An electronic throttle is a dc-motor-driven valve that regulates air inflow into the combustion system of the engine. The throttle control system should ensure fast and accurate reference tracking of the valve plate angle while preventing excessive wear of the throttle components by constraining physical variables to their normal-operation domains. These high-quality control demands are hard to accomplish since the plant is burdened with strong nonlinear effects of friction and limp-home nonlinearity. In this paper, the controller synthesis is performed in discrete time by solving a constrained time-optimal control problem for the piecewise affine (PWA) model of the throttle. To that end, a procedure is proposed to model friction in a discrete-time PWA form that is suitable both for simulation and controller design purposes. The control action computation can, in general, be restated as a mixed-integer program. However, due to the small sampling time, solving such a program online (in a receding horizon fashion) would be very prohibitive. This issue is resolved by applying recent theoretical results that enable offline precomputation of the state-feedback optimal control law in the form of a lookup table. The technique employs invariant set computation and reachability analysis. The experimental results on a real electronic throttle are reported and compared with a tuned PID controller that comprises a feedforward compensation of the process nonlinearities. The designed time-optimal controller achieves considerably faster transient, while preserving other important performance measures, like the absence of overshoot and static accuracy within the measurement resolution

We consider constrained finite-time optimal control problems for discrete-time linear time-invariant systems with constraints on inputs and outputs based on linear and quadratic performance indices. The solution to such problems is a time-varying piecewise affine (PWA) state-feedback law and can be computed by means of multiparametric programming. By exploiting the properties of the value function and the piecewise affine optimal control law of the constrained finite-time optimal control (CFTOC), we propose two new algorithms that avoid storing the polyhedral regions. The new algorithms significantly reduce the on-line storage demands and computational complexity during evaluation of the PWA feedback control law resulting from the CFTOC.