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Energetic Resilience of Linear Driftless Systems

Ram Padmanabhan, Melkior Ornik

TL;DR

The paper tackles energetic resilience in linear driftless systems subject to loss of authority over a subset of actuators, addressing finite-time regulation from a control-theoretic energy perspective. It introduces an energetic resilience metric that compares the nominal minimum input energy $E_N^{*}$ to the worst-case malfunctioning energy $ar{E}_M$, for a given final time $t_f$ and initial state $x_0$, across all admissible uncontrolled inputs. The authors derive closed-form expressions for the nominal energy and malfunctioning energy, establish a computable upper bound on the worst-case total energy, and obtain an exact expression and a bound for the special case of losing authority over one actuator, enabling a tangible bound on the resilience metric $r_M(t_f,R)$. A simulation with an underwater robot model validates the framework and shows that the resilience bound provides meaningful insight into energy penalties caused by actuator loss, informing energy budgeting and fault-tolerant design for driftless robotic systems.

Abstract

When a malfunction causes a control system to lose authority over a subset of its actuators, achieving a task may require spending additional energy in order to compensate for the effect of uncontrolled inputs. To understand this increase in energy, we introduce an energetic resilience metric that quantifies the maximal additional energy required to achieve finite-time regulation in linear driftless systems that suffer this malfunction. We first derive optimal control signals and minimum energies to achieve this task in both the nominal and malfunctioning systems. We then obtain a bound on the worst-case energy used by the malfunctioning system, and its exact expression in the special case of loss of authority over one actuator. Further considering this special case, we derive a bound on the metric for energetic resilience. A simulation example on a model of an underwater robot demonstrates that this bound is useful in quantifying the increased energy used by a system suffering such a malfunction.

Energetic Resilience of Linear Driftless Systems

TL;DR

The paper tackles energetic resilience in linear driftless systems subject to loss of authority over a subset of actuators, addressing finite-time regulation from a control-theoretic energy perspective. It introduces an energetic resilience metric that compares the nominal minimum input energy to the worst-case malfunctioning energy , for a given final time and initial state , across all admissible uncontrolled inputs. The authors derive closed-form expressions for the nominal energy and malfunctioning energy, establish a computable upper bound on the worst-case total energy, and obtain an exact expression and a bound for the special case of losing authority over one actuator, enabling a tangible bound on the resilience metric . A simulation with an underwater robot model validates the framework and shows that the resilience bound provides meaningful insight into energy penalties caused by actuator loss, informing energy budgeting and fault-tolerant design for driftless robotic systems.

Abstract

When a malfunction causes a control system to lose authority over a subset of its actuators, achieving a task may require spending additional energy in order to compensate for the effect of uncontrolled inputs. To understand this increase in energy, we introduce an energetic resilience metric that quantifies the maximal additional energy required to achieve finite-time regulation in linear driftless systems that suffer this malfunction. We first derive optimal control signals and minimum energies to achieve this task in both the nominal and malfunctioning systems. We then obtain a bound on the worst-case energy used by the malfunctioning system, and its exact expression in the special case of loss of authority over one actuator. Further considering this special case, we derive a bound on the metric for energetic resilience. A simulation example on a model of an underwater robot demonstrates that this bound is useful in quantifying the increased energy used by a system suffering such a malfunction.
Paper Structure (11 sections, 3 theorems, 27 equations, 1 figure)

This paper contains 11 sections, 3 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Facts
  3. Problem Formulation
  4. Nominal and Malfunctioning Energies
  5. Nominal Energy
  6. Malfunctioning Energy
  7. Resilience to the Loss of One Actuator
  8. Worst-case Total Energy
  9. Energetic Resilience Metric
  10. Simulation Example
  11. Conclusions and Future Work

Key Result

Lemma 1

Let $\mathcal{Z} \coloneqq \Bigl\{z:[0, t_f] \to \mathbb{R}^{n_z} : \frac{1}{t_f} \int_{0}^{t_f} z(t)\mathrm{d}t = \overline{z} \in \mathbb{R}^{n_z}\Bigr\}$ be the set of continuous, real vector-valued functions with given mean value $\overline{z}$ in the interval $[0, t_f]$. Let $z^*(t)$ denote the i.e., the function with minimum energy is a constant.

Figures (1)

  • Figure 1: Ratio of nominal and malfunctioning energies compared to the resilience metric. The thinner dashed lines plot $E_{M}^{+}-E_N$ for various uncontrolled inputs.

Theorems & Definitions (11)

  • Definition 1: Finite-time Stabilizing Resilience
  • Definition 2: Nominal Energy
  • Definition 3: Malfunctioning Energy
  • Definition 4: Total Energy
  • Definition 5: Worst-case Total Energy
  • Definition 6: Energetic Resilience
  • Lemma 1
  • Proposition 1: Worst-case Total Energy
  • Remark 1
  • Remark 2
  • ...and 1 more