You need to know what right-half-plane zeros are(jbconsulting.substack.com)
jbconsulting.substack.com
You need to know what right-half-plane zeros are
https://jbconsulting.substack.com/p/you-need-to-know-what-right-half
294 comments
There was nearly a disaster earlier in that flight too, when the pilots got sick from food poisoning and the cabin crew had to ask if any of the passengers knew how to fly a plane. As it turned out, one of the passengers had flown many years previously, and offered to try.
When he reached the cockpit, however, he took one look at all of the controls on the modern aircraft and realized that he wasn't going to be able to operate it. When the chief steward asked why, he replied, "I am just a simple Pole in a complex plane".
I'll follow lb1lf out.
When he reached the cockpit, however, he took one look at all of the controls on the modern aircraft and realized that he wasn't going to be able to operate it. When the chief steward asked why, he replied, "I am just a simple Pole in a complex plane".
I'll follow lb1lf out.
Why did the man from Gdansk marry the woman from Zakopane? 'Opposite Poles attract'.
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It has a grain of truth to it. Poland could donate MiGs to Ukraine, but other countries with US fighters could not, because Ukrainian soldiers didn't know how to fly them.
Actually not true. Many Ukrainian fighter pilots have received training in the US, where they flew F-16:s. (That is, before the current phase of the conflict.)
The hard part is actually the maintenance crews. A modern fighter plane requires constant intensive mainentance, they generally spend much more time being worked on than they spend actually in the air. And this maintenance work requires a lot of specific skills that don't necessarily translate well or at all between aircraft types.
The hard part is actually the maintenance crews. A modern fighter plane requires constant intensive mainentance, they generally spend much more time being worked on than they spend actually in the air. And this maintenance work requires a lot of specific skills that don't necessarily translate well or at all between aircraft types.
And a steady supply of parts.
Now this made me laugh like crazy
Did we have the same professor? Mine loved that joke too.
The students in the classroom next door always knew when Prof Lipovski told his joke because everybody along the hallway could hear the loud groan emanating from our room.
The students in the classroom next door always knew when Prof Lipovski told his joke because everybody along the hallway could hear the loud groan emanating from our room.
Heard a varient at Berkeley from a professor of Applied Math, which you would probably no longer hear.
Generic airline, specifically Polish people were asked to move to the left side of the plane. Because "Poles in the right half plane cause instability."
(As for application in economics, economist Steve Keen, for one, explicitly models time delays and dynamics, with results worth learning from, AFAIK.)
Generic airline, specifically Polish people were asked to move to the left side of the plane. Because "Poles in the right half plane cause instability."
(As for application in economics, economist Steve Keen, for one, explicitly models time delays and dynamics, with results worth learning from, AFAIK.)
I think all systems professors have the same joke book and there's only one page in it :)
The version my professor told was, “Why is politics in Poland so volatile? Because all the Poles on the right are unstable!”
And you know why the contour integral of Western Europe is zero? Because all the poles are in Central Europe.
(sorry)
(sorry)
As a resident of post-Brexit Britain, I was horrified as I read this expecting it to have the opposite punchline and to turn into an anti-immigration joke.
For those that don't know, Polish (and other Eastern European) migration was a major issue raised by some in the Brexit campaign.
For those that don't know, Polish (and other Eastern European) migration was a major issue raised by some in the Brexit campaign.
A more horrific and offensive version of the joke, which I actually saw recently as output from fortune [1], adds a part about the contour integral of the whole of Europe still being zero because the Poles are removable.
[1]: https://en.wikipedia.org/wiki/Fortune_(Unix)
[1]: https://en.wikipedia.org/wiki/Fortune_(Unix)
The article saying zeros (where the transfer function is 0) when it should be saying poles (where the denominator of the rational function is zero), right?
Controls engineer here. The author’s control theory knowledge is correct - a RHP zero indicates that corrective action will begin in the wrong direction (which the article compares well with countersteering on a bike). The technical term is a “non minimum phase system”. It’s possible this will lead to instability, but in general the long term stability of a system is determined by the location of its poles (which is more applicable to the author’s ice cream example). Poles in the RHP will cause a system to blow up.
