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A Constrained Tracking Controller for Ramp and Sinusoidal Reference Signals using Robust Positive Invariance

Geovana Franca dos Santos, Eugenio B. Castelan, Walter Lucia

TL;DR

The paper tackles constrained reference tracking for linear time-invariant systems aiming at offset-free convergence for ramp and sinusoidal references. It combines the Internal Model Principle with polyhedral robust positively invariant sets and Extended Farkas' Lemma to derive a single bilinear optimization that offline designs the PI-like controller gains, the admissible reference set, and the invariant domain. The method yields an augmented closed-loop model and an RPI constraint set that ensure constraint satisfaction while achieving asymptotic tracking for references in a polyhedral bound, demonstrated on a two-tank system. Two objective options trade off between expanding the admissible reference region and reducing integral error, with simulations illustrating robust performance under state and input constraints. This framework provides a practical, off-line design pathway for constrained ramp and sinusoid tracking in LTIC systems, with room for extension to multi-output configurations.

Abstract

This paper proposes an output feedback controller capable of ensuring steady-state offset-free tracking for ramp and sinusoidal reference signals while ensuring local stability and state and input constraints fulfillment. The proposed solution is derived by jointly exploiting the internal model principle, polyhedral robust positively invariant arguments, and the Extended Farkas' Lemma. In particular, by considering a generic class of output feedback controller equipped with a feedforward term, a proportional effect, and a double integrator, we offline design the controller's gains by means of a single bilinear optimization problem. A peculiar feature of the proposed design is that the sets of all the admissible reference signals and the plant's initial conditions are also offline determined. Simulation results are provided to testify to the effectiveness of the proposed tracking controller and its capability to deal with both state and input constraints.

A Constrained Tracking Controller for Ramp and Sinusoidal Reference Signals using Robust Positive Invariance

TL;DR

The paper tackles constrained reference tracking for linear time-invariant systems aiming at offset-free convergence for ramp and sinusoidal references. It combines the Internal Model Principle with polyhedral robust positively invariant sets and Extended Farkas' Lemma to derive a single bilinear optimization that offline designs the PI-like controller gains, the admissible reference set, and the invariant domain. The method yields an augmented closed-loop model and an RPI constraint set that ensure constraint satisfaction while achieving asymptotic tracking for references in a polyhedral bound, demonstrated on a two-tank system. Two objective options trade off between expanding the admissible reference region and reducing integral error, with simulations illustrating robust performance under state and input constraints. This framework provides a practical, off-line design pathway for constrained ramp and sinusoid tracking in LTIC systems, with room for extension to multi-output configurations.

Abstract

This paper proposes an output feedback controller capable of ensuring steady-state offset-free tracking for ramp and sinusoidal reference signals while ensuring local stability and state and input constraints fulfillment. The proposed solution is derived by jointly exploiting the internal model principle, polyhedral robust positively invariant arguments, and the Extended Farkas' Lemma. In particular, by considering a generic class of output feedback controller equipped with a feedforward term, a proportional effect, and a double integrator, we offline design the controller's gains by means of a single bilinear optimization problem. A peculiar feature of the proposed design is that the sets of all the admissible reference signals and the plant's initial conditions are also offline determined. Simulation results are provided to testify to the effectiveness of the proposed tracking controller and its capability to deal with both state and input constraints.
Paper Structure (6 sections, 3 theorems, 22 equations, 6 figures, 1 table)

This paper contains 6 sections, 3 theorems, 22 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Paper's contribution
  3. Constrained Control Problem
  4. Proposed Solution
  5. Simulation
  6. Conclusion

Key Result

Lemma 1

A polyhedron $\mathcal{L}$, eq:setL, is a RPI set of the system eq:closed, if and only if, exists a Meztler matrix $H \in \mathbb{R}^{l \times l}$, the matrix $H_r>0 \in \mathbb{R}^{l \times l_r}$ and a scalar $\gamma>0$, such that

Figures (6)

  • Figure 1: Projected RPI set $\mathcal{L}$ and state trajectory for reference \ref{['eq:ref_ramp']}
  • Figure 2: Output and reference tracking for reference \ref{['eq:ref_ramp']}
  • Figure 3: Error signal for reference \ref{['eq:ref_ramp']}
  • Figure 4: Projected RPI set $\mathcal{L}$ and state trajectory for sinusoidal reference
  • Figure 5: Output and reference tracking for sinusoidal reference
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Theorem 1
  • Example 1
  • Definition 2
  • Definition 3
  • ...and 1 more