Computation of lower bounds for the induced L2 norm of LPV systems
For control engineers analyzing LPV systems, this provides a practical lower bound that, when combined with upper bounds, can reduce unnecessary computation and yield insight via bad parameter trajectories.
This paper presents an algorithm to compute lower bounds for the induced L2 norm of LPV systems by restricting parameter trajectories to periodic signals, enabling exact norm calculation for periodic LTV systems. The lower bound complements existing upper bound techniques, and a small gap between bounds indicates no need for further computation.
Determining the induced L2 norm of a linear, parameter-varying (LPV) system is an integral part of many analysis and robust control design procedures. Most prior work has focused on efficiently computing upper bounds for the induced L2 norm. The conditions for upper bounds are typically based on scaled small-gain theorems with dynamic multipliers or dissipation inequalities with parameter dependent Lyapunov functions. This paper presents a complementary algorithm to compute lower bounds for the induced L2 norm. The proposed approach computes a lower bound on the gain by 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 linear 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 bad parameter trajectory for the LPV system that can be further analyzed to provide insight into the system performance.