pH Control for Control Valve Error

Up until now, we have considered that the control valve was ideal - i.e. the valve provides whatever reagent flow rate we desire (with saturation limits). In reality, things aren't that simple, with most valves displaying backlash characteristics. You should recall the discussion of backlash in the ball and plate control example.
For now, we will assume that the backlash error is 2% of the maximum flow rate allowed by the valve. Often, the valve errors can be as large as 5% or 10%. The block diagram of this new system is shown below:

With the influent flow variations discussed in the last example, there are now two sources of error in the system. In order to understand them better, we will examine them separately. Since the last example showed the effect of flow variations alone, we will now look at the effect of the valve errors alone.
The nature of the valve hysteresis is such that a simple exponential step response will be largely unaffected. Thus, to highlight the effect of the hysteresis, we will allow extremely small influent flow variations. The influent flow q(t) is then the nominal 10 L/min plus a small sinusoidal variation of amplitude 0.0003 L/min and period 10000 s.

Java Applet Simulation

Below is a Java applet which simulates the pH control system with control valve hysteresis. The blue plot shows the influent concentration and the green plot shows the effluent concentration. This process is extremely slow, so the time scale is accelerated somewhat: 1250 simulated seconds pass for each real second of the simulation, giving a horizontal scale of 2500 seconds per division
Pressing the "Change Parameters" button brings up the parameters window allowing you to change the controller values, the influent pH and the animation speed. The controller has been implemented in the anti-windup form to avoid the effects of the saturation of the valve. The animation speed can be 25, 12.5 or 6.25 frames per second. This speed is only the rate at which the screen is refreshed - the simulation is unaffected. The speed function is available for those with slower computers so that the simulations still appear in real-time (if somewhat jerkier). Note that the applet does some auto-detection of the speed of your computer, so if you select a speed that your computer cannot handle, the applet will reduce the speed accordingly.
The parameters window also allows you to select whether you want flow variations or valve errors. The possibilities are listed in the table below:

flow variations	valve error	description
off	on	The default state for the simulation, showing the valve error characteristics as discussed above.
off	off	No valve errors are present, allowing you to see the effect of the small sinusoidal flow variations alone.
on	off	Shows the effect of random flow variations, and is exactly the same as in the previous example.
on	on	Shows the effect of the valve errors in the presence of the random flow variations.

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Things to try	Things to notice
Look at the response of the system with valve errors and small sinusoidal flow variations.	Notice that the effluent pH varies between about 8.5 and 5.5
Turn off valve errors (and flow variations)	Observe the response of the small sinusoidal flow variations alone, and compare the output to the system with valve errors.
Turn off the valve errors, and turn on the flow variations	Recall the response from the previous example and note that this response is the same, with the output again varying between 8.5 and 5.5
Turn on both the valve errors and the flow variations	Note the effect of the both the valve hysteresis and the random flow variations.
Try the faster controller that you designed in the previous example	Notice the limited effect of this controller.

The essential point in this example is that both sources of error produce output variations of approximately the same size. If we were to construct a larger tank to reduce the error cause by the influent flow, then the valve error would start to dominate. Since the valve errors are of the same magnitude as the errors caused by the influent flow variations, increasing the tank volume would have virtually no effect on the output variations.
The next idea might be to try to reduce the valve errors. The valve error is a percentage of the valve's size, so to reduce the error, we need a smaller control valve.  However, a smaller control valve will not be able to provide the necessary reagent flow to neutralise the influent.
This highlights the need to consider both sources of error together, rather than trying to eliminating them individually. The next example presents a solution which considers these factors.