Method for adaptive automatic self-adjustment of multi-parameter automatic control systems for optimal conditions

FIELD: optimization technologies.

SUBSTANCE: method includes making a step in space of optimized parameters from starting state, in random direction in accordance to normal distribution law and in dependence from value of quality function in new state system is either returned to starting state and random step is formed, counted from old state, or new state is taken as starting state and random step is formed, counted from new starting state. Adaptation of distribution of direction of random steps is change of their mathematical expectation on basis of quality function increment sign, and adaptation of intensiveness of self-adjustment is performed on basis of value of current search step, i.e. recording coefficient is calculated as relation of number of unsuccessful attempts to total number of attempts, parameter of adjustment speed is calculated as relation of number of successful attempts to total number of attempts, performed up to current optimization moment.

EFFECT: higher search speed.

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The invention relates to automatic optimization of multivariable control objects with onextreme quality function is based on some optimality criteria.

Known methods of random search extremum of a function of the quality onexternal multivariate objects, which consists in forming a random input stepper impacts on site management [1].

The drawback of such methods is the low speed search control objects. Closest to the invention to the technical essence is a method based on the formation of normally distributed random input stepper impacts on site management and adaptation of their distribution and intensity of learning [2].

The disadvantage of this method is the low performance of the search. The speed of the search is achieved by adapting the intensity of self-learning based on the value of the current step.

The proposed method is that the create mode adaptive random search in the space of optimized parameters based on the random input of the stepper impacts on site management, distributed by the normal law, the expectation which automatically adapt depending on the signal the La, coming from the output of the object on the channel feedback, and adaptation of the intensity of study in the search process carried out on the basis of the value of the current step of the search, i.e. the coefficient memory is calculated as the ratio of the number of failed samples to the total number of samples held up to the current time optimization, parameter learning rates calculated as the ratio of the number of successful samples to the total number of samples held up to the current time optimization.

The drawing shows a block diagram of the algorithm that implements the proposed method for automatic tuning of multivariable control systems at optimal conditions (for definiteness, we present a case of function minimization quality).

The block diagram includes: 0 - block-initial initializations, 1 - unit forming step in a random direction in space optimized parameters in accordance with the normal distribution, 2 - block memory generated step 3 - unit value definition quality function at a new point in the space of optimized parameters, 4 - block determine the sign of the increment function quality, 5 - block conditions for successful step 8 - block memory function value quality at a good step, 6 and 9 blocks adapting the distribution of the directions of search steps and intensive the spine of learning in the configuration process, 7 - unit forming step in the opposite direction when the failed step, 10 - conditions for the end of optimization.

The proposed method is implemented as follows. In the space of optimized parameters from the initial condition X_itake a step in a random direction in accordance with the normal distribution law (blocks 1, 2, 3, 4).

If the value of the quality function in new condition Q(X_i+1) is greater than or equal to the value of the quality function at the origin Q(X_i), i.e. a random sample Ξⁿfailed (minimization problem), then the system returns to the initial condition X_ithen generate a random step counted from the old condition.

If the value of the quality function in new condition Q(X_i+1) less the value of the quality function at the origin Q(X_i), i.e. a random sample Ξⁿwas successful (minimization problem), then the initial state is taken to be the new Q(X_i+1), and then form a random step counted from the new initial state.

Recurrent expression for the search algorithm with normally distributed random samples becomes:

where Q⁰ _i=minQ(X_j), j=1,...,i is the smallest value of the quality function for the i previous steps of the search

a - the size of the working step.

The adaptation algorithm mean-square deviation of the direction of normally distributed random steps can be represented as the following expression:

where σ_I- the current value of the mean square deviation;

σ_minand σ_max- the minimum and maximum values of the mean-square deviation.

Adapting the distribution of the random directions of the steps is to change their expectations on the basis of the sign of the increment function quality (blocks 6 and 9 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_iΔX_i,

where W is the mathematical expectation of uniformly distributed random steps

k - coefficient memory (0≤k≤1),

δ - the speed parameter learning (0≤δ≤1).

The coefficients k and δ determines the rate of learning during the search.

When working on this algorithm adaptation vector W strives to reinvent itself in the direction opposite to the gradient of the function of the quality of optimized object, t is there are steps of the search will be on average directed toward the rapid decrease of the function.

