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The feature vector (X) is transmitted to a unit of polynomial converters (A-elements), which creates an mdimensional vector of secondary (polynomial) features Z = (a1(X ),a2 (X ),...aN (X )). These secondary features define a polynomial feature space a (X ), j = 1,2,...,n, a so-called rectifying space. Explicit function j form a (X ) is chosen in accordance with the given task and the training set, i.e. during the process of neural j network construction and self-organization [3-5].

The output tier contains solution threshold neuron-like elements:

N Yi = sign( w a (x)), i [1...,M].

j j j =The recurrent learning algorithm used in threshold polynomial neural network is a supervised learning algorithm, which offers a number of advantages over the frequently used back-propagation of error (BPE) [1], including the following:

It is not necessary to determine network structure in advance, since the algorithm adjusts itself during learning process.

This is a single-pass algorithm, i.e. the third layer (decision layer) neurons' weights are adjusted in the first pass through the training set.

The algorithm guarantees error-free classification of elements in the training set.

The algorithm constructs a neural network with a high degree of extrapolation to data beyond the training set.

The synthesized neural network allows creation of a neural knowledge base based on the source database.

The structural scheme of the distributed KDD system is shown on the Figure 2:

Web Server Apache Server with java applications PHP module, MySQL module Apache Tomcat Internet PHP scripts Request processing servlet JRE JDK 1.5 using English Library for work with neural Internet networks Data base server MYSQL D Internet Mail server Internet Mail client Web browser Figure 2. Structural scheme of the distributed neural KDD system.

Neural and Growing Networks At the first stage, an expert creates training and test sets in the form of database table. For this task, an expert query is constructed. The second stage concerns work with the knowledge base, at which a user query is constructed. As a reply, the system fills in the "Result" value corresponding to the certain feature cortege, which serves as an output generated by the system in response to the user query.

Conclusion The distributed KDD system proposed in the article allows remote usage of experience of experts in a certain field, and its implementation as a neural knowledge base. The system is a universal KDD tool, since it makes it possible to build decision-making models in any subject field. Its improvement to a web-service will allow thirdparty software developers to create specialized applications oriented on neural knowledge base usage. The system can be applied in research as well: as a tool for in-depth study of different effects in ecology, economics and other fields. It serves as means to integrate problem solving experience of geographically distributed users.

Acknowledgements The work has been done at partial support of RFBR-grant 05-01-08044-ofi and Program "GRID" of RAS Presidium.

Bibliography [1]. Lugger, G. F. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, 2003, p. 864.

[2]. Bagresan, A. A., Kuprianov, M. S., Stepanenko, V. V., Holod, I. I. Methods and Models of Data Analyzing: OLAP and Data Mining, 2004, p. 336.

[3]. Simon Haykin, Neural Networks a Comprehensive Foundation, 2006, p. 1104.

[4]. Timofeev A.V. Methods of Creation of Diophantine Neural Networks with Minimal Complexity RAS Report, 1995, Vol. 345 No.1, pp. 32-35.

[5]. Timofeev A.V. Parallelism and Self-Organization in Polynomial Neural Networks for Image Recognition Proceedings of the 7th International Conference on Pattern Recognition and Image Analysis: New Information Technologies (18October, 2004, St. Petersburg), pp. 97-100, 2004.

Authors Information Timofeev Adil Vasilievich Dr. Sc.,Professor, Honoured Scientist of Russian Federation, Saint-Petersburg Institute for Informatics and Automation of Russian Academy of Sciences, 199178, Russia, Saint-Petersburg, 14-th Line, 39, phone: +7-812-328-0421; fax: +7-812-328-4450, e-mail: tav@iias.spb.su Azaletsky Pavel Post Graduate Student, Saint-Petersburg State University of Aerospace Instrumentation, 190000, Russia, Saint-Petersburg, Bolshaya Morskaya, 67, phone: +7-812-328-0421; fax: +7-812-328-4450, e-mail: eaglenk2@mail.ru NEURAL NETWORK BASED OPTIMAL CONTROL WITH CONSTRAINTS Daniela Toshkova, Georgi Toshkov, Todorka Kovacheva Abstract: In the present paper the problems of the optimal control of systems when constraints are imposed on the control is considered. The optimality conditions are given in the form of Pontryagins maximum principle. The obtained piecewise linear function is approximated by using feedforward neural network. A numericak example is given.

