Controlling The Plucked String Model

Now you will gain some intuition into adjusting the PID control parameters $P_P,$ $P_I,$ and $P_D.$ Try to avoid adjusting the PID parameters in a manner that causes the system to become unstable. If instability occurs, the output from the pd patch will become louder and louder until the pd patch shuts itself off and sends the message UNSTABLE to the pd message window. If you ever see this message, close the patch and open it again. One way to avoid instability is to adjust the parameters slowly-backing off if ever the sound from the model starts becoming louder and louder. The blue button ``Reset parameters for no control'' will set $P_P=0$, $P_I=0$, and $P_D=0$ and may also be triggered by pressing the space bar.

The sliders for adjusting $P_I$, $P_D$ and $P_P$ are set up to make as much of the useful parameter space as available as possible. For example, the default position for the $P_I$ slider, which corresponds to $P_I=0$, leaves little room for making $P_I$ negative by moving the slider to the left. This is because even slightly negative values will cause the energy in the string to grow quite fast. Note that the actual values for each PID control parameter is shown in a box underneath the corresponding slider.

• Download the pd patch 4-1.pd, and open it in pd .

• Ensure that the patch is not in editing mode, and check the ``compute audio'' box in the main pd window.

• To virtually pluck the string, press any key or click the ``PLUCK'' button. You may optionally check the box ``pluck repeatedly.'' Adjust the ``Master volume'' slider on the right until the volume level is comfortable.

• The four sliders on the upper left calibrate the characteristics of the vibrating string model. Consider reviewing the digital waveguide model laboratory assignment to remind yourself of the meanings of these parameters.

• The fourth slider labeled ``Sensor and actuator position'' may be used to move the virtual sensor and actuator along the string. So that the string may be controlled successfully, the sensor and actuator must be located at the same position [1]. What sensor locations cause the sensor to measure no energy at the second harmonic? (Hint: If the sensor is placed near one of these locations, then the sensor will measure very little energy at the second harmonic. You may observe this effect by viewing the spectrum while adjusting the ``Sensor and actuator position'' slider.)

• How do you adjust $P_D$ so that the overall decay time constant becomes shorter?

• Click the button ``Reset parameters for no control'' and adjust $P_I$ for integral control damping. What differences do you observe in the spectrogram when integral control damping is applied?

• While damping behaves as predicted in the theory section, you will find that altering the pitch is more difficult. This might seem surprising given our derivation, but recall that we assumed that there was only one harmonic. Now $P_P$ alters the pitch of all of the harmonics simultaneously; however, it also alters the damping some. The way in which it alters the damping is complicated, so for larger values of $\vert P_P\vert$, one or more of the harmonics becomes unstable. This means that the pitch may be altered significantly only in conjunction with additional damping required to preserve stability. Try to see how far you can shift the lowest harmonic while preserving stability. Write down $\hat{f_0}$, $P_P$, $P_I$, and $P_D$ for making $\hat{f_0}$ both as large as possible and as small as possible.

• Compare your results with those shown in Figure 3 where $f_0$ is chosen so high that only the lowest harmonic appears on the spectra. Flatted notes have relatively lower pitch (see Figure 3, top), while sharpened notes have relatively higher pitch (see Figure 3, bottom). You will probably find that you cannot shift the pitch quite as far because your PID controller parameters are limited to a smaller space.

• Do you notice any problems in the sound of the controlled tones when the frequency is altered? One thing you may notice is inharmonicity. Can you surmise why this might occur?

• Challenge problem: Modify the subpatch pd controlled-string to induce tremolo in the modeled vibrating string. For our purposes here, we will let tremolo be defined as periodic variations in the amplitude of a signal. See the hint in the subpatch to help you decide where to make the required changes. Save the result into the file 4-1.tremolo.pd.

Figure 3: Example spectra for frequency shifting via PID control (top: flat (decreased pitch), middle: no control, and bottom: sharp (increased pitch))