# PID plant model: 1st order + integrating?

I’m trying to control the temperature of the heating surface of an electric cooking pan, filled with water (between room temp and boiling temp).

The heating power (controller output) is made by applying a duty cycle signal to an SSR heating, at 1Hz.

I heated the surface to 60°C and let the temperature stabilize. Then I made a step increase from 0% to 30% on the duty cycle.

I am unable to figure out what is the best plant model to use on my PID control since the step response (see attachment) shows a hybrid between 1) first-order plus dead time (FOPDT) and 2) Integrating plus dead time response. The ripples are caused by the SSR switching ON & OFF (effect of the duty cycle).

Reversing the problem unless you go for some form of model based predictive controller your stuck with PID anyway. It’s been a good many years since I studied control theory and looked at transfer functions , and I’ve never used any of it !

I don’t think your graphs tell you much - BUT it will be non linear as the heat required to go from say 30-50C will be less than that for 70-90C as evaporation and radiant heat loss is changing . There will be lag/overshoot due to the thermal mass of the heater and pan support, so you won’t see an immediate change in temperature for a change in power.
If it was me I’d tune up a PID loop for it and see how it goes , and if need be do a bit of gain/integral scheduling based on either set point or measured temperature!to assist with any non linearity . It will be suboptimal due to the lags or you might get a massive overshoot .

How accurate do you want the control to be anyway -? Where you measure the temperature has a big bearing as does stirring the water .

Make sure to measure the temperature in lockstep with the PWM cycle just before the heater switches on or you'll get artifacts from the point in the cycle changing between iterations. Your ripple in that diagram is not at 1Hz so I suspect its such an artifact caused by the iteration frequency differing from the PWM frequency.