On shrinking horizon move-blocking predictive control

Our theoretical developments are motivated by a real-world application under study in collaboration with a rail transport manufacturer, pertaining to the energy efficient operation of trains. Consider an electric train controlled by a digital control unit in discrete time, with sampling period Ts. Let us denote with k ∈Z the discrete time variable, with x(k) = [x1(k), x2(k)] the state of the train, where x1 is its position and x2 its speed (·T denotes the matrix transpose operator), and with u(k) ∈ [−1,1] a normalized traction force, where u(k) = 1 corresponds to the maximum applicable traction and u(k) =−1 to the maximum braking. The input u is the available control variable. The train has to move from one station with position x1 = 0 to the next one, with position x1 = x f , in a prescribed time t f . For a given pair of initial and final stations, the track features (slopes, curvature) are known in advance. Thus, in nominal conditions (i.e. with rated values of the train parameters, like its mass and the specifications of the powertrain and braking systems), according to Newton’s laws and using the forward Euler discretization method, the equations of motion of a reasonably accurate model of this system read: x1(k+1) = x1(k)+Tsx2(k) x2(k+1) = x2(k)+Ts ( FT (x(k),u(k))−FB(x(k),u(k))−FR(x(k)) M ) (1)