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SUMMARYStability analysis of a linear uncertain polynomial system in parameter space is one of the central themes of recent control theory. It often requires an enormous number of stability checks with conventional stability test methods. A delta operator polytopic polynomial, a convex combination of some delta operator vertex polynomials, is one of such cases. In this paper, robust stability is addressed for polytopic delta operator polynomials and several necessary and sufficient stability conditions are derived. The delta operator, an operator used to express discrete time systems, is known to have significant features: numerical advantage in implementation and ability to smoothly connect the z-operator with the Laplace operator. Based on this last feature and on the celebrated Edge theorem, we first derive three kinds of exact stability conditions for the uncertain delta operator polynomials. We then extend the directional stability radius method, which was developed for diamond polynomials, so that it can also be applied to polytopic polynomials. This extension gives rise to the fourth stability test. Furthermore, it is shown from the result of the numerical experiments with these stability analysis methods that one of these four methods, which uses eigenvalues of matrices, turns out to be most efficient for stability analysis of the polytope.

A linear discrete-time system represented by delta-operator is known to have an advantage of accuracy in numerical calculations over usual shift-operator systems. Analysis of such a system is a topic of recent times. In this paper. we are interested in stability analysis for delta-operator systems with parametric uncertainties represented by interval polynomials. Though the extreme point results hold for the stability of such poly nomials, computational cost becomes markedly enormous when the degree of the polynomials increases. We propose a new stability analysis method for the systems using stability margin in order to reduce the amount of work for stability analysis. We check if a hyperbox of an interval polynomial is contained in the stability region in the coefficient space. To do this, we propose 'directional stability radius' as a stability margin estimater. It is a stability radius where coeffient perturbations are supposed to be in certain constrained directions. The devised stability analysis method is to test which vertexes of the hyperbox are covered by certain stability hyperballs with the directional stability radius. By numerical examples, we show that the proposed method is more efficient than brute-force check of every vertex polynomial.

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