Nonsmooth Penalty Techniques in Control of the Continuous Casting Process
Timo Männikkö, Marko M. Mäkelä · Birkhäuser Basel eBooks · 1991
We introduce a mathematical model which is used to simulate the continuous casting process and to control the secondary cooling water sprays. The main object is to minimize the defects in the final products. The problem is formulated as an optimal control problem where the cost function is constructed according to certain metallurgical criteria and constraints. The temperature distribution of the strand is calculated by solving a nonlinear heat equation with free boundaries between solid and liquid phases. The state constraints are penalized by using (a) the usual quadratic penalty functions, which are differentiable, and (b) so called exact penalty functions, which are nonsmooth (nondifferentiable). Due to the phase change, the cost function is nonsmooth even in the case of quadratic penalties. Therefore, instead of the classical gradient-based algorithms, we use Proximal Bundle method, which is one of the most efficient methods of nonsmooth optimization. Some numerical results, calculated with the Cray supercomputer, are also presented.