Efficient Polynomial-Time Outer Bounds on State Trajectories for Uncertain Polynomial Systems Using Skewed Structured Singular Values

Outer bounds for the evolution of state trajectories of uncertain systems are useful for many purposes such as robust control, state and parameter estimation, model invalidation, safety evaluation, fault diagnosis, and experimental design. Obtaining tight outer bounds is, however, a challenging task. This technical note proposes a new approach to obtaining such bounds for discrete-time polynomial systems with uncertain initial state, uncertain parameters, and bounded disturbances. To obtain outer bounds, the nonlinear map describing the uncertain dynamical system is represented via linear fractional transformation. The bounds on the trajectories are obtained by computing polynomial-time upper and lower bounds for the skewed structured singular value of the linear fractional transformation. Algorithms with different tradeoffs between computational complexity and conservatism are outlined. The tradeoffs as well as efficiency of the approach are illustrated in a numerical example, which shows small conservatism of the obtained bounds.