On the unstable of continuous-time linearized nonlinear systems

It has been shown that quantifying the unstable in linear systems is important for establishing the existence of stabilizing feedback controllers. This paper addresses the problem of quantifying the unstable in continuous-time linearized systems obtained from nonlinear systems for a family of constant inputs, i.e., the largest instability measure for all admissible equilibrium points for all admissible constant inputs. It is supposed that the dynamics of the nonlinear system is polynomial in both state and input, 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), whose conservatism can be decreased by increasing the size of such LMIs. Some numerical examples illustrate the proposed results.