(642a) An Input/Output Approach to Control of Distributed Chemical Reactors
This work focuses on control of linear parabolic partial differential equations (PDEs) with state and control constraints which is formulated and solved using a state-space approach and an input/output approach.
In the state-space approach, the PDE is written as an infinite-dimensional system in an appropriate Hilbert space. Next, modal decomposition techniques are used to derive a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional system, and express the infinite-dimensional state constraints in terms of the finite-dimensional system state constraints. A number of model predictive control (MPC) formulations, designed on the basis of different finite-dimensional approximations, are then presented and compared. The closed-loop stability properties of the infinite-dimensional system under the low order MPC controller designs are analyzed, and sufficient conditions that guarantee stabilization and state constraint satisfaction for the infinite-dimensional system under the reduced order MPC formulations are derived. Other formulations are also presented which differ in the way the evolution of the fast eigenmodes is accounted for in the performance objective and state constraints. The impact of these differences on the ability of the predictive controller to enforce closed-loop stability and state constraints satisfaction in the infinite-dimensional system is analyzed. Finally, the MPC formulations are applied through simulations to the problem of stabilizing the spatially-uniform unstable steady-state of a linear parabolic PDE subject to state and control constraints.
In the input/output approach, the output is expressed as linear sums of discretized inputs or input gradients using the concept of impulse response (or step response) to construct an input/output model. Subsequently, an optimal control is formulated and solved as a standard least square problem with inequity constraints on the basis of the derived input/output model. The effectiveness of the proposed optimal control scheme is demonstrated through a continuous stirred tank reactor (CSTR) network and a chemical reactor with complex flow behavior. In the CSTR network example for which a state space model can be easily derived, the proposed control method yields the same control trajectory as the one obtained from Linear Quadratic Regulator (LQR) theory (designed based on the state space model) when there are no input constraints present. In the distributed chemical reactor example with complex geometry and flow behavior, where a low dimensional state-space model cannot be easily derived, the proposed optimal control method computes an optimal control trajectory and is shown to be advantageous over conventional control techniques.