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An Input-to-State Safety Approach Towards Safe Control of a Class of Parabolic PDEs Under Disturbances

Tanushree Roy, Ashley Knichel, Satadru Dey

TL;DR

This work tackles safe control of a class of linear parabolic PDEs under disturbances by integrating practical input-to-state safety (pISSf) with input-to-state stability (ISSt). A boundary-feedback control law is designed, and a pISSf barrier functional is constructed to keep state trajectories away from an unsafe set while an ISSt Lyapunov bound guarantees stability under disturbances. The gains are explicitly characterized by inequalities $[(\mu-1)+\frac{1}{4Lk}]\le0$ and $[(\beta-1)+\frac{1}{4Lk}]\le0$, with a barrier $\mathscr{B}(h)$ ensuring pISSf and a Lyapunov $V(h)$ ensuring ISSt. The methodology is demonstrated on a one-dimensional battery-thermal PDE with boundary cooling, where simulations show safety and stability are preserved under nominal operation and disturbance, including cyber-attacks; future work includes extending to higher-dimensional PDEs and incorporating input saturation.

Abstract

Distributed Parameter Systems (DPSs), modelled by partial differential equations (PDEs), are increasingly vulnerable to disturbances arising from various sources. Although detection of disturbances in PDE systems have received considerable attention in existing literature, safety control of PDEs under disturbances remains significantly under-explored. In this context, we explore a practical input-to-state safety (pISSf) based control design approach for a class of DPSs modelled by linear Parabolic PDEs. Specifically, we develop a control design framework for this class of system with both safety and stability guarantees based on control Lyapunov functional and control barrier functional. To illustrate our methodology, we apply our strategy to design a thermal control system for battery modules under disturbance. Several simulation studies are done to show the efficacy of our method.

An Input-to-State Safety Approach Towards Safe Control of a Class of Parabolic PDEs Under Disturbances

TL;DR

This work tackles safe control of a class of linear parabolic PDEs under disturbances by integrating practical input-to-state safety (pISSf) with input-to-state stability (ISSt). A boundary-feedback control law is designed, and a pISSf barrier functional is constructed to keep state trajectories away from an unsafe set while an ISSt Lyapunov bound guarantees stability under disturbances. The gains are explicitly characterized by inequalities and , with a barrier ensuring pISSf and a Lyapunov ensuring ISSt. The methodology is demonstrated on a one-dimensional battery-thermal PDE with boundary cooling, where simulations show safety and stability are preserved under nominal operation and disturbance, including cyber-attacks; future work includes extending to higher-dimensional PDEs and incorporating input saturation.

Abstract

Distributed Parameter Systems (DPSs), modelled by partial differential equations (PDEs), are increasingly vulnerable to disturbances arising from various sources. Although detection of disturbances in PDE systems have received considerable attention in existing literature, safety control of PDEs under disturbances remains significantly under-explored. In this context, we explore a practical input-to-state safety (pISSf) based control design approach for a class of DPSs modelled by linear Parabolic PDEs. Specifically, we develop a control design framework for this class of system with both safety and stability guarantees based on control Lyapunov functional and control barrier functional. To illustrate our methodology, we apply our strategy to design a thermal control system for battery modules under disturbance. Several simulation studies are done to show the efficacy of our method.
Paper Structure (10 sections, 4 theorems, 46 equations, 7 figures)

This paper contains 10 sections, 4 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Control Problem Formulation
  4. Input-to-State safety based control design
  5. Unsafe set and distance metric formulation
  6. pISSf via barrier functional characterization
  7. Construction of pISSf barrier functional and design requirements
  8. Input-to-State stabilizing safe control
  9. Simulation case studies
  10. Conclusion

Key Result

Proposition 1

Consider the PDE system given by h_system-input1, the prescribed unsafe set $\mathscr{U}\subset \mathbb{R}^+$ given by unsafe_set and the distance metric as defined in dist_metric. Suppose there exists a safety barrier functional $\mathscr{B}: \mathbb{H}_1 \to \mathbb{R}$ satisfying the following tw

Figures (7)

  • Figure 1: Temperature measurement at the two boundaries and the mid-section from the battery module under no-anomaly scenarios with Stability-Only Control (St-C), and Stability-and-Safety Control (StSf-C).
  • Figure 2: Coolant temperatures at the two boundaries of the battery module under no-anomaly scenarios with Stability-Only Control (St-C), and Stability-and-Safety Control (StSf-C).
  • Figure 3: State-of-Charge (SOC) of the battery module under nominal condition and under disturbance through battery overdischarge.
  • Figure 4: Heat generated in the battery module under nominal condition and under disturbance through battery overdischarge.
  • Figure 5: Spatiotemporal temperature distribution in the battery module under overdischarge with Stability-Only Control (St-C), and Stability-and-Safety Control (StSf-C).
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Lemma 1: A Version of Poincare Inequality
  • proof
  • Theorem 1: Design Requirements for pISSf
  • proof
  • Theorem 2: Design Requirements for both pISSf and ISSt
  • proof