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Asymptotic tracking by funnel control with internal models

Thomas Berger, Christoph M. Hackl, Stephan Trenn

TL;DR

This paper focuses on linear systems with linear reference internal models and shows that under mild adjustments of funnel control, it can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters).

Abstract

Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters).

Asymptotic tracking by funnel control with internal models

TL;DR

This paper focuses on linear systems with linear reference internal models and shows that under mild adjustments of funnel control, it can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters).

Abstract

Funnel control achieves output tracking with guaranteed tracking performance for unknown systems and arbitrary reference signals. In particular, the tracking error is guaranteed to satisfy time-varying error bounds for all times (it evolves in the funnel). However, convergence to zero cannot be guaranteed, but the error often stays close to the funnel boundary, inducing a comparatively large feedback gain. This has several disadvantages (e.g. poor tracking performance and sensitivity to noise due to the underlying high-gain feedback principle). In this paper, therefore, the usually known reference signal is taken into account during funnel controller design, i.e. we propose to combine the well-known internal model principle with funnel control. We focus on linear systems with linear reference internal models and show that under mild adjustments of funnel control, we can achieve asymptotic tracking for a whole class of linear systems (i.e. without relying on the knowledge of system parameters).
Paper Structure (8 sections, 3 theorems, 35 equations, 3 figures)

This paper contains 8 sections, 3 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System class
  3. Control objective
  4. Class of reference signals
  5. Internal models
  6. Controller design
  7. Funnel control -- main result
  8. Illustrative simulations

Key Result

Lemma II.1

Consider a system eq:System-lin with $(A,B,C)\in\Sigma_{m,r}$, let $\alpha(s)\in\mathbb{R}[s]$ be a monic polynomial and $(\tilde{A}, \tilde{B},\tilde{C}, I_m)$ be an internal model of the class $\mathcal{R}(\alpha)$. Then the serial interconnection of eq:System-lin and eq:int_mod, given by where is in class $\Sigma_{m,r}$ with $C_{\rm ic} A_{\rm ic}^{r-1} B_{\rm ic} = C A^{r-1} B = \Gamma$.

Figures (3)

  • Figure 1: Error evolution in a funnel $\mathcal{F}_{\varphi}$ with boundary $\varphi(t)^{-1}$.
  • Figure 2: Illustration of control system with internal model.
  • Figure 3: Simulation results for system \ref{['eq:example system']} using funnel controller \ref{['eq:fun-con']} without internal model [w/o IM] and funnel controller \ref{['eq:fun-con']} with internal model \ref{['eq:IM']} [].

Theorems & Definitions (8)

  • Definition I.2
  • Lemma II.1
  • proof
  • Lemma IV.1
  • proof
  • Theorem IV.2
  • proof
  • Remark V.1: Measurement noise