Synthesis for Constrained Nonlinear Systems using Hybridization and Robust Controllers on Simplices

In this paper, we propose an approach to controller synthesis for a class of constrained nonlinear systems. It is based on the use of a hybridization, that is a hybrid abstraction of the nonlinear dynamics. This abstraction is defined on a triangulation of the state-space where on each simplex of the triangulation, the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances. Except for the disturbances, this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade. We extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation. We show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system. Our approach, though conservative, can be fully automated and is computationally tractable. To show its effectiveness in practical applications, we apply our method to control a pendulum mounted on a cart.

💡 Research Summary

The paper introduces an automated synthesis framework for controlling constrained nonlinear systems by exploiting a hybrid abstraction based on a simplicial (triangulated) partition of the state space. The core idea is to replace the original nonlinear dynamics on each simplex with a conservative affine model plus a bounded disturbance set that captures the approximation error. This yields a piecewise‑affine hybrid system with explicit disturbance bounds, enabling the use of robust control techniques that have been developed for linear hybrid systems.

Methodology

1. State‑space triangulation – The user selects a region and a desired resolution; the region is partitioned into simplices (e.g., via Delaunay triangulation).
2. Affine approximation and disturbance construction – For each simplex, the nonlinear vector field is over‑approximated by an affine map ( \dot x = A_i x + b_i + w ) where ( w ) belongs to a convex polytope ( W_i ) that encloses the modeling error. The construction uses min‑max slopes of the nonlinear function on the simplex vertices.
3. Robust local controller synthesis – On each simplex a linear state‑feedback law ( u = K_i x + k_i ) is computed such that, for all disturbances ( w \in W_i ), the closed‑loop dynamics respect input and state constraints (e.g., actuator saturation, safety bounds). The synthesis is cast as a set of Linear Matrix Inequalities (LMIs) or a multi‑parametric linear program, allowing simultaneous minimisation of control effort and maximisation of admissible disturbance size.
4. Transition verification and coordination – Adjacent simplices share faces. The framework checks that trajectories exiting one simplex through a face can be continued by the controller of the neighboring simplex without violating constraints. If a transition fails, the algorithm either refines the mesh locally or tightens the disturbance set, iterating until all transitions are feasible.
5. Hybrid controller assembly – The collection of robust local controllers together with the transition logic forms a hybrid controller that can be applied directly to the original nonlinear system. Because the disturbance sets are guaranteed to contain the true approximation error, the hybrid controller inherits the robustness guarantees proved for the abstract model.

Experimental validation
The authors apply the approach to a classic cart‑pendulum system, which features a nonlinear pendulum dynamics, input saturation on the cart force, and safety limits on the pendulum angle. The four‑dimensional state space (cart position/velocity, pendulum angle/angular velocity) is triangulated into a modest number of simplices. For each simplex, LMIs are solved to obtain a robust feedback law that respects the force limit and keeps the pendulum angle within a prescribed safe interval. Simulations show that, regardless of the initial condition inside the admissible region, the hybrid controller drives the pendulum to the upright equilibrium while never exceeding the actuator limits or leaving the safety region. This demonstrates that the abstracted, disturbance‑inclusive model is sufficiently accurate for practical control.

Key contributions and insights

• Hybridization with explicit disturbance modeling provides a systematic way to bound the error introduced by linearizing a nonlinear system on a local region.
• Robust local synthesis extends existing piecewise‑affine control techniques to the disturbed case, guaranteeing constraint satisfaction for all admissible dynamics inside each simplex.
• Fully automated pipeline: mesh generation, affine approximation, robust synthesis, transition checking, and controller deployment are all algorithmic, making the method suitable for design‑time automation.
• Computational tractability: each local synthesis problem is a low‑dimensional convex program; the overall complexity scales linearly with the number of simplices, unlike global nonlinear optimization methods that often become intractable for real‑time use.

Limitations and future directions
The approach is inherently conservative because the disturbance sets must over‑approximate the true modeling error. Large disturbances can shrink the feasible region for the local controllers, leading to reduced performance (slower convergence, larger control effort). Mitigating this conservatism may involve adaptive mesh refinement, higher‑order (e.g., quadratic) local approximations, or data‑driven tightening of disturbance bounds. Scaling to high‑dimensional systems remains a challenge, as the number of simplices grows exponentially with dimension; exploiting system structure (e.g., sparsity, decoupling) or employing hierarchical decomposition could alleviate this. Finally, extending the framework to handle time‑varying or stochastic disturbances would broaden its applicability to real‑world scenarios where uncertainties evolve over time.

In summary, the paper presents a novel, robust, and automatable method for synthesising controllers for constrained nonlinear systems by hybridizing the dynamics on a simplicial partition and coordinating locally robust affine controllers. The successful cart‑pendulum case study validates the practicality of the approach and opens avenues for further research on scalability, reduced conservatism, and dynamic uncertainty handling.