Non-asymptotic Identification of LTI Systems from a Single Trajectory

Abstract

We consider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system's Markov parameters, from which a balanced realization is obtained using the classical Ho-Kalman algorithm. By proving a stability result for the Ho-Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data is needed to learn a balanced realization of the system up to a desired accuracy with high probability. ‡ Version 2 has two improvements: First, paper now uses spectral radius rather than largest singular value hence applies to a larger class of systems. Secondly, new sample complexity bounds are provided for approximating the system's Hankel operator via estimated Markov parameters. These bounds leverage stability and treat the system as if it has a logarithmic order.Hence, if we had access to infinitely many independent (y t , u t−k ) pairs, our task could be accomplished by a simple averaging. In this work, we will show that, one can robustly learn these matrices from a small amount of data generated from a single realization of the system trajectory. The challenge is efficiently using finite and dependent data points to perform reliable estimation. Observe that, our problem is identical to learning the concatenated matrix G defined asNext section describes our input and output data. Based on this, we formulate a least-squares procedure that estimates G. The estimateĜ will play a critical role in the identification of the system matrices.1 Balanced realizations give a representation of the system in a basis that orders the states in terms of their effect on the input/output behavior. This is relevant for determining the system order and for model reduction [23]. 2 While we assume diagonal covariance throughout the paper, we believe our proof strategy can be adapted to arbitrary covariance matrices.

Topology Identification of Heterogeneous Networks of Linear Systems

“…4] for consistency results (for infinite data) in this setup. There is also an interesting line of work [21] that studies the identification of the system's Markov parameters from finite data, and that provides statistical guarantees for the quality of estimation.…”

A Tutorial on Concentration Bounds for System Identification

“…In our companion paper [1], we show how these tools can be used to design and analyze self-tuning and adaptive control methods with finite-data guarantees. Although we focused on the full information setting, we note that many of the techniques described extend naturally to the partially observed setting [13], [14], [15].…”

“…An example of random variables that are sub-Gaussian but not Gaussian are bounded random variables -it can be shown that a random variable X taking values in [a, b] almost surely satisfies equation (13) with parameter…”

From Parameter Estimation to Dispersion of Nonstationary Gauss-Markov Processes

“…1]. The problem of estimating the parameter a from a block of outcomes of the Gauss-Markov source (1) is one of the simplest versions in recent studies of machine learning for dynamical systems [15,[25][26][27][28]. One objective of those studies is to obtain tight performance bounds on the least-squares estimates of the system parameters A, B, C, D from a single input / output trajectory {w i , y i } n i=1 in the state-space model: These studies on the limiting distribution and the LDP of the estimation error are asymptotic.…”

“…Both bounds (11) and (12) do not capture the dependence on a and n, and are the same for P + n and P − n . All the bounds in [15,[25][26][27][28] are either optimal only order-wise or involve implicit constants. Our main result on parameter estimation is a tight nonasymptotic lower bound on P + n and P − n .…”