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On parametric Hurwitz stability margin of real polynomials

Husek P.

Control and Cybernetics

2008

Vol. 37, no 3

549-563

The paper deals with the problem of determining Hurwitz stability of a ball of polynomials defined by a weighted lp norm in the coefficient space where p is an arbitrary positive integer including infinity. The solution of the case when the weights are supposed to be the same for coefficient being above and below its nominal value corresponding to symmetric ball has been given by Tsypkin and Polyak. However, sometimes it seems to be useful to have a possibility to consider these weights as different, resulting in the asymmetric ball. This is, for example, the situation where the weights express our level of confidence that the real value of a coefficient lies in some interval. Such approach is used if the value of a coefficient is estimated by an expert. Solution of the problem is based on frequency domain plot in the complex plane and on applying the Zero Exclusion Theorem. The main idea consists in separation of the original problem into four subproblems and using an appropriate coordinate transformation which makes the value set independent of frequency. This transformation makes it possible to move the relative value set into the origin of the complex plane and to easily formulate the necessary and sufficient condition of Hurwitz stability of asymmetric ball of polynomials with prescribed radius or determine the maximum radius preserving stability. The whole graphical procedure consists of four plots instead of one, needed in the symmetric case.

Non-quadratic performance design for Takagi-Sugeno fuzzy systems

Bernal M., Husek P.

International Journal of Applied Mathematics and Computer Science

2005

Vol. 15, no 3

383-391

This paper improves controller synthesis of discrete Takagi-Sugeno fuzzy systems based on non-quadratic Lyapunov functions, making it possible to accomplish various kinds of control performance specifications such as decay rate conditions, requirements on control input and output and disturbance rejection. These extensions can be implemented via linear matrix inequalities, which are numerically solvable with commercially available software. The controller design is illustrated with an example.