State-dependent Riccati equation feedback stabilization for nonlinear PDEs

Abstract

The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) is presented for H 2 and H ∞ control problems. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are proposed. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the Sine-Gordon, degenerate Zeldovich, and viscous Burgers' PDEs are presented, providing a thorough experimental assessment of the proposed methodology.

Convolutional Neural Networks for Very Low-Dimensional LPV Approximations of Incompressible Navier-Stokes Equations

“…For example, for affine-linearly parametrizable coefficients as in Equation ( 1), one can derive series expansions (see e.g., Beeler et al [9]) of the solution to the associated parameter-dependent Riccati equations and exploit them for efficient controller design; cp. [10]. Furthermore, if the image of ρ for the given system can be confined to a polygon, then one can provide a globally stabilizing controller (see e.g., Apkarian et al [11]) through the scheduling of a set of linear controllers.…”

Data-driven Tensor Train Gradient Cross Approximation for Hamilton-Jacobi-Bellman Equations

“…A similar idea is proposed in [2] where the authors propose a sub-optimal feedback law obtained via a feedforward neural network. In this case the training set is generated via the State-Dependent Riccati Equation (SDRE) strategy [6,10,3], an extension of the Riccati solution to nonlinear dynamics.…”