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Discrete-Time Implementation of Explicit Reference Governor

Mu'taz A. Momani, Mehdi Hosseinzadeh

TL;DR

The paper addresses constraint handling for pre-stabilized systems by introducing a discrete-time Explicit Reference Governor (ERG) that updates the applied reference via $v(kdt) = v((k-1)dt) + dt \cdot \kappa \cdot g(x(kdt), v((k-1)dt), r)$ using a Lyapunov-based Dynamic Safety Margin and an Attraction Field. It derives per-step bounds on the design gain $\kappa(kdt)$ to preserve the invariant admissible set and proves convergence to $r$ when strictly steady-state admissible or to the best admissible $r^*$ otherwise, through Lyapunov analysis with $W(v(kdt)) = \sum_{\omega=v(kdt)}^{r} \|\rho(\omega,r)\|^2$. The method features a simple, dynamic $\kappa$ without offline tuning and provides a pseudocode implementation, validated by simulations on a double integrator and aircraft longitudinal dynamics, and experimentally on a Parrot Bebop 2 drone. Results show dynamic $\kappa$ yields constraint satisfaction with improved convergence compared to fixed-gain schemes, making discrete-time ERG practical for safety-critical applications.

Abstract

Explicit reference governor (ERG) is an add-on unit that provides constraint handling capability to pre-stabilized systems. The main idea behind ERG is to manipulate the derivative of the applied reference in continuous time such that the satisfaction of state and input constraints is guaranteed at all times. However, ERG should be practically implemented in discrete-time. This paper studies the discrete-time implementation of ERG, and provides conditions under which the feasibility and convergence properties of the ERG framework are maintained when the updates of the applied reference are performed in discrete time. The proposed approach is validated via extensive simulation and experimental studies.

Discrete-Time Implementation of Explicit Reference Governor

TL;DR

The paper addresses constraint handling for pre-stabilized systems by introducing a discrete-time Explicit Reference Governor (ERG) that updates the applied reference via using a Lyapunov-based Dynamic Safety Margin and an Attraction Field. It derives per-step bounds on the design gain to preserve the invariant admissible set and proves convergence to when strictly steady-state admissible or to the best admissible otherwise, through Lyapunov analysis with . The method features a simple, dynamic without offline tuning and provides a pseudocode implementation, validated by simulations on a double integrator and aircraft longitudinal dynamics, and experimentally on a Parrot Bebop 2 drone. Results show dynamic yields constraint satisfaction with improved convergence compared to fixed-gain schemes, making discrete-time ERG practical for safety-critical applications.

Abstract

Explicit reference governor (ERG) is an add-on unit that provides constraint handling capability to pre-stabilized systems. The main idea behind ERG is to manipulate the derivative of the applied reference in continuous time such that the satisfaction of state and input constraints is guaranteed at all times. However, ERG should be practically implemented in discrete-time. This paper studies the discrete-time implementation of ERG, and provides conditions under which the feasibility and convergence properties of the ERG framework are maintained when the updates of the applied reference are performed in discrete time. The proposed approach is validated via extensive simulation and experimental studies.
Paper Structure (10 sections, 4 theorems, 35 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 4 theorems, 35 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Theoretical Analysis
  4. Discrete-Time ERG
  5. Feature of the Proposed Discrete-Time ERG
  6. Simulation Results
  7. Controlling a Double Integrator
  8. Controlling the Longitudinal Dynamics of an Aircraft
  9. Experimental Results
  10. Conclusion

Key Result

Lemma 1

Let $v((k-1)dt)\in\mathcal{D}$. At sampling instant $kdt$, let the design parameter $\kappa(kdt)$ be selected such that the following inequality is satisfied: where $\eta_2\in\mathbb{R}_{>0}$ is a smoothing factor, and $\bar{\vartheta}=\min_{i\in\{1,\cdots,n_c\}}\{\vartheta_i\}$ with $\vartheta_i\in\mathbb{R}_{\geq0}$ being the Euclidean distance between the equilibrium point $\bar{x}_{v((k-1)dt)

Figures (10)

  • Figure 1: Schematic structure of the ERG framework.
  • Figure 2: Violation of the invariance of the set $\mathcal{D}$ due to discretization errors.
  • Figure 3: Geometric illustration of the impact of the design parameter $\kappa$ on the invariance of the set $\mathcal{D}$.
  • Figure 4: Geometric illustration of $r^\ast$ and the attraction and repulsion terms.
  • Figure 5: Geometric illustration of changes in the threshold value and the Lyapunov function at sampling instant $kdt$. Note that the Lyapunov function is not necessarily ellipsoidal, while the threshold value is determined based on an ellipsoidal lower-bound of the Lyapunov function as in \ref{['eq:GammaOpt1']}.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 3
  • ...and 1 more