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Confidence-Aware Safe and Stable Control of Control-Affine Systems

Shiqing Wei, Prashanth Krishnamurthy, Farshad Khorrami

TL;DR

This work addresses the problem of synthesizing safe and stable control for control-affine systems via output feedback via output feedback (using an observer) while reducing the estimation error of the observer.

Abstract

Designing control inputs that satisfy safety requirements is crucial in safety-critical nonlinear control, and this task becomes particularly challenging when full-state measurements are unavailable. In this work, we address the problem of synthesizing safe and stable control for control-affine systems via output feedback (using an observer) while reducing the estimation error of the observer. To achieve this, we adapt control Lyapunov function (CLF) and control barrier function (CBF) techniques to the output feedback setting. Building upon the existing CLF-CBF-QP (Quadratic Program) and CBF-QP frameworks, we formulate two confidence-aware optimization problems and establish the Lipschitz continuity of the obtained solutions. To validate our approach, we conduct simulation studies on two illustrative examples. The simulation studies indicate both improvements in the observer's estimation accuracy and the fulfillment of safety and control requirements.

Confidence-Aware Safe and Stable Control of Control-Affine Systems

TL;DR

This work addresses the problem of synthesizing safe and stable control for control-affine systems via output feedback via output feedback (using an observer) while reducing the estimation error of the observer.

Abstract

Designing control inputs that satisfy safety requirements is crucial in safety-critical nonlinear control, and this task becomes particularly challenging when full-state measurements are unavailable. In this work, we address the problem of synthesizing safe and stable control for control-affine systems via output feedback (using an observer) while reducing the estimation error of the observer. To achieve this, we adapt control Lyapunov function (CLF) and control barrier function (CBF) techniques to the output feedback setting. Building upon the existing CLF-CBF-QP (Quadratic Program) and CBF-QP frameworks, we formulate two confidence-aware optimization problems and establish the Lipschitz continuity of the obtained solutions. To validate our approach, we conduct simulation studies on two illustrative examples. The simulation studies indicate both improvements in the observer's estimation accuracy and the fulfillment of safety and control requirements.
Paper Structure (13 sections, 5 theorems, 37 equations, 3 figures)

This paper contains 13 sections, 5 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. EKF-Based Nonlinear Observer
  3. Observer-Based Safe and Stable Control
  4. Local Exponential Stability of the Observer
  5. Control Lyapunov Functions
  6. Control Barrier Functions
  7. Confidence Optimization
  8. Combining CLFs and CBFs via Convex Optimization
  9. Tracking a Nominal Controller
  10. Simulation Studies
  11. A Second-Order Nonlinear System
  12. The Unicycle System
  13. Conclusion

Key Result

Proposition 1

Under Assumption ass:ass_p and assumptions on the boundedness of $\mathcal{ X }$, $\mathcal{ U }$, and $\hat{\mathcal{ X }}$ and the $\mathcal{ C }^2$-smoothness of $f$, $g$, and $q$, the EKF-based observer in eq:observer_recall is a local exponential observer. More specifically, there exist positiv for $t \geq 0$ with $\zeta(0) \in B_\epsilon$ where $B_\epsilon = \{v \in \mathbb{ R }^{n_x}: \lVer

Figures (3)

  • Figure 1: A second-order system stabilization problem. \ref{['fig:p1a_traj_wo_opt']} and \ref{['fig:p1a_traj_w_opt']}: Trajectories with and without confidence optimization. \ref{['fig:p1a_eigenvalues']}-\ref{['fig:p1a_control']}: Comparison of the eigenvalues (of $P(t)$), the state estimation error, and the control inputs, respectively.
  • Figure 2: A unicycle control problem. \ref{['fig:p2_trajs']}: Trajectories with and without confidence optimization. \ref{['fig:p2_control_wo_opt']} and \ref{['fig:p2_control_w_opt']}: Controls with and without confidence optimization. \ref{['fig:p2_eigenvalues']} and \ref{['fig:p2_ekf_error']}: Comparison of the eigenvalues (of $P(t)$) and state estimation errors given by the observer.
  • Figure 3: The unicycle system.

Theorems & Definitions (13)

  • Proposition 1: reif1998ekf
  • Definition 1
  • Theorem 1
  • proof
  • Definition 2: agrawal2022safe
  • Definition 3: Adapted from agrawal2022safe
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • ...and 3 more