Design of Sliding Mode Controllers using Reduced-order Koopman Mode Representations

Abstract

The Koopman operator framework allows for a linear, but infinite-dimensional, representation of the dynamics of a non-linear system. The Koopman modes, or observables, and the resulting linear dynamics are derived purely using a data-driven framework, where the data are system outputs measured at discrete samples; improving accuracy of the Koopman representation requires a large number of such modes to be considered. Recent results consider the system input as well, in the derivation of the discrete linear dynamics, thus enabling the design of controllers. Sliding mode controllers (SMCs), including the discrete-time versions, can handle parameter uncertainties and variations and also ensure that the control objective is satisfied in finite time. In this paper, a discrete-time SMC is designed for the output control of a dynamic system approximated by fewer Koopman modes; the SMC is expected to handle uncertainties introduced by the ignored modes. Conditions are identified for the closed-loop system to be stable, with the occurrence of sliding mode.