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Safety Control of Uncertain MIMO Systems Using Dynamic Output Feedback Barrier Pairs

Binghan He, Takashi Tanaka

Abstract

Safety control of dynamical systems using barrier functions relies on knowing the full state information. This paper introduces a novel approach for safety control in uncertain MIMO systems with partial state information. The proposed method combines the synthesis of a vector norm barrier function and a dynamic output feedback safety controller to ensure robust safety enforcement. The safety controller guarantees the invariance of the barrier function under uncertain dynamics and disturbances. To address the challenges associated with safety verification using partial state information, a barrier function estimator is developed. This estimator employs an identifier-based state estimator to obtain a state estimate that is affine in the uncertain model parameters of the system. By incorporating a priori knowledge of the limits of the uncertain model parameters and disturbances, the state estimate provides a robust upper bound for the barrier function. Comparative analysis with existing control barrier function based methods shows the advantage of the proposed approach in enforcing safety constraints under tight input constraints and the utilization of estimated state information.

Safety Control of Uncertain MIMO Systems Using Dynamic Output Feedback Barrier Pairs

Abstract

Safety control of dynamical systems using barrier functions relies on knowing the full state information. This paper introduces a novel approach for safety control in uncertain MIMO systems with partial state information. The proposed method combines the synthesis of a vector norm barrier function and a dynamic output feedback safety controller to ensure robust safety enforcement. The safety controller guarantees the invariance of the barrier function under uncertain dynamics and disturbances. To address the challenges associated with safety verification using partial state information, a barrier function estimator is developed. This estimator employs an identifier-based state estimator to obtain a state estimate that is affine in the uncertain model parameters of the system. By incorporating a priori knowledge of the limits of the uncertain model parameters and disturbances, the state estimate provides a robust upper bound for the barrier function. Comparative analysis with existing control barrier function based methods shows the advantage of the proposed approach in enforcing safety constraints under tight input constraints and the utilization of estimated state information.
Paper Structure (22 sections, 74 equations, 6 figures)

This paper contains 22 sections, 74 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Preliminaries
  5. Notations
  6. State Model
  7. Barrier Pairs
  8. Problem Statement
  9. Barrier Pair Synthesis
  10. LMIs for Invariance Property
  11. LMIs for State and Input Limits
  12. Barrier Pair Construction
  13. Barrier Function Composition
  14. Barrier Function Estimation
  15. Identifier-Based Estimator
  16. ...and 7 more sections

Figures (6)

  • Figure 1: In this paper, our switching system $\Sigma_s$ chooses either an original input $u = \hat{u}$ of $\Sigma_p$ or a known-to-be-safe input $u = \Sigma_k (y)$ based on a barrier function $\mathbf{B}$.
  • Figure 2: $\Sigma_s$ switches from the original input $u = \hat{u}$ to $u = \Sigma_k (y)$ if $\bar{\mathbf{B}}_\mathtt{C} \geq \bar{\upvarepsilon}$ and switches back to $u = \hat{u}$ if $\bar{\mathbf{B}}_\mathtt{C} < \underaccent{\bar{}}{\upvarepsilon}$.
  • Figure 3: In our first example, we consider an inverted pendulum system with $g = 9.8 \ \mathtt{m \cdot s ^ {-2}}$, $m = \frac{1}{9.8 ^ 2} \ \mathtt{kg}$, and $r = 9.8 \ \mathtt{m}$.
  • Figure 4: Simulation Results of Inverted Pendulum Example---(a)-(b) show the reference trajectories ($\mathtt{Ref.}$) of $y$ and $\dot{y}$ and the actual trajectories using the $\mathtt{CBF}$-$\mathtt{QP}$ method and our $\mathtt{OFB}$-$\mathtt{BP}$ method. (c) shows the actual input torque $u$ generated from the two methods, where we use the sigmoidal approximation in \ref{['eq:sigmoidal']} with $\underaccent{\bar{}}{\upvarepsilon}=0.98$ and $\bar{\upvarepsilon}=1.00$ for the $\mathtt{OFB}$-$\mathtt{BP}$ method.
  • Figure 5: In our second example, we consider a dual spring-mass system, where $m_1 = m_2 = 2$, $0.9 \leq k_1 \leq 1.0$ and $0.9 \leq k_2 \leq 1.0$.
  • ...and 1 more figures

Theorems & Definitions (6)

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