5.4
Labwork 4
Direct Digital Controller Design and Implementation: Swinging Crane
5.4.1
Apparatus
PCS1 apparatus, CODAS-II, Victor-II and Analogue I/O card
5.4.2
Introduction
When the PCS1 carriage module is inverted, ie in the crane position, the apparatus mimics an overhead
crane or a grab. In practice the grab is moved out over the material/goods to be raised, the grab is then
lowered and then raise the material. The problem is that if the top of the grab is moved suddenly, the
other end starts to swing violently and it takes a long time before the oscillations have died away and the
material can be hoisted safely. The object of this labwork is to devise a control law that will allow the grab
to reach a steady-state as fast as possible after a demand change in its horizontal position.
The dynamics of the system are essentially those of a pendulum and because of the very low damping,
the system shows a pronounced resonance at the natural frequency of the rod/mass combination.
Conventional analogue control of such a system using lead or lag compensators is very difficult. The best
approach is to use a 'notch' filter, ie a filter that has a anti-resonance. By placing the notch near the
natural frequency of the 'crane', the overall frequency response characteristic is smoothed out and then a
simple compensator can be employed to satisfy the design criteria. However this approach will not be
pursued here and it is left for students to explore this method for themselves. Rather than attempt to
control the system using analogue methods, discrete controllers will be designed and implemented. The
nature of the system lends itself very well to such methods which result in simple, robust and effective
control, for example - deadbeat controller.
5.4.3
Sample Time Selection and Design Objectives
Many of the arguments regarding sample time selection discussed in Labwork 3 for the inverted pendulum
are not applicable for the apparatus in this position. Although the system is very oscillatory, it is stable,
and even with a sluggish servo the system can be brought to rest. In the inverted pendulum mode,
however, it would not be possible to "catch" the pendulum before it topples over completely if the servo
behaviour were too slow or if the sample time were too long. In the case of the crane problem we can
design an effective controller with a much slower sampling rate.
For this system we can choose a more demanding design objective, ie dead-beat control with steady-state
set-point following, ie a compensator with integral action. Dead-beat control brings the system to rest in a
finite number of samples. As the system is second order, we can theoretically achieve dead-beat
behaviour in two sample intervals. Now, as the system oscillates naturally with a period of 0.94s, it seems
natural to try to bring it to rest in a half period. These arguments lead to a sample time of 0.94/4 = 0.235s.
In VICTOR-II the nearest sample time one can choose is 0.22s.
Page 5.17
In this application we can neglect the dynamics of the servo as it is so much faster and better behaved
than the crane. Furthermore as a first approximation we can neglect the damping in the pendulum. The
transfer function relating the centre of mass of the rod/mass assembly to the carriage position is assumed
to be Gp ( s ) 
1
1 s 2 /  n2

1
1  0. 0264s 2
Assuming that n is 6.16 rad/s. Enter this transfer function into CODAS-II and select a sample time of
0.22s. Transform the plant transfer function to the z-domain (<F5>). The resulting pulsed transfer
function is approximately Gp (s) 
n2
s2  n2
This is transformed again using CODASII as explained. Alternatively, you can do this by analysis following
similar lines to the example done for pg 5.13, ref 1.
Gp (z)  Z
R
|S1 e
|T s
 sT
U
|V
|W
R
|S
|T
n2
z  1 1 n2

Z
z
s s2  n2
s2  n2
U
|V
|W
Again a partial fraction is required of the term inside the curly brackets.
1 n2
1
s
  2
2
2
s s  n
s
s  n2
The
Z
 cos(  ))
l q z z 1  1z(z
2cos(  )z  z
Gp (z) 

2
1 2cosz  z2  (z  1)(z  cos )
1 2cos(  )z  z2
1  cos  z  zcos (1  cos )(z  1)

