Bounding the State Covariance Matrix for a Randomly Switching Linear System with Noise

TL;DR

This paper develops a method to bound the covariance matrix of a randomly switching linear system with noise by formulating a matrix optimization problem, enabling analysis of the system's behavior through invariant ellipsoids.

Contribution

It introduces a novel approach using Kronecker algebra to bound the covariance set of switching systems with noise, including an invariant ellipsoid computation.

Findings

Bounded the covariance set using an ellipsoid derived from an optimization problem

Provided a method to compute invariant ellipsoids for switching affine systems

Enhanced understanding of covariance dynamics in stochastic switching systems

Abstract

The propagation of a state vector is governed by a set of time-invariant state transition matrices that switch arbitrarily between two values. The evolution of the state is also perturbed by white Gaussian noise with a variance that switches randomly with the state transition relation. The behavior of this system can be characterized by the covariance matrix of the state vector, which is time varying. However, we can bound the set of covariances by comparing the switching system to an augmented system derived with Kronecker algebra. We formulate a matrix optimization problem to compute an ellipsoid that bounds the covariance dynamics, which in turn bounds the state covariance of the set of switching systems subject to white noise. In developing this approach, an invariant ellipsoid for a linear switching affine system is computed along the way.