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

FIELD: automatic optimization of multi-parameter controlled objects characterized with one-extreme quality function based upon any optimum criterion.

SUBSTANCE: in space of optimized parameters stroke is made from initial state in random direction according to normal distribution law in several separate parallel circuits; selecting the best line on base of value of quality function. If value of quality function in new state is equal to or exceeds quality function value in initial point, system returns to initial state. Then random stroke is again formed from old state in each parallel circuit. If quality function value in new state is less than value of said function in initial point, then initial state is taken as new one and again random stroke is formed from new initial state for each parallel circuit.

EFFECT: improved quick response of searching.

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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. A substantial increase in the speed of the search is achieved by the simultaneous operation of multiple independent parallel channels and adaptation 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 simultaneously in multiple parallel channels based on the random input of the stepper impacts n the control object, distributed by the normal law, the expectation which automatically adapt depending on the signal coming from the output of the object on the feedback channel. Make a selection of the best channel, which later will be the source channel to create adaptive random search. Adaptation of the intensity of study in the search process is performed based on the magnitude 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 about what transtv optimized parameters, 4 is a block for determining the sign of the increment function quality, 5 - unit select the best channel, 6 - block storage of the parameters of the best channel, 7 - block conditions for successful step 10 - block memory function value quality at a good step, 8 and 11 blocks adapting the distribution of the directions of search steps and intensity of self in the process of setting up, 9 - unit forming step in the opposite direction when the failed step, 12 - 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 in several independent parallel channels (blocks 1, 2, 3, 4).

Make a selection of the best channel, which later will be the original channel. If the value of the quality function (minimization) in all parallel channels is greater than or equal to zero, then choose the channel in which the value of the quality function is minimal. If there is a negative function value quality, then choose the channel, quality function which has a minimum negative value (block 5).

Q_{the HEA}(X)=Q_min(X), Q_n(X), n=1... k,

where k is the number of independent parallel channels,

Q_{the HEA}(X) is the value function is of the best quality of parallel channels

Q_min(X) is the minimum value of the function quality of all channels.

If the value of the quality function in new condition Q(X_i+1)=Q_{the HEA}(X) 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 state, in each parallel channel.

If the value of the quality function in new condition Q(X_i+1)=Q_{the HEA}(X) less than 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 take new Q(X_i+1), and then form a random step counted from the new initial state in each parallel channel.

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

where Q

Ξⁿ- a single random vector distributed according to a normal law,

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 8 and 11 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-δ Δ Q_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, i.e. the steps of the search will be on average directed toward the fastest smart the decision function.

Adapting the intensity of the self-study process optimization based on the value of current step (blocks 8 and 11 in the drawing) 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 Δ Q(X) moves to the extreme, the value of the working step and 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 Δ Q(X) is approaching the extreme, the value of the working step and decreases the learning rate δ reduced, i.e. there is no need to intensively study and CoE the rate of memorization k increases, ie remembers all previous experience.

The polynomial model is:

Focus model:

Quadratic model:

where [A, X] is the scalar product of the vector of parametric coefficients A=(a₁, a₂, a₃, a₄)^Tand the vector of the input coordinates X=(x₁x₂x₃x₄)^Tand_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 optimization. However, N should not be too small, otherwise the accuracy of finding the extremum is not high enough. The former is eriment on model functions showed 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 to significantly improve the performance of finding the extremum of the function of objects quality optimization: experimental studies conducted on model functions, showed that the performance gain of the proposed method in comparison with the prototype ranged from 25 to 40% depending on the type of model function and accuracy of finding the value of 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+k=kW_i-δ Δ Q_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 search for the children in several parallel channels, then, in accordance with condition Q_{the HEA}(X)=Q_min(X), Q_n(X), n=1... k, (for minimization), where k is the number of independent parallel channels; Q_{the HEA}(X) is the function value of the best quality of parallel channels; Q_min(X) is the minimum value of the function quality of all channels, select the best channel, which will be a source channel, and the adaptation of the intensity of study in the search process carried out on the basis of the value of the current step of a search in accordance with the expression

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