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Data-Driven Dynamic Controller Synthesis for Discrete-Time General Nonlinear Systems

Behrad Samari, Abolfazl Lavaei

TL;DR

This work tackles the problem of safety controller synthesis for discrete-time general nonlinear systems with unknown dynamics and input constraints. It introduces a direct data-driven framework that uses an adding-one-integrator augmentation to form an augmented dt-GNS, enabling learning of augmented CBCs and dynamic safety controllers from a single input-state trajectory via data-dependent LMIs (and an alternative scenario-based program). The key contributions are the design of augmented CBCs (A-CBCs) from data, a quadratic form for the CBC, and both deterministic and probabilistic guarantees, demonstrated on challenging nonlinear and high-dimensional case studies. The approach yields finite-horizon safety certificates without requiring full model identification and avoids bilinearities in the optimization, offering scalable, runtime-applicable safety guarantees for unknown nonlinear systems under input constraints.

Abstract

Synthesizing safety controllers for general nonlinear systems is a highly challenging task, particularly when the system models are unknown, and input constraints are present. While some recent efforts have explored data-driven safety controller design for nonlinear systems, these approaches are primarily limited to specific classes of nonlinear dynamics (e.g., polynomials) and are not applicable to general nonlinear systems. This paper develops a direct data-driven approach for discrete-time general nonlinear systems, facilitating the simultaneous learning of control barrier certificates (CBCs) and dynamic controllers to ensure safety properties under input constraints. Specifically, by leveraging the adding-one-integrator approach, we incorporate the controller's dynamics into the system dynamics to synthesize a virtual static-feedback controller for the augmented system, resulting in a dynamic safety controller for the actual dynamics. We collect input-state data from the augmented system during a finite-time experiment, referred to as a single trajectory. Using this data, we learn augmented CBCs and the corresponding virtual safety controllers, ensuring the safety of the actual system and adherence to input constraints over a finite time horizon. We demonstrate that our proposed conditions boil down to some data-dependent linear matrix inequalities (LMIs), which are easy to satisfy. We showcase the effectiveness of our data-driven approach through two case studies: one exhibiting significant nonlinearity and the other featuring high dimensionality.

Data-Driven Dynamic Controller Synthesis for Discrete-Time General Nonlinear Systems

TL;DR

This work tackles the problem of safety controller synthesis for discrete-time general nonlinear systems with unknown dynamics and input constraints. It introduces a direct data-driven framework that uses an adding-one-integrator augmentation to form an augmented dt-GNS, enabling learning of augmented CBCs and dynamic safety controllers from a single input-state trajectory via data-dependent LMIs (and an alternative scenario-based program). The key contributions are the design of augmented CBCs (A-CBCs) from data, a quadratic form for the CBC, and both deterministic and probabilistic guarantees, demonstrated on challenging nonlinear and high-dimensional case studies. The approach yields finite-horizon safety certificates without requiring full model identification and avoids bilinearities in the optimization, offering scalable, runtime-applicable safety guarantees for unknown nonlinear systems under input constraints.

Abstract

Synthesizing safety controllers for general nonlinear systems is a highly challenging task, particularly when the system models are unknown, and input constraints are present. While some recent efforts have explored data-driven safety controller design for nonlinear systems, these approaches are primarily limited to specific classes of nonlinear dynamics (e.g., polynomials) and are not applicable to general nonlinear systems. This paper develops a direct data-driven approach for discrete-time general nonlinear systems, facilitating the simultaneous learning of control barrier certificates (CBCs) and dynamic controllers to ensure safety properties under input constraints. Specifically, by leveraging the adding-one-integrator approach, we incorporate the controller's dynamics into the system dynamics to synthesize a virtual static-feedback controller for the augmented system, resulting in a dynamic safety controller for the actual dynamics. We collect input-state data from the augmented system during a finite-time experiment, referred to as a single trajectory. Using this data, we learn augmented CBCs and the corresponding virtual safety controllers, ensuring the safety of the actual system and adherence to input constraints over a finite time horizon. We demonstrate that our proposed conditions boil down to some data-dependent linear matrix inequalities (LMIs), which are easy to satisfy. We showcase the effectiveness of our data-driven approach through two case studies: one exhibiting significant nonlinearity and the other featuring high dimensionality.