Adapting the intensity of the self-study process optimization based on the value of current step (blocks 6 and 9 in Fig.) by using the following expressions for the coefficients k and δ:

i.e.

where k_beatsand k_bad- the number of successful and unsuccessful trials, conducted to the current time optimization.

The point of this algorithm is to adapt the intensity of the self-study is the following. If in the process of finding a growing number of successful steps k_beatsand the number of failed steps k_baddecreases, i.e. the quality function ΔQ(X) moves to the extreme, the value of the working step α increases the speed of learning δ also increases, i.e. there is intensive training based on the new experience, while the coefficient memorizing k decreases, i.e. there is no need to remember all previous experience.

If in the process of finding the number of successful steps k_beatsdecreases, and the number of failed steps k_badincreases, i.e. the quality function ΔQ(X) is approaching the extreme, the value of the working step α decreases the learning rate δ reduced, i.e. there is no need for intensive study, and the coefficient memory k increases, i.e. memorized all previous experience.</>

The polynomial model is:

Q(X)=x₁+x₁ ²+x₂+x₂ ²+x₃+x₃ ²+x₄+x₄ ²+1

Focus model:

Quadratic model:

where [A, X] is the scalar product of the vector of parametric coefficients A=(a₁and₂and₃and₄)^Tand the vector of the input coordinates X=(x₁x₂x₃x₄)^t, a_i=1, i=1...4, b_ijthe matrix elements:

The function of the Powell:

Q(X)=(x₁+10x₂)²+5(x₃-x₄)⁴+(x₂-2x₃)⁴+10(x₁-x₄)⁴.

The function of Powell simulates the object optimization pronounced brainstew quality function.

This method has the versatility that allows you to use it regardless of the form of the quality function. Moreover, if there is some a priori information about the structure of the quality function, it is not necessary to use all the information about failed and successful samples, and can be limited to data for the last N steps of the optimization process is AI. However, N should not be too small, otherwise the accuracy of finding the extremum is not high enough. Experiments on model functions showed that for most cases the optimum is N=100. It is enough for high mobility algorithm adaptation step and finding the extremum with high precision.

For comparative testing were identical for all models set of 100 initial search points, the coordinates of which was a uniformly distributed random number in the range from minus 10 to plus 10. Cycle extremum for each starting point was repeated 100 times.

From the tables it is seen that the maximum gain in performance occurred when optimizing the model object that has a quality function with a distinct brainstew (Function of Powell). Because the real objects of automatic control very often have a ravine quality function, the application of the proposed method is suitable for efficient optimization.

Achievable technical effect of the application of the proposed method allows to reduce losses on the search and improve the performance of finding the extremum of the function of objects quality optimization: experimental studies conducted on model functions, showed that the gain in fast is deistvii of the proposed method in comparison with the prototype ranged from 5 to 9 percent, depending on the type of model function and accuracy of finding the extremum (see table).

References

1. Rastrigin L.A. System extreme control. M.: Nauka, 1974, s-446.

2. Merinov A.V., Kravchenko, A. Adaptive random search Devices and systems. Management, monitoring, diagnostics. - 2001. No. 6. - P.39-42 (prototype).

Method of adaptive automatic self-tuning multivariable control systems at optimal conditions by creating adaptive random search in the space of optimized parameters based on the random input of the stepper effects on the control object, the mathematical expectation which automatically adapt depending on the signal coming from the output of the object on the feedback channel in accordance with the ratio of W_i+1=kW_i-δΔQ_iΔX_i, where W is the mathematical expectation of uniformly distributed random steps, k is the coefficient memory δ - the speed parameter self-learning, Q - quality function, X is the state of the object in space optimized parameters, the random input of the stepper influences adapt in accordance with a ratio of

where k_beatsand k_bad- the number of successful and unsuccessful trials, conducted to the current time optimization, characterized in that the adaptation intensity of halogen with exceptiona the self-study in the search process carried out on the basis of the value of the current step of the search, i.e. the coefficient memory is calculated as the ratio of the number of failed samples to the total number of samples held up to the current time optimization, parameter learning rates calculated as the ratio of the number of successful samples to the total number of samples held up to the current time optimization, in accordance with expressions