Region-of-Attractor Planners

• Region-of-Attractor planners are frameworks that use mathematical or data-driven certificates, such as Lyapunov functions and SOS relaxations, to define and verify invariant sets for system stability.
• They integrate both offline planning (e.g., funnel libraries and certificate synthesis) and online strategies (trajectory and switching logic) to guarantee convergence and robust performance across hybrid, polynomial, and model-free systems.
• Empirical evaluations demonstrate that these planners achieve significant computational gains and reduced failure rates, enabling fast and reliable control in complex dynamical environments.

A Region-of-Attractor (RoA) planner is a planning framework that leverages explicit, mathematically or data-driven inner approximations to the region of attraction of equilibria, invariant sets, or trajectories to ensure stability guarantees in autonomous and controlled dynamical systems. RoA planners algorithmically certify, estimate, or learn sets such that, if trajectories are kept within these regions, convergence to the target behavior is guaranteed. This approach directly embeds the region of attraction concept—rooted in Lyapunov theory or reachable set analysis—into both offline planning (funnel libraries, certificate synthesis) and online planning (trajectory or switching logic) to enforce formal stability, robustness, and performance criteria across a broad range of control and motion planning applications, especially in challenging settings such as hybrid systems, polynomial systems, and model-free scenarios.

1. Formal Problem Setting and Motivation

The key objects in RoA planning are controlled or autonomous dynamical systems of the following form:

• Hybrid systems with discrete modes and continuous dynamics, defined by

$$\dot x = f_i(x,u;p_i), \quad x \in \mathcal C_i,\quad x^+ = h_i(x,u;p_i, p_j),\ x \in \mathcal D_{i,j}$$

where $i$ ranges over mode indices, $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^m$ is the input, and $p_i$ represents mode configurations (Meng et al., 2023).

• Polynomial systems under semialgebraic constraints and optimal control, formulated as:

$$\dot{x}(t) = f(t, x(t), u(t)),\quad x(t) \in X,\ u(t) \in U$$

and with a target set $X_T$ at terminal time (Cibulka et al., 2021).

• Discrete or continuous-time autonomous systems for which the true vector field is unknown but can be sampled from execution (Shen et al., 2022).

The typical objective is to design a planning or control law such that, starting from an initial state within a certified subset $S \subseteq$ RoA, the trajectory converges to a stable equilibrium or target set, while possibly respecting switching logic or state constraints.

RoA planners address specific challenges: (i) instability introduced at discrete mode switches in hybrid systems, (ii) computational tractability for complex nonlinear systems where large-scale direct computations are infeasible, and (iii) model uncertainty or absence of explicit dynamics, necessitating data-driven invariance certification.

2. Methodologies for Region-of-Attractor Certification

RoA planners critically depend on efficient and correct inner approximations of the region of attraction. Several methodologies are established:

• Lyapunov-based certificates (hybrid systems): Learn or synthesize, per mode, a Control Lyapunov Function (CLF) $V_i(x)$ satisfying $V_i(x) > 0$ for $x \ne x_i^*$, $V_i(x_i^*) = 0$, and $\dot V_i(x) < 0$ along closed-loop trajectories. Sublevel sets $\mathcal{R}_i(c_i) = \{x : V_i(x) \le c_i\}$ serve as RoA estimates (Meng et al., 2023).
• Sum-of-squares (SOS) relaxations (polynomial systems): Formulate the search for invariant or reachable sets as a hierarchy of semidefinite programs (SDPs) over polynomial storage and Lyapunov functions, extracting feasible outer-approximations to the true RoA (Cibulka et al., 2021).
• Recurrence-based, model-free sets: Identify sets $S$ for which every trajectory initiated at $x_0 \in S$ returns to $S$ within a bounded number of steps or time $\tau$, leveraging only sampled trajectories. Insights from recurrence imply that such sets are strict inner approximations to the true RoA under mild conditions (Shen et al., 2022).

These approaches differ in their assumptions, expressivity, and computational tradeoffs but ultimately serve to generate certifiable, invariant sets suitable for direct integration into planning frameworks.

3. Constructing and Embedding RoA-Based Planners

The embedding of RoA approximations into planners follows architectural patterns tailored to the systems’ modalities and problem structure.

Hybrid Systems with Learned RoA:

• For each mode, learn a neural-network CLF-controller pair via exponential decrease loss

$$\mathcal{L}_{clf} = \sum_t \mathrm{ReLU} \left(\gamma V_i(x_t, p_i) + \frac{V_i(x_{t+1}, p_i) - V_i(x_t, p_i)}{\Delta t}\right)$$

• Estimate maximal $c_i^*$ for which the sublevel set is indeed in the RoA. Use a neural estimator $\hat c_i(p_i) \approx c_i^*$.
• At mode switches, use a differentiable planner to solve for configuration parameters $p_i$ so that post-jump states land in the next mode’s RoA, ensuring cross-mode $\epsilon$-stability via a Lyapunov decrease condition (Meng et al., 2023).

Spatio-Temporally Split SOS Funnel Chains:

• Decompose the high-dimensional SDP problem along time and spatial coordinates, yielding smaller, parallelizable SDPs on partitioned cells.
• Compute certificates $(v_{i,k}, w_i)$ on each cell and time interval, with coupling conditions ensuring monotonicity of the value function across cell and time boundaries.
• Assemble funnel libraries—directed graphs where each node is an invariant funnel, and edges correspond to verified set inclusions—and plan sequences of funnels from start to goal (Cibulka et al., 2021).

