Abstraction-based Control of Unknown Continuous-Space Models with Just Two Trajectories

TL;DR

This work addresses controlling unknown nonlinear polynomial systems by constructing a data-driven symbolic model and an alternating simulation function (ASF) that relates the unknown plant to its abstraction, e.g., $x^+ = A \mathcal{M}(x) + B u$ vs. the symbolic model with $\hat{x}^+ = \hat{f}(\hat{x},\hat{u})$. It shows how to build the ASF and a hybrid interface from only two input-state trajectories, supported by a data-driven sum-of-squares (SOS) program and a rank condition ensuring persistent excitation. The main contribution is a closed-form data-based representation that yields a valid ASF with guaranteed tracking error, enabling correct-by-design discrete controllers on the abstract model to be implemented on the unknown system. The approach avoids explicit system identification and provides formal guarantees for properties such as safety and reach-while-avoid in continuous space, demonstrated on a case study.

Abstract

Finite abstractions (a.k.a. symbolic models) offer an effective scheme for approximating the complex continuous-space systems with simpler models in the discrete-space domain. A crucial aspect, however, is to establish a formal relation between the original system and its symbolic model, ensuring that a discrete controller designed for the symbolic model can be effectively implemented as a hybrid controller (using an interface map) for the original system. This task becomes even more challenging when the exact mathematical model of the continuous-space system is unknown. To address this, the existing literature mainly employs scenario-based data-driven methods, which require collecting a large amount of data from the original system. In this work, we propose a data-driven framework that utilizes only two input-state trajectories collected from unknown nonlinear polynomial systems to synthesize a hybrid controller, enabling the desired behavior on the unknown system through the controller derived from its symbolic model. To accomplish this, we employ the concept of alternating simulation functions (ASFs) to quantify the closeness between the state trajectories of the unknown system and its data-driven symbolic model. By satisfying a specific rank condition on the collected data, which intuitively ensures that the unknown system is persistently excited, we directly design an ASF and its corresponding hybrid controller using finite-length data without explicitly identifying the unknown system, while providing correctness guarantees. This is achieved through proposing a data-based sum-of-squares (SOS) optimization program, enabling a systematic approach to the design process. We illustrate the effectiveness of our data-driven approach through a case study.

Figures (2)

• Figure 1: The left-hand side subfigure illustrates the trajectories of $x_1$ ( ), $x_2$ ( ), $\hat{x}_1$ ( ), and $\hat{x}_2$ ( ). As shown, it is clear that the safety property is fulfilled. Moreover, the right-hand side subfigure depicts the norm of the error between the trajectories of the original system and those of its symbolic model. As illustrated, the error is always lower than the $\epsilon$ reported.
• Figure 2: The trajectories must originate from the initial set , reach the target set , while avoiding the obstacle . As shown, the trajectories of the unknown system \ref{['sys_org']} ( ) and its data-driven symbolic model ( ) closely align, demonstrating compliance with the system's reach-while-avoid property.

Theorems & Definitions (17)

• Definition 2.1: dt-IANSP
• Remark 2.2: On $\mathcal{M}(x)$
• Definition 2.3: Symbolic Model Construction
• Definition 2.4: ASF
• Remark 2.5: Interface Function
• Remark 2.6: Interpretation of Definition \ref{['def: ASF']}
• Theorem 2.7: Closeness Guarantee
• proof
• Remark 2.8: Special Case $\rho \equiv 0$
• Lemma 3.1: Data-based Representation