Singular Infinite Horizon Quadratic Control of Linear Systems with Known Disturbances: A Regularization Approach

TL;DR

This paper addresses a singular infinite horizon quadratic control problem for linear systems with known disturbances by introducing a regularization method, enabling the derivation of optimal controls where traditional methods fail due to singularity.

Contribution

The paper proposes a regularization approach combined with perturbation analysis to solve singular control problems that cannot be addressed by classical optimal control techniques.

Findings

Derived the infimum of the cost functional for the singular control problem.

Designed a sequence of state-feedback controls approaching optimality.

Provided an example demonstrating singular trajectory tracking.

Abstract

An optimal control problem with an infinite horizon quadratic cost functional for a linear system with a known additive disturbance is considered. The feature of this problem is that a weight matrix of the control cost in the cost functional is singular. Due to this singularity, the problem can be solved neither by application of the Pontriagin's Maximum Principle, nor using the Hamilton-Jacobi-Bellman equation approach, i.e. this problem is singular. Since the weight matrix of the control cost, being singular, is not in general zero, only a part of the control coordinates is singular, while the others are regular. This problem is solved by a regularization method. Namely, it is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an infinite horizon integral of the squares of the singular control coordinates with a small positive weight. Due to a smallness of this coefficient, the new problem is a partial cheap control problem. Using a perturbation technique, an asymptotic analysis of this partial cheap control problem is carried out. Based on this analysis, the infimum of the cost functional in the original problem is derived, and a minimizing sequence of state-feedback controls is designed. An illustrative example of a singular trajectory tracking is presented.