Mathematics/5. Mathematic modeling

D.t.s. Syzdykov D.J., c.t.s. Shiryeva O.I., PhD candidate Samigulina Z.I.

KazNTU after K.I. Satpaev, Kazakhstan

Adaptive control systems on basis of the general parameter method

Abstract. The paper addressed a problem of adaptive control system synthesis for linear objects with uncertainties in the time domain on basis of the general parameter method.

1. Introduction

It the real conditions the system can be controlled on a basis of a priory information in the form of a program on the whole period of the system’s functioning or by the procedures adaptive and recursive estimation to eliminate the priory parameter uncertainty with the using the principles of feedback control. In this case, solution can not be reduced to a single act, but continued during the observation to the controlled object [1]. Among the methods of parameter’s estimation of the dynamical system the measurements of input and output signals the special meaning have the general parameter method [2]. This method is answer by the following requirements: simplicity, low computational efficiency, possibility of using in the normal operating mode, the guarantee precision to the decision of the identification’s problem for the time’s finite interval of the observation.

The using of estimates on based of the general parameter method in the theory of adaptive systems it will be solve the following problems: low degree of convergence (it’s property of robust estimates in the theory of adaptive systems); deficiency of the property of the consideration external disturbance (optimal estimation algorithms) [3].

2. The general parameters method for the adaptive control systems

Let the control object is described by a difference equation [4]

(1)

where – -dimensional vector of measurement; –-dimensional vector of unknown parameters; the output of the object .

The model for object (10) defines a similar structure in the form

, (2)

where – estimate of the output values of the object; –- dimensional estimates vector of object unknown parameters.

Assuming that the process parameters are estimated according to the recurrence algorithm:

(3)

Let the input signals satisfy the conditions:

(4)

Let the optimal sequence is defined as Then the mean squared norm of the parametric errors after the steps of estimation is [3]

(5)

where – the average squared norm of the parametric errors; – the vector of parametric errors; – Euclidean norm; – number of estimated parameters.

If the optimal value for this case is defined as then

(6)

There are values from (14), (15):

(7)

which characterize the convergence rate of the algorithm (12) for the optimal sequence .

Assume that in estimate of vector object (10) are configured, not all components of the model (11), and you can take, for example one parameter, which is an integral part of their overall [3,4].

Then the model (11) can be written as

(8)

where I – unite vector; – a general parameter, customizable according to the additive principle; – N -dimensional vector of initial values of the parameter estimates.

Set the parameters according to the algorithm:

, (9)

will be adequate for the algorithm (12).

When the algorithm (9) is realized we need setting one parameter of the model (8) instead the model parameters (2).

The degree of convergence of the algorithm of the General parameter method (9) describe by the expression (from [2]).

(10)

where the sign of the expectation; ; an average value of general parameter in the steady state; – vector of parameter errors in the steady state.

Equation (10) shows that the rate of convergence for the algorithm doesn’t depend on the number of estimated parameters. In expression (19) defines the total variance of the parameter in the steady state as:

Thus, the variance of the general parameter is a measure of the accuracy of parameter identification process. So, if you ask some accuracy parameter estimates, then the condition parameters of the object will be defined as

In the situation when, i.e. accuracy of the method of assessment doesn’t satisfy, need to use the methods of identification which can be reduced to the required limits [3].

References:

1. Syzdykov D., Akhmetov D., Dote Y. Fuzzy System Identification with General Parameter Radical Basis Function Neural Network / Smart Engineering Systems. New York; Asme Press, 1998, v.8, pp.199–204.

2. Ashimov AA Syzdykov D.Zh. Identification of system by the general parameter method: A Guide to the Automatic Control Theory / Ed. A.A. Krasovsky. - Moscow: Nauka, 1987. –pp.263–271.

3. Ashimov, A.A. and D.J. Syzdykov (1981). Identification of high dimensional system by the general parameter method. In: Preprints 8^th Triennial World Congress of IFAC, Kyoto, Japan, pp. 32–37.

4. Akhmetov D.F. and Y. Dote (1999). General parameter radial basis function neural network based adaptive fuzzy systems. In: Advances in Soft Computing, Engineering Design and Manufacturing (R. Roy, T. Furuhashi and P. K. Chawdhry, Eds.), Springer-Verlag, London, UK. –pp. 260–277.