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Consistency analysis of refined instrumental variable methods for continuous-time system identification in closed-loop

Rodrigo A. González, Siqi Pan, Cristian R. Rojas, James S. Welsh

TL;DR

This work analyzes the generic consistency of refined instrumental variable methods for closed-loop continuous-time system identification, focusing on SRIVC and CLSRIVC. It proves that a continuous-time controller in the loop with sampled data leads to non-consistency for both estimators, while a discrete-time controller enables generic consistency; oversampling can mitigate bias in the continuous-time case, and a discrete-time CLSRIVC variant is shown to be generically consistent. Theoretical results are supported by Monte Carlo simulations, which demonstrate the practical implications for when these estimators can be trusted in closed-loop identification. Overall, the study provides actionable guidance on selecting sampling strategies and controller types to ensure reliable continuous-time system identification in closed-loop settings.

Abstract

Refined instrumental variable methods have been broadly used for identification of continuous-time systems in both open and closed-loop settings. However, the theoretical properties of these methods are still yet to be fully understood when operating in closed-loop. In this paper, we address the consistency of the simplified refined instrumental variable method for continuous-time systems (SRIVC) and its closed-loop variant CLSRIVC when they are applied on data that is generated from a feedback loop. In particular, we consider feedback loops consisting of continuous-time controllers, as well as the discrete-time control case. This paper proves that the SRIVC and CLSRIVC estimators are not generically consistent when there is a continuous-time controller in the loop, and that generic consistency can be achieved when the controller is implemented in discrete-time. Numerical simulations are presented to support the theoretical results.

Consistency analysis of refined instrumental variable methods for continuous-time system identification in closed-loop

TL;DR

This work analyzes the generic consistency of refined instrumental variable methods for closed-loop continuous-time system identification, focusing on SRIVC and CLSRIVC. It proves that a continuous-time controller in the loop with sampled data leads to non-consistency for both estimators, while a discrete-time controller enables generic consistency; oversampling can mitigate bias in the continuous-time case, and a discrete-time CLSRIVC variant is shown to be generically consistent. Theoretical results are supported by Monte Carlo simulations, which demonstrate the practical implications for when these estimators can be trusted in closed-loop identification. Overall, the study provides actionable guidance on selecting sampling strategies and controller types to ensure reliable continuous-time system identification in closed-loop settings.

Abstract

Refined instrumental variable methods have been broadly used for identification of continuous-time systems in both open and closed-loop settings. However, the theoretical properties of these methods are still yet to be fully understood when operating in closed-loop. In this paper, we address the consistency of the simplified refined instrumental variable method for continuous-time systems (SRIVC) and its closed-loop variant CLSRIVC when they are applied on data that is generated from a feedback loop. In particular, we consider feedback loops consisting of continuous-time controllers, as well as the discrete-time control case. This paper proves that the SRIVC and CLSRIVC estimators are not generically consistent when there is a continuous-time controller in the loop, and that generic consistency can be achieved when the controller is implemented in discrete-time. Numerical simulations are presented to support the theoretical results.
Paper Structure (12 sections, 60 equations, 5 figures, 1 table)

This paper contains 12 sections, 60 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Closed-loop settings
  3. The SRIVC and CLSRIVC estimators
  4. Analysis of the SRIVC and CLSRIVC estimators for Setting 1
  5. Reducing the bias of refined instrumental variable methods in closed-loop via oversampling
  6. Analysis of the SRIVC and CLSRIVC estimators for Setting 2
  7. Consistency of the SRIVC estimator for Setting 2
  8. The CLSRIVC estimator for Setting 2 and its consistency analysis
  9. Simulations
  10. Consistency tests
  11. Bias behavior of the SRIVC estimator in Setting 2
  12. Conclusions

Figures (5)

  • Figure 1: Block diagrams for the closed-loop Settings 1 and 2.
  • Figure 2: Mean of the estimated parameters for an increasing sample size, Setting 1.
  • Figure 3: Mean of the estimated parameters for an increasing sample size, Setting 2.
  • Figure 4: Variance of the estimated parameters for an increasing sample size, Setting 2.
  • Figure 5: Normalized bias of the SRIVC estimator versus SNR, Setting 2.