Multiplicative Vs Additive (PID) Feedback Loops

Question

I normally think of PID feedback loops as being additive. This might not be quite standard, but, e.g. for a simple discrete-time proportional gain loop,

$$ C_{t+1}=C_{t}+G(S-M_t) \tag 1 $$

Where $C_{t}$ is the input at time stamp $t$, $G$ is my loop gain, $S$ is my setpoint or desired measured value, and $M_t$ is the measured value. Then of course one could add integrator and derivative terms, but the way I've written it we are adding the PID response to our control variable.

However sometimes in the right context it makes more physical sense to implement a multiplicative feedback loop, e.g.

$$ C_{t+1} = \frac{S}{M_t} C_{t} \tag 2 $$

However while there are books written about PID feedback loops I realized that I'm not sure I've ever heard of a formal description or discussion of feedback loops that operate like this, such that I'm not sure i even know the right words to search for. Does this have a name? Are there any good references for this?

Answer 1

In the case of your example, you can take the log of both sides to get a homologue of your (1) that is linear:

$$\log C_{t+1} = \log C_t + \log S - \log M_t \tag a$$

(a) is linear in $\log C_t$ and $\log M_t$. It also immediately inspires an equivalent to proportional gain: modifying (a) to

$$\log C_{t+1} = \log C_t + G \left (\log S - \log M_t \right) \tag b $$

is the same as modifying (2) to

$$C_{t+1} = \left( \frac {S}{M_t} \right)^G C_t \tag c$$

and may prove useful.

I have implemented AGC loops for communications receivers in log-amplitude space, using a PI controller to maintain a constant average amplitude*.

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* This isn't advised for voice communication -- at least, the guy giving me advise says it sounds unnatural, and at low signal-to-noise ratio the static would get extra irritating. But it works well for a data receiver, which is what I was building.