Variable structure observer design in matrix second-order formulation for linear & nonlinear vibrating systems using velocity measurements

Abstract

This paper addresses the problem of Variable Structure Systems (VSS) observer design based on the natural Matrix Second Order (MSO) model that represents a large class of linear and nonlinear mechanical vibrating structures. In this natural MSO form, the symmetric and definiteness properties of the system matrices are exploited to search for a suitable Lyapunov function and an effective VSS estimation law. The proposed observers can be used to robustly estimate oscillations in other degrees-of-freedom (dof) of a multiple-degrees-of-freedom (mdof) linear vibrating system by processing measured velocity signals and Control inputs from one or more dof in the presence of matched uncertainties. The method is then extended to cover a class of mdof nonlinear vibrating structures with Lipschitz non-linearities. The benefits of this approach are that it does not require an initial modal transformation, and the observer design problem is solved without resorting to the solution of a nonlinear matrix Riccati equation.