A Method to Robustify Exact Linearization Against Parameter Uncertainty

Abstract

This paper addresses the issue of uncertain parameters in the case of the control of nonlinear systems which are exact linearizable by state feedback. It is shown that the linearizing feedback may be complemented by an additional robustifying compensator, designed to ensure robust stability and performance against the uncertainty of some model parameters. This allows to bridge two state-of-the-art design methodologies such as exact linearization and robust control synthesis. Exact linearization allows the transformation of nonlinear dynamics into linear ones by an eventually dynamic state feedback and by a change of coordinates. However, due to the uncertain nature of some model parameters, their nominal values used in the transformation may be different from their real values. This parameter misfit implies that the resulting transformed dynamics may still include non-linearities or may be a linear system, but different from the one that results for the nominal parameter values. The paper proposes a procedure to cover the uncertainties remaining after exact linearization and to design an additional linear compensator, denoted by K(s), to ensure robust performance and stability. The design of the compensator K(s) involves standard $\mathcal{H}_\infty$ techniques, based on an output multiplicative uncertainty structure. The weighting matrices of the output multiplicative structure are obtained such that they cover a model set obtained by linearizing the transformed nonlinear system over a sufficiently fine grid above the uncertain parameter range. The suggested approach is illustrated by multiple (SISO and MIMO) examples, including a two-degrees-of-freedom robotic arm. It is shown by simulation that the additional robustifying compensator may stabilize the system for parameter values that would result in unstable behavior without its application and may also result in a better tracking performance.