Lie symmetries of nonlinear systems with unknown inputs

Abstract

It has been revealed that a dynamical system being unobservable or unidentifiable for a given set of observations is fundamentally related to the existence of Lie symmetries. Lie symmetries thus have the potentiality of providing useful tools to analyze and improve the observability and identifiability properties of dynamical systems from a fundamental perspective. This work proposes a computational framework for finding general Lie symmetries of nonlinear systems in the presence of unknown inputs. The occurring framework does not rely on mathematical Ansatz as the typical setting in the existing methods, and it is therefore capable of computing all the groups of Lie symmetries admitted by the system of interest. To alleviate the computation burden of the general framework, an alternative method is developed which is based on a priori assumptions on the functional forms of Lie symmetries to be calculated. The two proposed computation methods can be used as complementary tools to handle real engineering systems efficiently and robustly. Furthermore, utilization of the calculated Lie symmetries to detect and improve observability properties (with identifiability being the observability of unknown model parameters) is systematically discussed. Effective strategies of changing sensor placement, transforming system model and adding model assumptions are introduced to improve observability, and optimally render an unobservable system observable. The concept, computation and application of Lie symmetries are illustrated through several examples of linear, nonlinear and non-smooth dynamical systems.