Recursive Experiment Design for Closed-Loop Identification with Output Perturbation Limits

TL;DR

The paper tackles the problem of identifying ARMAX models while operating under a known closed-loop controller and strict bounds on output perturbations. It introduces a recursive, time-domain design that perturbs the input in a way that remains informative for parameter estimation yet keeps the induced output perturbation within user-specified limits, and it yields a closed-form one-step solution for the perturbation. A sensitivity-based impulse-response framework is developed to translate perturbation bounds into linear constraints and to enable online updates of the constraint and objective using current parameter estimates. Numerical experiments show that the method effectively bounds \\delta_t$ while maintaining competitive identification accuracy, offering a practical alternative to unconstrained pseudo-random perturbations in safe closed-loop operation.

Abstract

In many applications, system identification experiments must be performed under output feedback to ensure safety or to maintain system operation. In this paper, we consider the online design of informative experiments for ARMAX models by applying a bounded perturbation to the input signal generated by a fixed output feedback controller. Specifically, the design constrains the resulting output perturbation within user-specified limits and can be efficiently computed in closed form. We demonstrate the effectiveness of the method in a numerical experiment.

Figures (4)

• Figure 1: Closed-loop system where $G$ and $H$ are unknown and $K$ is a known output feedback controller with a set-point $r$. A perturbation $d$ is added and the resulting the output is $\widetilde{y} = y + \delta$, where $y$ is the nominal (unperturbed) output and $\delta$ is the output perturbation. We seek a recursive design of $d$ that yields information about $G$ and $H$ while limiting $\delta$.
• Figure 2: Top: Frequency response $|G(\omega)|^2$ and sensitivity function $|G_d(\omega)|^2$. Bottom: Power spectrum of designed $d_{t}$ for varying constraints $\delta_{\max}$.
• Figure 3: Magnitude of output perturbation $\mathbb{E}[|\delta_{t}|]$ when using prbs and the designed input perturbations for various constraints $\delta_{\max}$. The magnitude can be compared to the reference signal $r_{t} \equiv 1$. Dotted lines show user-specified limits $\delta_{\max}$.
• Figure 4: Mean squared error \ref{['eq:MSEparameter']} when using prbs and the designed input perturbations for various constraints $\delta_{\max}$. To be considered with Figure \ref{['fig:perturbationmagnitude']} in mind.

Theorems & Definitions (3)

• Remark 1
• Theorem 1
• proof