Linear vs. nonlinear robustness analysis: a case study

We present a case study designed to highlight some of the practical issues that can arise when using linear robustness analysis techniques such as the structured singular value mu to analyse the robustness of uncertain nonlinear systems. The problem considered in the case study is the familiar ball and beam position control task, where the mass of the ball and the time-delay in the beam actuator are assumed to be uncertain. Using a symbolic linear fractional transformation (LFT)-based modelling approach, it is shown how both the original nonlinear and linearised plants may be represented in the form of LFT's. A linear controller is designed for the uncertain linearised plant using a mu-synthesis approach. The robustness of the linear and nonlinear closed-loop systems is then checked using mu-analysis and the Popov criterion, respectively. It is shown that as the degree of nonlinearity in the plant is increased, a sharp fall in the robustness properties of the controller from those predicted by the linear analysis is observed. The results of the study highlight the need for additional nonlinear analysis to confirm robustness analysis results derived using linearised models of nonlinear systems, as is usually the case in practice.