1  2cosz  z2
1  2cosz  z2
For the pendulum
 = T = 6.155 x 0.22 = 1.354
 cos = 0.215
Hence
Gp ( z) 
0.785 ( z 1)
1  0. 43 z  z2
Select the root-locus domain and you will observe that there are a pair of poles present on the unit-circle.
These poles represent the oscillatory dynamics of the pendulum. A sample rate of four times the natural
frequency would make these poles on the unit circle pure imaginary.
Page 5.18
5.4.4
Dead-Beat Controller Design Using Simplified Plant Model
To design this compensator, we allow Gc(z) to cancel the two poles on the unit circle. This effectively
produces a notch filter that cancels the pendulum resonance (see later). Cancelling the plant zero on the
unit circle, would result in a marginally stable controller (pg 311, ref 1). For steady-state set point
following, the closed-loop transfer function will be F ( z) 
0. 5 ( z  1)
z2
The compensator, Gc(z) is given by -
Gc ( z) 
1 F ( z)
Gp 1 F ( z)
Gc ( z) 
0. 637 (1 0. 43 z  z2 )
( z  1)( z  0. 5 )
Hence
Enter the plant and compensator transfer functions into CODAS-II and examine the closed-loop time
response. Also examine the control effort (<U>). Save the CODAS model to file as "crane1".
5.4.5
Procedure
Place the PCS1 carriage module into the 'crane' position, ie where the pendulum is hanging down
vertically. Connect the 26 way cable to the PCS1 control console. Switch on the power to the PCS1 unit
and start up VICTOR-II
Select the DDC mode with a sample time of 0.22s. Change the manual output to 40%, the set-point to
40% and the excitation to 20%. Enter the compensator transfer function. Save the settings to a file as
"crane1". Before switching to Auto press <G> to remind yourself of how oscillatory this system is. Press
<G> again and wait for the system to settle, or stop it swinging with your hand. Now switch VICTOR-II to
Auto and repeat. Freeze the display and measure the settling time. Is it the same as the simulation?
How does the observed response compare with the simulation?
5.4.6
Frequency Response of Plant and Compensator
Using CODAS-II load the file "crane1". Select the frequency domain <F8> and the Bode Gain View (<V>).
Press <F3> to select compensator and the plant. Draw the frequency response of the overall open-loop
system. Notice that the curve is smooth with no sharp resonances or anti resonances below the Nyquist
frequency.
Now press <F4> to look at the plant alone and draw its open-loop frequency response. Now change the
plant denominator to unity, bring in the compensator (<F3>). This time when you press <G> you will see
the frequency response of the compensator on its own. You will observe that the compensator has an
anti-resonance that cancels the plant resonance.
Page 5.19
5.4.7
Dead-Beat Controller Design Using Plant Model with Damping
Modify the plant transfer function in CODAS-II to incorporate damping, ie
Gp ( s ) 
1
1  0. 0011s  0. 0264s 2
In the next sections we shall design a controller that takes into account the damping in the actual system.
5.4.7.1 Ringing Pole Problem
When damping is introduced in the plant transfer function, the poles and zeros all move inside the
unit circle, ie
Gp ( z ) 
0.76(z+0.97)
z  0. 41z  0. 91
2
If a dead-beat controller is designed using the above model where the zero at z=-0.9 is cancelled,
the compensator will have a pole at z=-0.9. The resulting compensator is
Gc ( z ) 
1. 32( z2  0. 41z  0. 91)
( z 1)( z  0. 97 )
Enter the above compensator into VICTOR, make sure the deviation is zero and switch to Auto.
Do not at this stage press <G> to excite the system. You will already observe an oscillatory mode
occurring even in this quiescent state. We will introduce a step change in the set-point, but be
ready to switch back to manual if the oscillations in the rig become too violent. Press <G> now
and observe the response. You probably are back in manual by now!. Clearly, cancelling a zero
on the real negative axis is not advisable. In fact there is very little advantage to be gained by
cancelling any plant zeros at all.
5.4.7.2 Correct Design Approach to Avoid Ringing Pole Problem
This time the compensator is designed without cancelling the zero at z=-0.97. The resulting
compensator is
Gc ( z) 
0. 67 ( z2  0. 41z  0. 91)
( z 1)( z  0. 5 )
Try the response with this compensator and compare the results with those obtained with the
simpler plant model. Is the behaviour closer to the modelled response?. Try refining the plant
model so that the correlation between the actual crane and the simulation is close.
5.4.8
References
Ref 1: Golten JW, Verwer AA, "Control System Design and Simulation", McGraw Hill, 1991
Page 5.20