The generalised discrete algebraic Riccati equation arising in LQ optimal control problems: Part I

Abstract

Abstract-A geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation arising from the classic infinite-horizon linear quadratic (LQ) control problem and the output-nulling and reachability subspaces of the underlying system. This analysis reveals the presence of a subspace that plays a crucial role in the solution of the related optimal control problem.

The role of the generalised continuous algebraic Riccati equation in impulse-free continuous-time singular LQ optimal control

“…The proof of this result can be carried out along the same lines of the proof of Theorem 2.1 in [6], and is omitted.…”

“…In the discrete time, the solution of finite and infinite-horizon LQ problems can be found resorting to the so-called generalised discrete algebraic Riccati equation. In particular, the link between the solutions of LQ problems and the solutions of generalised discrete algebraic/difference equations have been investigated in [9,4] for the finite horizon and in [3] for the infinite horizon. A similar generalisation has been carried out for the continuous-time algebraic Riccati equation in [7], where the generalised Riccati equation was defined in such a way that the inverse of R appearing in the standard Riccati equation is replaced by its pseudo-inverse.…”

“…We begin by observing that an important difference between the continuous and the discrete-time generalised Riccati equations is the fact that in the continuous case, differently from the discrete case [6], it is not true that all symmetric and positive semidefinite solutions of GCARE(Σ) are also solutions of CGCARE(Σ). Consider for example the following example, where…”

“…Most of these concepts are adaptation of several results that were presented in [6] to the continuous time. First, since as aforementioned the Popov matrix Π is assumed symmetric and positive semidefinite, we can consider a factorisation of the form…”

“…Nevertheless, to date the role of this equation in relation to the solution of optimal control problems in the continuous time has not been fully explained. The goal of this paper is to fill this gap, by providing a counterpart of the results in [6] for the continuous case. In particular, we describe the role that the generalised continuous algebraic …”