An iterative method for the finite-time bilinear-quadratic control problem
Journal of Optimization Theory and Applications
Matrix computations (3rd ed.)
Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition
Journal of Optimization Theory and Applications
Existence and Uniqueness of Optimal Control for a Distributed-Parameter Bilinear System
Journal of Dynamical and Control Systems
Nonlinear optimal tracking control with application to super-tankers for autopilot design
Automatica (Journal of IFAC)
H∞ stabilization of the current profile in tokamak plasmas via an LMI approach
Automatica (Journal of IFAC)
Journal of Control Science and Engineering
Hi-index | 22.15 |
We present a framework to solve a finite-time optimal control problem for parabolic partial differential equations (PDEs) with diffusivity-interior actuators, which is motivated by the control of the current density profile in tokamak plasmas. The proposed approach is based on reduced order modeling (ROM) and successive optimal control computation. First we either simulate the parabolic PDE system or carry out experiments to generate data ensembles, from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained reduced order model corresponds to a bilinear control system. Based on quasi-linearization of the optimality conditions derived from Pontryagin's maximum principle, and stated as a two boundary value problem, we propose an iterative scheme for suboptimal closed-loop control. We take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers. The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative schemes for optimal control of bilinear systems, the proposed scheme avoids repeated numerical computation of the Riccati equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical simulation study shows the effectiveness of this approach.