Robustness analysis of uncertain, nonlinear systems

Based on the notion of positively invariant tubes, we present a simple and novel methodology for quantitative analysis of uncertain nonlinear systems described by a parametrized family of ordinary differential equations with uncertain initial values. Both the uncertain parameters and the sets they take values in are allowed to be piecewise continuous in time. The proposed methodology can be used to verify and generate pointwise in time upper bounds on the worst-case deviation of the actual system state from a nominal state trajectory. This is useful in verifying limits on the worst-case state behaviour and in the estimation of the worst-case transient response and steady state error. Two examples demonstrate that this methodology compares favourably with alternative techniques like simulation and Lyapunov-like analysis with regard to computational speed and tractability. A third example shows that the proposed method can improve upon one of the best known bounds on the solutions of linear systems. The fourth example involving speed control of a PM synchronous motor highlights the method´s potential in the synthesis of robust nonlinear controllers