Keywords: optimal control, constraints, neural networks XII-th International Conference "Knowledge - Dialogue - Solution" Introduction The optimal control problem with constraints is usually solved by applying Pontryagins maximum principle. As is known the optimal control solution can be obtained computationally. Even in the cases when it is possible an analytical expression for optimal control function to be found, the form of this function is quite complex. Because of that reason the possibilities of using neural networks for solving the optimal control problem are studied in the present paper.

The ability of neural networks to approximate nonlinear function is central to their use in control. Therefore it can be effectively utilized to represent the regulator nonlinearity. Other advantages are their robustness, parallel architecture.

Lately, different approaches are proposed in the literature treating the problem of constrained optimal control for using neural networks. In [Ahmed 1998] a multilayered feedforward neural network is employed as a controller.

The training of the neural network is realized on the basis of the so called concept of Block Partial Derivatives. In [Lewis 2002] a closed form solution of the optimal control problem with constraints is obtained solving the associate Hamilton-Jacobi-Bellman (HJB) equation. The solution of the value function of HJB equation is approximated by using neural networks.

In the present paper the problem of finding the optimal control with constraints is considered. A numerical example is given.

Problem Statement The control system, described by following differential equations is considered:

dxi n = xi + biu (i = 1, 2, , n) a (1) ij dt j=where x are phase coordinates of the system, function u describes the control action and a are constant j ij coefficients. The admissible control u belonging to the set U of piecewise linear functions is constrained by the condition u(t) (2) Following problem for finding the optimal control is formulated. To find such a control function u(x,, x ) for the 1 n system (1) among all the admissible controls that the corresponding trajectory (x (t),, x (t)) of the system (1) 1 starting from any initial state (x (0),, x (0)) to tend to zero at t and the performance index 1 n J = xi + ru2 dt q 2 (3) i i=to be converging and to take its smallest possible value. The coefficient qi and r are positive weight constants.

Optimality Conditions The notation is introduced [Pontryagin 1983]:

n f0 (x1,..., xn, u) = q x2 + ru (4) j j j=n fi(x1,..., xn, u)= aijx2 + biu (i = 1,,n) (5) j j=One more variable is added to the state variables (x,, x ) of the system (1) [Chjan 1961]. It is a solution of 0 1 n the following equation dx= f0(x1,..., xn, u) (6) dt Neural and Growing Networks and initial condition x (0) = 0. Then the quantity J according to (9) becomes equal to the boundary of x(t) when t. The system of differential equation, which are adjoint to the system (7) is composed with new variables ={,,..., }:

0 1 n d0 n f = - = (7) dt x=n di n f = - = -2qii aijj (i = 1,,n) (8) dt x=0 j=After that the Hamilton function is composed:

n n n n dx n H(,,u)= dt = f(x1,...,xn,u)=0qx2 +ru2+a xj +biu (9) i i i ij j==0 =0 i=i= In the right-hand side of Eq. (9) the quantity u is contained in the expression n H1 = r0(t)u2(t) + u(t) i(t) b (10) i i=Because of that the condition for maximum of H coincide with the condition n max H1 = maxr0 (t)u2(t) + u(t) i (t) = bi |u|1 |u| i=2 (11) n n 1 = maxr0(t)u(t) + i (t) - i (t) bi bi |u|1 2r0 i=1 4r i= Having in mind condition (7) the quantity is a constant. As its value can be any negative number it is set to = -1.

After placing this value in Eq. (11) the maximum of the expression in the square brackets will be reached when the first negative addend becomes zero if it is possible or takes its minimal absolute value. The expression n (12) bii(t) u(t) 2r i=will take its minimal absolute value if on condition |u| 1 a value of the following kind is chosen for u n n 1 b b 2r i at 2r i i i i=1 i= n u(t) = 1 at bii 1 (13) 2r i= n -1 at bii - 2r i= The values of (t) can be determined if the adjoint equations (7), (8) are solved. This leads to the requirement the initial values of (0) to be found beforehand.

