Robust Stability Analysis for a Class of Extended

We consider analysis of nonlinear systems that can be brought into a state dependent representation known as extended linearization. Under this formulation, conventional linear analysis techniques may be adapted to study the stability, optimality, and robustness properties of nonlinear systems. When subject to system uncertainty, estimating the radius of stability for systems under extended linearization is difficult since the closed-loop system equations are not available explicitly. A method for obtaining the upper bound for the radius of stability in this class of systems is proposed. It is shown that the stability radius around a suitable domain can be obtained by computing the largest singular value of an overvalued matrix with special properties. Additionally, a property of extended linearization is that it relies on a non-unique factorization of the system dynamics to bring the nonlinear system into a psuedo linear form referred to as the State Dependent Coefficient (SDC) parameterization. Under system uncertainty, each SDC parameterization will produce its own radius of stability in a region of interest in the state space. We propose a method to obtain the SDC parameterization which results in the maximum radius of stability for the original nonlinear system in the region of interest. It is shown that the problem of finding the maximum radius of stability from a hyperplane of SDC parameterizations can be reduced to constrained minimization of the spectral norm of a comparison system.