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Intelligent acceleration adaptive control of linear $2\times2$ hyperbolic PDE systems

Xianhe Zhang, Yu Xiao, Xiaodong Xu, Biao Luo

Abstract

Traditional approaches to stabilizing hyperbolic PDEs, such as PDE backstepping, often encounter challenges when dealing with high-dimensional or complex nonlinear problems. Their solutions require high computational and analytical costs. Recently, neural operators (NOs) for the backstepping design of first-order hyperbolic partial differential equations (PDEs) have been introduced, which rapidly generate gain kernel without requiring online numerical solution. In this paper we apply neural operators to a more complex class of $2\times2$ hyperbolic PDE systems for adaptive stability control. Once the NO has been well-trained offline on a sufficient training set obtained using a numerical solver, the kernel equation no longer needs to be solved again, thereby avoiding the high computational cost during online operations.Specifically, we introduce the deep operator network (DeepONet), a neural network framework, to learn the nonlinear operator of the system parameters to the kernel gain. The approximate backstepping kernel is obtained by utilizing the network after learning, instead of numerically solving the kernel equations in the form of PDEs, to further derive the approximate controller and the target system. We analyze the existence and approximation of DeepONet operators and provide stability and convergence proofs for the closed-loop systems with NOs. Finally, the effectiveness of the proposed NN-adaptive control scheme is verified by comparative simulation, which shows that the NN operator achieved up to three orders of magnitude faster compared to conventional PDE solvers, significantly improving real-time control performance.

Intelligent acceleration adaptive control of linear $2\times2$ hyperbolic PDE systems

Abstract

Traditional approaches to stabilizing hyperbolic PDEs, such as PDE backstepping, often encounter challenges when dealing with high-dimensional or complex nonlinear problems. Their solutions require high computational and analytical costs. Recently, neural operators (NOs) for the backstepping design of first-order hyperbolic partial differential equations (PDEs) have been introduced, which rapidly generate gain kernel without requiring online numerical solution. In this paper we apply neural operators to a more complex class of hyperbolic PDE systems for adaptive stability control. Once the NO has been well-trained offline on a sufficient training set obtained using a numerical solver, the kernel equation no longer needs to be solved again, thereby avoiding the high computational cost during online operations.Specifically, we introduce the deep operator network (DeepONet), a neural network framework, to learn the nonlinear operator of the system parameters to the kernel gain. The approximate backstepping kernel is obtained by utilizing the network after learning, instead of numerically solving the kernel equations in the form of PDEs, to further derive the approximate controller and the target system. We analyze the existence and approximation of DeepONet operators and provide stability and convergence proofs for the closed-loop systems with NOs. Finally, the effectiveness of the proposed NN-adaptive control scheme is verified by comparative simulation, which shows that the NN operator achieved up to three orders of magnitude faster compared to conventional PDE solvers, significantly improving real-time control performance.

Paper Structure

This paper contains 31 sections, 4 theorems, 73 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Neural operator
  4. Adaptive control of hyperbolic PDEs
  5. NO approximations for PDE control
  6. Paper structure and contributions
  7. Notation
  8. Problem statement
  9. Basic adaptive control using swapping design
  10. Filter equation
  11. Parameter estimation law
  12. Adaptive control
  13. Approximated kernel by neural operator
  14. Operator learning with DeepONet
  15. The network structure of DeepONet
  16. ...and 16 more sections

Key Result

Lemma 1

For every time $t \ge 0$, we introduce the following properties for the operators defined in region T

Figures (8)

  • Figure 1: The intelligent acceleration adaptive control scheme combined with DeepONet. The approximated kernel obtained from the off-line trained operator network instead of the analytic kernel in numerical computation.
  • Figure 2: The structure of the DeepONet network used for the simulation. Three structures of the branch network are set up for comparison.
  • Figure 3: Training loss of the approximate kernels: the horizontal axis represents the number of iterations of the network, and the vertical axis represents the value of the loss function.
  • Figure 4: Backstepping kernels. Exact solution obtained by the finite difference method (Top row). Predicted solution of the trained DeepONet (Middle row). Deviation of estimated value from actual value of corresponding kernel (Buttom row).
  • Figure 5: NO-approximate kernels for backstepping design in adaptive stabilization control for a class of 2$\times$2 hyperbolic PDE systems.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Theorem 2