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?”

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.

—

“I had to come

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

that made the linear model stable also made the nonlinear

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.

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

that made the linear model stable also made the nonlinear

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.

https://nadertg.github.io/MechanicsData/double_pend.gif

====