Position Control using Lead Compensators Bill Barraclough Sheffield

Position Control using Lead Compensators Bill Barraclough Sheffield Hallam University

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

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

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

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

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

The Motor u The laboratory motors have a transfer function approximately

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

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

Method 1: “Trial and Error” u Controller transfer function:

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

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

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

The Simple Method u We will do the conversion by the simple method using an interval Ts of 0. 1 s. u 1. 33(s+2. 5)/(s+7) becomes. . . u 1. 33[(1 -z-1)/0. 1 + 2. 5]/[(1 -z-1)/0. 1 + 7] u which by algebra gives u (0. 9782 - 0. 7824 z-1)/(1 - 0. 5882 z-1)

The Tustin Method u Now the sum becomes (since 2/Ts = 20) u 1. 33[20(1 -z-1)/(1+z-1)+2. 5]/[20(1 -z-1)/(1+z 1) + 7] u giving by unreliable Barraclough mathematics u (1. 1085 - 0. 8619 z-1)/(1 - 0. 4815 z-1)

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

Designing for a particular closed-loop performance u Suppose we decide we require an undamped natural frequency of 5 rad/s. . . u. . . and a damping ratio of 0. 8. u This means that the closed-loop transfer function needs to be u 25/(s 2 + 8 s + 25)

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

The sum continues. . . u This means that u D(s)G(s)/[1 + D(s)G(s)] = 25/(s 2 + 8 s + 25) u and as G(s) = 12/[s(s + 2. 5)] u we will show that u D(s) must be 2. 08(s + 2. 5)/(s + 8) u to produce the required performance.

Your turn ! u If we use a sampling interval of 0. 1 s again u What will the digitised transfer functions be u using the simple method. . . u. . . and the Tustin method ? u We can check the Tustin one by MATLAB u using the “c 2 dm” command.

“Your Turn” continued u The syntax is u [nd, dd]=c 2 dm(num, den, ts, ’tustin’) u num and den represent the T. F. in s u ts is the sampling interval u nd and dd represent the T. F. in z.

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