of the 70-m axis servos is described. Linear quadratic optimal control and Kalman filter theory are reviewed, and model development and verification are discussed. Families of optimal controller and Kalman filter gain vectors were generated by varying weight parameters. Performance specifications were used to select final gain vectors