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L1 Adaptive Output Feedback for Non-square Systems with Arbitrary Relative Degree

Hanmin Lee, Venanzio Cichella, Naira Hovakimyan

TL;DR

The paper tackles output-feedback control for non-square MIMO systems with arbitrary vector relative degree under matched uncertainties. It introduces an $cL_1$ adaptive controller that employs a right interactor, a state-output predictor, a low-pass filter, and projection-based adaptive laws to achieve tracking of a reference signal with semi-global stability and arbitrarily small steady-state error. Key contributions include handling underactuated systems with time-varying uncertainties without requiring regressor-based uncertainty parameterization, and providing rigorous transient and steady-state performance guarantees validated by academic and inverted pendulum experiments. The work broadens the applicability of $cL_1$ methods to non-square, highly relative-degree systems with practical robustness and implementability implications for complex MIMO plants.

Abstract

This paper considers the problem of output feedback control for non-square multi-input multi-output systems with arbitrary relative degree. The proposed controller, based on the L1 adaptive control architecture, is designed using the right interactor matrix and a suitably defined projection matrix. A state-output predictor, a low-pass filter, and adaptive laws are introduced that achieve output tracking of a desired reference signal. It is shown that the proposed control strategy guarantees closed-loop stability with arbitrarily small steady-state errors. The transient performance in the presence of non-zero initialization errors is quantified in terms of decreasing functions. Rigorous mathematical analysis and illustrative examples are provided to validate the theoretical claims.

L1 Adaptive Output Feedback for Non-square Systems with Arbitrary Relative Degree

TL;DR

The paper tackles output-feedback control for non-square MIMO systems with arbitrary vector relative degree under matched uncertainties. It introduces an adaptive controller that employs a right interactor, a state-output predictor, a low-pass filter, and projection-based adaptive laws to achieve tracking of a reference signal with semi-global stability and arbitrarily small steady-state error. Key contributions include handling underactuated systems with time-varying uncertainties without requiring regressor-based uncertainty parameterization, and providing rigorous transient and steady-state performance guarantees validated by academic and inverted pendulum experiments. The work broadens the applicability of methods to non-square, highly relative-degree systems with practical robustness and implementability implications for complex MIMO plants.

Abstract

This paper considers the problem of output feedback control for non-square multi-input multi-output systems with arbitrary relative degree. The proposed controller, based on the L1 adaptive control architecture, is designed using the right interactor matrix and a suitably defined projection matrix. A state-output predictor, a low-pass filter, and adaptive laws are introduced that achieve output tracking of a desired reference signal. It is shown that the proposed control strategy guarantees closed-loop stability with arbitrarily small steady-state errors. The transient performance in the presence of non-zero initialization errors is quantified in terms of decreasing functions. Rigorous mathematical analysis and illustrative examples are provided to validate the theoretical claims.

Paper Structure

This paper contains 11 sections, 9 theorems, 126 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Problem formulation
  4. Problem statement
  5. Parametrization of uncertain function
  6. $\mathcal{L}_1$ adaptive controller design
  7. Stability Analysis
  8. Illustrative Examples
  9. Academic example
  10. Inverted pendulum on a cart
  11. Conclusions

Key Result

Theorem 1

Let $M_0(s) = C \left( s{\mathbb{I}}_n - A \right)^{-1} B + D$, where $A \in {\mathbb{R}}^{n \times n}$, $B \in {\mathbb{R}}^{n \times m}$, $C \in {\mathbb{R}}^{p \times n}$, and $D \in {\mathbb{R}}^{p \times m}$. Assume that $(A,C)$ and $(A,B)$ are observable and controllable pairs, respectively. S with where $A_z \in {\mathbb{R}}^{n_z \times n_z}$ is Hurwitz, $B_z \in {\mathbb{R}}^{n_z \times m

Figures (6)

  • Figure 1: System responses with $\Gamma=500$ and $f_1(x,t)$
  • Figure 2: Tracking and estimation errors
  • Figure 3: System responses with $\Gamma=500$ and $f_2(x,t)$
  • Figure 4: Inverted pendulum: position, angle, and control input for scenario 1.
  • Figure 5: Inverted pendulum: position, angle, and control input for scenario 2.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Corollary 2
  • proof
  • ...and 24 more