Programmed Control

control of an object’s mode of operation according to a predetermined program. Programmed control can be achieved both with feedback (closed-loop control systems) and without feedback (open-loop control systems). Closed-loop programmed control systems can function with or without optimization of the mode of work of the controlled object. The process of programmed control with optimization may be viewed as the minimization of the function that characterizes the difference between the desired and actual states of the object. For example, the programmed control of aircraft dictates a course of flight that will pinpoint the location of the craft at any given point in time.

The term “programmed control” with optimization arose in the theory of control of systems having stochastic inputs, that is, systems subject to random perturbations. For example, suppose that the movement of an object is described by a system of differential equations of the type ẋ = f(x, u, ζ), where x is the phase vector, ζ is a random vector-function, and u(t) is the control vector. Let us also assume that the goal of control is to conduct the object (system) from an initial state x₀ to a certain final state x_T. Because the system is stochastic, there can be no question of a precise attainment of the final state x_T. It is possible to speak only of a choice of control that minimizes a certain function of the final state J[x(T)]. Here, the norm J[x(T)) = ║x(T) - x_T║ is taken as such a function.

The method of investigation outlined below is widespread in the theory of systems similar to those with stochastic inputs, among which are control systems for rockets and many technological processes. Let us assume that ζ = 0, that is, the system is determined. It is then possible to try to find control U(t), which will conduct the system exactly to state x_T along a certain path—function x(t). If the goal of control is attainable, quite a large number of such paths can be determined. Therefore, it becomes possible to select control U(t) (a program) that will ensure an optimal value for some criterion. For example, if we are speaking of bringing a rocket into a given orbit, the criterion may be fuel consumption. In this way, the concept of optimal program arises, which in addition usually encompasses the concept of optimal path x̄(t) and optimal control Ũ (t). The concept of optimal program pertains only to idealized systems. Therefore, after determining the optimal program, the designer also designs a system to control the program and the path. We may write U = Ũ + u, where Ũ is a fixed function of time and u is a correcting control, which acts through a feedback circuit. The system of control possesses the means of measuring the actual path; the task of the correcting control is to ensure a minimal discrepancy between the real path x(t) and the optimal one x̄(t), which achieves the goal of control x_T.