Controllability Test for Nonlinear Datatic Systems

TL;DR

This work tackles controllability in nonlinear datatic systems by introducing ε-controllability, a point-to-region relaxation that is well-suited to discrete data descriptions. Leveraging a Lipschitz continuity assumption, the authors prove a one-step controllability backpropagation theorem that enables expanding controllable regions backward through state neighborhoods. They present two algorithms: MECS, a tree-search method that iteratively expands ε-controllable subsets, and FERF, a fixed-radius shortest-path variant that reduces computation while preserving a mutual-controllability assumption. Complexity analyses and experiments on a mass-spring system, a Van der Pol oscillator, and a tunnel-diode circuit demonstrate the approach's ability to identify ε-controllable states and quantify controllability via the degree-of-controllability metric. This framework provides a practical tool for data-driven controllability analysis and informs controller design when exact state-to-state trajectories are unavailable.

Abstract

Controllability is a fundamental property of control systems, serving as the prerequisite for controller design. While controllability test is well established in modelic (i.e., model-driven) control systems, extending it to datatic (i.e., data-driven) control systems is still a challenging task due to the absence of system models. In this study, we propose a general controllability test method for nonlinear systems with datatic description, where the system behaviors are merely described by data. In this situation, the state transition information of a dynamic system is available only at a limited number of data points, leaving the behaviors beyond these points unknown. Different from traditional exact controllability, we introduce a new concept called $ε$-controllability, which extends the definition from point-to-point form to point-to-region form. Accordingly, our focus shifts to checking whether the system state can be steered to a closed state ball centered on the target state, rather than exactly at that target state. On its basis, we propose a tree search algorithm called maximum expansion of controllable subset (MECS) to identify controllable states in the dataset. Starting with a specific target state, our algorithm can iteratively propagate controllability from a known state ball to a new one. This iterative process gradually enlarges the $ε$-controllable subset by incorporating new controllable balls until all $ε$-controllable states are searched. Besides, a simplified version of MECS is proposed by solving a special shortest path problem, called Floyd expansion with radius fixed (FERF). FERF maintains a fixed radius of all controllable balls based on a mutual controllability assumption of neighboring states. The effectiveness of our method is validated in three datatic control systems whose dynamic behaviors are described by sampled data.

Figures (23)

• Figure 1: Comparison of control paradigms. In the modelic control paradigm, the first step is to establish a dynamic model through system identification. This model offers a continuous but inaccurate description of state transition information. In the datatic control paradigm, data is used directly for system analysis and controller synthesis, providing a discrete yet precise description of state transition.
• Figure 2: The difference between exact controllability and $\epsilon$-controllability: The graph depicts four state trajectories from collected data. Stars denote target states; blue points denote states within the neighborhood of the target $A$ with radius $\epsilon$. Pink points denote exact controllable states w.r.t. target $B$, as they can be precisely steered to the target state. Conversely, red points denote $\epsilon$-controllable states w.r.t. target $A$, which are only capable of being steered into a neighborhood of that target state.
• Figure 3: One-step controllability backpropagation. Lipschitz continuity restricts the subsequent states of the expanded ball to be within the reachable ball. If the reachable ball is contained by an $\epsilon$-controllable subset, i.e., the selected ball, all states in the expanded ball are also $\epsilon$-controllable. Consequently, controllability backpropagates from the selected ball to the expanded ball.
• Figure 4: Four key steps of MECS algorithm. Selection: Choose the leaf node with the maximum radius. Expansion: Find data points with subsequent states within the selected ball. Evaluation: Compute the radii of the expanded balls using the one-step controllability backpropagation theorem. Pruning: Remove the leaf nodes that are contained by others.
• Figure 5: A example of MECS. In the first iteration, the target ball is selected, and four $\epsilon$-controllable balls are expanded. In the second iteration, two more $\epsilon$-controllable balls are expanded, and one of them is pruned.

Theorems & Definitions (4)

• Definition 1: $\epsilon$-controllability
• Definition 2: $\epsilon$-controllable set
• Theorem 1: one-step controllability backpropagation
• proof