Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unifying Direct and Indirect Learning for Safe Control of Linear Systems

Amir Modares, Niyousha Ghiasi, Bahare Kiumarsi, Hamidreza Modares

TL;DR

The paper addresses safe control of linear systems with model uncertainty and disturbances by unifying direct (closed-loop) and indirect (open-loop) learning within a CMZ framework. It introduces a data-driven closed-loop representation that constrains models to those explainable by data and prior knowledge using conformal equality constraints, and a CMZ-based set-membership identification to refine a prior open-loop model into a posterior FOLMS. A data-driven linear programming approach enforces $\ abla$-contractivity (λ-contractivity) of a polyhedral safe set, guaranteeing robust invariance under disturbances. Simulation results show dramatic reductions in uncertainty and improved feasibility for the safety guarantees as data accumulates, validating the approach and its safety-stability properties. The framework lays groundwork for extensions to nonlinear systems via polynomial zonotopes and highlights practical considerations for data excitation and safety during online updates.

Abstract

This paper develops learning-enabled safe controllers for linear systems subject to system uncertainties and bounded disturbances. Given the disturbance zonotope, the databased closed-loop dynamics (CLDs) are first characterized using a matrix zonotope (MZ), and refined through several steps to yield a constrained matrix zonotope (CMZ). This refinement is achieved by introducing conformal equality constraints that eliminate incompatible disturbance realizations. More precisely, prior knowledge and observed data are used separately to construct CMZ representations of disturbance sequences that conform to both data and prior knowledge, and are intersected by the initial MZ of the disturbance sequence, producing a refined CMZ. This approach reduces conservatism. To further reduce the conservativeness, we unify open-loop learning with closed-loop learning by presenting a novel set-membership identification method that models open-loop dynamics as a CMZ. The prior knowledge serves as an initial feasible open-loop model set (FOLMS) of this CMZ, which is refined into a posterior set whenever new informative online data becomes available. This posterior FOLMS then adaptively replaces the prior knowledge set employed in the disturbance elimination of the closed-loop learning process. The resulting refined parameterized set of CLD is subsequently leveraged to directly and adaptively learn a controller that robustly enforces safety. Toward this goal, we formulate a linear programming problem that guarantees λcontractiveness of a polyhedral safe set. A simulation example is provided to validate the effectiveness of the proposed approach and support the theoretical results.

Unifying Direct and Indirect Learning for Safe Control of Linear Systems

TL;DR

The paper addresses safe control of linear systems with model uncertainty and disturbances by unifying direct (closed-loop) and indirect (open-loop) learning within a CMZ framework. It introduces a data-driven closed-loop representation that constrains models to those explainable by data and prior knowledge using conformal equality constraints, and a CMZ-based set-membership identification to refine a prior open-loop model into a posterior FOLMS. A data-driven linear programming approach enforces -contractivity (λ-contractivity) of a polyhedral safe set, guaranteeing robust invariance under disturbances. Simulation results show dramatic reductions in uncertainty and improved feasibility for the safety guarantees as data accumulates, validating the approach and its safety-stability properties. The framework lays groundwork for extensions to nonlinear systems via polynomial zonotopes and highlights practical considerations for data excitation and safety during online updates.

Abstract

This paper develops learning-enabled safe controllers for linear systems subject to system uncertainties and bounded disturbances. Given the disturbance zonotope, the databased closed-loop dynamics (CLDs) are first characterized using a matrix zonotope (MZ), and refined through several steps to yield a constrained matrix zonotope (CMZ). This refinement is achieved by introducing conformal equality constraints that eliminate incompatible disturbance realizations. More precisely, prior knowledge and observed data are used separately to construct CMZ representations of disturbance sequences that conform to both data and prior knowledge, and are intersected by the initial MZ of the disturbance sequence, producing a refined CMZ. This approach reduces conservatism. To further reduce the conservativeness, we unify open-loop learning with closed-loop learning by presenting a novel set-membership identification method that models open-loop dynamics as a CMZ. The prior knowledge serves as an initial feasible open-loop model set (FOLMS) of this CMZ, which is refined into a posterior set whenever new informative online data becomes available. This posterior FOLMS then adaptively replaces the prior knowledge set employed in the disturbance elimination of the closed-loop learning process. The resulting refined parameterized set of CLD is subsequently leveraged to directly and adaptively learn a controller that robustly enforces safety. Toward this goal, we formulate a linear programming problem that guarantees λcontractiveness of a polyhedral safe set. A simulation example is provided to validate the effectiveness of the proposed approach and support the theoretical results.

Paper Structure

This paper contains 8 sections, 7 theorems, 58 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Operations on Constrained Matrix Zonotopes
  3. Problem Formulation
  4. Closed-loop System Representation Using Data and Prior Knowledge
  5. Unifying Direct and Indirect Learning
  6. Robust Safe Control Design
  7. Simulation Results
  8. conclusion

Key Result

Lemma 1

Consider two CMZs $\mathscr{M}^v=<G^v,C^v,A^v,B^v >$, $G^v \in \mathbb{R}^{n \times p s_v}$, $C^v \in \mathbb{R}^{n \times p}$, $A^v \in \mathbb{R}^{k_{v} \times q_{v} s_v}$, and $B^v \in \mathbb{R}^{k_{v} \times q_{v} }$, $v=1,2$. Then, where $\bar{A}^v=[\bar{A}^v_{:,1} \quad ... \quad \bar{A}^v_{:,s_v}]$, $\bar{G}^v=[\bar{G}^v_{:,1} \quad ... \quad \bar{G}^v_{:,s_v}]$ with and $b=\max(p,q_1,

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 5: Phase portrait of the trajectories from a single run, originating from different initial points on the boundary of the safe set.
  • Figure 6: Feasibility rate of the optimization problem \ref{['LPf']} as a function of (a) the scaling factor $\alpha$ and (b) the contraction factor $\lambda$, averaged over 100 Monte Carlo simulations.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • ...and 2 more