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Genetic synthesis algorithms of neural networks for control systems

In the solution of optimising problems of combinatory type use specific algorithms among which the basic methods deposit some:

a) Cutting off

b) Combinatory

c) Confidants.

Genetic algorithms (HECTARES) refer to - to the third type.

But HECTARES cannot be applied in the pure state, use of specificity of a problem, as well as various heuristics of local martempering, as greedy algorithms is inherent in them. Problems which are observed in research-it problems with restrictions. Successfully to solve the given problems by means of G it is necessary to use methods of leaving from restrictions. The given methods are divided into three class-rooms:

a) straight lines,

b) indirect,

c) the representing.

The circuit design of HECTARES is presented in drawing 1.9.

Drawing 1.9 - the Circuit design of genetic algorithm

Have well proved in the solution of a various sort of problems of optimisation adaptive properties of HECTARES. HECTARES have a property of implicit parallelism that allows it dynamically, both to find, and to investigate areas which contain either local, or global optimum. Given properties G allow at its work in dynamically changing medium to optimise funktsional Q (x).

The main objective of a problem of discrete optimisation is a finding from final, but enough the big number, the best solution. A problem of discrete optimisation (1.1) difficult, both for classical algorithms, and for HECTARES that is connected with features of problems of this class-room.

1. Problems have a reboric character, but with growth of parametres of a problem search becomes basic impossible.

2. These problems refer to to irregular and they can have some optimum, including local in which value of function does not coincide with the global.

3. Besides, was possiblly, that areas of feasible solutions are inconsistent and are not convex. A measure of affinity of solutions to specify inconveniently, as at "close" values x1χ2? Gмогут to be available as much as far values Q (x1) and Q (x2). To define affinity of the solution in these problems various technicians who in the core depend on an aspect of problems use.

To overcome these difficulties, various modifications G which successfully enough cope with a problem solving of such class-room are offered. One of these modifications - algorithm of genetic programming (GP) [28, 41].

In the solution of not trivial problems use of a method of a machine-aided programming does not lead to success if in it is not present there is nobody hierarchical

The mechanism considering various regularity, symmetries, and also uniformity or templates which are present at space of problems. Such mechanism in usual computer programs are subroutines in the form of procedures and functions. This idea in the form of automatically defined functions was introduced in GP by John Koza. It has shown, that at use of automatically defined functions efficiency of the solution of a problem method GP essentially raises. Therefore it will be reasonable to assume, that the functions, which specific in a solved problem, will be known even before start of algorithm GP and will be switched on in functional assemblage.

The given steps will raise efficiency of searches of solutions. The given functions can appear those functions which meet in population more often. For definition of the given functions we use concept of a template.

Template is функцияf (x1..., xn, p1..., pm), in which x1..., xn - it is independent arguments, and p1..., pmявляются in parametres instead of which substitute arbitrary functions g1 (x1..., xn)..., gm (x1..., xn). Such, as for example: f (x, p) = x + p*sin (p), f (x, x*x) = x + x*x*sin (x*x).

Function can fulfil to some template if in it there will be such parametres when at their substitution in a template identical function can be gained. A template with such parametres name transfer in function. For example, function g (x) = x*x*sin (x*x) fulfils a template f (p) = p*sin (p), and fx*x) - transfer fв g.

The template f ι dominates a template f2 provided that f1и f2различны, and f2 fulfils fl. For example: f1 (p1, p2) = p1*sin (p2), f2 (p) = p*sin (p) where f dominates to value f2: f2 (p) = f1 (p, p).

Intersection of functions or templates g1..., gnназывают assemblage of templates, which as F = {f1., fm}:

1. A parametre giудовлетворяется fдля all i, j;

2. Is not present such iили j, that fдоминируетfj;

3. Would be not present шаблонаfкоторый it was fulfilled giи it would be dominated by any template from F.

Let's observe algorithm of search of templates. For algorithm the source information is an assemblage of expressions, or trees. On the basis of the given assemblage we make the list unique podderevev which is arranged in sequence up the dip depths. For example, assemblage podvyrazheny: x2+x3, x1*x2, x1 * (x2+x3), sin (x2+x3), exp (x1*x2) or assemblage of expressions: x1 * (x2+x3), sin (x2+x3), exp (x1*x2) ∙

The algorithm purpose is search of a template which fulfils these podderevja. And search according to such rules, as is carried out:

1. To each template should fulfil though two podvyrazhenija.

2. If at f1доминируется f2и is not present podvyrazhenija with which it is fulfilled f1, but it is not fulfilled f2, тоf1нужно to kick.

