Minimizing transients via the Kreiss system norm

TL;DR

This paper introduces new system norms to evaluate and minimize transient responses in stable LTI systems, using structured feedback controllers that improve stability and prevent nonlinear undesirable behaviors.

Contribution

It proposes novel system norms for transient assessment and develops controllers that minimize these norms, enhancing system stability and robustness.

Findings

Controllers effectively reduce transient responses.

Method prevents transition to nonlinear regimes.

Validated through Lyapunov techniques and simulations.

Abstract

We introduce system norms which assess transient behavior of stable Linear Time-Invariant (LTI) systems. This allows us to address undesired responses to initial conditions, finite resource consumption signals, or persistent perturbations. We then consider the challenging problem of minimizing these norms in closed loop using structured linear feedback. The computed controllers mitigate transients in a linearized closed loop, with the potential side effect of enlarging the region of stability of the underlying non-linear controlled system. In applications this helps to prevent transition to undesired nonlinear regimes, limit cycles or chaotic behavior. The success of our approach is certified a posteriori using Lyapunov-like techniques and simulations, as we demonstrate through a variety of applications.