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Adaptive Observers for MIMO Discrete-Time LTI Systems

Anchita Dey, Shubhendu Bhasin

TL;DR

This work addresses online identification of the state and system matrices of an unknown MIMO discrete-time LTI system with transfer matrix $G_{sp}(s)$ by designing an adaptive observer that separates state estimation from parameter learning. It adopts a recursive least squares estimator with covariance resetting to guarantee boundedness of the parameter and state estimates without requiring persistent excitation, and extends the approach to the general LTI form using a filtered regressor representation with Kronecker-product constructions. The key contributions are (i) the first discrete-time MIMO adaptive observer applicable to general LTI form, (ii) a rigorous boundedness analysis that holds irrespective of excitation, and (iii) demonstration via a numerical example and comparison to existing SISO designs. The results provide a robust online identification framework for MIMO LTI systems, with convergence enhanced by a $PE$ regressor when available.

Abstract

In this paper, an adaptive observer is proposed for multi-input multi-output (MIMO) discrete-time linear time-invariant (LTI) systems. Unlike existing MIMO adaptive observer designs, the proposed approach is applicable to LTI systems in their general form. Further, the proposed method uses recursive least square (RLS) with covariance resetting for adaptation that is shown to guarantee that the estimates are bounded, irrespective of any excitation condition, even in the presence of a vanishing perturbation term in the error used for updation in RLS. Detailed analysis for convergence and boundedness has been provided along with simulation results for illustrating the performance of the developed theory.

Adaptive Observers for MIMO Discrete-Time LTI Systems

TL;DR

This work addresses online identification of the state and system matrices of an unknown MIMO discrete-time LTI system with transfer matrix by designing an adaptive observer that separates state estimation from parameter learning. It adopts a recursive least squares estimator with covariance resetting to guarantee boundedness of the parameter and state estimates without requiring persistent excitation, and extends the approach to the general LTI form using a filtered regressor representation with Kronecker-product constructions. The key contributions are (i) the first discrete-time MIMO adaptive observer applicable to general LTI form, (ii) a rigorous boundedness analysis that holds irrespective of excitation, and (iii) demonstration via a numerical example and comparison to existing SISO designs. The results provide a robust online identification framework for MIMO LTI systems, with convergence enhanced by a regressor when available.

Abstract

In this paper, an adaptive observer is proposed for multi-input multi-output (MIMO) discrete-time linear time-invariant (LTI) systems. Unlike existing MIMO adaptive observer designs, the proposed approach is applicable to LTI systems in their general form. Further, the proposed method uses recursive least square (RLS) with covariance resetting for adaptation that is shown to guarantee that the estimates are bounded, irrespective of any excitation condition, even in the presence of a vanishing perturbation term in the error used for updation in RLS. Detailed analysis for convergence and boundedness has been provided along with simulation results for illustrating the performance of the developed theory.
Paper Structure (11 sections, 32 equations, 5 figures)

This paper contains 11 sections, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations:
  3. Problem Formulation
  4. MIMO Adaptive Observer
  5. Design of the Parameter Estimation Law
  6. Covariance resetting:
  7. Convergence and Boundedness
  8. A Numerical Example
  9. Conclusion
  10. Proof of Lemma \ref{['a0']}
  11. Proof of Lemma \ref{['a1']} part (2)

Figures (5)

  • Figure 1: Normalized parameter estimates obtained using (a) non-PE regressor (b) PE regressor.
  • Figure 2: $2$-norm of state estimation error with and without a PE regressor.
  • Figure 3: (a) Input without PE (b) Input with PE.
  • Figure 4: Normalized parameter estimates without using a PE regressor for the observer (a) suzuki1980discrete and (b) proposed.
  • Figure 5: $2$-norm of state estimation error without using a PE regressor for the observer (a) suzuki1980discrete and (b) proposed.