Vladimír Kučera

Fundamental theorem of state feedback for singular systems

Abstract The limits of state feedback in altering the dynamics of singular systems are studied. A necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system. The condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials. The development is based on the properties of polynomial matrices. A procedure is given for the calculation of a feedback gain which achieves the desired dynamics.

Stability of Discrete Linear Feedback Systems

Abstract The feedback system which consists of a system F to be compensated and a dynamical compensator R is considered. Both systems are assumed discrete, linear, constant and defined over an arbitrary field. Given an F , a necessary and sufficient condition of stability for the feedback system is expressed in algebraic form and all compensators R stabilizing the feedback system are found. An interesting physical interpretation of the stability condition is given in terms of mode cancellations between the F and R . The results can be applied to system stabilizing, synthesizing optimal feedback control systems, and decoupling by dynamical feedback.

Exact model matching, polynomial equation approach

Abstract A detailed analysis of the exact model matching problem in single-input—output linear systems is presented. The method of attack is based on polynomial algebra. The solvability conditions under various constraints are given and a simple design procedure is proposed. It consists in solving a linear polynomial equation which directly yields a compensator in a form suitable for realization.

1972 IFAC congress paper: On nonnegative definite solutions to matrix quadratic equations

The problem of finding all nonnegative definite solutions to the Riccati algebraic equation PA + AP - PBBP + CC = 0 is solved in general. The class of such solutions is shown to constitute a distributive lattice. Finally an algorithm to compute those solutions without checking all general solutions is proposed.

Constant solutions of polynomial equations

Abstract A necessary and sufficient condition is given for the equation AX + BY = C in polynomial matrices to have a constant solution pair X, Y and also for X to be non-singular. A sufficient condition is then established under which the equation with A and B fixed has a constant solution for each C from a given class. Applications to the construction of static state feedback or static output injection in linear systems are discussed.

Model matching of descriptor systems by proportional state feedback

Abstract Given a regular, linear descriptor system E x = Fx + Gu , y = Hx with transfer function T ( s ) we consider the problem of designing a proportional state feedback u = Lx + Mv with M non-singular such that the resulting system is regular and its transfer function equals a prespecified rational matrix W ( s ). Necessary and sufficient conditions are given for the existence of such a feedback, as well as for the internal properness and/or stability of the resulting system. These conditions are interpreted in terms of the zeros of T ( s ) and W ( s ). A simple design procedure is proposed which consists of finding a constant solution to a polynomial matrix equation.

Realizing the Action of a Cascade Compensator by State Feedback

Abstract The class of dynamic cascade compensators is characterized whose effect on a given linear generalized state-space system can be represented by a linear static state feedback. Internal properness and stability of the resulting closed-loop system are investigated. A simple procedure is proposed for the calculation of any and all feedback gains given the system and the compensator.

Towards a Fundamental Theorem of State Feedback for Singular Systems

Abstract The limits of state feedback in altering the dynamics of singular systems are studied. A major step towards this end is solved here, namely, a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a regular system obtained by state feedback from the completely singular system. The condition consists of inequalities which involve impulse controllability indexes, a new notion introduced here, and the degrees of the invariant polynomials. The development is based on the properties of polynomial matrices. A procedure is given for the calculation of a feedback gain which achieves the desired dynamics.

The standard ℋ2-optimal control problem: a polynomial solution

In this paper we present a solution to the ‘standard’ ℋ2 control problem. The solution presented here is based upon the algebra of polynomials, and in particular we use the polynomial equation approach. In considering the scalar version of the standard ℋ2 problem we derive a couple of linear polynomial (diophantine) equations which define the optimal regulator. The coefficients of these equations are obtained from polynomial spectral factorization.

On Nonnegative Definite Solutions to Matrix Quadratic Equations

Summary The problem of finding all nonnegative definite solutions to the “Riccati” algebraic equation PA + AP - PBBP + CC = 0 is solved in general. The class of such solutions is shown to constitute a distributive lattice. Finally an algorithm to compute those solutions without checking all general solutions is proposed.