RU2189068C2 - Method for adaptive automatic adjustment of multi-parameter automatic control systems to optimum conditions - Google Patents

Abstract

FIELD: automatic optimization of multiparameter controlled members possessing a single-extreme quality function based on some optimality criterion. SUBSTANCE: the method is realized as follows. In a space of optimized parameters a step is made from the original state in a random distribution in accordance with the uniform distribution law. If the value of the quality function in the new state exceeds or equals the value of the quality function in the original state (i.e. the random chosen sample turned out to be an unsuccessful one), the system returns to the original state, after which a random step counted off from the old one is formed again. Adaptation of distribution of the direction of random steps consists in variation of their mathematical expectation on the basis of the sign of increment of the quality function, and adaptation of the value of random steps - on the basis of the relationship of successful and unsuccessful samples in the process of adaptation. EFFECT: enhanced speed of response of self-adaptation. 1 dwg

Description

 The invention relates to automatic optimization of multi-parameter control objects having a one-extreme quality function based on some optimality criterion.

 Known methods for random search of the extremum of the quality function of one-extremal multi-parameter objects, which consist in the formation of random input step effects on the control object [1].

 The disadvantage of such methods is the lack of adaptation of the magnitude of random input step effects in the search process, which leads to a significant decrease in their efficiency and speed.

 Closest to the invention in technical essence is a method based on the formation of random evenly distributed input step-by-step actions on the control object and adaptation of their distribution and magnitude [2].

 The disadvantage of this method is the low speed of the search and the lack of universality of application, since the algorithm for adapting the magnitude of the input step effects in this method was obtained for a particular case of the optimization object.

 The proposed method consists in creating an adaptive random search mode in the space of optimized parameters, based on the formation of random input step-by-step actions on the control object, distributed according to a uniform law, the mathematical expectation of which is automatically adapted depending on the signal coming from the object’s output via the feedback channel communication, and the adaptation of the magnitude of random input step effects is based on the ratio of successful and unsuccessful samples in the process of oyki.

 The drawing shows a block diagram of an algorithm that implements the proposed method for automatically tuning multi-parameter automatic control systems to optimal conditions (a variant of the algorithm with recounting, for definiteness, the case of minimizing the quality function is presented).

The method is implemented using an algorithm, a block diagram of which includes:
1 - block for forming a step in a random direction in the space of optimized parameters in accordance with the uniform distribution law, 2 - block for storing the formed step, 3 - block for determining the value of the quality function at a new point in the space of optimized parameters, 4 - block for determining the sign of the increment of the quality function, 7 - a unit for storing the value of the quality function with a successful step, 5 and 8 - blocks for adapting the distribution of direction and magnitude of random steps in the space of optimized parameters, 6 - the block is formed I step in the opposite direction in an unsuccessful step.

 The proposed method is implemented as follows.

In the space of optimized parameters from the initial state X _i , a step is taken in a random direction in accordance with the uniform distribution law. If the value of the quality function in the new state Q (X _{i + 1} ) is greater than or equal to the value of the quality function at the starting point Q (X _i ), that is, a random test turned out to be unsuccessful (minimization task), then the system returns to the initial state X _i , after which again forms a random step, counted from the old state.

где Q_i ⁰=min Q(X_j), j=1,..., i - наименьшее значение функции качества за i предыдущих шагов поиска;
F(Ξ,W) - единичный вектор, определяющий направление случайного шага:

где Ξ- единичный случайный вектор, равномерно распределенный по всем направлениям пространства оптимизируемых параметров с нулевым математическим ожиданием;
W - вектор памяти (математическое ожидание направления случайных шагов), реализующий адаптацию распределения направления шагов,
a_i - величина рабочего шага на i-м шаге поиска.The recurrence formula for the displacement in space of optimized parameters by this algorithm has the following form:

where Q _i ⁰ = min Q (X _j ), j = 1, ..., i is the smallest value of the quality function for i of the previous search steps;
F (Ξ, W) is the unit vector that determines the direction of the random step:

where Ξ is a unit random vector uniformly distributed in all directions of the space of optimized parameters with zero mathematical expectation;
W is a memory vector (the mathematical expectation of the direction of random steps) that implements the adaptation of the distribution of the direction of steps,
a _i is the value of the working step at the i-th search step.

Adaptation of the distribution of the direction of random steps consists in changing their mathematical expectation based on the sign of the increment of the quality function (blocks 5 and 8 in the drawing). The algorithm for the continuous adaptation of the mathematical expectation of random steps can be represented as the following vector recurrence relation:
W _{i + 1} = kW _i -δΔQ _i ΔX _i ,
where W is the mathematical expectation of uniformly distributed random steps;
k is the memorization coefficient (0≤k≤1);
δ is the parameter of the learning speed (0≤δ≤1).

где l_i и s_i - соответственно число удачных и неудачных проб, совершенных до i -го шага оптимизации.When working on this adaptation algorithm, the vector W tends to be rebuilt in the direction opposite to the gradient of the quality function of the optimized object, that is, the search steps will, on average, be directed towards the fastest decrease in the quality function. The algorithm for adapting the magnitude of random steps can be represented as the following expression:

where l _i and s _i are, respectively, the number of successful and unsuccessful samples completed before the i-th optimization step.

 The meaning of this algorithm is as follows. If in the search process there is a significant number of successful samples, then the step size increases. However, if unsuccessful search steps begin to prevail, then obviously the system has approached the extremum of the quality function, and the magnitude of the input step effects decreases.

 If there is certain a priori data on the form of the quality function of the optimization object, then it is not necessary to use all the information about unsuccessful and successful tests; you can limit yourself to data from the last N optimization steps. However, N should not be too small, otherwise the accuracy of finding the extremum will not be high enough. As experiments on model functions have shown, for most cases the optimal number is N = 100, which is sufficient for high mobility of the algorithm for adapting the step size and for finding an extremum with high accuracy.

 The achieved technical effect of the application of the proposed method allows to reduce search losses and increase the speed of finding the extremum: experimental studies conducted on model functions showed that the gain in speed of the proposed method compared to the prototype was from 32 to 55% depending on the type of model function (see table). In addition to the algorithmic simplicity of the proposed method, it also has universality of application, which allows it to be used regardless of the specific type of quality function.

Центральная модель является моделью сепарабельного объекта управления, у которого отсутствует перекрестное влияние входных параметров.Central model:

The central model is a model of a separable control object in which there is no cross-influence of input parameters.

где [A, X] - скалярное произведение вектора параметрических коэффициентов А=(а₁, а₂, а₃, а₄)^T и вектора входных координат Х = (x₁, x₂, x₃, x₄)^T, а_i=1, i=1...4, b_ij - элементы матрицы:

Квадратичная модель имеет более сложную структуру по сравнению с центральной и моделирует несепарабельный объект с перекрестным влиянием параметров.Quadratic model:

where [A, X] is the scalar product of the vector of parametric coefficients A = (a ₁ , a ₂ , a ₃ , and ₄ ) ^T and the vector of input coordinates X = (x ₁ , x ₂ , x ₃ , x ₄ ) ^T , and _i = 1, i = 1 ... 4, b _ij are the elements of the matrix:

The quadratic model has a more complex structure than the central one and models an inseparable object with the cross influence of parameters.

Powell Function:
Q (X) = (х ₁ + 10х ₂ ) ² +5 (х ₃ -х ₄ ) ⁴ + (х ₂ -2х ₃ ) ⁴ +10 (х ₁ -х ₄ ) ⁴
The Powell function models an optimization object with a pronounced ravine of a quality function.

All model functions have one minimum at x _i = 0, i = 1 ... 4.

 When conducting comparative testing, we used the same set of one hundred search starting points for all models, the coordinates of which were random uniformly distributed numbers ranging from minus 10 to plus 10. The extremum search cycle for each starting point was repeated 1000 times. For all three models, the search parameters were the same: memory coefficient k = 0.7; learning rate parameter δ = 0.1. The search for the extremum was carried out with an accuracy of 0.01.

LITERATURE
1. Rastrigin L.A. Extreme control systems. M .: Nauka, 1974, p. 422-432.

 2. Rastrigin L.A. Adaptation of complex systems. Riga: Zinatie, 1981, pp. 88-91, pp. 106-107 (prototype).

Claims (1)

A method of adaptive automatic tuning of multi-parameter automatic control systems to optimal conditions by creating an adaptive random search mode in the space of optimized parameters, based on the formation of random input step-by-step actions on the control object, the mathematical expectation of which is automatically adapted depending on the signal from the object output via the feedback channel communication in accordance with the ratio
W _{i + 1} = kW _i -δΔQ _i ΔX _i ,
where W is the mathematical expectation of uniformly distributed random steps;
k is the memorization coefficient;
δ is the parameter of the learning speed;
Q - quality function;
X is the state of the object in the space of optimized parameters,
characterized in that the magnitude of the random input step actions during adjustment to optimal conditions is automatically adapted based on the ratio of successful and unsuccessful samples in accordance with the expression

where a is the step size of the random input step action;
l _i is the number of successful random samples completed before the i-th search step;
s _i - the number of unsuccessful random samples committed before the i-th search step.

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