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Electrical
I'll draw heavily on Wikipedia in this post. Here's a diagram of a feedback amplifier:Engineers think of how this circuit would operate on sinusoids, so the open loop gain A_{OL} and the feedback factor β may be frequency dependent. The formula for closed loop gain is:
If β is positive, in this convention, the feedback is negative, because it is fed back to the negative input port. And the closed loop gain is less than open-loop. But note that the statement that it is negative is frequency dependent.
The closed loop gain is the factor by which you multiply a frequency component of the input I(f) to get the output O(f).\[O(f)=A_{fb}(f) I(f)\] And if you go to the time domain by inverse Fourier Transform, this becomes a convolution: \[O(t)=\int_{-\infty}^{t}\hat{A}(t-\tau)I(\tau) d\tau\] where \(\hat{A}\) is the inverse Fourier Transform of the closed loop gain A_{fb}
Factor: For consistency, set α = -β. Then the ranges are:
- α < 0 : negative feedback - stable
- 0 < α < 1/A_{OL} : positive feedback - stable (but getting less so)
- α > 1/A_{OL} : positive feedback - unstable (oscillation)
Z-transform
EE's often use a Z-transform to represent time series in a kinda frequency domain way. I think of it as a modified discrete-time Fourier transform. It renders the transfer function as, often, a rational function and allows you to study the stability and other aspects of the response in terms of the location of its poles. More laterTime Series
Feedback is the AR part of ARMA. The transfer function is analogous to a version with exogenous inputs (ARMAX). Or you can look at the original version as formally analogous with the noise acting as input.Anyway, if you treat the lag operator as a symbolic variable, you see again a rational function acting as a transfer function. That is developed into a Z-transform equivalence here.
Factor: In time series, there isn't a commonly used equivalent of the boundary between positive and negative feedback, in the amplifier sense. There is a stability criterion, which can be got from the Z-transform expression. It requires locating the roots of the denominator polynomial (poles). Then, in this formulation, the roots have to have magnitude >1 for stability. For 1st order:
\[y_n+a_1 y_{n-1}=...\]
the requirement is that |a_{1}| < 1.
Climate - equilibrium
This goes back at least to the Cess definition used by Zhang et al 94, for example. Suppose you have equilibrium T=0 and flux=0. Then you impose a forcing G W/m2. The system incorporates a number of temperature dependent flux terms F_{i}, which, with the sign convention of G, each vary with T asΔF_{i} = -λ_{i} ΔT.
So the new equilibrium temperature is
T = G/λ,
where λ is the sum of the λ_{i}. The nett feedback is negative, else runaway, but the components of the sum could be positive or negative.
Globally, for example, there is a base "no-feedback sensitivity" feedback of about 3.3 W/m2/C, based just on BB radiation. The exact amount is not important here. That gives the often quoted 1.2 C per CO2 doubling. Although not usually thought of as a feedback, it goes into the sum of feedback factors. The λ_{i} that add to it are called negative feedbacks, because they add to the stabilization. Those that subtract are positive.
Comparison with Elec and Time Series - much simpler.It's equilibrium (DC) - there are no dynamics. Time does not appear.
Climate - non-equilibrium
DS11 and SB11 add some thermal dynamics, with the equation \[C dT/dt = G - \lambda T\] measuring the time response to the perturbation provided by G. Their time scales are too short to assume equilibrium. This approaches steady state as the temperature approaches G/λ, the sensitivity value. C is the thermal capacity. Adding the gross dynamics does not change the feedback concepts (though it expresses the potential for thermal runaway). In EE terms, C is a single capacitance,and you could think of it as a resistance (1/λ)-capacitance(C) circuit. But I don't think that changes any of our feedback issues.
[Added:]
The DE shows an ARMA analogue if you convert the derivative to a difference:
\[T_n-T_{n-1} = -\lambda {\delta}t T_n + {\delta}t F(t_n) \] or
\[T_n = (T_{n-1}+ {\delta}t F(t_n))/(1+\lambda {\delta}t) T_n \]
which starts to look like the closed loop gain equation. But it's also very like ARMAX(1,0,1)
The differential equation has the solution:
\[O(t)=\int_{-\infty}^{t}e^{\lambda*(t-\tau)}F(\tau) d\tau\]
very like the iFT time domain expression of the closed loop gain. It has, however, a restricted transfer function, corresponding in EE terms to a single pole. This can be seen by Laplace Transform:
\[s \hat{T} - T(0)=-\lambda \hat{T} +\hat{F}\]
or \[\hat{T}= (T(0) + \hat{F})/(s +\lambda)\]
There are plenty of ways to generalise the de approach. T could be a vector and λ a square matrix, which would give multiple poles. Or λ could be a function of t, perhaps with a convolution operation.
