Duty Cycle Adjustment Profile

The graph at the left shows a control profile for my smoker. The vertical scale represents the duty cycle from 0-100%. The horizontal scale is the temperature reading. 200F (B) is the set point with a control range from 180F (A) to 220F (C) The initial heater element duty cycle (a guess) is 50% (B).

Throughout the iteration process the set point (oven temperature) remains constant and the estimated duty cycle is adjusted until a stable temperature reading is within acceptable limits of the set point.

The black line represents the temperature reading vs the heating element duty cycle settings. As the temperature reading rises to 180F (A) the duty cycle remains at 100%. When the temperature rises above 180F the heater element duty cycle will be changed continuously for temperature readings indicated by the black curve line. Should the temperature rise above 220F (C), the duty cycle will be set to 0% (powered off) until the temperature again drops below 220F. This will continue until the temperature readings stablize, somewhere between 180F and 220F.

If the 50% duty cycle was a good guess the temperature will stablize at 200F (B). The next section details what happens when the temperature readings stablize at a temperature other than the set point.

Stable Temperature Adjustment

If the temperature readings stablizes at a temperature of, for example, 190F (A) the heater element duty cycle needs to be adjusted to reach the desired set point of 200F.

The estimated duty cycle (B) is set to the heater element duty cycle indicated by (A), 63%, and the adjustment profile is regenerated. The new profile is shown in the next section.

New Adjustment Profile

The regenerated curve for the new estimated duty cycle of 63% (B) yields a new heating element duty cycle of 73% (A) for a temperature of 190F. The heating element duty cycle is adjusted accordingly and the next stable temperature reading will trigger a new adjustment. This is a conservative adjustment and most likely can be improved on.

This stabilize/adjust process is repeated until the temperature readings stabilizes at the set point.

Calculating the heating element duty cycle

Based on the following settings:
Set Point: 200F
Estimated duty cycle: 50% (R1)
Control range: 40 degrees (B1)
Temperature reading: 190F (R2)

A2 to B2 is the same ratio as A1 to B1:
A2 = B2 * (A1 / B1)

C2 = COT(A2)

C1 to D1 is the same ratio as C2 to D2:
C1 = D1 * (C2 / D2)

The heater element duty cycle is:
50 + C1

The next section shows how B2 is derived

Determining the shape of the duty cycle adjustment curve

The behaviour of the adjustment curve is modified by selecting different per centages (1-99) of the cotangent cycle. This per cent becomes a factor of PI centered on PI/2.

The 76% curve is what was used in the examples above. This is the value I am using to test my implementation.

The 98% curve has very steep gradients. This would require a very precise output control system.

50% and below become essentially a straight line and would be better suited to a simple linier equation.

The main factor limiting how high you can set this percentage is the precision of the control system. If set too high a dead space will be created around the set point. This is where the temperature reading is above/below the set point but the indicated duty cycle is the same as the current duty cycle.

Current Temperature Reading

One of the things I observed on my PID controller is the variance of the temperature readings. The readings bounced around with a 4-5 degree swing.

I expect to take an average of the last 10 readings and use this as the current reading.

Stable Temperature Determination

This may be the most difficult calculation of the algorithm. Initially, I will base it on a sample of the latest 100 readings. The stability will be determined by the calculated slope and the average will be used for the actual temperature reading.

Adjustment curve and range

In many systems the positive and negative control are not symetrical. To compensate for this different control curves could be generated for adjustment above and below the set point. The control range above and below the set point don't necessarily have to be the same value.

Zero / 100% Duty Cycles

Conditions may reuire that a zero duty cycle may not be completely off or that 100% may not actually imply full power.

Embedded systems

Test Application

The heading line displays the current time and elapsed time.
First column - Displays temperature and duty cycle information. I used large numbers so I could monitor progress without having to have the PC directly in front of me.
Second column - Detailed data about what is going on internally.
Third column - The adjustment curve. A graph of the figures used to calculate the current smoking temperature. A graph of the figures used to determine the smoking temperature stability. The history of the duty cycle.

Control Testing Set Up

Duty Cycle

Arduino Mega 2650

My goal here is to build a completely self-contained smoker controller.