The H2-optimal Control Problem of CSVIU Systems: Discounted, Counter-discounted and Long-Run Solutions -- Part I: The Norm

TL;DR

This paper investigates H2-norm measures for CSVIU systems, linking stochastic stability, energy growth, and long-term behavior, with a focus on discounting effects and steady-state error constraints.

Contribution

It introduces a unified framework for H2-norms in CSVIU systems, connecting stochastic stability, energy measures, and long-run performance under discounting.

Findings

Finiteness of H2-norms relates to stochastic stability.

Counter-discounting imposes stricter steady-state error constraints.

Unified vanishing discount approach links transient and long-term behaviors.

Abstract

The paper deals with the H2-norm and associated energy or power measurements for a class of processes known as CSVIU (Control and State Variation Increase Uncertainty). These are system models for which a stochastic process conveys the underlying uncertainties, and are able to give rise to cautious controls. The paper delves into the non-controlled version and fundamental system and norms notions associated with stochastic stability and mean-square convergence. One pillar of the study is the connection between the finiteness of one of these norms or a limited energy measurement growth with the corresponding stochastic stability notions. A detectability concept ties these notions, and the analysis of linear-positive operators plays a fundamental role. The introduction of various H2-norms and energy measurement performance criteria allows one to span the focus from transient to long-run behavior. As the discount parameter turns into a counter-discount, the criteria enforce stricter requirements on the second-moment steady state errors and on the exponential convergence rate. A tidy connection among this H2-performance measures cast employs a unifying vanishing discount reasoning.