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Initial Excitation-based Adaptive Observers for Discrete-Time LTI Systems

Anchita Dey, Soutrik Bandyopadhyay, Shubhendu Bhasin

TL;DR

The paper addresses online simultaneous estimation of the state and unknown parameters $A$ and $B$ in a discrete-time LTI system using only input–output data, without relying on the persistent excitation (PE) condition. It introduces an IE-based adaptive observer with a two-layer filtering structure and a normalized gradient-descent update, augmented by regressor modification to extract richer information and achieve faster convergence under a finite-time IE (and SIE) condition. Theoretical analysis proves exponential convergence of both parameter and state estimates under IE/SIE, and simulations validate the method against PE-based approaches, demonstrating practical feasibility for stabilization tasks. This work broadens adaptive observer applicability to discrete-time systems by relaxing PE requirements and providing online verifiability of the excitation condition, with potential extensions to nonlinear and time-varying settings.

Abstract

In practical applications, the efficacy of a control algorithm relies critically on the accurate knowledge of the parameters and states of the underlying system. However, obtaining these quantities in practice is often challenging. Adaptive observers address this issue by performing simultaneous state and parameter estimation using only input-output measurements. While many adaptive observer designs exist for continuous-time systems, their discrete-time counterparts remain relatively unexplored. This paper proposes an initial excitation (IE)-based adaptive observer for discrete-time linear time-invariant systems. In contrast to conventional designs that rely on the persistence of excitation condition, which requires continuous excitation and infinite control effort, the proposed method does not require excitation for infinite time, thus making it more practical for stabilization tasks. We employ a two-layer filtering structure and a normalized gradient descent-based update law for learning the unknown parameters. We also propose modifying the regressors to enhance information extraction, leading to faster convergence. Rigorous theoretical analysis guarantees bounded and exponentially converging estimates of both states and parameters under the IE condition, and simulation results validate the efficacy of the proposed design.

Initial Excitation-based Adaptive Observers for Discrete-Time LTI Systems

TL;DR

The paper addresses online simultaneous estimation of the state and unknown parameters and in a discrete-time LTI system using only input–output data, without relying on the persistent excitation (PE) condition. It introduces an IE-based adaptive observer with a two-layer filtering structure and a normalized gradient-descent update, augmented by regressor modification to extract richer information and achieve faster convergence under a finite-time IE (and SIE) condition. Theoretical analysis proves exponential convergence of both parameter and state estimates under IE/SIE, and simulations validate the method against PE-based approaches, demonstrating practical feasibility for stabilization tasks. This work broadens adaptive observer applicability to discrete-time systems by relaxing PE requirements and providing online verifiability of the excitation condition, with potential extensions to nonlinear and time-varying settings.

Abstract

In practical applications, the efficacy of a control algorithm relies critically on the accurate knowledge of the parameters and states of the underlying system. However, obtaining these quantities in practice is often challenging. Adaptive observers address this issue by performing simultaneous state and parameter estimation using only input-output measurements. While many adaptive observer designs exist for continuous-time systems, their discrete-time counterparts remain relatively unexplored. This paper proposes an initial excitation (IE)-based adaptive observer for discrete-time linear time-invariant systems. In contrast to conventional designs that rely on the persistence of excitation condition, which requires continuous excitation and infinite control effort, the proposed method does not require excitation for infinite time, thus making it more practical for stabilization tasks. We employ a two-layer filtering structure and a normalized gradient descent-based update law for learning the unknown parameters. We also propose modifying the regressors to enhance information extraction, leading to faster convergence. Rigorous theoretical analysis guarantees bounded and exponentially converging estimates of both states and parameters under the IE condition, and simulation results validate the efficacy of the proposed design.

Paper Structure

This paper contains 12 sections, 2 theorems, 28 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. The Adaptive Observer Design
  5. The First Layer of Filter
  6. Modification of the Linear Regression
  7. The Second Layer of Filter
  8. SIE Condition
  9. Parameter Estimation Law
  10. Convergence Analysis
  11. Simulation Result
  12. Conclusion

Key Result

Lemma 1

The regressor $W_t$ defined in MatReg is SIE if and only if the regressor $S_t$ defined in Sreg is positive definite for some $t=t_{SIE}$, where $t_{SIE}\geq qn+mn+n$, and $n,\;q,\;m$ are the number of states, outputs and inputs of the plant mentioned in sys1-ytrue.

Figures (5)

  • Figure 1: 2-norm of the estimation error $\tilde{\psi}_t$ with the proposed method.
  • Figure 2: 2-norm of the state estimation error with the proposed method.
  • Figure 3: Comparison of the 2-norm of the parameter estimation errors using (a) the proposed method with an IE input, (b) DEY20238708 with a PE input, and (c) DEY20238708 with a non-PE input.
  • Figure 4: Comparison of the 2-norm of the state estimation errors using (a) the proposed method with an IE input, (b) DEY20238708 with a PE input, and (c) DEY20238708 with a non-PE input.
  • Figure 5: Input signal used for (a) the proposed method, (b) DEY20238708 with PE, and (c) DEY20238708 with non-PE.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof