Udc 17. 977+644. Bashkir State University, Republic of Bashkortostan
Algorithm for solving the optimal control problem with terminal constraints
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Algorithm for solving the optimal control problem with terminal constraints. Let us formulate a combined algorithm
for solving the optimal control problem with terminal constraints based on the penalty method and genetic algorithms. The main idea of the penalty method is to replace the problem with constraints by the problem , where H i At each k-th iteration, age k is determined using genetic algorithms . Nay Hi(u) = . is the penalty parameter determined at the kth iteration (the larger z, the greater the penalty for non-compliance with the constraints). The penalty function S(u, zk ) is defined to be equal to zero if the constraints are satisfied and greater than zero if The work of the genetic algorithm is to sequentially change the set of vectors, called the population. Each vector is an analogue of an individual (chromosome) in a population and consists of a set of numerical values, which are called genes or traits [15, 16]. According to a certain rule, two individuals are selected from the population (by the selection operator), which are crossed with each other, i.e., a new vector-individual is generated, which is subsequently subjected to the action of the mutation operator, as a result of which one or several genes (the coordinate values of the vector ). The fittest individuals from the current population pass to the next generation. The fitness of an individual is determined by calculating the fitness function, which is the criterion of optimality. This procedure for changing the generations of the population continues until the criterion for terminating the search is reached [17, 18]. The main idea of genetic algorithms is to imitate the evolutionary processes of living organisms: the fittest individuals pass to the next generation, while retaining the best properties of their parents and acquiring new useful properties under the influence of changing environmental conditions. ) = max g0(x(t1)) = ÿ min [ÿg0(x(t1))]. the torus u ÿ (t) is used as the initial torus at the next iteration for the withdrawn vector u of the personal value of the penalty parameter z ÿ (t), which provides the minimum of the optimality criterion (5) for a given z 2 uÿU m k k uÿU ÿ i=1 134 AUTOMETRY. 2020. V. 56, No. 6 k Machine Translated by Google Rp . ÿ . ci = ÿai + (1 ÿ ÿ)bi , where ÿ is a random number from the range (0, 1), i = 1, r. l and r are the numbers of individuals with the highest selection probabilities. ÿ i = 1, Rp. u(t) = . (With The selection operator "tournament selection" performs two tournaments. The first tournament is a random selection of two individuals from the population. At the second tournament, an individual is selected at random from two individuals selected at the first tournament. Q(u j Step 3. Crossing - generation of offspring individuals c = (c1, c2, . . . , cr) and d = . ÿ ÿÿÿÿ . as a population of individuals, a set of vectors uj (t) = (uj1(t), uj2(t), . . . , ujr(t)), where j = 1, Rp, Rp is the size of the population. The coordinates of the vector uj (t) are called genes. We will determine the fitness of individuals by calculating the optimality criterion (5). Then the algorithm for solving the optimal control problem with terminal constraints will consist of the following steps. Step 1. Set the initial parameters of the algorithm: N is the number of partition points of the time interval [0, t1], Rp is the population size, Max P is the maximum number of populations, the number of the current iteration of the genetic algorithm q = 0, initial value = 0, 01; 0.1; 1) [13], a constant G ÿ [4, 10] to increase pa Step 2. Selection - selection of a pair of individuals from the current population for the next ) = Q(u d = (b1, . . . , bs, as+1, . . . , ar) , ÿÿÿÿ ur(t) . (t), u0 di = ÿbi + (1 ÿ ÿ)ai , u1(t) u2(t) , = (d1, d2, . . . , dr) from the parent pair. Arithmetic crossover generates children according to the rule . . . ), (t) = (u j1 on the set of admissible control values U, determine fitness for each individual. To do this, it is necessary to find a numerical solution to the system of differential equations (1) with initial conditions (2). ÿ ÿÿÿÿ . crossing. The panmixia selection operator performs a random equiprobable selection. The selection operator "roulette" selects two individuals with the best value of the fitness function. For the minimum problem, this best value is the minimum value of the optimality criterion (5). For each individual, the probability of selection is calculated by the formula ) q and b = u r, where ÿÿÿÿ = As individuals of living organisms, we will consider the vector of control functions . . p(u Two individuals are selected with the highest probabilities a = u The simplest crossover creates children according to the rule c = (a1, . . . , as, bs+1, . . . , br) , where s is a random number from the range [1, r ÿ 1]. 0 0 of the penalty parameter z of the penalty parameter, the number of the current iteration of the penalty method algorithm k = 0, the parameter ÿ > 0 to terminate the algorithm. Randomly set the initial population u j q u0 jr(t)) (t), . . . , j2 u21 . . . u2N ur1 . . . urN u11 . . . u1N q j=1 0 q i i 0 q l E. V. Antipina, S. I. Mustafina, A. F. Antipin, S. A. Mustafina 135 Machine Translated by Google k+1 X1 ÿ X2, X2 ÿ X4, X1 ÿ X3, X1 ÿ X4, X2 ÿ X3, X3 ÿ X5, 0 u y kj (T) = k0je ÿEj/(RT) 136 AUTOMETRY. 2020. V. 56, No. 6 Random mutation transforms a randomly selected gene of each of the descendants c, d anhydride. dx4 = k2x2 + k4x1; , population to a local extremum. dx2 = k1x1 ÿ k2x2 ÿ k5x2; dt dx3 = k3x1 + k5x2 ÿ k6x3; dt Download 338.38 Kb. Do'stlaringiz bilan baham: |
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