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Let me rephrase that, the spike from the current (controlled by the PID) in the previous "cycle" will cause the voltage in the next cycle to "ramp up" in a sporadic linear manner (think of an irregular saw tooth waveform moving up linearly).
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this sounds suspicious. Im assuming these cycles are separate steps in a process, and that something else (or nothing) is happening between cycles.
If the PID loop is remembering data from a previous cycle, and that is affecting the start of the next cycle, it sounds like a SW variable, or a sensor input is not getting cleared or initialized when the next cycle starts.
Digital PID control loops do have memory built into them, you compare the present reading to the last reading to get the D component - and you sum the error signal to get the I part - so when you are starting a new cycle (or a new process) if all those variables are not zeroed out (or initialized to a starting value) then yes, data will be carried over from the last cycle to the next and screw up the PID control.