In the course of algorithm performance there is a search to depth of all combinations podderevev. For each of combinations there is an intersection podderevev by which required templates are formed. We will more low instance search for 4­х podderevev:

If any combination forms empty intersection other combinations "containing" this intersection are not observed.

The most important stage of algorithm of search of templates-it process of a finding of intersections of functions, or podderevev. And, operations in root knot that intersection was not empty, should be equal since observe podderevja in ascending order depths. Therefore by the moment when it is necessary to find intersection any podderevev, already should be those templates which will simultaneously fulfil any steams from daughter podderevev (the such are known touches podderevev depth from 2 and further).

Stages of process of a finding of intersection of functions:

1. Process definition komplementarnyh operands.

2. Process formulation of all possible combinations komplementarnyh operands.

3. A stage formulation of a template and appointment of parametres for all combinations.

4. Process on removal of doubling templates.

5. A stage on removal of templates, before the template found except for that which coincide with that template which participates in intersection.

6. A stage of removal of dominating templates.

7. Process of formulation of transfers as the gained templates participating in intersection.

At 1st stage make comparison of operands with the account of their commutativity, and also associativity of operations and definitions so-called komplementarnyh operands - operands which have equal root knots. As generally operands-functions represent by means of different templates it is required to find intersection of the given operands.

We admit, that operands and and bподдеревьев are functions, and they fulfil to assemblage of templates F1, and an operand b - F2. In this case intersection and and bбудет subset F1∩F2 from which have expelled dominating templates.

At a situation when the operand and a template is a transfer there is nobody a template f, the operand bподдерева is a function, and it fulfils to assemblage of templates F, F1 - subset F, therefore fудовлетворяет to all volume of elements F1. Then intersection and and bбудет assemblage F1 from which dominating templates are expelled.

Reception of several combinations komplementarnyh operands is is possible at a finding of intersection of operands. Initial podderevja for each of combinations it is possible to present in so:

f1 = t1 (a11, a12...) * t2 (b11, b12...) *... * C1 * C2 *... * d11 * d12*...

f2 = t1 (a21, a22...) * t2 (b21, b22...) *... * C1 * C2 *... * d21 * d22 *...,

Where value t - is templates of intersection of operands, value C are variables, value d is nekomplementarnye operands.

Template basis make komplementarnye operands (tи C), and other operands replace with parametres:

f3 = t1 (p11, P12...) * t2 (p21, P22, ∙∙∙) * ∙∙∙ * C1 * C2 *... * p31 * P32 * ∙∙∙

Steps of process of replacement of operands initial podderevev in parametres: Definition of unique operands. For example,

1. f = t1 (x1, x1) *t2 (x1, x1+x2) *x1 * (x2+x3) * (x2+x3), f = t1 (p1, p1) *t2 (p1, P2) *P1*P3*P3.

2. Comparison of matching operands to a template t, for example,

R = {(P1, P1), (P1, P2), (P1, P2), (P2, p3)}, T = {(P1, P1), (p2, P1), (p3, p2)},

P = {1, 2, 2, 3}.

3. Comparison with elements of assemblage Tпар nekomplementarnyh the operands, gained on a step № 2.

4. Appointment of parametres to operands which have not formed pair on steps №2 and №3.

azhdoe from podderevev on the first step observe separately, make comparison of arguments of templates, and also other operands among themselves and replacement in their parametres. Further make analogous operation, but observe already steams which match to arguments of templates. On a step №3 appoint parametres nekomplementarnym to operands, whose steams coincide with the assemblage elements, which gained on a step №2. In this case creation of several alternatives of appointment of parametres is is possible. Comparison of other operands of functions is made on last step with the account of that, how many time they repeat. For example, if initial functions: f1=t1 (x1,

^1) * ^2

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A source: Al-Bareda Ali JAhja Senan. MODELS And OPTIMUM CONTROL SYNTHESIS ALGORITHMS In BIOENGINEERING SYSTEMS of REHABILITATION TYPE ON THE BASIS OF PRODUCTION ENGINEERING of NEURAL NETWORKS. The dissertation on competition of a scientific degree of a Cand.Tech.Sci. Moscow - 2018. 2018

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