Optimal control of a voice-coil-motor with Coulombic friction

The voice-coil-motor is a widely used mechatronic device, which represents a typical electrodynamic actuator for machine tool axes, bonding machines and hydraulic/pneumatic valve drives. One principal task consists in steering the system precisely to a prescribed target in minimal time or with minimal energy. To achieve this goal, we formulate an optimal control problem using a dynamical system derived in Zirn [19]. Since Coulombic friction is modelled by a jump function depending on the sign of the velocity, the optimal control problem belongs the class of nonsmooth optimization problems. We show that time-optimal controls are bang-bang for all physically reasonable control bounds. Switching times are directly optimized by nonlinear programming methods, which also allow to compute parametric sensitivity derivatives. Energy-optimal solutions are presented for several fixed final times.