speed and position can be controlled precisely. An example, based on a computer simulation, is given. The notation used is summarized in the last section of this paper. Derivation of the Controller Gains In general, the drag force on a vehicle is a quadratic function of its speed Thus D(V) V. (1) in which and are road-resistance coefficients and is the air-drag parameter a b c (2) in which is the air density, is the drag coefficient, and is the frontal area. CD A If is the time-varying thrust applied to a vehicle of mass the equation of T(t) m, motion is (3) There is generally a time lag between commanded and actual thrust. If the thruster time constant is, actual thrust in terms of the commanded thrust is given by the differential equation T Tc (4) Consider small disturbances around some specific set point denoted by a bar over each variable. Thus, let (5) (6) (7) in which (8) Let the control equation be (9) in which is the position gain constant and is the velocity gain constant. Then, assuming the bared values correspond to the command condition, substitute equation (9) and then equations (5) and (6) into equation (4) and differentiate, assuming that the command speed is relatively slowly varying. The result is
speed and position can be controlled precisely. An example, based on a computer simulation, is given. The notation used is summarized in the last section of this paper. Derivation of the Controller Gains In general, the drag force D(V) on a vehicle is a quadratic function of its speed V. Thus (1) in which a and b are road-resistance coefficients and c is the air-drag parameter (2) in which is the air density, CD is the drag coefficient, and A is the frontal area. If T(t) is the time-varying thrust applied to a vehicle of mass m, the equation of motion is (3) There is generally a time lag between commanded and actual thrust. If the thruster time constant is, actual thrust T in terms of the commanded thrust Tc is given by the differential equation (4) Consider small disturbances around some specific set point denoted by a bar over each variable. Thus, let (5) (6) (7) in which (8) Let the control equation be (9) in which is the position gain constant and is the velocity gain constant. Then, assuming the bared values correspond to the command condition, substitute equation (9) and then equations (5) and (6) into equation (4) and differentiate, assuming that the command speed is relatively slowly varying. The result is