Guaranteed simulation error bounds for linear time invariant systems identified from data

TL;DR

This paper presents a method to derive guaranteed simulation error bounds for linear time-invariant systems identified from data, ensuring reliable model predictions within a finite horizon using set membership techniques.

Contribution

It introduces a novel approach combining data-driven estimation and set membership to obtain guaranteed multi-step prediction error bounds for stable LTI systems.

Findings

The method accurately estimates noise bounds, system order, and decay rate from data.

It refines feasible parameter sets to ensure guaranteed error bounds.

Numerical simulations validate the effectiveness of the proposed approach.

Abstract

This is a technical report that extends and clarifies the results presented in [1]. The model identification problem for asymptotically stable linear time invariant systems is considered. The system output is affected by an additive noise with unknown bound, and a finite set of data is available for parameter estimation. The goal is to derive a model with guaranteed simulation error bounds for all predicted time steps, up to a finite horizon of choice. This is achieved in three steps. At first, the noise bound, system order, and impulse response decay rate are estimated from data. Then, the estimated quantities are used to refine the sets of all possible multi-step predictors compatible with data and prior assumptions (Feasible Parameter Sets, FPSs). The FPSs allow one to derive, in a Set Membership framework, guaranteed error bounds for any given multi-step predictor, including the one obtained by simulating the system model. Finally, the wanted model parameters are identified by numerical optimization, imposing the constraints provided by the FPSs and using one of two proposed optimality criteria, namely to minimize the worst-case error bound over the considered simulation horizon, or to minimize a standard simulation error criterion. Numerical simulations illustrate the validity of the approach.