A Linear Programming Approach for the Worst-Case Norm of Uncertain Linear Systems Subject to Disturbances with Magnitude and Rate Bounds

This paper presents methods to compute the worst-case norm (WCN) of uncertain linear time-invariant systems. The system input is modelled by the magnitude and rate bounds, and the impulse response of uncertain linear systems lies inside response bounds. Since the computation of the exact WCN is an NP-hard problem, we develop two methods, namely, the simplicial method and the tetrahedral method, to compute upper bounds of the WCN. Computing these upper bounds is equivalent to solving linear programming (LP) problems. In the solving process, we take into account of sparsity structures in the LP problems, and apply a primal interior-point method in the software implementation. Numerical examples reveal that the tetrahedral method outperforms the simplicial method. In particular, the tetrahedral method gives a tighter bound of the WCN and uses fewer flops. Hence, the tetrahedral method is applicable and efficient to approximate the WCN