Udc 17. 977+644. Bashkir State University, Republic of Bashkortostan

NUMERICAL ALGORITHM FOR SOLVING THE PROBLEM
OF OPTIMAL CONTROL WITH TERMINAL CONSTRAINTS FOR DYNAMIC
SYSTEMS
Most of the numerical methods for solving optimization problems face a number of difficulties
associated with the nonlinearity of the models describing the dynamic pro-
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E. V. Antipina, S. I. Mustafina, A. F. Antipin, S. A. Mustafina
133
dx
where x(t) = (x1(t), x2(t), . . . , xn(t)) is the vector of phase variables; u(t) = (u1(t), u2(t), . . . , ur(t)) ÿ U is the vector of
control functions; U is the set of admissible control values; t is an independent variable; f(x(t), u(t), t) = (f1(x(t), u(t), t),
f2(x(t), u(t), t), . . . , fn(x(t), u(t), t)) is a continuous vector function together with its partial derivatives. Let t be time, then
the interval [0, t1] is the time of system operation. Initial phase vector x(0) = x
Q0(u) = g0(x(t1)).
Formulation of the problem. Let us formulate the problem of optimal control of a dynamic process with terminal
constraints. Let the controlled process be defined on the interval [0, t1] by the system of differential equations [1]
,
(3)
(2)
high-dimensional processes of the problems to be solved. Thus, nonlinearity does not allow using linear programming
methods [6]. The presence of restrictions on the control and phase variables makes it difficult to apply the methods of
the calculus of variations [7]. The large dimension of problems due to high computational costs complicates the use of
dynamic programming [8]. To apply the maximum principle, an additional check of the found solution for optimality is
required [9]. In addition, solutions to optimal control problems obtained using most numerical methods start from the
starting point of the search for a solution, which, in turn, requires the researcher to know some approximation of this
starting point, at least from the physical considerations of the problem posed. Currently, when solving problems of
modeling and optimizing dynamic processes, genetic algorithms are becoming increasingly popular [10, 11]. They make
it possible to overcome the difficulties that arise in solving optimal control problems. During the operation of genetic
algorithms, many alternative solutions are processed in parallel, while the search is concentrated on the most promising
of them. Genetic algorithms make it possible to solve optimization problems for many parametric multiextremal
functions, including those for non-linear objective functions. Genetic algorithms are a direct optimization method, i.e.,
their operation does not require the calculation of the derivative of the objective function. Important advantages of genetic
algorithms are the independence of the found solution from the initial approximation, as well as the absence of the
requirement for the continuity of the objective function and its derivatives [12, 13]. At the same time, genetic algorithms
can be easily modified with a change in the number of phase variables, which allows them to be used and easily adjusted
when solving optimization problems for various processes.
where gi(t) are continuously differentiable functions with respect to all arguments. Let us introduce
the optimality criterion
Qi(u) = gi(x(t1)) 0, i = l + 1, m,
x(0) = x
(4)
with initial conditions
Qi(u) = gi(x(t1)) = 0, i = 1, l;
Let the restrictions be imposed on the phase variables
(1)
dt
beat
= f(x(t), u(t), t)
It is required on the interval [0, t1] to find the control vector u ÿ (t) from the set ÿ (t) and the corresponding solutions
of the system (1) x(t) of nable controls U such that conditions (3) are
satisfied for u and the criterion optimality (4) takes the minimum value.
0
0
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(5)
2
S(u, zk ) =
Q0(u
where S(u, zk ) is the penalty function, z
|gi(x(t1))|, i = 1, l, max{0,
gi(x(t1))}, i = l + 1, m.
Q(u) = Q0(u) + S(u, zk ) ÿ min,
The problem of finding the maximum is reduced to the problem of finding the minimum by changing the sign to the
opposite in front of the function:
to z
without restrictions, the sequence of solutions of which gives a solution to the original problem.
restrictions are violated:
Consider the auxiliary function [14]
(in),