Position Control using Lead Compensators

Bill BarracloughBill Barraclough

Sheffield Hallam University

Technology considered

A small d.c. motor actually to drive the A small d.c. motor actually to drive the systemsystem

Torque (and therefore acceleration) depends Torque (and therefore acceleration) depends on applied voltageon applied voltage

Back e.m.f. of the motor means the T.F. is Back e.m.f. of the motor means the T.F. is of the form K/[s(1 + Ts)]of the form K/[s(1 + Ts)]

So it inherently contains integration !So it inherently contains integration !

Possible Controllers

Proportional + Derivative (Stability Proportional + Derivative (Stability problems will arise if we include integration)problems will arise if we include integration)

Velocity feedback using a tachogeneratorVelocity feedback using a tachogenerator Lead CompensatorLead Compensator We will concentrate on the lead compensator We will concentrate on the lead compensator

but we will also mention the other but we will also mention the other possibilitiespossibilities

The lead compensator

These controllers often provide good These controllers often provide good performance without some of the performance without some of the drawbacks of the p.i.d.drawbacks of the p.i.d.

We will obtain the transfer function of a We will obtain the transfer function of a suitable lead compensator for a small d.c. suitable lead compensator for a small d.c. motor used to control position ...motor used to control position ...

... and produce a digital version.... and produce a digital version.

The Motor

We will base the work on a motor type We will base the work on a motor type which we have in the laboratory ...which we have in the laboratory ...

... and on which you will have the ... and on which you will have the opportunity to try out the resulting opportunity to try out the resulting controllers !controllers !

The Lead Compensator

Its transfer function (and that of the lag Its transfer function (and that of the lag compensator) is of the formcompensator) is of the form

K s as b( )

The Motor

The laboratory motors have a transfer The laboratory motors have a transfer function approximatelyfunction approximately

122 5s s( . )

The Procedure

Obtain the TF in “s” of the lead Obtain the TF in “s” of the lead compensatorcompensator

Digitise itDigitise it Implement it !Implement it !

Two Approaches

Decide to replace the motor’s “pole” by a Decide to replace the motor’s “pole” by a faster one. This determines “a” ...faster one. This determines “a” ...

... and use trial and error to find “K” and ... and use trial and error to find “K” and “b”.“b”.

Or decide the closed-loop T.F. we require Or decide the closed-loop T.F. we require and deduce the controller T.F. needed to and deduce the controller T.F. needed to achieve it.achieve it.

Method 1: “Trial and Error”

Controller transfer function:Controller transfer function:

K ss b( . )

2 5

MATLAB/SIMULINK to the rescue!

Use of MATLAB and SIMULINK Use of MATLAB and SIMULINK suggested that good performance would suggested that good performance would result from the following controller:result from the following controller:

133 2 57

. ( . )ss

We have two methods of digitising this T.F.

The “simple” methodThe “simple” method

The “Tustin” methodThe “Tustin” method

szTs

1 1

sT

zz

s

2 1

1

1

1

Which is better ?

The simple method is easier algebraicallyThe simple method is easier algebraically but ...but ... The Tustin method leads to a controller The Tustin method leads to a controller

which performs more nearly like the which performs more nearly like the analogue version. analogue version.

The Simple Method

We will do the conversion by the simple We will do the conversion by the simple method using an interval Tmethod using an interval Tss of 0.1 s. of 0.1 s.

1.33(s+2.5)/(s+7) becomes ...1.33(s+2.5)/(s+7) becomes ... 1.33[(1-z1.33[(1-z-1-1)/0.1 + 2.5]/[(1-z)/0.1 + 2.5]/[(1-z-1-1)/0.1 + 7])/0.1 + 7] which by algebra giveswhich by algebra gives (0.9782 - 0.7824z(0.9782 - 0.7824z-1-1)/(1 - 0.5882z)/(1 - 0.5882z-1-1))

The Tustin Method

Now the sum becomes (since 2/TNow the sum becomes (since 2/Tss = 20) = 20)

1.33[20(1-z1.33[20(1-z-1-1)/(1+z)/(1+z-1-1)+2.5]/[20(1-z)+2.5]/[20(1-z-1-1)/(1+z)/(1+z-1-1) ) + 7]+ 7]

giving by unreliable Barraclough giving by unreliable Barraclough mathematicsmathematics

(1.1085 - 0.8619z(1.1085 - 0.8619z-1-1)/(1 - 0.4815z)/(1 - 0.4815z-1-1))

How do the controllers perform ?

Both digital versions have slightly more Both digital versions have slightly more overshoot than the analogue version.overshoot than the analogue version.

The Tustin one is nearer to the analogue The Tustin one is nearer to the analogue version than is the “simple” one.version than is the “simple” one.

Both digital versions give a reasonably Both digital versions give a reasonably good performance.good performance.

Designing for a particular closed-loop performance

Suppose we decide we require an Suppose we decide we require an undamped natural frequency of 5 rad/s ...undamped natural frequency of 5 rad/s ...

... and a damping ratio of 0.8.... and a damping ratio of 0.8. This means that the closed-loop transfer This means that the closed-loop transfer

function needs to befunction needs to be 25/(s25/(s22 + 8s + 25) + 8s + 25)

The required controller T.F. ?

We have:We have:

So forward path = D(s) x G(s) ..So forward path = D(s) x G(s) .. and the CLTF is D(s)G(s)/[1 + D(s)G(s)]and the CLTF is D(s)G(s)/[1 + D(s)G(s)]

D(s)D(s) G(s)G(s)++

__

The sum continues ...

This means thatThis means that D(s)G(s)/[1 + D(s)G(s)] = 25/(sD(s)G(s)/[1 + D(s)G(s)] = 25/(s22 + 8s + 25) + 8s + 25) and as G(s) = 12/[s(s + 2.5)]and as G(s) = 12/[s(s + 2.5)] we will show thatwe will show that D(s) must be 2.08(s + 2.5)/(s + 8)D(s) must be 2.08(s + 2.5)/(s + 8) to produce the required performance.to produce the required performance.

Your turn !

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

“Your Turn” continued

The syntax isThe syntax is [nd,dd]=c2dm(num,den,ts,’tustin’)[nd,dd]=c2dm(num,den,ts,’tustin’) num and den represent the T.F. in snum and den represent the T.F. in s ts is the sampling intervalts is the sampling interval nd and dd represent the T.F. in z.nd and dd represent the T.F. in z.

Summary

Lead compensators are often useful in Lead compensators are often useful in position control systems using a d.c. motorposition control systems using a d.c. motor

with a “Type 1” transfer function.with a “Type 1” transfer function. We have examined two methods of doing the We have examined two methods of doing the

digitisation.digitisation. The Tustin method gives the best The Tustin method gives the best

approximation to the analogue performance approximation to the analogue performance for a given sampling interval.for a given sampling interval.