Identification

DC Motor : Step Responses


1.2

1

Amplitude (Volt)
0.8

0.6

0.4

0.2

0
0 2 4 6 8 10 12 14 16 18 20
Time (t/T_s)


Figure 7.19. Step responses for the identified DC motor models (Output error with extended prediction model and output error with fixed predictor, d = 0, n_A = 1, n_B = 2)

4. 
   Flexible Transmission
   

In this example the dynamic model of a flexible transmission, with low damped resonant modes, will be identified for control design purposes. A view of the flexible transmission is given in Figure 7.20 and the schematic representation of the control system is shown in Figure 7.21.

The flexible transmission is made of three pulleys linked by two elastic belts (see Figure 7.20). One of these pulleys is constrained to the axis of a DC motor. The motor position is controlled by a local servo (speed and position feedback). The dynamics of the local position control is very fast if compared to the mechanical system.
The control problem is to get the desired position of the third pulley, by modifying the input voltage of the position control of the motor that drives the first
pulley. The output y(t) of the system is the axis position of the third pulley, and the command signal u(t) is the reference for the first pulley axis position. The mechanical loads that can be added on the third pulley modify the system inertia and, consequently, also the resonant modes of the mechanical system.
We are concerned in the following with the open loop identification of the model for this process (between u(t) and y(t)) for the case without additional load (the controller in Figure 7.21 is not connected).
The sampling frequency is 20 Hz (T_s = 50ms). The excitation signal is a PRBS of small magnitude generated by a shift register with N = 7, and with a frequency divider p = 2. In the file poulbo1.c 254 I/O samples are stored after removal of DC component. The data stored in the file poulbo1.c¹¹are plotted in Figure 7.22.
The system has two resonant modes and then a possible choice is n_A = 4. However, an estimation of the order for the model can be carried on by using techniques presented in Chapter 6, Section 6.5.
The results obtained (with the functions estorderiv.m (MATLAB^®) or estorderiv.sci (Scilab)) are shown in Figure 7.23, thus confirming that the order for the model to be chosen is


n  max(n_A , n_B  d )  4

A detailed order estimation (using WinPim) gives the values n_A = 4, n_B = 2, d = 2.

The estimation of the delay can be verified by identifying with the RLS, with n_A = n_B = 4, d = 0.


S = 1 M = 1(RLS) A = 1 FILE: POULBO1.C DELAY D = 0 A(1) = -1.5752 B(1) = 0.0104
A(2) = 1.8384 B(2) = 0.0098
A(3) = -1.4788 B(3) = 0.3077
A(4) = 0.8896 B(4) = 0.4146
The results obtained are
Note that |B(1)|, |B(2)| << 0.15|B(3)|, and that clearly implies d = 2.

¹¹ Available from the website: http://landau-bookic.lag.ensieg.inpg.fr.