Petr Zagalak

Brief Analysis of nonlinear time-delay systems using modules over non-commutative rings

The theory of non-commutative rings is introduced to provide a basis for the study of nonlinear control systems with time delays. The left Ore ring of non-commutative polynomials defined over the field of meromorphic function is suggested as the framework for such a study. This approach is then generalized to a broader class of nonlinear systems with delays that are called generalized Roesser systems. Finally, the theory is applied to analyze nonlinear time-delay systems. A weak observability is defined and characterized, generalizing the well-known linear result. Properties of closed submodules are then developed to obtain a result on the accessibility of such systems.

Assigning the Kronecker invariants of a matrix pencil by row or column completions

Abstract The challenge consists in describing the relationships between the Kronecker invariants of a matrix pencil and one of its subpencils. For a given subpencil, an algorithm for constructing a matrix pencil with prescribed Kronecker invariants should also be proposed.

The row-by-row decoupling via state feedback: a polynomial approach

Necessary and sufficient conditions for the row-by-row and integrator decoupling of a linear system x = Ax + Bu, y = Cx via nonregular static state feedback are established. A procedure is outlined to calculate one such gain.

Proper solutions of polynomial equations

Abstract The linear equation AX + BY = C is studied, where A, B, and C are given polynomial matrices such that A−1 B is a strictly proper rational matrix. All polynomial matrix solution pairs X, Y such that YX−1 is a proper rational matrix are parametrized. The study is motivated by the polo placement techniques in the design of linear control systems.

A method for determining the non-existence of a common quadratic Lyapunov function for switched linear systems based on particle swarm optimisation

The existence of a common quadratic Lyapunov function (CQLF) for a switched linear system guarantees its global asymptotic stability. Even if progress in finding the conditions for the existence/non-existence of a CQLF is significant, especially in switched linear systems consisting of N second-order systems or two systems of order n, the general case of N systems of order n still remains open. In this article, a sufficient condition for the non-existence of a CQLF for N systems of order n is derived. Based on the condition, a new method for determining the non-existence of a CQLF, using particle swarm optimisation, was designed and is described. Examples illustrating the proposed method are introduced at the end of this article.

On pole structure assignment in linear systems

The problem of pole structure assignment (PSA) by state feedback in implicit, linear and uncontrollable systems is discussed in the article. It is shown that the problem is solvable if the system is regularisable. Then necessary and sufficient conditions for characteristic polynomial assignment are established. In the case of PSA (invariant polynomials assignment) just necessary conditions have been obtained. But it turns out that these conditions are also sufficient in some special cases. This happens, for example, when the system does not possess any non-proper controllability indexes. A possible application of the achieved results to modelling a constrained movement of a robot arm is outlined, too.

A semi-canonical form for a class of right-invertible linear systems

We present a semi-canonical form for a class of right invertible systems under the action of the transforma- tion group (T, F, G,@P), where @P is a permutation matrix acting on the outputs. Under certain additional conditions, this form is canonical, in particular when the system is nonsingular and controllable.

On a Special Case of the Morgan problem

Abstract A special case of the problem of decoupling by state feedback is considered. Using the so-called polynomial approach the necessary and sufficient conditions of its solvability stated in (Eldem et al., 1997) are reestablished.

State feedback in linear control theory

Abstract The role of system invariants in solutions of classical control problems when regular state feedback is used is reviewed. The structural modifications that arise when these problems are extended to the case of non-regular state feedback are presented. An interpretation of state feedback problems in terms of matrix pencils completion is also discussed.

On the Problem of Decoupling

Abstract The row-by-row decoupling of linear multivariable systems with stability is considered. An implicit necessary and sufficient condition for this problem to have a solution is provided. The approach used here takes advantage of the properties of the ring of proper and stable rational functions.