In last month's column I outlined a simple circuit that could control temperature by turning a heater on and off. One primary drawback of this binary controller circuit is that even in a steady-state condition, the controlled process temperature would oscillate a small amount around the desired set point temperature in a limit cycle. While this isn't a problem for many applications, such as the furnace in my house, there are many circumstances, such as aircraft control surfaces, where even well-defined and well-controlled oscillation would be completely unacceptable.
Proportional Control Systems
One way to get more control over process variables is to move to a proportional control system. Because of the increased complexity of such a controller, and the space available in this column, I will be touching on only some of the key features of such a device.
The main difference between the ON-OFF control system of last month's column and a proportional control system is that the controller provides a continuously variable output. For a temperature controller, this means that instead of being able to turn the heater only on or off, it could command that heater to turn on to 37% of maximum output, or any other value between completely off and completely on.
Error and Instability
The simplest controller circuit that can provide a continuously variable output is an amplifier. In the most basic case, the controller amplifies the error signal and uses the result to drive the actuator. In practical applications, however, this has two drawbacks. The first is that under steady-state conditions, the process variable will nearly always mismatch the set point by some small amount. This steady-state error will be inversely proportional to the controller's DC gain. While one obvious fix would be to crank up the controller gain to some arbitrarily high value (you can buy op amps with gains >1,000,000), this results in the appearance of the second drawback—dynamic instability.
The limit cycle seen in the binary controller is one form of instability, but often a form you can learn to live with. The kind of instability you get from excessive gain in a linear controller is not usually so well-behaved—in the worst case it can send the process variable oscillating wildly across some range of values, limited only by the actuator's output drive (a.k.a. control authority). This is usually a really bad thing.
Instability, at least in simple linear systems, results from two characteristics of the control loop. The first is that there is a finite delay between the time when a command is applied to the actuator and when the system and feedback sensor respond. There is also a delay in the controller. When this delay has the right characteristics, it can invert the phase of a periodic response, for example a sin(t) signal can appear to be –sin(t). This has the effect of turning the control loop's negative feedback into positive feedback at some frequencies.
The next culprit is gain. If the loop's gain is >1 at a frequency where the delay causes positive feedback, it can (and typically will) oscillate at that frequency. Loop gain is the product of all the gains in the loop—controller, actuator, controlled system, and sensor. Because of the variety of quantities and units (volts, amps, °C), a control system's loop gain may sometimes be less than intuitively obvious.
Loop stability can be improved by decreasing the gain, especially at those frequencies where it is prone to oscillation. Gain reduction should be familiar to anyone who has experienced feedback in a PA system; the two usual cures are to turn down the amplifier gain, or point the microphone away from the speakers (also reducing loop gain). Because as previously noted, steady-state error is inversely proportional to gain, gain reduction—at least at DC—is often not feasible. The alternative is to design a controller that has a high DC gain, but attenuated gain at higher frequencies. In many respects, designing a linear controller is similar to creating an analog filter.
Depending on the system you want to control, it can be quite difficult to design an effective controller. On the low end of the complexity scale are dominant pole and the ever-popular PID control algorithms. Complexity goes up from there to handle more complex controls, such as missile guidance systems.
A few months ago, an example of a more down-to-earth potential need for well-designed controls presented itself at work. We acquired an inexpensive toaster oven to perform some experiments. While I don't expect laboratory-grade accuracy from a kitchen appliance, I was genuinely surprised to measure temperature swings of more than 100°F when the device was set to 450°F. We had even loaded it with a brick for temperature stabilization. If this is typical behavior for such appliances, I think I now know why frozen pizza never tastes like the real thing.
Ed Ramsden, BSEE , a member of the Sensors Editorial Advisory Board, designs sensors for the heavy-truck industry in Portland, OR.