On the geometry of the continuous-time generalized algebraic Riccati equation arising in LQ optimal control

Ferrante, Augusto; Ntogramatzidis, Lorenzo

doi:10.1109/ecc.2015.7330801

Cited by 1 publication

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“…(B). There exists a symmetric and positive semidefinite solution of CGCARE(Σ); 1 We make this remark since, if the cost is unbounded for every control, one might alternatively say that all controls are optimal since they all lead to the same value of the performance index. coincide, and and for each initial state x 0 ∈ R n , there exists u 0 (t) such that J ∞ (x 0 , u 0 )…”

Section: Resultsmentioning

confidence: 99%

“…We consider u to be a solution of Problem 1 only if the corresponding value of the performance index is finite. 1 Moreover, we say that a solution u * of Problem 1 is regular if u * ∈ C ∞ [0, ∞).…”

Section: A Preliminariesmentioning

confidence: 99%

“…This link has been examined both from an algebraic and a geometric angle. Now that this relationship has been clarified and explained, an important direction of future research aims at obtaining a full characterization of the set of solutions of the constrained generalized continuous algebraic Riccati equation that parallels the discrete time counterpart in [1], [2].…”

Section: A Geometric Conditionsmentioning

confidence: 99%

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Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

Ferrante

Ntogramatzidis

2016

Systems & Control Letters

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The purpose of this paper is to close the remaining gaps in the understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear-quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, it is only in a recent paper that a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system. I. INTRODUCTION This paper addresses the continuous-time linear quadratic (LQ) optimal control problem when the matrix weighting the input in the cost function, traditionally denoted by R, is possibly singular. This problem has a long history. It has been investigated in several papers and with the use of different techniques, see [5], [12], [9], [8], [6] and the references cited therein. In particular, in the classical contributions [5] and [12] it was proved that i) an optimal solution of the singular LQ problem exists for all initial conditions if the class of allowable controls is extended to include distributions; ii) the regular part of the optimal control can still be written as a static state feedback u = −K x as in the regular case. In the discrete time, the solution of regular and singular finite and infinite-horizon LQ problems can be found resorting to the so-called constrained generalized discrete algebraic Riccati equation, see [2], [1] and also [10]. A similar generalization has been carried out for the continuous-time algebraic Riccati equation in [7], where the constrained generalized 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. On Augusto Ferrante is with the Dipartimento

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Section: Resultsmentioning

confidence: 99%

Section: A Preliminariesmentioning

confidence: 99%