Jaracz, Kazimierz
Dr eng., Institute of Technology, WSP Kraków, Podchorążych 2, 30084 Kraków
The paper deals with the formulation of the method of realization of the mathematical description of the sensitivity function of state variable of a d.c. motor to the parameters undetermined by random disturbances.
The possibility of examining the driving motor operation at the disturbance of supply voltage is based on the extended model of the motor. Owning to that the matrices of both state and input matrices are the functions of principle parameters describing the disturbance, they are based on the approximation of the autocovariance torque in the internal of stacionarity. This method in the theory of stochastic control is known as the method of expanding of the control object [4].
By enlarging the definition of the sensitivity function of motor current and its angular velocity (owing to the variation of deterioration function parameters) and their influence on the random functions, one receives the model, which enables the determination of behaviour of motor state variables (at the every interval of observation) on the variation of deterioration.
The method of construction of the motor sensitivity may be presented as follows:
(i)  it is necessary to determine the approximate form of the autocovariance time of voltage deterioration momentum, 
(ii)  to construct the forming filter, which will change the white noise into coloured one, described by the approximated form of the autocovariance momentum, 
(iii)  to determin the enlarged motor model, with the hipermatrix coefficients being the functions of parameters of both deterioration and the drive system, 
(iv)  to define the averaged sensitivity function of current and angular velocity to the variance of parameters of probabilistic deterioration, 
(v)  to buildup the accompanying sensitivity motor model, 
(vi)  to calculate the averaged functions of motor sensitivity of discussed motor state variables aided by microcomputer. 
As the approximation form of the autocovariance deterioration torque in its stationary, interval in practice, one of the forms of standard expressions, determining the probabilistic characteristic second momentum of the coloured noise with the dominant frequency , i.e. the expression of the form may be applied.
,  (1) 
In it the 
  variance, 
  irregularity constant,  
  oscillatary constant,  
  time shift. 
All above is compatible with the empiric results of measurements and approximation of the above function for the voltage deterioration of the drive of the rolling mill [2]. The coloured noise neceived form the white noise as the result of synthesis by the forming filter (of the state [x_{1}, x_{2}]^{T}) of the intensivity S will be described by the equation [4]:
,  (2) 
where  [b_{1}, b_{2}]^{T}   input matrix of the filter, 
w( )   white noise. 
Applying the basic model of the d.c. motor basing on the usually applied form:
,  (3) 
where M = L_{a} ; the index f values concerns the parameters and values of the excitation circuit, a  parameters and values of the armature circuit. Symbol L denotes the inductivity, R denotes the resistance, u  voltages, i  currents,  angular velocity, J  effective intertia moment reduced to the motor shaft, m_{s}  load torque.
Owning to that, that extended model of the d.c. motor is as follows:
,  (4) 
where
,  (5) 
We will start with the extension of the definition of the sensitivity of the trajectory introduced by Cruz'ea on the random stationary processes. In the disturbance conditions the averaged random sensitivity functions of the armature current and angular velocity will influence the variance of the disturbance vector parametery will have the form, respectivity:
,  (6) 
, i = 1, 2  (7) 
where E  average value operator, p = [p_{1},p_{2}]^{T} = [, ]^{T}
By differentiating the vectorhipermatrix state equation (4) in
relation to p_{i}, with the continuity conditions respected
the model of the sensitivity of the discussed model of motor to
variation of the coefficients of irregularity
and oscillatory :
, i = 1, 2  (8) 
, i = 1, 2  (9) 
The Exemplary Results of Calculations.
The averaged sensitivity functions were calculated by means of SIPRO 3.4 language oriented on to the function graphs.
The calculations have been realized for a d.c. motor driving the long profile rollingmill. The parameters of the motorfare as follows:
Normal power rating  P_{N} = 6 MW 
Normal armature voltage  U_{aN} =1350 V 
Normal armature current  I_{aN} =4750 A 
Normal velocity  n [65,165]

Excitation current  I_{f} [50,210] A 
Excitation voltage  U_{fN} = 220 V 
Armature circuit  R_{a} = 0,024 
Armature circuit inductivity  L_{a} = 0,067 mH 
Excitation circuit resistance  R_{f} = 0,66 
Extation circuit inductivity  L_{f} = 2,8 H 
Effective inertia  J = 95410 kgm^{2} 
Number of excitation winding turns  Z_{f} = 972 
Dissipation coefficient  =1,2 
Motor voltage constant  c_{e} = 174,46 
Motor torque constatnt  c_{m} =174,46 
The main flux  = 0,43 Wb 
Electromechanical time constant  T_{cm} = 0,186 s 
The disturbances and filtr parameters:
= 47,62V
= 144 s^{1}
= 1783 s^{1}
b_{2} = 16,9
= 0,059
=4,5 10^{5}
The examples of calculation results are presented in Fig. 1.
Fig. 1a present the averaged function of sensitivity of armature current on the variation of the irregularity; 2b  as before, but on the variation of oscillatory coefficient.
In the paper the methods of construction of a sensitivity model of selected electric machine were presented, on the variation of random disturbances of parameters.
In order to that it is necessary to obtain the approximation of the author covariance momentum of the disturbances. As the approximation model the representative form of the autocovarince function of the discussed the noise component appearing in the armature supply voltage.
The approximation may differ from the above mentioned, but regarding in a general form, the characteristic parameter of the random disturbance.
The extended model of the machine regards these parameters of disturbances and enables the construction of associated averaged sensitivity model of the discussed machine. The microcomputer aided calculations enables receiving the quantitive evaluation the variation of current and angular velocity parameters of the motors (Fig.1a and 1b).
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