Analytical Construction Robust Optimal Control Systems by the Criterion of Quick Action with Infinitely High Gain

Abstract

The problem of the synthesis of robust control systems with a high gain and, in particular, optimal by the criterion of quick action, which allow optimal control by the accuracy of regulation of multidimensional non-linear dynamic objects with functional uncertainties, is discussed. A method is proposed for the analytical construction of optimal control systems by the criterion of quick action for a wide class of multidimensional nonlinear dynamic objects with functional uncertainties, unstable objects; no minimal-phase objects, neutral object and objects with differentiation properties. Simplicity and universality, mathematical rigor and physical validity of this method consists in usingR.R Be llman’s method and decomposing the optimal by the criterion of quick action problem into a series of simple first-order simple problems of the same type. A theoretically comprehensive solution to the robust control problem is given by the idea of constructing systems that are stable with an unlimited increase in gain. In this case, optimal systems have stability properties. Such systems are synthesized using quadratic quality functionals that are not explicitly dependent on the control signal and the restriction on the control signal. It is significant that, in contrast to continuous systems with unmeasurable perturbations and a little-known object, in which the conditions of invariance require the use of infinitely large gains, in relay (discontinuous) systems, the equivalent effect is achieved using finite control actions. Since the performance problem is a particular problem of the accuracy of reproducing the input action on the control object, the established control error (including all error coefficients: by position, speed, acceleration, jerk, etc.) is theoretically strictly equal to zero if external and internal interference, acting only on the control object, but not on the control system, including sensors of state variables of the control object or the input signal of the task. However, due to the inertia of the object, there can be no talk of accuracy in the transient process of working out the input signal of the task, even if it is optimal in terms of the criterion of fast action.