Michael Šebek

Brief An LMI condition for robust stability of polynomial matrix polytopes

A sufficient LMI condition is proposed for checking robust stability of a polytope of polynomial matrices. It hinges upon two recent results: a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NP-hard robust stability results.

AN ALGORITHM FOR STATIC OUTPUT FEEDBACK SIMULTANEOUS STABILIZATION OF SCALAR PLANTS

Abstract Static output feedback design and simultaneous stabilization are difficult control tasks for which no general efficient algorithm has been designed so far. In this note we show that, in the special case of scalar plants, the problem of simultaneous stabilization by static output feedback can however be solved in polynomial time using standard tools of numerical algebra.

An LMI Condition for Robust Stability of Polynomial Matrix Polytopes 1

Abstract A sufficient LMI condition is proposed for checking robust stability of a polytope of polynomial matrices. It hinges upon two recent results: a new approach to polynomial matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NP-hard robust stability results.

Rank-One LMI Approach to Robust Stability of Polynomial Matrices

Necessary and sufficient conditions are formulated for checking robust stability of an uncertain polynomial matrix. Various stability regions and uncertainty models are handled in a unified way. The conditions, stemming from a general optimization methodology similar to the one used in μ-analysis, are expressed as a rank-one LMI, a non-convex problem frequently arising in robust control. Convex relaxations of the problem yield tractable sufficient LMI conditions for robust stability of uncertain polynomial matrices.

Efficient algorithms for discrete-time symmetric polynomial equations with complex coefficients☆

Discrete-time symmetric polynomial equations with complex coefficients are studied in the scalar and matrix case. New theoretical results are derived and several algorithms are proposed and evaluated. Polynomial reductionalgorithms are first described to study theoretical properties of the equations. Sylvester matrix algorithms are then developed to solve numerically the equations. The algorithms are implemented in the Polynomial Toolbox for MATLAB.

A Linear Programming Approach for Ventilation Control in Tunnels

A control scheme for highway tunnels is designed based on a static model of a highway tunnel. The controller is designed to keep the exhaust levels inside the tunnel below given limits. During operation, the model is being modified by adaptation. The control itself is achieved by linear programming. Finally, an example of the control is given

Numerical Methods For Polynomial Matrix Rank Evaluation

Abstract Two algorithms are proposed for evaluating the rank of an arbitrary polynomial matrix. They rely upon constant matrix rank evaluations and therefore are more reliable than already existing elementary polynomial operations techniques. Some applications are mentioned, such as polynomial null-space extraction or conditions of existence of solutions to matrix polynomial equations arising in control problems.

Use of Delta Distributions in Space-Time Control Systems

Abstract The paper deals with continuous-space-time technological processes and describes, how the approach of Laplace transform transfer functions can be used. It appears that in that way, the mathematical device of delta-distributions is very useful. Some new extensions of it are presented.

Robust Stability of Polynomial Matrices

Abstract A method to analyze robust stability of parameterized scalar polynomials is generalized to cover parameterized polynomial matrices. A direct solution by calculating the determinant of the parameterized polynomial matrices and then to handling the subsequent scalar problem is not recommended. Appart from possible numerical difficulties, the determinant coefficients are quite complicated functions of those in the given matrix. If the matrix coefficients are not yet fully determined (e.g., if unknown controller parameters are to be fixed as yet by a design procedure), their further “scrambling” makes the analysis more difficult if not impossible. The alternative procedure proposed here is based on a new type of guardian map developed for the matrix case. It completely avoids the need to enumerate the determinant. This feature makes it more suitable if the analysis is to be used in robust control design.