Hardy-Schatten Norms of Systems, Output Energy Cumulants and Linear Quadro-Quartic Gaussian Control

TL;DR

This paper explores the relationship between output energy cumulants in linear stochastic control systems and Schatten norms, proposing a novel control criterion that balances average energy and variance, and deriving optimal controllers.

Contribution

It introduces a new performance criterion combining mean and variance of output energy, linking Schatten norms to control design, and provides equations for optimal controllers with a homotopy solution method.

Findings

Derived equations for optimal controllers incorporating Schatten norms.

Linked output energy cumulants to Hardy space norms and risk-sensitive indices.

Outlined a homotopy method for solving the control problem numerically.

Abstract

This paper is concerned with linear stochastic control systems in state space. The integral of the squared norm of the system output over a bounded time interval is interpreted as energy. The cumulants of the output energy in the infinite-horizon limit are related to Schatten norms of the system in the Hardy space of transfer functions and the risk-sensitive performance index. We employ a novel performance criterion which seeks to minimize a combination of the average value and the variance of the output energy of the system per unit time. The resulting linear quadro-quartic Gaussian control problem involves the H2 and H4-norms of the closed-loop system. We obtain equations for the optimal controller and outline a homotopy method which reduces the solution of the problem to the numerical integration of a differential equation initialized by the standard linear quadratic Gaussian controller.