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# Time-proportional control: more from an on/off switch

09 June 2009

### Time-proportional control, a form of pulse-width modulation, is a mathematical technique that allows a feedback controller to use an on/off or discrete actuator as if it were a continuous actuator capable of generating control efforts anywhere between 0 per cent and 100 per cent.

##### Figure 1

The trick is to turn the actuator on and off for periods proportional to the desired control effort.

Fig 1: By cycling a discrete actuator on and off, a time-proportional controller can emulate the effects of a continuous actuator. In the top example, the controller is attempting to achieve a 50 per cent control effort by keeping the actuator in the "on" position 50 per cent of the time. In the bottom example, the controller is trying to achieve a 75 per cent control effort by turning the actuator on for three duty cycles then off for one.

Time-proportional control, a form of pulse-width modulation, is a mathematical technique that allows a feedback controller to use an on/off or discrete actuator as if it were a continuous actuator capable of generating control efforts anywhere between 0 per cent and 100 per cent. The trick is to turn the actuator on and off for periods proportional to the desired control effort.

Consider a home cooling system, for example. Most thermostats use a “bang-bang” control algorithm that compares the actual room temperature with the setpoint specified by the room’s occupants, then turns the air conditioner fully on or fully off if the temperature is more than a few degrees too high or too low. This technique causes the room temperature to fluctuate around the setpoint, but in most homes, that’s good enough.

The thermostat could achieve tighter control with a continuous actuator such as a motorised damper that would continuously allow a measured amount of chilled air into the room. Those are common in commercial HVAC applications but are typically too expensive for home use.

But with time-proportional control, a home thermostat wouldn’t need a continuous actuator to emulate its effects. It could use the air conditioner’s on/off switch to regulate not the amount of cool air being dumped into the room but the duration of each blast. To achieve an X per cent control effort, the thermostat would simply turn the air conditioner on for X units of time then off for 100 minus X units of time.

If those units are small compared to the time it takes to cool the room (a few minutes or so), then the average effect of turning the air conditioner fully on for X per cent of the time will be identical to running the air conditioner at X per cent of full capacity continuously. In the short term, the room temperature would still fluctuate around the setpoint, but typically not as much as with bang-bang control.

Other applications of time-proportional control might require minimising those fluctuations, in which case the minimum time between “on” and “off” commands — the controller’s duty cycle — would have to be reduced. Unfortunately, that would also increase the wear and tear on the actuator by increasing the frequency with which it switches states.

A 50 per cent control effort would be the worst case since the actuator would have to switch states at the end of every duty cycle. In the home cooling example, a duty cycle less than several minutes long would quickly wear out the air conditioner’s motor starter.

At the other extreme, a duty cycle on the order of hours would help extend the life of the motor starter, but the air conditioner would end up running for hours on end, thereby amplifying the room temperature’s fluctuations to an uncomfortable degree.

Time-proportional control works best on relatively slow processes and processes that provide a mechanism for smoothing out the effects of the actuator’s flip-flopping. In addition to temperature control, applications suitable for this technique include level and pressure control of large volumes and applications for which a continuous actuator would be prohibitively expensive.

Vance VanDoren is consulting editor for Control Engineering. Reach him at controleng@msn.com