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Neural Network-based Universal Formulas for Control

Pol Mestres, Jorge Cortés, Eduardo D. Sontag

TL;DR

The paper tackles enforcing an arbitrary number of state-dependent affine input inequalities in nonlinear control-affine systems by introducing a smooth universal controller defined as the minimizer of a strictly convex function over a feasible set. It proves strict convexity and smoothness of the universal formula, and then replaces real-time minimization with a neural network approximation, exhibiting a normalization that allows training on bounded parameter sets and reuse across problems with fixed input dimension and constraint count. The NN-based approach offers substantial reductions in online computation and can be used to warmstart exact optimization, with simulations demonstrating safe stabilization tasks and significant speedups. The work advances practical real-time control under multiple objectives (stability, safety, input limits) and provides a pathway for scalable, dimension-agnostic implementation, while outlining avenues for handling higher dimensions and non-affine constraints.

Abstract

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the fact that a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. This is why we refer to such NN approximation as a NN-based universal formula for control. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a bounded set in the state space, which significantly simplifies the training process. Various simulations showcase the performance of the proposed solution.

Neural Network-based Universal Formulas for Control

TL;DR

The paper tackles enforcing an arbitrary number of state-dependent affine input inequalities in nonlinear control-affine systems by introducing a smooth universal controller defined as the minimizer of a strictly convex function over a feasible set. It proves strict convexity and smoothness of the universal formula, and then replaces real-time minimization with a neural network approximation, exhibiting a normalization that allows training on bounded parameter sets and reuse across problems with fixed input dimension and constraint count. The NN-based approach offers substantial reductions in online computation and can be used to warmstart exact optimization, with simulations demonstrating safe stabilization tasks and significant speedups. The work advances practical real-time control under multiple objectives (stability, safety, input limits) and provides a pathway for scalable, dimension-agnostic implementation, while outlining avenues for handling higher dimensions and non-affine constraints.

Abstract

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the fact that a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. This is why we refer to such NN approximation as a NN-based universal formula for control. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a bounded set in the state space, which significantly simplifies the training process. Various simulations showcase the performance of the proposed solution.

Paper Structure

This paper contains 7 sections, 4 theorems, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem Statement
  4. A Universal Formula for an Arbitrary Number of Affine Constraints
  5. Neural Network Approximation of the Universal Formula
  6. Simulations
  7. Conclusions, Limitations, and Future Work

Key Result

Proposition 3.1

(Strict convexity): Let ${\mathbf{p}} \in \mathbb{R}^{N+mN}$. The function $J_{{\mathbf{p}}}$ is strictly convex in the convex domain $\mathcal{K}_{{\mathbf{p}}}$.

Figures (2)

  • Figure 1: (left) Trajectories of the closed-loop system obtained from the neural network based controller for Example \ref{['ex:safe-stabilization-single-integrator']}. (right) Trajectories of the closed-loop system obtained from numerically finding the controller $u^*$ online, warmstarting the solver with the NN-based controller for Example \ref{['ex:safe-stabilization-single-integrator']}. Initial conditions are denoted by red crosses, the origin is the black dot, and the green region denotes the unsafe set.
  • Figure 2: (left) Projection in the $(x,y)$ plane of trajectories of the closed-loop system obtained from the neural network based controller for Example \ref{['ex:safe-stabilization-unicycle-drift']}. (right) Projection in the $(x,y)$ plane of trajectories of the closed-loop system obtained from numerically finding the controller $u^*$ online and warmstarting the solver with the NN-based controller for Example \ref{['ex:safe-stabilization-unicycle-drift']}. Initial conditions are denoted by red crosses (and all have an initial orientation $\theta_0 = \pi+0.1$), the origin is the black dot, and the green region denotes the unsafe set. Black arrows indicate the orientation of the unicycle (i.e., the $\theta$ variable) at that point of the trajectory. Observe that the velocity $v$ could be negative, so that at points near the right of the target, the vehicle is "backing up". We note also that due to the presence of a drift term in the $y$ dynamics, the orientation of the unicycle might not be tangent to the trajectory.

Theorems & Definitions (18)

  • Example 2.1
  • Example 2.2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 8 more