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A POD approach to identify and control PDEs online through State Dependent Riccati equations

Alessandro Alla, Agnese Pacifico

TL;DR

This work addresses the control of partial differential equations (PDEs) with unknown parameters using the state-dependent Riccati equation (SDRE) approach and employs model order reduction through the proper orthogonal decomposition (POD) method to enhance the efficiency of the method.

Abstract

We address the control of Partial Differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. The method works as follows, given an estimated parameter configuration, we compute the corresponding control using the State-Dependent Riccati Equation (SDRE) approach. Subsequently, after computing the control, we observe the trajectory and estimate a new parameter configuration using Bayesian Linear Regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. We also focus on the computational cost of the algorithm, since we deal with high dimensional systems. To enhance the efficiency of the method, indeed, we employ model order reduction through the Proper Orthogonal Decomposition (POD) method. The considered problem's dimension is notably large, and POD provides impressive speedups. Further, a detailed description on the coupling between POD and SDRE is also provided. Finally, numerical examples will show the accurateness of our method across two test cases.

A POD approach to identify and control PDEs online through State Dependent Riccati equations

TL;DR

This work addresses the control of partial differential equations (PDEs) with unknown parameters using the state-dependent Riccati equation (SDRE) approach and employs model order reduction through the proper orthogonal decomposition (POD) method to enhance the efficiency of the method.

Abstract

We address the control of Partial Differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. The method works as follows, given an estimated parameter configuration, we compute the corresponding control using the State-Dependent Riccati Equation (SDRE) approach. Subsequently, after computing the control, we observe the trajectory and estimate a new parameter configuration using Bayesian Linear Regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. We also focus on the computational cost of the algorithm, since we deal with high dimensional systems. To enhance the efficiency of the method, indeed, we employ model order reduction through the Proper Orthogonal Decomposition (POD) method. The considered problem's dimension is notably large, and POD provides impressive speedups. Further, a detailed description on the coupling between POD and SDRE is also provided. Finally, numerical examples will show the accurateness of our method across two test cases.
Paper Structure (16 sections, 25 equations, 9 figures, 8 tables, 4 algorithms)

This paper contains 16 sections, 25 equations, 9 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Control of nonlinear problems via State Dependent Riccati Equations
  3. Proper Orthogonal Decomposition for SDRE
  4. Discrete Empirical Interpolation Method
  5. Numerical Example: control of the 2D Allen-Cahn
  6. Online Identification and Control of Unknown Nonlinear Dynamics using POD
  7. Choose an initial parameter configuration.
  8. Compute the snapshots (step \ref{['step: snap']}).
  9. Project the problem into the reduced subspace (step \ref{['step: proj']}).
  10. Compute the reduced control (steps \ref{['step: pi']}-\ref{['step: u']}).
  11. Observe the trajectories (step \ref{['step: traj']}).
  12. Update the parameter configuration based on the observations (steps \ref{['step: BLR1']}-\ref{['step: BLRcond']}).
  13. Numerical experiments
  14. Test 1: Allen-Cahn 2D
  15. Test 2: 2D Nonlinear advection equation
  16. ...and 1 more sections

Figures (9)

  • Figure 1: System observation can be seen as a black box: given a control $u(t)$ and an initial state $x_0$, we can observe the trajectory $x(t;u(t),\mu^*)$ obtained with the provided inputs. Here $\mu^*$ is an unknown parameter of the PDE. System observations allow us to see the system evolution even if $\mu^*$ is unknown. With these observations we will approximate the unknown parameter. In principle any input $u$ could be used for observing the system, but we will look for a $u$ that minimizes a given cost functional.
  • Figure 2: The control, or input, acts in the black region $\Omega_c$ and the output, i.e. the cost, is considered in the blue region $\Omega_o=\bigcup_{i=1}^z\Omega_{o_i}$.
  • Figure 3: Solutions at time $t=3$ of \ref{['ex1']} for the uncontrolled problem $u(t)=0$ (left), and the stabilized solution through Algorithm \ref{['alg: sdre']} (right).
  • Figure 4: Results of Algorithm \ref{['alg: podsdre']} for different snapshots set. POD error $\mathcal{E}(r)$ (left), Difference for the cost functional $\mathcal{E}_j(r)$ (middle), CPU time (right). The time needed to solve Alg. \ref{['alg: sdre']} was 135s. We can observe an impressive speed up of the POD-DEIM method.
  • Figure 5: Test 1. Solution at time $t=3$ from Alg. \ref{['alg:RL']} (left) and Alg. \ref{['alg:RL POD red']} (middle). See the right panel of Figure \ref{['fig:ex1']} for a comparison with algorithm \ref{['alg: sdre']}. The right panel of this picture shows the controls found by the three mentioned algorithms. No noise was added to the data in this test.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 1