5.1Labwork 1
Static and Dynamic Characteristics of the Pendulum Control System (PCS1)
5.1.1 Object
In this experiment you will -
1. Familiarise yourself with the PCS1 rig
2. Examine the characteristics of the carriage servo. Tune its gain and the degree of velocity feedback to optimise its performance. Obtain a dynamic model of the servo.
3. Examine the dynamic characteristics of the pendulum and hence model it dynamically
5.1.2 Apparatus
Two voltmeters, Signal Generator and an Oscilloscope
5.1.3 Carriage Servo Alone
Place the rig in the inverted pendulum position and unscrew the pendulum rod. Disconnect all the links on the console. Connect the output of the set-point potentiometer (socket A) to the input of the carriage servo (socket H). Position the set-point potentiometer (P1) so that it is at its mid-range. Adjust the servo gain potentiometer (P3) and the velocity feedback potentiometer (P4) so that they are about mid-range. Switch on the power to the apparatus.
Move the set-point potentiometer to and fro and observe the carriage change its position. Using the two voltmeters, measure the set-point voltage and the carriage position voltage, Vx, (socket J). Increase the set-point voltage uniformly from 0 to +10V and then back down to -10V and finally back to 0V in steps of 2V and plot a graph of the servo position against set-point voltage. Make sure that you always move the pot in the required sense. If you move it too far one way, move the set-point potentiometer back and then approach the required value slowly.
a) What is the useful linear range of the carriage servo?
b) What is the hysteresis of the carriage servo expressed in volts?
c) What is the sensitivity of the carriage servo expressed in volts/volt?
The non-linear behaviour (clipping characteristic) has been introduced deliberately to prevent the carriage hitting the end stops too hard and so damaging the rig.
5.1.4 Effect of Servo Gain on Hysteresis
Disconnect the servo input (socket H) from the set-point potentiometer and connect the servo input to ground. Push the carriage to the right with your hand and gently release it. Make a note of the carriage position voltage. Repeat, pushing the carriage in the opposite direction and letting it go again. Hence calculate the hysteresis of the carriage servo in volts. Compare this figure with that obtained above in 3b. Reduce the servo gain pot to about 25% of its range and repeat the test. Finally put the servo gain to maximum and measure the carriage hysteresis once more.
What conclusion do you draw about the effect of servo gain on the amount of hysteresis present in the carriage servo? Can you explain the reason for your observation?
5.1.5 Transient Response of the Servo Subsystem
Reduce the velocity feedback to zero. Use a signal generator to apply a +1V square wave of frequency 1Hz to the input of the carriage servo. Observe the carriage position on an oscilloscope (socket J). Adjust the servo gain until the response shows a small overshoot of about 15% (damping ratio of about 0.7). Now increase the servo gain to a maximum and adjust the velocity feedback pot (P4) so that the response is similar to the one observed previously.
Which of these modes of operation is preferable? Explain your reasons.
From now on leave the gain of the carriage servo at a maximum and the amount of velocity feedback at the value that produced a step response with a slight overshoot.
5.1.6 Modelling of the Carriage Servo
In this experiment we will obtain a transfer function of the carriage servo. To do this properly, you really needs a frequency response analyser, but the method described here is a simple method for obtaining an approximate second order model.
First we are going to determine the bandwidth of the carriage servo. Apply a 1 Hz sinusoidal signal with a peak to peak amplitude of 1 volt to the input of the servo (socket H). Observe the carriage position voltage on an oscilloscope. You may see a slight distortion because of the hysteresis that is present, but the amplitude will be nearly the same as the applied signal, ie the servo-subsystem has unity low frequency gain which confirms the result obtained in 5.1.3c. Increase the frequency of the applied signal until the observed voltage has reduced to 0.7V, ie of the low frequency value. The frequency at which this occurs is the bandwidth of the system b. For a system with a damping ratio of 0.7, the natural frequency, n, is equal to the bandwidth. Based on this approximation, calculate the natural frequency, n. (Don't forget to work in rad/s). Hence deduce the transfer function of the servo-subsystem assuming that the damping ratio is 0.7
As an example, a system whose bandwidth is 9.6Hz and a damping ratio of 0.7 has the approximate transfer function -
5.1.7 Dynamic Model of the Pendulum
In this part of the experiment we will model the dynamic behaviour of the "pendulum". We shall do this by observing the transient response of the system from an initial value. This really requires a storage 'scope or a pen-recorder, but if VICTOR-II* is available, it can be used like a storage oscilloscope with a digitising cursor.
Before fitting the pendulum rod into the carriage, position the mass at the end of the rod. Estimate the position of the centre of mass of the rod/mass assembly by trying to balance it on your finger. (The centre of mass is located just below the bottom of the mass.) We shall call this length, the effective pendulum length, L.
Screw the pendulum rod firmly into the carriage. Tip the rig upside down so that it is in the "crane" position. Connect the pendulum angle signal, V, to the oscilloscope. If VICTOR-II is being used to monitor the angle as the measured value, (analogue channel 3). Select a sample time of 0.0137s for maximum resolution (Controller menu <F3>).
Displace the pendulum by about 30o, release it and record the transient response. After about 20 cycles freeze the display. Estimate the period of oscillation, T, and the logarithmic decrement using equation 1 (see ref 1, pg 73 and pg 80).
The value of the period obtained can be calculated theoretically from the basic dynamics of a simple pendulum. The well known formula for the period is , where L is the effective length of the pendulum (ie from the pivot to the centre of mass). Do this calculation using the effective centre of mass obtained earlier and compare the period obtained with that observed experimentally.
The logarithmic decrement, , is defined as -
= ln(mk/mk+2)(1)
where mk is the magnitude of the kth overshoot. From the logarithmic decrement, , the damping ratio, may be calculated using the formula -
(2)
This system is very lightly damped, so in order to arrive at a figure for the damping ratio, it is best to plot a graph of ln(mk) versus k. Use the cursor to extract data from the frozen display in the VICTOR recorder window. The logarithmic decrement, , is twice the slope of the graph.
Suppose for example the logarithmic decrement is 0.02, then the damping ratio figure works out to be 0.0033. You may need to modify the damping ratio figure slightly to obtain a better overall fit to the observed response.
From the period, T, the natural frequency can be deduced assuming that the damping ratio is very small, ie n ~ 2/T. Suppose, T is 1.02s, then n is approximately 6.16 rad/s. Thus, with a natural frequency value of 6.16 rad/s and a damping ratio figure of 0.0033, the pendulum can be modelled approximately as a second order system -
Use CODAS to compare the response of the model with that of the actual system. To do this, enter the transfer function of the system, switch to open-loop, define an initial condition of unity (<Z>) and choose a user defined input of 0. Figure 1.1 shows the trace obtained.
* VICTOR-II Virtual Instrument Controller software package.
5.1.8 References
Ref 1 - Golten JW, Verwer AA, "Control System Design and Simulation", McGraw Hill, 1991
Page 5.1
Figure L1.1
Page 5.1