A CORRELATION-BASED METHOD FOR ONLINE SYSTEM IDENTIFICATION
3.4 Impulse Response
A Correlation-Based Method for Online System Identification
correlation between the PRBS signal and the system’s normal working input is mitigated by repeating the PRBS sequence.
While large values of A result in better impulse response curves, a trade-off needs to be made between how much the system can tolerate in terms of distortion from the PRBS-induced component and how clean the impulse response needs to be. Since this method ultimately requires perturbing the motor’s torque control input with the PRBS signal, it is important to ensure the amplitude of the PRBS signal does not exceed the tolerable limit for the system, a factor which needs to be specified before testing begins.
A Correlation-Based Method for Online System Identification
maintained. Comparing the impulse function to a pulse function enables one to determine the Laplace transform of the impulse function, using the Laplace transform of a pulse function.
The pulse function in Figure 3-5 may be defined as follows
0 0
0
0, 0 ( ) , 0
0, t
d t A t t
t
t t
(3.11)
Analysis of the Laplace transform of d(t) is made simpler by considering the pulse function to be the superposition of a step function of height A/t0 beginning at time t=0 and a second step function, this time negative, with a height also A/t0 and beginning at time t=t0. Thus,
0
0 0
( ) A ( ) A ( )
d t t t t
t t
(3.12)
Here, ó(t) represents the unit step function and is defined as, 0, 0 ( ) 1, 0
t t
t
(3.13)
The Laplace transform of d(t) is shown as,
0
0
0 0
0 0
[ ( )] [ ( )] [ ( )]
st
A A
L d t L t L t t
t t
A A
st st e
(3.14)
0
0
( ) A (1 st )
D s e
st
(3.15)
By comparison, the Laplace transform of the impulse function is given by,
A Correlation-Based Method for Online System Identification
0
0 0
0
0
0
0 0
0 0
0 0
[ ( )] lim [ ( )]
lim (1 )
(1 )
lim
t
st t
st
t
L h t L d t
A e
st
d A e
dt
d st dt As A
s
(3.16)
Equation (3.16) shows that the Laplace transform of an impulse response is equal to the area under the impulse. For the special case of a unit impulse response where A=1, the area is also unity. Thus, for a system P(s) excited with a unit impulse input, the Laplace transform of the output Y(s) of the system is
( ) ( ) ( ) ( )
Y s D s P s P s (3.17)
Now, taking the inverse Laplace transform on both sides of Equation (3.17) yields
1 1
[ ( )] [ ( )]
( ) ( ) L Y s L P s
y t p t
(3.18)
By noting that y(t) is actually the response of the system to a unit impulse (and is therefore called the impulse response function) it is shown in Equation (3.18) that the Laplace transform of y(t) gives the system transfer function P(s). Consequently, the impulse response and the system transfer function contain the same information about the dynamics of the system, and it is thus possible to identify the system dynamics by measuring and analysing the impulse response of the system. It has been shown that the mathematical correlation between a system’s input and output will produce the system’s impulse response curve if the input has an impulsive autocorrelation. It has also been shown that PRBS signals have an impulsive autocorrelation, hence they are suitable signals for determining a system’s impulse response and from this the system dynamics.
A Correlation-Based Method for Online System Identification 3.5 Effect of varying PRBS parameters on numerical impulse response
An investigation was undertaken into what effect varying each of the PRBS parameters L (length of sequence), N (number of repetitions), and A (amplitude), has on the numerical impulse response that results from the correlation.
In working toward a practical technique for determining the impulse response of a system using correlation techniques, simulations were performed using Matlab and its associated simulation software Simulink. Refer to Appendix B for relevant Matlab script files and Simulink models.
The system under investigation is defined by the transfer function P(s), where ( ) 10
P s 1
s
(3.19)
The Simulink model used in the simulations is shown in Figure 3-6.
Figure 3-6 Simulink model used for simulation of PRBS-based tests to determine numerical impulse response
The traces in Figure 3-7 show the normal system working input, the PRBS input, the PRBS superimposed on the system input and the system output. The values used in this simulation were exaggerated in order to display clearly the system input with the superimposed PRBS component. When testing on a real system, the PRBS signal amplitude is limited so as to not cause such a noticeable effect.
A Correlation-Based Method for Online System Identification Figure 3-7 Oscilloscope traces for simulations.
The results for various values of L and N are shown in Figure 3-8 and Figure 3-9. For the following simulation results, all the amplitudes of the PRBS and random noise (where added) are given in percentage. This refers to the percentage amplitude of the system’s normal working input.
A Correlation-Based Method for Online System Identification Figure 3-8 Effect as L is changed on impulse response curve
Figure 3-9 Effect of changing N on impulse response curve (L=127)
A Correlation-Based Method for Online System Identification
From Figure 3-8 it can be seen that as L is increased, the numerical impulse response curve better approximates the theoretical curve. A similar phenomenon occurs as N is increased, as seen in Figure 3-9. The disadvantage of increasing L and/or N is that it takes longer to capture the data as well as to compute the cross correlation.
By increasing A, the response also improves, as shown in Figure 3-10.
Figure 3-10 Effect of changing A, the PRBS amplitude.
Note that as A is increased, the peak of the impulse response curve increases by a factor of A2, since the peak of the autocorrelation of the PRBS signal increases by the same factor. It is therefore imperative that the impulse response that is calculated from the correlation be correctly scaled, especially if it is being used for parameter extraction.