Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient model predictive control for nonlinear systems modelled by deep neural networks

Jianglin Lan

TL;DR

This work addresses controlling discrete-time systems whose nonlinear dynamics are represented by deep neural networks under state and input constraints. It introduces a dual-mode MPC approach combining a steady-state pre-stabilizing feedback with a constraint-enforcing MPC, and develops three solving strategies for the nonlinear MPC: an exact MIP encoding and two relaxations (LR and enhanced LR, eLR) to balance accuracy and computation. The enhanced LR approach achieves tracking performance close to the exact MIP while maintaining substantially lower computation, demonstrated on an inverted pendulum model with NN-based dynamics. These methods enable real-time control of NN-driven nonlinear systems and lay the groundwork for extensions to robust/tube-MPC and other activation functions. The results highlight a practical trade-off between optimality and speed, with eLR offering a scalable path toward real-time NN-assisted MPC.

Abstract

This paper presents a model predictive control (MPC) for dynamic systems whose nonlinearity and uncertainty are modelled by deep neural networks (NNs), under input and state constraints. Since the NN output contains a high-order complex nonlinearity of the system state and control input, the MPC problem is nonlinear and challenging to solve for real-time control. This paper proposes two types of methods for solving the MPC problem: the mixed integer programming (MIP) method which produces an exact solution to the nonlinear MPC, and linear relaxation (LR) methods which generally give suboptimal solutions but are much computationally cheaper. Extensive numerical simulation for an inverted pendulum system modelled by ReLU NNs of various sizes is used to demonstrate and compare performance of the MIP and LR methods.

Efficient model predictive control for nonlinear systems modelled by deep neural networks

TL;DR

This work addresses controlling discrete-time systems whose nonlinear dynamics are represented by deep neural networks under state and input constraints. It introduces a dual-mode MPC approach combining a steady-state pre-stabilizing feedback with a constraint-enforcing MPC, and develops three solving strategies for the nonlinear MPC: an exact MIP encoding and two relaxations (LR and enhanced LR, eLR) to balance accuracy and computation. The enhanced LR approach achieves tracking performance close to the exact MIP while maintaining substantially lower computation, demonstrated on an inverted pendulum model with NN-based dynamics. These methods enable real-time control of NN-driven nonlinear systems and lay the groundwork for extensions to robust/tube-MPC and other activation functions. The results highlight a practical trade-off between optimality and speed, with eLR offering a scalable path toward real-time NN-assisted MPC.

Abstract

This paper presents a model predictive control (MPC) for dynamic systems whose nonlinearity and uncertainty are modelled by deep neural networks (NNs), under input and state constraints. Since the NN output contains a high-order complex nonlinearity of the system state and control input, the MPC problem is nonlinear and challenging to solve for real-time control. This paper proposes two types of methods for solving the MPC problem: the mixed integer programming (MIP) method which produces an exact solution to the nonlinear MPC, and linear relaxation (LR) methods which generally give suboptimal solutions but are much computationally cheaper. Extensive numerical simulation for an inverted pendulum system modelled by ReLU NNs of various sizes is used to demonstrate and compare performance of the MIP and LR methods.
Paper Structure (9 sections, 1 theorem, 21 equations, 5 figures)

This paper contains 9 sections, 1 theorem, 21 equations, 5 figures.

Table of Contents

  1. Problem description and preliminary
  2. Model predictive control design
  3. Methods for solving the MPC problem
  4. MIP method
  5. Linear relaxation (LR) method
  6. Enhanced linear relaxation (eLR) method
  7. Discussions
  8. Numerical example
  9. Conclusion

Key Result

Proposition 4.1

The solution quality of the proposed methods satisfy: $S_\text{LR} \leq S_\text{eLR} \leq S_\text{MIP} = S_\text{origin}.$

Figures (5)

  • Figure 1: Illustration of (a) the ReLU neuron $z_{i,j} = \phi(\hat{z}_{i,j})$ with $\hat{l}_{i,j} \leq \hat{z}_{i,j} \leq \hat{u}_{i,j}$ and its (b) MIP encoding (the neuron is inactive if $\delta_{i,j} = 0$ and active if $\delta_{i,j} = 1$) and (c) linear relaxation (the area between the blue and red lines).
  • Figure 2: Tracking performance and runtime for 3-layer NNs with different number of neurons at the hidden layer.
  • Figure 3: Example output tracking and control input for the 3-layer NN with 50 neurons in Case 1.
  • Figure 4: Tracking performance and runtime for NNs with different number of hidden layers each having 10 neurons.
  • Figure 5: Tracking performance and runtime for 3-layer NNs with 50 neurons under different prediction horizons.

Theorems & Definitions (1)

  • Proposition 4.1