Paper Structure

This paper contains 12 sections, 7 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General Nonlinear Framework
  3. Notation
  4. Discrete-Time General Nonlinear Systems
  5. Control Barrier Certificates
  6. Augmented dt-GNS
  7. Data-Driven Design of A-CBC and Safety Controller
  8. Alternative computation for A-CBCs and safety controllers
  9. Deterministic correctness guarantee
  10. Probabilistic correctness guarantee
  11. Simulation Results
  12. Conclusion and Discussion

Key Result

Theorem 1

Consider the dt-GNS $\Upsilon = (A, f, \mathcal{X}, U)$, as described in Definition def: dt-GNS, where ${\mathcal{X}_{\mathrm{o}}}$ and ${\mathcal{X}_{\mathrm{1}}}$ represent the initial and unsafe sets, respectively. Let $\pmb{\mathds{B}}$ be a CBC for $\Upsilon$ as defined in Definition def: CBC.

Figures (3)

  • Figure 1: As shown in Subfigure \ref{['fig:subfig1']}, if one selects the entire admissible control input set as the initial set for $\zeta_2$ (i.e., $\mathcal{X}^{\zeta_2}_\mathrm{o}$), it would be impossible to find a level set that distinguishes the inadmissible set (i.e., the unsafe set for $\zeta_2$ denoted by $\mathcal{X}^{\zeta_2}_\mathrm{1}$) from $\mathcal{X}^{\zeta_2}_\mathrm{o}$. However, as illustrated in Subfigure \ref{['fig:subfig2']}, with the proposed adjustments, it becomes feasible to find a level set that effectively separating $\mathcal{X}^{\zeta_2}_\mathrm{o}$ from $\mathcal{X}^{\zeta_2}_\mathrm{1}$ .
  • Figure 2: Starting from various initial conditions in the initial set $\mathcal{X}^\zeta_{\mathrm{o}}$, $1000$ state and control input trajectories of the unknown dt-GNS (\ref{['eq: ex1_sys']}) are generated using the designed dynamic safety controller (\ref{['des_u']}). As illustrated, neither the states enter the unsafe set nor is the control input constraint violated, i.e., neither of them enters the unsafe sets . The unsafe sets marked by $(a)$ are to ensure the input constraints are met, while the unsafe sets labeled by $(b)$ are to ensure the safety of the system's states. Moreover, $\pmb{\mathds{B}_a} (\zeta) = \eta_a$ and $\pmb{\mathds{B}_a} (\zeta) = \gamma_a$ are indicated by colors and , respectively.
  • Figure 3: The heatmap shows the values of $\pmb{\mathds{B}_a}(\mathcal{A} \mathcal{F}(\zeta) + \mathcal{B}\vartheta) - \pmb{\mathds{B}_a}(\zeta) - c_a$, with $\mathcal{A}$ and $\mathcal{B}$ reported in (\ref{['app3']}), over the state set considered. As can be seen, these values are negative all over the state set, which means that condition (\ref{['eq: A-con_bar_3']}) is fulfilled.

Theorems & Definitions (22)

  • Definition 1: dt-GNS
  • Remark 1: Dictionary Availability
  • Definition 2: CBC
  • Remark 2: Special Case $c \rightarrow 0$
  • Remark 3: Empty Intersection between ${\mathcal{X}_{\mathrm{o}}}$ and ${\mathcal{X}_{\mathrm{1}}}$
  • Theorem 1: Safety Guarantee for dt-GNS
  • Remark 4: On Choosing $\epsilon_1$ and $\epsilon_2$
  • Definition 3: A-CBC
  • Corollary 1: Safety Guarantee for A-dt-GNS
  • Remark 5: Safety Guarantee and Input Constraint Enforcement for dt-GNS
  • ...and 12 more