Finite-Time Output Energy Measure for Polynomial Systems With Applications in Observability Analysis

This paper presents a set-based output energy measure for constrained polynomial systems with parameter uncertainties. Output energy is measured in terms of the L2-norm on a finite-time interval while the initial conditions and parameters are allowed to take values from a set. By specifying a bound on the output norm, the measure allows further to determine the set of initial conditions and parameters which lead to satisfaction of this bound. Furthermore, this set characterizes whether an uncertain system can be estimated by a norm-observer and, therefore, can be applied for observability analysis. The derivation of the set is based on recasting a nonlinear program with embedded differential equations into an infinite-dimensional linear program. This is achieved by reformulating the system dynamics in terms of occupation measures. The chosen relaxation approach of the linear program generically guarantees that the obtained outer-approximation converges, for increasing relaxation order, to the true set of initial conditions and parameters satisfying the specified bound on the output norm.