Computation of a lower bound for the induced ℒ2 norm of LPV systems

Determining the induced ℒ₂ norm of a linear, parameter-varying (LPV) system is an integral part of many analysis and robust control design procedures. In general, this norm cannot be determined explicitly. Most prior work has focused on efficiently computing upper bounds for the induced ℒ₂ norm. This paper presents a complementary algorithm to compute lower bounds for this norm. The proposed approach is based on restricting the parameter trajectory to be a periodic signal. This restriction enables the use of recent results for exact calculation of the ℒ₂ norm for a periodic time varying system. The proposed lower bound algorithm has two benefits. First, the lower bound complements standard upper bound techniques. Specifically, a small gap between the bounds indicates that further computation, e.g. upper bounds with more complex Lyapunov functions, is unnecessary. Second, the lower bound algorithm returns a worst-case parameter trajectory for the LPV system that can be further analyzed to provide insight into the system performance. Numerical examples are provided to demonstrate the applicability of the proposed approach.