Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations
The work addresses the challenging problem of robust control for nonlinear PDEs, which is important for applications in engineering and physics, but the approach is limited to moderate dimensions and is incremental in nature.
The paper proposes a method for synthesizing robust and optimal feedback controllers for nonlinear PDEs by approximating the infinite-dimensional system with a pseudospectral collocation method and solving the associated high-dimensional Hamilton-Jacobi-Isaacs equation using a separable representation and polynomial approximation. The method is effective for robust stabilization of nonlinear dynamics up to dimension 12 and demonstrates optimal stabilization and disturbance rejection on nonlinear parabolic PDEs.
We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct a robust feedback control based on the $\cH_{\infty}$ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension $d\approx 12$. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modelling framework for the robust control of PDEs under parametric uncertainties.