An Efficient Algorithm to Compute Norms for Finite Horizon, Linear Time-Varying Systems

TL;DR

This paper introduces a new, efficient algorithm for computing the induced norms of finite-horizon Linear Time-Varying systems, combining existing methods to achieve faster convergence and guaranteed bounds.

Contribution

A novel algorithm that combines power iteration and Riccati Differential Equation approaches to efficiently compute bounds on system norms with provable guarantees.

Findings

The new algorithm converges monotonically.

It provides both upper and lower bounds within a specified tolerance.

Numerical examples demonstrate improved efficiency and accuracy.

Abstract

We present an efficient algorithm to compute the induced norms of finite-horizon Linear Time-Varying (LTV) systems. The formulation includes both induced $\mathcal{L}_2$ and terminal Euclidean norm penalties. Existing computational approaches include the power iteration and bisection of a Riccati Differential Equation (RDE). The power iteration has low computation time per iteration but overall convergence can be slow. In contrast, the RDE condition provides guaranteed bounds on the induced gain but single RDE integration can be slow. The complementary features of these two algorithms are combined to develop a new algorithm that is both fast and provides provable upper and lower bounds on the induced norm within the desired tolerance. The algorithm also provides a worst-case disturbance input that achieves the lower bound on the norm. We also present a new proof which shows that the power iteration for this problem converges monotonically. Finally, we show a controllability Gramian based simpler computational method for induced $\mathcal{L}_2$-to-Euclidean norm. This can be used to compute the reachable set at any time on the horizon. Numerical examples are provided to demonstrate the proposed algorithm.