It depends. While true when the signal is periodic, it is not so in general.
I skipped some details. A crucial condition is that the voltage be bounded in the long term, which excludes the exponential example. Or for finite intervals, if the voltage is the same at the beginning and the end, then over that interval there will be zero correlation with its first derivative. This is true regardless of periodicity. It can be completely random (but differentiable, and well-behaved enough for the correlation coefficient to exist), and the zero correlation will still hold.
Of course, if the control system is good enough, in practice the correlation will drown in the noise.
For every control system that works well enough to be considered a control system at all, the correlation will totally drown in the noise. It will be unmeasurably small, and no investigation of the system using statistical techniques can succeed if it is based on the assumption that causation must produce correlation.
For example, take the simple domestic room thermostat, which turns the heating full on when the temperature is some small delta below the set point, and off when it reaches delta above. To a first approximation, when on, the temperature ramps up linearly, and when off it ramps down linearly. A graph of power output against room temperature will consist of two parallel lines, each traversed at constant velocity. As the ambient temperature outside the room varies, the proportion of time spent in the on state will correspondingly vary. This is the only substantial correlation present in the system, and it is between two variables with no direct causal connection. Neither variable will correlate with the temperature inside. The temperature inside, averaged over many cycles, will be exactly at the set point.
It’s only when this control stystem is close to the limits of its operation—too high or too low an ambient outside temperature—does any measurable correlation develop (due to that approximation of the temperature ramp as linear breaking down). The correlation is a symptom of its incipient lack of control.
no control system without complete future knowledge of inputs is perfect.
Knowledge of future inputs does not necessarily allow improved control. The room thermostat (assuming the sensing element and the heat sources have been sensibly located) keeps the temperature within delta of the set point, and could not do any better given any information beyond what it has, i.e. the actual temperature in the room. It is quite non-trivial to improve on a well-designed controller that senses nothing but the variable it controls.
The capacitor is just a didactic example. Connect it across a laboratory power supply and twiddle the voltage up and down, and you get uncorrelated voltage and current signals.
Somewhere at home I have a gadget for using a computer as a signal generator and oscilloscope. I must try this.
On the other hand, I’d guess that 99% of actual capacitors are the gates of digital FETs (simply due to the mindbogglingly large number of FETs). Given just a moment’s glimpse of the current through such a capacitor, you can deduce quite a bit about its voltage.
For every control system that works well enough to be considered a control system at all, the correlation will totally drown in the noise.
False. Here (second graph) is an example of a real-life thermostat. The correlation between inside and outside temperatures is evident when the outside temperature varies.
The thermostat isn’t actually doing anything in those graphs from about 7am to 4pm. There’s just a brief burst of heat to pump the temperature up in the early morning and a brief burst of cooling in the late afternoon. Of course the indoor temperature will be heavily influenced by the outdoor temperature. It’s being allowed to vary by more than 4 degrees C.
I skipped some details. A crucial condition is that the voltage be bounded in the long term, which excludes the exponential example. Or for finite intervals, if the voltage is the same at the beginning and the end, then over that interval there will be zero correlation with its first derivative. This is true regardless of periodicity. It can be completely random (but differentiable, and well-behaved enough for the correlation coefficient to exist), and the zero correlation will still hold.
For every control system that works well enough to be considered a control system at all, the correlation will totally drown in the noise. It will be unmeasurably small, and no investigation of the system using statistical techniques can succeed if it is based on the assumption that causation must produce correlation.
For example, take the simple domestic room thermostat, which turns the heating full on when the temperature is some small delta below the set point, and off when it reaches delta above. To a first approximation, when on, the temperature ramps up linearly, and when off it ramps down linearly. A graph of power output against room temperature will consist of two parallel lines, each traversed at constant velocity. As the ambient temperature outside the room varies, the proportion of time spent in the on state will correspondingly vary. This is the only substantial correlation present in the system, and it is between two variables with no direct causal connection. Neither variable will correlate with the temperature inside. The temperature inside, averaged over many cycles, will be exactly at the set point.
It’s only when this control stystem is close to the limits of its operation—too high or too low an ambient outside temperature—does any measurable correlation develop (due to that approximation of the temperature ramp as linear breaking down). The correlation is a symptom of its incipient lack of control.
Knowledge of future inputs does not necessarily allow improved control. The room thermostat (assuming the sensing element and the heat sources have been sensibly located) keeps the temperature within delta of the set point, and could not do any better given any information beyond what it has, i.e. the actual temperature in the room. It is quite non-trivial to improve on a well-designed controller that senses nothing but the variable it controls.
Exponential decay is a very very ordinary process to find a capacitor in. Most capacitors are not in feedback control systems.
The capacitor is just a didactic example. Connect it across a laboratory power supply and twiddle the voltage up and down, and you get uncorrelated voltage and current signals.
Somewhere at home I have a gadget for using a computer as a signal generator and oscilloscope. I must try this.
On the other hand, I’d guess that 99% of actual capacitors are the gates of digital FETs (simply due to the mindbogglingly large number of FETs). Given just a moment’s glimpse of the current through such a capacitor, you can deduce quite a bit about its voltage.
False. Here (second graph) is an example of a real-life thermostat. The correlation between inside and outside temperatures is evident when the outside temperature varies.
The thermostat isn’t actually doing anything in those graphs from about 7am to 4pm. There’s just a brief burst of heat to pump the temperature up in the early morning and a brief burst of cooling in the late afternoon. Of course the indoor temperature will be heavily influenced by the outdoor temperature. It’s being allowed to vary by more than 4 degrees C.
OK, maybe I misunderstood your original point.