Stability analysis of nonlinear systems via locally defined density functions

This paper considers local stability analysis of nonlinear systems with deriving a positively invariant set based on Rantzer's stability theory by using density functions. We define notion of locally defined density functions around an equilibrium that give monotonously increasing positive measures near the equilibrium of a nonlinear system. Under several assumptions, it is shown that some level set of a locally defined density function is a positively invariant set where almost all of the system trajectories converge to the equilibrium.

1. INTRODUCTION

One of the recent topic in stability analysis of nonlinear systems is Rantzer's density functions [10], which can be regarded as a dual of the Lyapunov's stability theory (e.g., [2]). Density functions have been paid a considerable attention [[1], [3]–[5], [7]–[9], [12]–[15]]. In particular, convexity of synthesis problems of state feedback gains [7] is a remarkable feature of density functions, which allows us to synthesize of nonlinear gains by sums-of-squares [6] which reduce the synthesis problem to a semidefinite program with standard LMIs.