Model-Free Planners with Recurrent Sets:

• Iteratively refine a candidate set $S$ (e.g., a ball/polyhedron around $x^*$) by excluding counterexamples (points whose sampled finite-horizon trajectories do not return to $S$).
• Use the final $k$-recurrent set as an explicit constraint for motion planners, restricting the planner’s admissible state-space to ensure stability even in the absence of a model.
• Integration into sampling-based motion planning (e.g., RRT*) is achieved by only admitting expansions that remain inside the certified set (Shen et al., 2022).

4. Theoretical Guarantees and Convergence Properties

RoA planners provide formal stability guarantees drawing directly from the properties of the underlying certificates.

• Lyapunov-based planners: Under per-mode CLF conditions and explicit Lyapunov decrease at mode transitions, $\epsilon$-exponential stability is established. Theoretical results state that if $V_i$ satisfies

$$\alpha_i \|x - x_i^*\| \le V_i(x, p_i) \le \beta_i \|x - x_i^*\|,\ \dot V_i \le -\gamma_i V_i$$

and appropriate cross-mode conditions, then all trajectories remain inside the chain of estimated RoAs and converge to the equilibria with bounded error (Meng et al., 2023).

• Split SOS approaches: The spatio-temporal decomposition provably yields outer-approximations to the true region of attraction, with convergence in Lebesgue volume as degree $d \to \infty$ and under Archimedean conditions for each cell. Explicit coupling ensures monotonic decrease along all trajectories, with graph structure in the funnel library guaranteeing sequential invariance (Cibulka et al., 2021).
• Recurrence-based model-free methods: If the certified set $S^*$ is $k$-recurrent and contains the equilibrium in its interior with no other recurrent dynamics, it is necessarily contained within the true region of attraction. Sample complexity and almost-sure convergence of the learning algorithm are established, with the total number of counterexamples sharply bounded (Shen et al., 2022).

5. Algorithmic Implementations and Computational Aspects

RoA planners employ diverse algorithmic pipelines:

• Neural RoA planners: Use batch or trajectory sampling for data generation, small two-layer ReLU controllers, and backpropagation through time for parameter updates. Monte Carlo estimation refines RoA boundaries, and minimal gradient-descent steps are performed online at mode transitions (Meng et al., 2023).
• Spatio-temporal SOS planners: Partition the state and time domains, assemble multiblock sparse SDPs with standard solvers (such as MOSEK or ADMM variants), and extract piecewise-polynomial certificates for use in runtime planning. Complexity scales linearly with the number of partition cells rather than combinatorially, enabling efficient real-time updates (Cibulka et al., 2021).
• Recurrent set learning: Employ interior-point random sampling and short trajectory simulations; counterexamples drive the iterative contraction of candidate sets. Both sphere and polyhedron representations are available, with multi-center variants better capturing nonconvexity. The algorithms operate both offline and in continual online refinement (Shen et al., 2022).

Observed computational performance can be significant: certain neural RoA planners run 10–50 times faster than model predictive control baselines (~1 ms per step), spatio-temporal SOS split methods recover ~90% of analytical RoA volume in fractions of monolithic SDP solve time, and model-free recurrence algorithms terminate with a small number of counterexamples per run.

6. Empirical Evaluation and Comparative Analysis

RoA planners have been systematically evaluated on canonical hybrid, nonlinear, and black-box systems:

Benchmark Domain	RoA Planner Results	Baseline Comparison
Car tracking (hybrid) (Meng et al., 2023)	67–75% reduction in failures, 10–50× faster than MPC	Outperform RL, MPC, CLF, LQR, QP, HJB
Pogobot maze (hybrid) (Meng et al., 2023)	Maintained safe jumps, <10% collisions, 1/70× MPC runtime	Only RoA planner maintained invariance across all segments
Bipedal walking (hybrid) (Meng et al., 2023)	Near-HJB performance for reference gait, generalized to arbitrary gaits in 6 h	HJB: 36 h per gait; RoA planner: 1 ms/step vs 20 ms/step
Cubic/Integrator (polynomial) (Cibulka et al., 2021)	Piecewise SOS splits exactly match analytical RoA at low degree; superior volume-time tradeoff	Monolithic SDP comparatively less precise/tractable
2D nonlinear (model-free) (Shen et al., 2022)	1–3 counterexamples sufficed; final sets covered most true RoA	Standard planners without RoA constraint are not invariant

Ablation studies confirm the robustness of planner hyperparameters, with minimal effects when parameters are varied within reasonable ranges. Time discretization as fine as 0.16 ms per step leads to less than 3% variation in metrics (Meng et al., 2023).

7. Extensions, Practical Considerations, and Research Directions

Region-of-attractor planners provide a principled template adaptable to uncertaintly, model misspecification, and real-time planning constraints. The architecture supports both fully offline (library-based) and online (adaptive, receding horizon) planning, and is compatible with deterministic, stochastic, and data-driven domains.

Current research focuses on improving scalability in high-dimensional or high-switching-rate settings, integrating richer function representations (e.g., deep architectures and piecewise polynomials), as well as relaxing or adapting certificate assumptions to encompass broader classes of attractors and system uncertainties. A plausible implication is that, when combined with contemporary learning methods, RoA planners will enable provably stable, sample-efficient, and reliable planning for autonomous systems in highly uncertain, multi-modal, and partially observable environments (Meng et al., 2023, Cibulka et al., 2021, Shen et al., 2022).