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Numerical operations among rational matrices: standard techniques and interpolation

Petr HušekMichael ŠebekJan Štecha — 1999


Numerical operations on and among rational matrices are traditionally handled by direct manipulation with their scalar entries. A new numerically attractive alternative is proposed here that is based on rational matrix interpolation. The procedure begins with evaluation of rational matrices in several complex points. Then all the required operations are performed consecutively on constant matrices corresponding to each particular point. Finally, the resulting rational matrix is recovered from the...

Discrete-time symmetric polynomial equations with complex coefficients

Didier HenrionJan JežekMichael Šebek — 2002


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 reduction algorithms 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.

Robust pole placement for second-order systems: an LMI approach

Didier HenrionMichael ŠebekVladimír Kučera — 2005


Based on recently developed sufficient conditions for stability of polynomial matrices, an LMI technique is described to perform robust pole placement by proportional-derivative feedback on second-order linear systems affected by polytopic or norm-bounded uncertainty. As illustrated by several numerical examples, at the core of the approach is the choice of a nominal, or central quadratic polynomial matrix.

Rank-one LMI approach to robust stability of polynomial matrices

Didier HenrionKenji SugimotoMichael Šebek — 2002


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...

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