Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

Abstract

In this paper, a new methodology is developed for the closed-form solution of a generalized version of the finite-horizon linear-quadratic regulator problem for LTI discrete-time systems. The problem considered herein encompasses the classical version of the LQ problem with assigned initial state and weighted terminal state, as well as the so-called fixed-end point version, in which both the initial and the terminal states are sharply assigned. The present approach is based on a parametrization of all the solutions of the extended symplectic system. In this way, closed-form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions. By taking advantage of standard software routines for the solution of the algebraic Riccati and Stein equations, our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the Riccati difference equation

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

“…Therefore, applying Lemma 2.1 we find (4), that can be rewritten as ker R X ⊆ ker S X and also as S X G X = 0.…”

On solving boundary value problems associated with generalised LQ control of 2-D systems

“…First, we present a generalised version of the optimality conditions established in [6] in order to accomodate the northwest affine constraints (4)(5). Note that the conditions presented here are not only necessary, but also sufficient for optimality.…”

“…Conversely, if (7-13) admit solutions h i,j , v i,j , λ i,j , μ i,j , u i,j , ξ j and ζ i , then the corresponding h i,j , v i,j and u i,j minimize J subject to the constraints (1), (3), (4)(5). The proof of this theorem can be carried out along the same lines of that in [8].…”

“…Here we generalise those conditions to accomodate possible north-west affine constraints on the horizontal and vertical semi-states. This case can be considered as the two-dimensional counterpart of the problem formulated and solved in [5]. The solution of the equations which characterise the optimality condition is discussed, and a simple method to compute the optimal control law is described.…”

On the definition of negative imaginary system for not necessarily rational symmetric transfer functions

“…We also recall that Positive Real Lemma may be viewed as the "father" of the algebraic Riccati equation (ARE) which plays a fundamental role in almost any area of systems theory, both in the discrete in the continuos time cases, see e.g. [8], [9], [3], [7], [10], [4], [13], [12] for a sample of different aspects, properties and applications of ARE and for an extensive reference list. This is the conference version of the journal paper [11] submitted by the same authors to Automatica and currently in press.…”