A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems

Abstract

A closed-form expression parameterizing the solutions of the extended symplectic difference equation over a finite time interval is given under the mild assumption of modulus-controllability. This representation is expressed in terms of the strongly unmixed solution of a discrete ARE and of an algebraic Stein equation. The most important application of this result is a generalized version of the finite-horizon LQ regulator: In particular our framework enables different kind of boundary conditions to be treated in a unified fashion, without resorting to the Riccati difference equation for the computation of the optimal control function

Generalized Finite-Horizon Linear-Quadratic Optimal Control

“…A detailed proof of the main result can be found in Ferrante and Ntogramatzidis (2007b); see also Zattoni (2008) and Ferrante and Ntogramatzidis (2012). The extended symplectic pencil is defined by…”

“…The set P defined in (17) is the set of for which these trajectories satisfy the boundary conditions. All the details of this construction can be found in Ferrante and Ntogramatzidis (2007b …”

“…This problem has been the object of a large number of contributions in the recent literature, under different names: Parameter varying systems, jump linear systems, switching systems, and bumpless systems are definitions extensively used to denote different classes of systems affected by sensible changes in their parameters or structures (Balas and Bokor 2004). In recent years, a new unified approach emerged in , , Ferrante and Ntogramatzidis (2007a), and Ferrante and Ntogramatzidis (2007b) that solves the finite-horizon LQ optimal control problem via a formula which parameterizes the set of trajectories generated by the corresponding Hamiltonian differential equation in the continuous time and the extended symplectic difference equation in the discrete case. Loosely, we can say that the expressions parameterizing the trajectories of the Hamiltonian differential equation and the extended symplectic difference equation using this approach hinge on a pair of "opposite" solutions of the associated algebraic Riccati equations.…”

Comments on "Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems

“…The expressions parameterizing the trajectories of HDE and ESDE given in these first contributions hinge on particular solutions of the associated algebraic Riccati equations (AREs). While controllability of the given system was required in these first papers, because both the stabilizing and anti-stabilizing solutions of the ARE were involved, in more recent publications it was shown that generalizations of the same technique are possible under much milder assumptions: namely, sign-controllability in the continuous case, see [2,Section 1.4] and [4], and modulus controllability in the discrete case, see [2, Section 2.4] and [3]. These assumptions are to date the weakest conditions that guarantee existence of solutions of an ARE.…”

“…This active stream of research not only produced the theoretical background which is necessary for the application of these techniques to more general types of systems, but it also considerably enlarged the range of optimization problems that can be successfully addressed. In particular, in [4] and [3] it is shown that the parameterization technique described above can be applied to (continuous and discrete) finite-horizon LQ problems with the most general form of affine constraints at the end-points (thus encompassing the standard, the fixed end-point and the point-to-point The purpose of this comment paper is to show that the parameterization of the trajectories of the ESDE presented as the main and original result in [1] is a particular case of the results presented in the Ph.D. thesis [2] and subsequently published in [3]. Such a result is established in [1] only under unnecessary restrictive assumptions.…”