# FOUND IT! I knew a thing called the Lyapunov function was key to a lot of the magic of non-linear calculations and wondered “did anybody remodel phase locked loops using Lyapunov?” Yup. “Lyapunov Redesign of Classical Digital Phase-Lock Loops” http://dabramovitch.com/pubs/lpll_cdig3.pdf I don’t know if I’ll understand it but I’m gonna try to understand some of it.

What I’m thinking is : they’re both forms of stablility. One (logic) is more determinate than the other (metastable).But if the metastable has primacy with the logic function carved out as a “special case” of metastability, these run conditions would no longer be error but an integral part of system functioning.
—–
FOUND IT! I knew a thing called the Lyapunov function was key to a lot of the magic of non-linear calculations and wondered “did anybody remodel phase locked loops using Lyapunov?”

Yup.

“Lyapunov Redesign of Classical Digital Phase-Lock Loops”

http://dabramovitch.com/pubs/lpll_cdig3.pdf

I don’t know if I’ll understand it but I’m gonna try to understand some of it.
Oh I like this guy:
https://ieeexplore.ieee.org/abstract/document/7320636
” Trying to keep it real: 25 Years of trying to get the stuff I learned in grad school to work on mechatronic systems “

What works from school and what fails.
to terms with the third theme, that is that many, many
practical control problems are solved quite simply with
very little deep consideration of unmodeled dynamics.”

You don’t have to know WHY it works.
—-
‘
VI. PHASE-LOCKED LOOPS: SO MUCH FEEDBACK,
SUCH SIMPLE ANALYSIS
Early in my career, a fellow engineer at Ford Aerospace
named Dan Witmer walked into my cubicle and asked me
how I would do nonlinear analysis for a phase-locked loop.
“What’s a phase-locked loop?” Once he patiently explained
it to me, I blindly stated that I would simply use Lyapunov
redesign [6], [31]. While nonlinear analysis of phase-locked
loops was an interesting subject, it was not the main learning
point of these devices. I have claimed that PLLs are the most
ubiquitous feedback loops built by humans [34], showing
up in all of our smart phones, digital watches, and every
other computational device we have, and yet the feedback
analysis done in textbooks is only of the simplest type
[35], [36], [37]. Discussions with PLL experts at several
companies also showed that while they knew intricacies of
the circuits and the envelope behavior of the phase detector,
they generally used only very simple linear feedback analysis
in their designs. They paid surprisingly little attention to the
stability of the PLL.It took a while to understand that since most PLLs were
ﬁrst or second order, and for most of these one could show
that even the most basic rules of ﬁlter design led to closed-
loop responses for the phase-space where the parameters
model stable [6], [38]. (In this case phase space refers to the
modulation domain or the baseband or envelope behavior of
the PLL.) These loops were simple and stable, because the
“plant” to be controlled was always an integrator, as shown
in the lower drawing of Figure 5, and the loop ﬁlter was
either a gain or a ﬁrst-order lag, which resulted in stable
closed-loop behavior of the system. The Lyapunov analysis
showed that in these cases the parameters that made the
second-order linear model stable also made the second-order
nonlinear model stable. For the circuit designers creating
PLLs, they knew from experience that even the simplest,
dumbest controller (a.k.a. loop ﬁlter) would produce a stable
response and so they gave it no mind. Higher order PLLs
failed this, and thus were harder to analyze [5].If our open-loop system is adequately modeled by an
integrator, then feedback control of that open-loop system
becomes trivial (Section XII). It is easy to show that the ﬁrst
and second-order analog PLLs are always stable, even with
the sinusoidal phase-detector nonlinearity [6]. Furthermore,
with the right discretization, even the classical discrete-time
PLL was stable [38]. What about harmonics? Didn’t the
designers need to throw in some ﬁlters to get rid of signal
at the carrier frequency and its harmonics? Why were these
not in the textbooks? The answer came from HP/Agilent
PLL expert, Rick Karlquist, who laughed and said (more or
less), “Well, no RF engineer worth their salt wouldn’t know
to put in those ﬁlters! It goes without saying. That’s why it
doesn’t have to be put in the textbooks.” In this moment I
realized that any PLL that was not second order was likely
to be beaten into a form that looked second order. All those
analog RF ﬁlters were there, but they were never considered
part of the analysis. A lot of work was done to make the
system look like an integrator, and then control was done
from there. What kind of control was done? Both kinds, lag
ﬁlter and PI control. 4 The designers were left to work out the
more taxing PLL design problems of having a good phase
detector and an oscillator with minimal phase noise.
https://arxiv.org/abs/1606.06570
-===
“No physical implementation of a digital circuit can reliably avoid, resolve, or detect metastability; any digital circuit, including “detectors,” producing different outputs for different input signals can be forced into metastability”
—-
Kleene’s 3-valued logic can model metastability

—–
. In addition, fractal structure is considered a hallmark of metastability (Friston, 1997; Freeman and Holmes, 2005; Werner, 2010; Tognoli and Kelso, 2014) in the brain. Metastability is the ability of neural systems to integrate functionally segregated entities (e.g., brain areas, end effectors) in a task relevant manner in space and time. Thus, it is an indicator of our ability to adapt and coordinate to the external world.
When I research how researchers research: I look for their shortcuts, proxies, substitutions, name changes, and tricks.

Looking at a 2020 paper about metastability in the brain, I find this:

“Finally, we apply a proxy for metastability-the standard deviation of the Kuramoto order parameter”.

ah ha! What’s Kuramoto? A nice shortcut.