Robust control of particulate processes using uncertain population balances

A general method is proposed for the synthesis of robust, nonlinear controllers for spatially homogeneous particulate processes described by population balances including time-varying uncertain variables. The controllers are synthesized via Lyapunov's direct method on the basis of finite-dimensional approximations of the population balances which are obtained by using the method of weighted residuals. The controllers enforce stability in the closed-loop system and attenuation of the effect of uncertain variables on the output, and achieve particle-size distributions with desired characteristics. The robustness of the controllers with respect to unmodeled dynamics is also addressed within the singular perturbations framework. The controllers enforce the desired stability and performance specifications in the closed-loop system, provided that the unmodeled dynamics are stable and sufficiently fast. The proposed control method is applied to a continuous crystallizer with fines trap in which the nucleation rate and the crystal density change arbitrarily with time, and the actuator and sensor dynamics are explicitly considered in the process model, but not included in the model used for controller synthesis. Simulation runs of the closed-loop system clearly demonstrate that the controller attenuates uncertainty; achieves a crystal-size distribution with desired characteristics, and is superior to nonlinear controllers that do not account for the presence of uncertainty.