An iterative procedure for solving the discrete terminal control problem

In this paper a technique is proposed for solving the discrete terminal control problem. The plant is assumed to be linear with bounded input. The cost function is a quadratic function of the terminal states. The approach that is used is to change the original minimization problem with differential constraints into a mathematical programming formulation. Due to the linearity of the problem this is a simple operation. Since the cost function is quadratic, the resulting mathematical programming problem becomes a quadratic programming problem. A computationally efficient solution to this quadratic programming problem is presented which uses the properties of the solution of the discrete optimal terminal control problem. For instance it is shown that if the desired terminal point lies outside the reachable set, the optimum control will have at most n-1 control intervals, where n is the order of the system, when the control is not at its maximum magnitude. For the case where the desired point lies within the reachable set, there exists at least one control sequence with at most n intervals when the control is not at its maximum magnitude. The algorithm which is developed uses these properties to avoid the manipulation of any matrices of order greater than n.