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This paper provides the exact, complete, and possibly non connected robust-stability domain for linear uncertain dynamical systems, which polynomially depend upon a given number of real parameters. Derived in terms of a pencil of multiple degrees pertaining to the perturbed system’s dynamics and to the characteristic function of the considered quadratic matrix inequality (QMI) region, the bounds of these robust-stability domains, in the parameter space, turn out to be delimited by the finite, real and unduplicated generalised eigenvalues of this pencil, indicating when the system’s eigenvalues cross the boundaries of the considered QMI region. The paper also shows that these generalised eigenvalues coincide with those of an augmented pencil of degree just one, thus rendering the computation of these generalised eigenvalues, in general, a standard and relatively simple operation.

Exact robust D-stability analysis for linear dynamical systems with polynomial parameter perturbation