On the Stability of Datatic Control Systems

TL;DR

The paper tackles stability verification for datatic control systems lacking explicit models by introducing η-testing, a Lyapunov-based method that leverages Lipschitz continuity to bound the unknown dynamics in unmeasured regions. By bounding the time derivative of a Lyapunov function as the solution to a quadratically constrained linear program over the intersection of data-driven balls, it yields a sufficient condition for stability that can be checked pointwise on the data. The approach unifies continuous- and discrete-time, as well as nonlinear and linear (including data-driven linear) settings, and includes practical steps for local Lipschitz constant estimation and complexity-aware implementation. Empirical results on oscillator, vehicle, and pendulum systems demonstrate the method’s ability to verify stability, instability, and critical stability from data alone, with clear guidance on when conclusions are definitive or inconclusive.

Abstract

The development of feedback controllers is undergoing a paradigm shift from $\textit{modelic}$ (model-driven) control to $\textit{datatic}$ (data-driven) control. Stability, as a fundamental property in control, is less well studied in datatic control paradigm. The difficulty is that traditional stability criteria rely on explicit system models, which are not available in those systems with datatic description. Some pioneering works explore stability criteria for datatic systems with special forms such as linear systems, homogeneous systems, and polynomial systems. However, these systems imply too strong assumptions on the inherent connection among data points, which do not hold in general nonlinear systems. This paper proposes a stability verification algorithm for general datatic control systems called $η$-testing. Our stability criterion only relies on a weak assumption of Lipschitz continuity so as to extend information from known data points to unmeasured regions. This information restricts the time derivative of any unknown state to the intersection of a set of closed balls. Inside the intersection, the worst-case time derivative of Lyapunov function is estimated by solving a quadratically constrained linear program (QCLP). By comparing the optimal values of QCLPs to zero in the whole state space, a sufficient condition of system stability can be checked. We test our algorithm on three datatic control systems, including both linear and nonlinear ones. Results show that our algorithm successfully verifies the stability, instability, and critical stability of tested systems.

Figures (22)

• Figure 1: Two types of control paradigms. Modelic control (on the upper path) first performs system identification and then synthesizes controllers. Datatic control (on the lower path) directly solves controllers using data.
• Figure 2: System descriptions in two control paradigms. Modelic description is continuous but maybe erroneous. Datatic description is discrete but error-free.
• Figure 3: Schematics of different system dynamics. The dashed lines stand for possible system dynamics. The blue circles represent data points where the true values of dynamics are measured. For a linear system, the value of system dynamics between data points can be easily determined given sufficient linearly independent points, while this does not hold for a nonlinear system. Nevertheless, with the assumption of continuity, we are able to extend system information from data points to unknown regions, leading to an estimation of the range of system dynamics (see the red shaded region in Fig.\ref{['fig: system dynamics schematic-c']}).
• Figure 4: Intersection of closed balls given by Lipschitz continuity.
• Figure 5: QCLP for finding the maximum value of time derivative of Lyapunov function.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3