Vlad Ionescu

Continuous and discrete-time Riccati theory: A Popov-function approach

Abstract Necessary and sufficient conditions for the existence of the stabilizing solution of the algebraic Riccati equation are derived for both the continuous and the discrete-time case under the weakest possible assumptions imposed on the initial data. Both frequency-domain conditions involving an associated Popov function and time-domain conditions involving an associated matrix pencil are investigated.

On computing the stabilizing solution of the discrete-time Riccati equation

Abstract A necessary and sufficient condition is given for the existence of the stabilizing solution of the discrete-time Riccati equation without any invertibility or positivity assumptions. The result involves a special associated matrix pencil.

Generalized continuous-time Riccati theory

Abstract A Riccati-like equation which incorporates as special cases the standard continuous-time algebraic Riccati equation and the constrained continuous-time algebraic Riccati equation is introduced and studied. A characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil.

Generalized discrete-time Popov-Yakubovich theory

Abstract This paper deals with the problem of general conditions for the existence of a stabilizing solution to the so-called Kalman-Szego-Popov-Yakubovich system. The well known Popov-Yakubovich ‘positivity condition’ is replaced with a more general one expressed in terms of the invertibility of an adequate operator. This allows one to consider also game-theoretical situations.

Time-varying discrete Riccati equation: some monotonicity results

Abstract Using a Fre´chet derivative based approach some monotonicity and comparison results concerning the solutions of the time-varying discrete time Riccati equation are obtained. Connections with the existence of the semistabilizing solution are made explicit as well.

Infinite horizon disturbance attenuation for discrete-time systems. A Popov-Yakubovich approach

It is proved that for the discrete-time linear systems with time-varying coefficients the existence of a controller which simultaneously stabilizes and provides prescribed disturbance attenuation for the resultant closed-loop system, implies the existence of global solutions to several Kalman-Szegö-Popov-Yakubovich systems. It is also proved that this fact is equivalent to the existence of the positive semidefinite stabilizing solutions to corresponding game-theoretic Riccati equations. The family of all controllers with the above mentioned properties is constructed in terms of the solutions to the cited Kalman-Szegö-Popov-Yakubovich systems. The main tool is the generalized Popov-Yakubovich theory which is essentially developed in an operator-theoretic framework.

Time-varying discrete Riccati equation in terms of Ben Artzi-Gohberg dichotomy

Abstract Necessary and sufficient conditions for the existence of the stabilizing solution to the time-varying discrete Riccati equation are derived in terms of the so-called Ben Artzi-Gohberg dichotomy. It is shown that the problem of such existence conditions is strongly related to that of unique solvability, in vector-valued l 2 -spaces, of a system of singular difference equations which, in this case, is termed the extended Hamiltonian system (EHS). Connections between the existence of the stabilizing solution to the Riccati equation and the bounded invertibility of an appropriate l 2 -operator are pointed out via the notion of disconjugacy.

The class of suboptimal and optimal solutions to the discrete Nehari problem: characterization and computation

Simple and explicit formulae, in a linear fractional transformation (LFT) form, for all suboptimal and optimal solutions to the discrete-time version of the one-block Nehari problem, are derived under no restrictive conditions imposed on the initial data. Using the so-called ‘signature condition’—a generalized Popov-theory type argument—general solvability conditions in terms of a certain type of discrete-time Kalman-Szego-Popov-Yakubovich system are obtained. For the class of optimal approximations, derived via singular perturbation techniques, the LFT coefficients are all of McMillan degree n - r, where n and r are the McMillan degree and the multiplicity of the largest Hankel singular value of the system to be approximated, respectively. The results also lead directly to a simple numerical algorithm.

The time-varying discrete four block Nehari problem: A generalized Popov-Yakubovich type approach

The paper provides necessary and sufficient solvability conditions for the time-variant discrete four block Nehari problem in terms of the existence of the stabilizing solutions to two coupled Riccati equations. A parametrization of the class of all solutions is also given. The results are easily obtained from a signature condition — a generalized Popov Yakubovich type argument-imposed on an appropiate rational node. The present development may be seen as an alternative of the theory developed by Gohberg, Kaashoek and Woerdeman [15].

The four-block Adamjan–Arov–Krein problem for discrete-time systems☆

Abstract We consider the problem of approximating in H∞ norm a given discrete inverse-time stable system (possibly improper) T= T 11 T 12 T 21 T 22 by a discrete-time system S with no more than r poles outside the unit disk (possibly at infinity) such that T 11 +S T 12 T 21 T 22 ∞ Here r⩾0 is an integer and γ>0 is a prescribed tolerance. If r=0, this is the four-block Nehari problem while if we consider the “one block case” where T=T11 we obtain the well-known Hankel norm approximation problem. The theoretical developments are based on a frequency domain signature condition while the class of solutions is constructed in state-space in terms of the solutions to two Riccati equations.