Quantifying the unstable in linearized nonlinear systems

It has been shown that quantifying the unstable in linear systems is important for establishing the existence of stabilizing feedback controllers in the presence of communications constraints. In this context, the instability measure is defined as the sum of the real parts (continuous-time case) or the product of the magnitudes (discrete-time case) of the unstable eigenvalues. This paper addresses the problem of quantifying the unstable in linearized systems obtained from nonlinear systems for a family of constant inputs, i.e., quantifying the largest instability measure over all admissible equilibrium points and all admissible constant inputs. It is supposed that the dynamics of the nonlinear system is polynomial in both state and input, either continuous-time or discrete-time, and that the set of constant inputs is a semialgebraic set. Two cases are considered: first, when the equilibrium points are known polynomial functions of the input, and, second, when the equilibrium points are unknown (polynomial or non-polynomial) functions of the input. It is shown that upper bounds of the sought instability measure can be established through linear matrix inequalities (LMIs) by searching for polynomially-dependent Lyapunov function candidates. Moreover, it is shown that these upper bounds are nonconservative for a sufficiently large degree of the Lyapunov function candidates under some conditions. Lastly, necessary and sufficient conditions are provided for establishing whether the obtained upper bounds are nonconservative. Some numerical examples also show the advantages of the proposed method with respect to grid techniques.