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An ANN-Enhanced Approach for Flatness-Based Constrained Control of Nonlinear Systems

Huu-Thinh Do, Ionela Prodan, Florin Stoican

TL;DR

The paper addresses constraint handling for nonlinear differentially flat systems by leveraging ReLU-ANNs to approximate the nonlinear constraint mapping introduced by flatness-based feedback linearization. It derives a polyhedral, union-of-polytopes representation of the distorted constraint and encodes it via mixed-integer programming to integrate with control Lyapunov function-based control and model predictive control, including explicit MPC possibilities. Through a detailed cell-enumeration procedure and error-bounding strategies, the approach enables safe constraint satisfaction while preserving linearized dynamics, as demonstrated on aircraft longitudinal dynamics, a planar UAV, and a PMSM model. The work advances practical constrained control for nonlinear flat systems by bridging neural network constraint representations with tractable optimization-based controllers, albeit with computational overhead that motivates future complexity-reduction and formal stability study.

Abstract

Neural networks have proven practical for a synergistic combination of advanced control techniques. This work analyzes the implementation of rectified linear unit neural networks to achieve constrained control in differentially flat systems. Specifically, the class of flat systems enjoys the benefit of feedback linearizability, i.e., the systems can be linearized by means of a proper variable transformation. However, the price for linearizing the dynamics is that the constraint descriptions are distorted geometrically. Our results show that, by using neural networks, these constraints can be represented as a union of polytopes, enabling the use of mixed-integer programming tools to guarantee constraint satisfaction. We further analyze the integration of the characterization into efficient settings such as control Lyapunov function-based and model predictive control (MPC). Interestingly, this description also allows us to explicitly compute the solution of the MPC problem for the nonlinear system. Several examples are provided to illustrate the effectiveness of our framework.

An ANN-Enhanced Approach for Flatness-Based Constrained Control of Nonlinear Systems

TL;DR

The paper addresses constraint handling for nonlinear differentially flat systems by leveraging ReLU-ANNs to approximate the nonlinear constraint mapping introduced by flatness-based feedback linearization. It derives a polyhedral, union-of-polytopes representation of the distorted constraint and encodes it via mixed-integer programming to integrate with control Lyapunov function-based control and model predictive control, including explicit MPC possibilities. Through a detailed cell-enumeration procedure and error-bounding strategies, the approach enables safe constraint satisfaction while preserving linearized dynamics, as demonstrated on aircraft longitudinal dynamics, a planar UAV, and a PMSM model. The work advances practical constrained control for nonlinear flat systems by bridging neural network constraint representations with tractable optimization-based controllers, albeit with computational overhead that motivates future complexity-reduction and formal stability study.

Abstract

Neural networks have proven practical for a synergistic combination of advanced control techniques. This work analyzes the implementation of rectified linear unit neural networks to achieve constrained control in differentially flat systems. Specifically, the class of flat systems enjoys the benefit of feedback linearizability, i.e., the systems can be linearized by means of a proper variable transformation. However, the price for linearizing the dynamics is that the constraint descriptions are distorted geometrically. Our results show that, by using neural networks, these constraints can be represented as a union of polytopes, enabling the use of mixed-integer programming tools to guarantee constraint satisfaction. We further analyze the integration of the characterization into efficient settings such as control Lyapunov function-based and model predictive control (MPC). Interestingly, this description also allows us to explicitly compute the solution of the MPC problem for the nonlinear system. Several examples are provided to illustrate the effectiveness of our framework.

Paper Structure

This paper contains 19 sections, 5 theorems, 84 equations, 19 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statement
  3. Constrained flatness-based feedback linearization
  4. Rectified linear artificial neural networks
  5. Constraint reformulation with ReLU-ANN
  6. Enumeration of cells formed by ReLU-ANN
  7. Constraint characterization
  8. Discussion on error bound estimation approaches
  9. Control implementation with ANN-based constraints
  10. CLF implementation
  11. MPC implementation
  12. Simulation study
  13. Aircraft longitudinal dynamics
  14. Planar UAV model
  15. PMSM model
  16. ...and 4 more sections

Key Result

Proposition 1

For a ReLU activation function $\sigma(s)$, denote its activation status with a binary variable $\alpha$: $\alpha=+1$ when the function is activated (i.e., $\sigma(s) = s$) and $\alpha=-1$ when the function is not activated (i.e., $\sigma(s) = 0$). Then, the output of a neuron $y = \sigma(\boldsymbo

Figures (19)

  • Figure 1: Constrained feedback linearization control scheme based on ReLU-ANN.
  • Figure 2: Single hidden layer ReLU-ANN approximating the linearizing mapping.
  • Figure 3: Illustrative example of Proposition \ref{['prop:oneNeuron']} with a neuron of $\boldsymbol{w}=20.5^\top, b=-1$.
  • Figure 4: PWA representation of ReLU-ANN with the approximation of $y=\cos x_1 + \sin x_2.$
  • Figure 5: Cell enumeration algorithm (Algorithm \ref{['algo:cellEnum']}) applied for the approximation in \ref{['eq:tildeV_zv']}.
  • ...and 14 more figures

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3: Convoluted constraint enforcement with ReLU-ANN do2024_reluANN
  • proof
  • Remark 2
  • Proposition 4: Grid-based error estimation
  • proof
  • Proposition 5
  • ...and 2 more