Comparison with Elec and Time Series - simple dynamics of heat storage. But there are no time-scales associated with the individual feedbacks. They are not reactive. I don't think there is any need to consider phase shift in the feedback.
That's useful. In my experience lapsed EEs are often thrown by positive feedback, assuming it must inevitably lead to runaway.
ReplyDeleteThis was initially incomprehensible to me, as your feedback equation shows - positive feedback < 1 does not lead to runaway. And indeed this is the basis of many early (super-regenerative) radio receivers, which used a single amplifier stage with positive feedback to achieve high gains with few components.
I think the problem may spring from the fact that many electronic components vary in gain with temperature, and so the gain can vary significantly over time (including going > 1), as a result of which positive feedback tends to be avoided in practice.
Kevin C
Kevin,
ReplyDeleteYes, you can hear the super-regenerative effect in an acoustic feedback loop just below where it starts howling. Very frequency-selective - it sounds awful.
There's the same gap in climate. WV feedback is positive but does not presently lead to runaway.
Nick Stokes: WV feedback is positive but does not presently lead to runaway.
ReplyDeleteGood discussion, NIck.
Only comment I have to add is that in general when you have net amplification (positive feedback > 1), this is unphysical unless there is a corresponding stabilizing nonlinearity (mathematically the nonlinearity needs to appear in the "damping" sector of the system of differential equations. Nonlinearities that just make the system stiffer for example, won't stabilize runaway oscillation.)
In general effect of the nonlinearity will be to "dampen" the amplification, so that at a sufficiently highly amplitude operating point, the system will remain stable, in spite of the underlying instability.
If you start the system at a low enough temperature, the system will 'run away' until it reaches a "just stable" operating point. Similarly, start it at too high of a temperature, and it will drop until it oscillates about the stability point.
Likely one can extend these concepts to the case for example of a "step function" in CO2 forcing (pick either sign).
Another point to make is the fact that water can change state in our climate, and the frozen version of it drastically affects the net albedo of the Earth, there is a sort of net-amplification/nonlinearity rolled into one here.
Get enough warming, and you will get a runaway condition, in which most of the ice melts (as the ice albedo decreases, that acts as a stabilizing nonlinearity). Similar idea if you get enough cooling, the increase in albedo can lead to a runaway precipitation event (e.g., an ice age).
I'll let you take a stab at that example, if you want, as to what that would be classified as, in EE literature. I suspect chemistry may have language better suited for this type of scenario.
Thanks, Carrick.
ReplyDeleteYes, people talk too freely about positive feedback leading to infinite heating etc. It's really just a transition to a state where non-linearities change the gradient so the same feedback doesn't apply. I think the classic electric circuit with high positive feedback is the bistable multivibrator. It takes a pulse to get from it's quasi-stable state, then quickly passes through the region where it is a positive feedback amplifier, then to its other state.
In climate, one time where we might have had instability was in the recovery from the Younger Dryas period. A sudden temperature change (maybe) but soon limited by the non-linear response.
In chemistry, I think the language is "explosion" :)
Or maybe "runaway chemical reaction"? Or thermal runaway?
ReplyDeleteIt's also instructive to look at historical examples of
runaway climate change and look at the reasons they were thought to occur. (This doesn't include ice ages, which I think should be considered too as well as sudden shifts from glaciation to near ice free conditions.)
These seem to be the main ones:
Water freezing (response to cooling, negative feedback of frozen water)
Ice melting (response to heating from other forcings, usually)
Methane released (associated with thawing or tectonic activity)
Massive volcanic eruptions (tectonic activity).
Asteroid impacts.
Biosphere disruptive events (introduction of new species, either through evolution or tectonic plates.)
Nice post Nick. You sent me around the internet reading up on Z transforms.
ReplyDeleteThanks, Jeff,
ReplyDeleteI've always liked Z-transforms - polynomials seem to look more natural.
And β of the climate system is -0.4 because the ~200 W/M2 solar absorbed by the system is amplified to ~500 W/m2 to the surface.
ReplyDelete