My gripe with the article is that the author tries to wow you with some obscure technical points about a system which is unmodeled and he does not understand, to wave his hands at a vague conclusion. If he had made the same point using common English phrases that encapsulate the idea (“positive feedback loop”, “we have to let it get worse before it gets better”, etc), then it would be a lot clearer how wispy his argument is.
My gripe with the article is that the author tries to wow you with some obscure technical points about a system which is unmodeled and he does not understand, to wave his hands at a vague conclusion. If he had made the same point using common English phrases that encapsulate the idea (“positive feedback loop”, “we have to let it get worse before it gets better”, etc), then it would be a lot clearer how wispy his argument is.
The real answer is Ilyushin made terribly finicky engines and people skimped on maintenance but that’s another story.
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I hate to be the guy to ask you to explain the joke, but why would that make the plane crash?
I do know what a “pole” refers to, but I don’t follow the crash bit.
I do know what a “pole” refers to, but I don’t follow the crash bit.
A little bit broader than you asked for, but:
A dynamic control system is modeled by a set of dynamic equations, usually expressed as partial derivatives. To analyze the behaviour of such a system, the equation is solved or approximated in the complex time domain. The relevant part of the solution is where the real part of time is positive, i.e. the right-half plane.
A pole is a coordinate for which the dynamic equations have no solution (y = 1/z has a pole at z=0), which results in undefined or uncontrolled behaviour.
(also see https://en.wikipedia.org/wiki/Zeros_and_poles)
A dynamic control system is modeled by a set of dynamic equations, usually expressed as partial derivatives. To analyze the behaviour of such a system, the equation is solved or approximated in the complex time domain. The relevant part of the solution is where the real part of time is positive, i.e. the right-half plane.
A pole is a coordinate for which the dynamic equations have no solution (y = 1/z has a pole at z=0), which results in undefined or uncontrolled behaviour.
(also see https://en.wikipedia.org/wiki/Zeros_and_poles)
Honestly, this comment better than the article.
I don't think a vague but more precise mathematical explanation of the terms zero and pole are even that difficult to understand, x has a zero, 1/x has a pole, people kind of know what that means if you look at a graph of a pole, I don't think a rigorous definition of pole is that far off - a pole of f is just a zero of 1/f.
Instead we get waffle like:
> Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
I know it's "roughly" speaking, but isn't it too rough?
Instead we get waffle like:
> Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
I know it's "roughly" speaking, but isn't it too rough?
A control system with a pole (in the control systems sense) in the right hand half of the plane will be unstable. If the system changes during operation, the pole can move around the coordinate plane.* As it goes from the left (stable) towards the right (unstable) the system will begin to oscillate as the pole crosses the Y axis.
In the joke, a Pole (person of Polish origin) was in the left of the (air) plane. When they crossed the aisle to look out the window, the (air) plane became unstable and crashed (pole in the right hand plane).
*My control systems professor loved to explain using an example of a driver as a control system. The system (car + human driver) seeks to minimize error against the lines on the road. If the driver starts drinking, one of the system's poles will move right. The car will start overshooting first, then will start weaving, before finally crashing when the driver is too drunk.
In the joke, a Pole (person of Polish origin) was in the left of the (air) plane. When they crossed the aisle to look out the window, the (air) plane became unstable and crashed (pole in the right hand plane).
*My control systems professor loved to explain using an example of a driver as a control system. The system (car + human driver) seeks to minimize error against the lines on the road. If the driver starts drinking, one of the system's poles will move right. The car will start overshooting first, then will start weaving, before finally crashing when the driver is too drunk.
tl; dr: The system is unstable as it has positive feedback loops. You can "first day of class" think of it as the implied series 1/(s-p) -- p the pole, s the Laplace variable -- exists and converges.
And in particular you want that series to converge on the imaginary axis which means it does not diverge in the frequency domain (Fourier transform). Essentially, that means that you have regions of frequencies where your system diverges/amplifies them excessively and thus breaks down: is unstable. Filters do the opposite.
