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From a Single Trajectory to Safety Controller Synthesis of Discrete-Time Nonlinear Polynomial Systems

Behrad Samari, Omid Akbarzadeh, Mahdieh Zaker, Abolfazl Lavaei

TL;DR

The proposed approach leverages measured data acquired through an input-state observation, referred to as a single trajectory, collected over a specified time horizon, to design a CBC and its corresponding safety controller directly from the finite-length observed data, without explicitly identifying the unknown dynamical system.

Abstract

This work is concerned with developing a data-driven approach for learning control barrier certificates (CBCs) and associated safety controllers for discrete-time nonlinear polynomial systems with unknown mathematical models, guaranteeing system safety over an infinite time horizon. The proposed approach leverages measured data acquired through an input-output observation, referred to as a single trajectory, collected over a specified time horizon. By fulfilling a certain rank condition, which ensures the unknown system is persistently excited by the collected data, we design a CBC and its corresponding safety controller directly from the finite-length observed data, without explicitly identifying the unknown dynamical system. This is achieved through proposing a data-based sum-of-squares optimization (SOS) program to systematically design CBCs and their safety controllers. We validate our data-driven approach over two physical case studies including a jet engine and a Lorenz system, demonstrating the efficacy of our proposed method.

From a Single Trajectory to Safety Controller Synthesis of Discrete-Time Nonlinear Polynomial Systems

TL;DR

The proposed approach leverages measured data acquired through an input-state observation, referred to as a single trajectory, collected over a specified time horizon, to design a CBC and its corresponding safety controller directly from the finite-length observed data, without explicitly identifying the unknown dynamical system.

Abstract

This work is concerned with developing a data-driven approach for learning control barrier certificates (CBCs) and associated safety controllers for discrete-time nonlinear polynomial systems with unknown mathematical models, guaranteeing system safety over an infinite time horizon. The proposed approach leverages measured data acquired through an input-output observation, referred to as a single trajectory, collected over a specified time horizon. By fulfilling a certain rank condition, which ensures the unknown system is persistently excited by the collected data, we design a CBC and its corresponding safety controller directly from the finite-length observed data, without explicitly identifying the unknown dynamical system. This is achieved through proposing a data-based sum-of-squares optimization (SOS) program to systematically design CBCs and their safety controllers. We validate our data-driven approach over two physical case studies including a jet engine and a Lorenz system, demonstrating the efficacy of our proposed method.
Paper Structure (12 sections, 4 theorems, 34 equations, 2 figures, 1 algorithm)

This paper contains 12 sections, 4 theorems, 34 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Discrete-Time Nonlinear Polynomial Systems
  5. Control Barrier Certificates
  6. Data-Driven Construction of CBCs and Safety Controllers
  7. Computation of CBCs and Safety Controllers
  8. Simulation Results
  9. Conclusion
  10. Appendix: Collected Data and Designed Matrices
  11. Jet engine
  12. Lorenz

Key Result

Theorem 2.3

Consider a dt-NPS $\Sigma$ as introduced in Definition def: dt-NPS with $X_0$ and $X_u$ being its initial and unsafe sets, respectively. Suppose $\mathds{B}$ is a CBC for $\Sigma$ as in Definition def: CBC. The dt-NPS is safe in the sense that states' trajectories of dt-NPS never reach the unsafe se

Figures (2)

  • Figure 1: $100$ state trajectories of unknown jet engine with the designed controller in \ref{['controller']} starting from different initial conditions in $X_0 \in [0, 2] \times [-2, 2]$. Initial and unsafe regions are depicted by green and red boxes, while $\mathds B(x) = \alpha_1$ and $\mathds B(x) = \alpha_2$ are indicated by and .
  • Figure 2: $250$ state trajectories of unknown Lorenz system and a representative trajectory under the designed controller in \ref{['eq:controller Lorenz']}, starting from different initial conditions in $X_0 \in [0, 2] \times [-2, 2]^2$. Initial and unsafe regions are depicted by blue and red boxes, while $\mathds B(x) = \alpha_1$ and $\mathds B(x) = \alpha_2$ are indicated by and , respectively.

Theorems & Definitions (14)

  • Definition 2.1: dt-NPS
  • Definition 2.2: CBC
  • Theorem 2.3: Infinite-horizon safety
  • Remark 3.1
  • Lemma 3.2: Data-based dt-NPS
  • proof
  • Remark 3.3: Rank condition
  • Remark 3.4
  • Theorem 3.5: Data-driven CBC and safety controller
  • proof
  • ...and 4 more