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Distributed Stability Certification and Control from Local Data

Surya Malladi, Nima Monshizadeh

Abstract

Most data-driven analysis and control methods rely on centralized access to system measurements. In contrast, we consider a setting in which the measurements are distributed across multiple agents and raw data are not shared. Each agent has access only to locally held samples, possibly as little as a single measurement, and agents exchange only locally computed signals. Consequently, no individual agent possesses sufficient information to identify the entire system or synthesize a controller independently. To address this limitation, we develop distributed dynamical algorithms that enable the agents to collectively compute global system certificates from local data. Two problems are addressed. First, for stable linear time-invariant (LTI) systems, the agents compute a Lyapunov certificate by solving the Lyapunov equation in a fully distributed manner. Second, for general LTI systems, they compute the stabilizing solution of the algebraic Riccati equation and hence the optimal linear-quadratic regulator (LQR). An initially proposed scheme guarantees practical convergence, while a subsequent augmented PI-type algorithm achieves exact convergence to the desired solution. We further establish robustness of the resulting LQR controller to uncertainty and measurement noise. The approach is illustrated through distributed Lyapunov certification of a quadruple-tank process and distributed LQR design for helicopter dynamics.

Distributed Stability Certification and Control from Local Data

Abstract

Most data-driven analysis and control methods rely on centralized access to system measurements. In contrast, we consider a setting in which the measurements are distributed across multiple agents and raw data are not shared. Each agent has access only to locally held samples, possibly as little as a single measurement, and agents exchange only locally computed signals. Consequently, no individual agent possesses sufficient information to identify the entire system or synthesize a controller independently. To address this limitation, we develop distributed dynamical algorithms that enable the agents to collectively compute global system certificates from local data. Two problems are addressed. First, for stable linear time-invariant (LTI) systems, the agents compute a Lyapunov certificate by solving the Lyapunov equation in a fully distributed manner. Second, for general LTI systems, they compute the stabilizing solution of the algebraic Riccati equation and hence the optimal linear-quadratic regulator (LQR). An initially proposed scheme guarantees practical convergence, while a subsequent augmented PI-type algorithm achieves exact convergence to the desired solution. We further establish robustness of the resulting LQR controller to uncertainty and measurement noise. The approach is illustrated through distributed Lyapunov certification of a quadruple-tank process and distributed LQR design for helicopter dynamics.
Paper Structure (17 sections, 11 theorems, 150 equations, 4 figures)

This paper contains 17 sections, 11 theorems, 150 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Problem statement
  4. Data-based splitting scheme
  5. Distributed algorithms to find a Lyapunov certificate
  6. Distributed algorithm to solve LQR problem of LTI system
  7. Robustness
  8. Partially known input matrix
  9. Noisy data
  10. Case studies
  11. Distributed data-driven computation of Lyapunov function: Quadruple tank
  12. Distributed data-driven LQR: Helicopter hover dynamics
  13. Conclusions
  14. Appendix A: Distributed share computation
  15. Appendix B: Proofs of technical lemmas
  16. ...and 2 more sections

Key Result

Lemma 1

jiang2021trends Consider a networked system, whose dynamics are given by where $\mathcal{N}_i$ is the set of in-neighbours of agent $i$. The averaged dynamics or blended dynamics are given by Assume that the equilibrium $s^*$ is uniformly asymptotically stable for the blended dynamics eqn: Blended gen, and let $\mathcal{D}_b$ be an open subset of the domain of attraction of $s^*$. Define Then,

Figures (4)

  • Figure 1: Quadruple tank: Asymptotic convergence to the solution of the Lyapunov equation for $\gamma=10^3$.
  • Figure 2: Helicopter dynamics: Practical convergence to the solution of the Riccati equation for different values of $\gamma$
  • Figure 3: Helicopter dynamics: Asymptotic convergence to the solution of the Riccati equation for $\gamma=500$.
  • Figure 4: Helicopter dynamics: Expected LQR cost vs. size of the uncertainty (left) and vs. noise energy level (right).

Theorems & Definitions (11)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • Theorem 4
  • Lemma 4
  • Lemma 5
  • Proposition 1
  • ...and 1 more