Identification of the parameters of a DC motor state space model
Abstract
A method for the identification of the DC state space model parameters based on the minimization of the error function using the least squares method is described in this paper. The algorithm is practically applied in the laboratory environment on an industrial DC motor. The verification of the results was performed by comparing the characteristic signals of real and modeled systems. The results show that the quality of the identification is satisfactory.
Introduction
The identification of system parameters is the first step in the analysis and synthesis of control systems. Identification Quality strongly impacts on the results of all other computations. In the theory of automatic control, many methods of identification are developed. Which method will be applied depends on the characteristics of the system. In this paper, we described an identification algorithm based on the least squares method. A practical test of this algorithm of estimation is done on a DC motor.
parameter estimation with the least squares method
A DC motor is a second-order system described with two differential equations: one which describes electrical and one which describes mechanical parts of the motor. The idea is to analyse the motor as two first-order systems. The main signals are responses of two first order sub-systems on appropriate inputs. Using a discrete state-space model of the motor and applying the least square method on the recorded signals, we get straightforward equations for the computation of all the necessary parameters: Rr, Lr , Je , Fe , Kme and Kem (Eykhoff, Wilsoons, 1974).
Experimental results
The practical application was realized in the laboratory where a DC middle-power motor was used as a control object. It is coupled with a DC generator which serves as a load. Generation of the input signals and measure of the responses were performed with the acquisition system based on the appropriate acquisition card and the MATLAB-SIMULINK software. The three following experiments were described:
- Determination of the linear mode of the DC motor: determination of the interval of the armature voltage in which the rotor velocity of the rotor shaft changes linearly. This experiment is of great importance for further work because all derived equations, describing the operation of the DC motor, are valid if the motor is treated as a linear component.
- Estimation of the value of Kem and Kme : the DC motor was in the generator mode with the coupled DC generator working in the motor mode. For different velocities of the rotor shaft, the induced voltage on the motor armature is measured. By forming the two column vectors based on the measurement values and using the least squares algorithm formula we get Kem. It should be noted that the coefficients Kem and Kme have the same numerical values when they are represented in appropriate SI units.
- 3. Estimation of the value of Rr, Lr, Je and Fe : the armature winding is supplied with the appropriate input, and the armature current and the rotor velocity (with tahogenerator) were measured with the acquisition card. Applying the least squares estimation algorithm, we get the values of the required parameters of the DC motor.
Validation of the results
Based on the estimated parameters, the state space model and the transfer function were formed as appropriate models for the DC motor. The simulation was performed in the MATLAB-SIMULINK programme. The simulated and real signals were compared. The close agreement of these signals, as shown in Figs. 7 and 8, confirms the quality of the described algorithm.
Conclusion
The identification process is the first and very important step in the control system analysis and synthesis. In this paper, the algorithm of the identification of the state space model parameters of the DC motor is presented. The algorithm is based on the least squares method and all the necessary experiments in the laboratory conditions were described. The obtained results show good agreement between simulated and real systems.
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