Linear System Identification Under Multiplicative Noise from Multiple Trajectory Data

Abstract

We study identification of linear systems with multiplicative noise from multiple trajectory data. A least-squares algorithm, based on exploratory inputs, is proposed to simultaneously estimate the parameters of the nominal system and the covariance matrix of the multiplicative noise. The algorithm does not need prior knowledge of the noise or stability of the system, but requires mild conditions of inputs and relatively small length for each trajectory. Identifiability of the noise covariance matrix is studied, showing that there exists an equivalent class of matrices that generate the same second-moment dynamic of system states. It is demonstrated how to obtain the equivalent class based on estimates of the noise covariance. Asymptotic consistency of the algorithm is verified under sufficiently exciting inputs and system controllability conditions. Non-asymptotic estimation performance is also analyzed under the assumption that system states and noise are bounded, providing vanishing high-probability bounds as the number of trajectories grows to infinity. The results are illustrated by numerical simulations.

Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories

“…In this section, we will present our main results on bounding the estimation error Ĝ − G from (10). We will show that the term (Y − Y * − ) −1 decreases with a rate of O( 1 N ), and then upper bound the growth rate of other terms in (10) separately.…”

“…In this section, we will present our main results on bounding the estimation error Ĝ − G from (10). We will show that the term (Y − Y * − ) −1 decreases with a rate of O( 1 N ), and then upper bound the growth rate of other terms in (10) separately. Using recent results on the balanced realization algorithm with the adjustments to accommodate the nonsteady state Kalman filter, we then show that the estimation error of the system matrices A, C, K p−1 will also be bounded when the error Ĝ − G is small enough.…”

“…Furthermore, only historical snippets of data about the system may be available, without the ability to easily observe long-run behavior. Additionally, in settings where one has the ability to restart the system or have several copies of the system running in parallel, one may obtain multiple trajectories generated by the system dynamics [10]. The paper [11] studies the sample complexity of identifying a system whose state is fully measured using only the final data points from multiple trajectories.…”

Learning Optimal Controllers for Linear Systems With Multiplicative Noise via Policy Gradient

“…Then it suffices to take the variances α i and β j and matrices A i and B j as the eigenvalues and (reshaped) eigenvectors of Σ A and Σ B , respectively, after a projection onto a set of orthogonal real-valued vectors [43]. The goal is to determine a closed-loop state feedback policy π * with u t = π * (x t ) from a set Π of admissible policies which solves the optimization in (1).…”

“…When there was high multiplicative noise, the noise-ignorant controller K actually destabilized the system Many practical networked systems can be approximated by diffusion dynamics with losses and stochastic diffusion constants (edge weights) between nodes; examples include heat flow through uninsulated pipes, hydraulic flow through leaky pipes, information flow between processors with packet loss, electrical power flow between generators with resistant electrical power lines, etc. A derivation of the discrete-time dynamics of this system is given in [43]. We considered a particular four-state, four-input system, and open-loop mean-square stable with the following parameters:…”