On linear-quadratic optimal control of implicit difference equations

Abstract

In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyze a discrete-time Lur'e equation and a corresponding Kalman-Yakubovich-Popov inequality. We show that solvability of the Kalman-Yakubovich-Popov inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context. a discrete-time version of the inequality introduced in [36], where V (E, A, B) denotes an inequality projected on a certain subspace V (E, A, B) , i. e., V * M(P )V 0 holds for any basis matrix V of V (E, A, B) . We show statements which relate the solvability of this inequality to the non-negativity of the Popov function on the unit circle, a certain rational matrix function defined bywhere G ∼ (z) := G z −1 * for a rational matrix G(z) ∈ K(z) n×n . In Sections 4 and 5 we introduce the notion of inertia for palindromic matrix pencils evaluated on the unit circle and provide spectral characterizations regarding positivity of the Popov function, similar to the characterizations which were obtained in [35] and [41] for even matrix pencils in the continuous-time case.In Section 6 we investigate the Lur'e equation for the discrete-time optimal control problem which is a generalization of the algebraic Riccati equation (2). This means that we seek solution triples (X, K, L) ∈ K n×n × K q×n × K q×m fulfilling

On discrete-time port-Hamiltonian (descriptor) systems

“…Observe that the solvability of ( 12) as well as the complete causality and stability are invariant under equivalence transformations of the system. Hence we may assume that (E, A) is in Weierstraß form (3). If the pair is completely causal and stable, then the index of (E, A) is at most one by Proposition 2.4.…”

“…The discussed passivity notions lead to the following variants of the Kalman-Yakubovich-Popov (KYP) inequalities. Different generalizations of KYP inequalities for discrete-time descriptor systems were introduced in [3].…”

“…From passivity with respect to a certain supply rate one can only conclude that the corresponding KYPs hold on the system space, see [15] for the continuous-time case. The solvability of such KYP inequalities restricted to subspaces was studied in [13,50,51] for continuous-time systems and in [3] for discrete-time systems.…”

“…., E, A ∈ K n×n , B ∈ K n×m , C ∈ K m×n , with possibly singular E, are well studied, see e.g. [3,10,16,40]. While for general linear descriptor systems many system properties, like controllability, reachability, observability, passivity, and stability, can be characterized via linear algebra calculations, in the last 20 years for continuous-time systems a new model class has received a lot of attention.…”

Optimal control of DAEs with unconstrained terminal costs

“…Besides positive semi-definiteness of the weight matrix P , no further assumptions are made on the terminal cost. This is in contrast to the assumption commonly made in the literature that the weight matrix only penalizes differential states, i.e., is of the form x(t f ) E P Ex(t f ) [1]- [3] or no terminal cost is considered with an infinite time horizon [4]- [8]. Also note that whereas commonly a closed interval is of interest, in this paper a half open interval is considered.…”