Exact Robust Stability Domain for Polynomially Parameter Dependent Dynamical Systems

This paper provides, in a first step, the exact, complete and possibly non connected robust stability domain for linear uncertain dynamical systems which polynomially depend on a single scalar parameter. Derived in terms of a pencil of multiple degree pertaining to the Hurwitz matrix of the uncertain system characteristic polynomial, the bounds of this robust stability domain turn out to be delimited by the finite, real, and unduplicated generalized eigenvalues of this pencil. These eigenvalues indicate precisely when the system eigenvalues cross the imaginary axis from or into of the stability domain. The paper also shows that these generalized eigenvalues coincide with those of an associated augmented pencil of degree just one, thus rendering the computation of these generalized eigenvalues a standard and simple operation. The decision on stability of every connected subdomain is very easily made by checking whether, for a single and arbitrary parameter value inside this subdomain, stability holds. The resulting algorithm is extremely simple and computationally appealing, even for large scale systems. In a second step, an extension to the multi-parameter case is also made wherein dynamics, which polynomially depend on a finite number of parameters, are treated in the same context via a random gridding procedure that discovers the exact stability bounds to any desired resolution. Several examples demonstrate the power of this method when compared to existing classical results.