First u(t) is assumed not to reach its boundary values. Then after placing the upper expression from (13) instead of u(t) in Eqs. (1), (7) (8) one obtains dxi n bi n = + j (i = 1,...,n) aijx j 2r b j dt j=1 j=(14) n di = 2qixi - j a ji dt j=XII-th International Conference "Knowledge - Dialogue - Solution" This system of equations has to be solved with the initial conditions x (0),, x (0) as well as with the final 1 n (boundary) conditions lim x1(t) = lim x2(t) =... = lim xn(t) = (15) t t t It is necessary the appropriate initial conditions (0), , (0) to be selected in such a way that the initial and 1 n the final conditions for x (t),, x (t) to be satisfied.

1 n The relationship between x(0) and (0) has the following form [4]:

i i n i (0) = xi (0) (i =1, ,n) (16) ij j=These relationships have to be kept in any time, for which one can always assume to be the initial one. Therefore the optimal control u within the boundaries is determined and it has the following form:

n u = xi k (17) i 2r i=n where ki = ji b j j=The expression (17) holds only in the cases when the absolute value of the sum (k1x1 +... + knxn ) is not 2r greater than one. When (k1x1 +...+knxn) >1 the optimal control passes on the boundary i.e. |u| = 1, if the right 2r hand boundary conditions are satisfied i.e. the solution of the system (1), which became nonlinear in connection to the nonlinear relationship between u and x,, x, tends to zero. In other words the solution of the system has 1 n to be asymptotically stable. Thus the optimal control is defined by the expression n n 1 k k 2r xi at 2r xi i i i=1 i= n u(t) = 1 at (18) kixi 2r i= n -1 at kixi -2r i=Structure and Training of the Neural Network For the control function realization a feed forward neural network with one hidden layer is used. Thus the necessity of solving a large number of equations for determining the coefficients k drops off.

i The neural network consists of three layers an input, output and hidden one. The input and hidden layers have five neurons and the output layer one. The activation function of the output neuron is piecewise linear. The neural network output is +1 () y = () () (19) -1 () where = wTz. The neural network input is denoted z and w is the neural network weight. The neural network output represents the control u, x the state vector and weights are the coefficient k.

The neural network is trained according to the back-propagation algorithm. Let the training sample {z(n), d(n)}N n=be given where z(n) are the system states and d(n) is the corresponding control, which are known preliminarily.

The neural network is trained according to the back-propagation algorithm [Haykin 1999].

Neural and Growing Networks Simulation Results In order to verify the suggested approach for solving the optimal control problem following system is considered:

dx= xdt dx= -x1 - 2x2 + u dt and the control is constrained by |u| The performance index to be minimized is of the form:

(t)+x2(t)+ u2 (t)dt [x1 The problem is solved by using Pontryagins principle and neural networks. The results, which are obtained by both approaches, are compared. In Fig. 1 the optimal control, obtained by using neural networks is shown. Fig. depicts the corresponding states trajectory. In Fig. 3 and Fig. 4 the optimal control, obtained by applying the maximum principle and the corresponding trajectory are given respectively. By 1 and 2 are denoted x and x 1 respectively.

1.5 1.1 0.5 0.0 -0.5 -0.-1 --1.5 -1.0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 time,s time,s Fig. 1. Optimal control, obtained by using the Fig. 2. Optimal control, obtained by applying the suggested neural network based approach maximum principle 15 10 5 0 -5 --10 --15 -0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 time,s time,s Fig.3 Optimal trajectory Fig.4 Optimal trajectory (neural network based approach) (Pontryagins maximum principle) control control state state XII-th International Conference "Knowledge - Dialogue - Solution" Conclusion In the present paper an approach for optimal constrained control based on using of neural networks is suggested.

On the basis of the simulation experiments one can say that the proposed approach for optimal control is accurate enough for the engineering practice. The suggested approach can be applied for optimal control in real time, where the control is constrained.

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