ON COMPUTATIONAL REDUNDANCY OF THE DICHOTOMOUS SEARCH AND CONDITIONAL MINIMIZATION OF UNIMODAL FUNCTIONS BY THE ECONOMICAL DICHOTOMOUS SEARCH

The article presents the synthesis of a functional diagram of an adaptive automatic control system (ACS) for controlling an aircraft with an automatically reconfigurable multidimensional PI controller, which provides the minimum static and minimum mean square error of control with minimal energy consumption for the formation of the control exposure. The synthesis of ACS algorithms is performed as a result of solving the problem of conditionally minimizing the quadratic functional of the generalized work (taking into account restrictions on state variables and control actions given by differential equations of the control object (CO) and inequalities). The mathematical description of the multidimensional CO is carried out using the CO model in the state space, which automatically takes into account the mutual influence of individual control loops on each other. As the state variables of the aircraft, linear displacements, speeds and accelerations of the center of mass of the aircraft, and angular displacements, speeds and accelerations of the rotational movement of the aircraft relative to the center of mass are used. The matrix equation of dynamics of the aircraft is formed by a system of nonlinear differential equations of the first order of forces and moments of forces acting on the aircraft. To ensure the minimum static control error, integrators are included in the ACS (for each control action). The algorithm for the formation of control actions of the extended CO, providing the declared properties of the ACS, is obtained as a result of solving the problem of conditional minimization of the generalized work functional. The task of conditional minimization of a functional with constraints is performed by the maximum principle. The resulting two-point boundary value problem is transformed by the invariant immersion method into a Cauchy problem for optimal values of state variables. The evaluation of the characteristics of a specific adaptive ACS for the spacecraft is expected to be obtained as a result of further research by mathematical modeling.

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Binary Scanning Search (Biscan) for Conditional Minimization of Weakly Unimodal Functions

Bulletin of the South Ural State University Series Computational Mathematics and Software Engineering ◽

10.14529/cmse180404 ◽

2018 ◽

Vol 7 (4) ◽

Keyword(s):

Conditional Minimization

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OPTIMIZATION OF UNIMODAL FUNCTIONS USING PARABOLIC PREDICTOR search

VESTNIK OF ASTRAKHAN STATE TECHNICAL UNIVERSITY SERIES MANAGEMENT COMPUTER SCIENCE AND INFORMATICS ◽

10.24143/2072-9502-2018-3-101-108 ◽

2018 ◽

pp. 101-108

Author(s):

Vladimir Aleksandrovich Kodnyanko

Keyword(s):

Golden Section ◽

Similar Nature ◽

Smooth Functions ◽

Unimodal Function ◽

Direct Function ◽

Monotonic Functions ◽

Golden Section Search ◽

Conditional Minimization ◽

Function Predictor ◽

Predictor Function

A combined parabolic predictor search is proposed for the conditional minimization of the unimodal function using the predictive-based selective application of phases of extremum search by golden section search and parabolic search. The formula for calculating the value of parabolic predictor function is given, with its help it is possible to work out the forecast and tactics of extremum search of the minimized function. Predictor includes forecasting extremeness, monotony and constancy of function on a segment of uncertainty. Identification forecast for a direct function is described, using which allows to find a solution in three calculations. The assertion is made that if three successive computations of a function give points with similar ordinates, then abscissa of each point can be a solution of the problem. The procedure of identifying non-direct monotonic functions is described. It is shown that the reliability of monotonicity forecast can be determined by five calculations of the function. There has been described the procedure of using phases of parabolic method, which can be performed at favorable prediction of detecting the internal extremum of function. It has been stated that carrying out these phases, even with favorable forecast, can be considered inexpedient for cases when it is recognized that the problem is weakly sensitive or insensitive to the parabolic forecast. Block diagrams of algorithms implementing the method are given. It is shown that, compared to golden section search, the predictor has 3-5 times faster response for smooth functions and is comparable by this criterion to Brent method. The predictor achieves the greatest speed when minimizing monotonic functions. The method works somewhat slower than golden section search, however, it is much faster than Brent method when searching for the minimum of piecewise, flat, planar and other functions of a similar nature for which approximation of parabola does not give the expected effect. In comparison with Brent method, parabolic predictor has 1.5-4 times more speed in solving problems of such type.

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