On solution of the problem of correcting scalar terminal state of an aircraft for an arbitrary distribution of a multiplicative perturbation

System analysis, control and data processing


Аuthors

Ignatov A. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: alexei.ignatov1@gmail.com

Abstract

We investigate the problem of correcting scalar terminal state of the aircraft. We consider the case of one correction. The correction is carried out by a high-thrust engine. The initial state of the aircraft is random. Distribution of the multiplicative perturbation connected with implementation error of rated specific impulse is arbitrary and independent fr om the initial state of the aircraft. Optimal control is selected in the class of piecewise constant functions, depending on the initial position of the aircraft. Optimal control is searched by maximizing the probability that the terminal state will be in a certain zone. We find an analytical expression for criterial function by the law of total probability and propose an algorithm of searching for optimal solution. Since we should optimize the function of unknown origin on open sets for searching for optimal solution, then we approximate criterial function by the method of middle rectangles. To find optimal solution of obtained function we discretize the initial probability measure and we replace the continuous random variable characterizing the multiplicative perturbation by the discrete random variable in approximating function. Then we reduce the optimization problem to the mixed-integer linear programming problem. We consider an example. We investigate the dependence of the optimal value of the criterion on the number of the segments of the partition and the number of realizations for a fixed set of input data for calculations using the IBM ILOG CPLEX package. We show that the accuracy of solution is more influenced by the number of segments of the partition, rather than the number of realizations, if the number of realizations is large. In the example we give a form of optimal control, which is close to control, obtained using the optimal confidence set in the case wh ere the initial position and the multiplicative perturbation are normally distributed random variables. The resulting control has a dead zone and is nonlinear.

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