Online policy iteration algorithm for optimal control of linear hyperbolic PDE systems

In this paper, the linear quadratic (LQ) optimal control problem is considered for a class of linear distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs). Reinforcement learning (RL) technique is introduced for adaptive optimal control design from the design-then-reduce (DTR) framework. Initially, a policy iteration (PI) algorithm is proposed, which learns the solution of the space-dependent Riccati differential equation (SDRDE) online without requiring the internal system dynamics of the PDE system. To prove its convergence, the PI algorithm is shown to be equivalent to an iterative procedure of a sequence of space-dependent Lyapunov differential equations (SDLDEs). Then, the convergence is established by showing that the solutions of SDLDEs are a monotone non-increasing sequence that converges to the solution of the SDRDE. For implementation purpose, an online least-square method is developed for the approximation of the solutions of the SDLDEs. Finally, the proposed design method is applied to the distributed control of a steam-jacketed tubular heat exchanger to illustrate its effectiveness.