Bounding the State Covariance Matrix for a Randomly Switching Linear System with Noise
It offers a theoretical tool for analyzing stability and performance of switching systems with noise, relevant to control theory and signal processing.
The paper bounds the state covariance matrix for a randomly switching linear system with noise by formulating a matrix optimization problem that computes an ellipsoid bounding the covariance dynamics. This provides a guaranteed bound for the set of possible covariances.
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.