1. Position Control using Lead Compensators
Bill Barraclough
Sheffield Hallam University

2. Technology considered
A small d.c. motor actually to drive the system
Torque (and therefore acceleration) depends on applied voltage
Back e.m.f. of the motor means the T.F. is of the form K/[s(1 + Ts)]
So it inherently contains integration !

3. Possible Controllers
Proportional + Derivative (Stability problems will arise if we include integration)
Velocity feedback using a tachogenerator
Lead Compensator
We will concentrate on the lead compensator but we will also mention the other possibilities

4. The lead compensator
These controllers often provide good performance without some of the drawbacks of the p.i.d.
We will obtain the transfer function of a suitable lead compensator for a small d.c. motor used to control position ...
... and produce a digital version.

5. The Motor
We will base the work on a motor type which we have in the laboratory ...
... and on which you will have the opportunity to try out the resulting controllers !

6. The Lead Compensator
Its transfer function (and that of the lag compensator) is of the form

7. The Motor
The laboratory motors have a transfer function approximately

8. The Procedure Obtain the TF in “s” of the lead compensator Digitise it
Implement it !

9. Two Approaches
Decide to replace the motor’s “pole” by a faster one. This determines “a” ...
... and use trial and error to find “K” and “b”.
Or decide the closed-loop T.F. we require and deduce the controller T.F. needed to achieve it.

10. Method 1: “Trial and Error”
Controller transfer function:

11. MATLAB/SIMULINK to the rescue!
Use of MATLAB and SIMULINK suggested that good performance would result from the following controller:

12. We have two methods of digitising this T.F.
The “simple” method
The “Tustin” method

13. Which is better ? The simple method is easier algebraically but ...
The Tustin method leads to a controller which performs more nearly like the analogue version.

14. The Simple Method
We will do the conversion by the simple method using an interval Ts of 0.1 s.
1.33(s+2.5)/(s+7) becomes ...
1.33[(1-z-1)/ ]/[(1-z-1)/ ]
which by algebra gives
( z-1)/( z-1)

15. The Tustin Method Now the sum becomes (since 2/Ts = 20)
1.33[20(1-z-1)/(1+z-1)+2.5]/[20(1-z-1)/(1+z-1) + 7]
giving by unreliable Barraclough mathematics
( z-1)/( z-1)

16. How do the controllers perform ?
Both digital versions have slightly more overshoot than the analogue version.
The Tustin one is nearer to the analogue version than is the “simple” one.
Both digital versions give a reasonably good performance.

17. Designing for a particular closed-loop performance
Suppose we decide we require an undamped natural frequency of 5 rad/s ...
... and a damping ratio of 0.8.
This means that the closed-loop transfer function needs to be
25/(s2 + 8s + 25)

18. The required controller T.F. ?
We have:
So forward path = D(s) x G(s) ..
and the CLTF is D(s)G(s)/[1 + D(s)G(s)]
D(s)
G(s) + _

19. The sum continues ... This means that
D(s)G(s)/[1 + D(s)G(s)] = 25/(s2 + 8s + 25)
and as G(s) = 12/[s(s + 2.5)]
we will show that
D(s) must be 2.08(s + 2.5)/(s + 8)
to produce the required performance.

20. Your turn ! If we use a sampling interval of 0.1 s again
What will the digitised transfer functions be
using the simple method ...
... and the Tustin method ?
We can check the Tustin one by MATLAB
using the “c2dm” command.

21. “Your Turn” continued The syntax is [nd,dd]=c2dm(num,den,ts,’tustin’)
num and den represent the T.F. in s
ts is the sampling interval
nd and dd represent the T.F. in z.

22. Summary
Lead compensators are often useful in position control systems using a d.c. motor
with a “Type 1” transfer function.
We have examined two methods of doing the digitisation.
The Tustin method gives the best approximation to the analogue performance for a given sampling interval.