P.S. The other note is that for real linear time invariant systems the region of convergence of the series/Laplace transform of the system must be positive for the system to be causal -- and thus real and implementable. So the joke could also have been modified to get a magical and unstable plane.
P.S. The other note is that for real linear time invariant systems the region of convergence of the series/Laplace transform of the system must be positive for the system to be causal -- and thus real and implementable. So the joke could also have been modified to get a magical and unstable plane.
> You can "first day of class" think of it as the implied series 1/(s-p) -- p the pole, s the Laplace variable -- exists and converges. And in particular you want that series to converge on the imaginary axis which means it does not diverge in the frequency domain (Fourier transform)
If this was the first day of any class I took, I would have dropped it before the second day.
If this was the first day of any class I took, I would have dropped it before the second day.
That makes me think of calculus in freshman's year. The first week the prof explained all maths we had learned in highschool, and then seemingly continued that same pace every week, it was rough. Especially for some of the smart kids who had never experienced learning material coming at them faster than they could take it in. The types who opened their textbook the night before the exam and would ace it in highschool got a real test of character.
That's why letting kids cruise in high school is a terrible mistake. So many school systems are uninterested in making sure everyone is challenged, especially in maths.
University is good because you'll meet lots of people who are smarter than you are.
University is good because you'll meet lots of people who are smarter than you are.
I don't even know how kids cruised in high school and still got their diplomas.
I flunked out of high school despite getting A's and B's on all my tests because I never did my homework which was always at least 50% of our grade.
Trigonometry was so interesting to me as a 15 year old that I decided to make it my internet alias. 95+% on every test (even trig identities!), still got a C in the class, with the teacher taking me aside 1-on-1 to tell me "I'm breaking the rules to give you this C when I'm supposed to be giving you an F because you only did 2 of the 30 homework assignments."
I flunked out of high school despite getting A's and B's on all my tests because I never did my homework which was always at least 50% of our grade.
Trigonometry was so interesting to me as a 15 year old that I decided to make it my internet alias. 95+% on every test (even trig identities!), still got a C in the class, with the teacher taking me aside 1-on-1 to tell me "I'm breaking the rules to give you this C when I'm supposed to be giving you an F because you only did 2 of the 30 homework assignments."
In my highschool maths grades were always 100% based on tests.
I get the idea of motivating students to do their homework failing them when they test perfect doesn't make sense.
I like it when the homework allows the students to skip questions on the test. That way you reward the work but still let's the students catch up if they didn't do the homework.
I get the idea of motivating students to do their homework failing them when they test perfect doesn't make sense.
I like it when the homework allows the students to skip questions on the test. That way you reward the work but still let's the students catch up if they didn't do the homework.
I like what my Calculus teacher in college did. Homework was only 3% of your grade, but if you did at least 80% of the homework, then you could redo any questions you got wrong on the tests and midterm.
That's why I feel it's good to learn a bit about Laplace transform for anybody doing anything technical. It has so many applications you can hardly get away from it.
Acausal systems are implementable. They're common in signal processing and even in some rather exotic control systems:
https://www.researchgate.net/publication/253627368_Feedforwa...
https://www.researchgate.net/publication/253627368_Feedforwa...
Because if your poles are not in the left half [of the] plane, the system is unstable.
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The weight imbalance from everybody rushing to the right made the plane unstable.
Interestingly, I recently spoke to a loadmaster who told me that left/right side weight distributions are far less important than forward/aft, to the point that for the majority of aircraft they're loading, imbalance to the left or right side of the plane aren't accounted for at all.
Yes you want the center of mass to be more forward than the center of lift. Otherwise, a small deviation in the pitch becomes hard to impossible to recover -- the system is unstable and stalls extremely easily. If that is a fighter plane of course you want that, so you make sure the opposite is true, and have a computer counteract that during normal flight.
I laughed the first time I made an airplane with the centre of mass behind the centre of lift in Kerbal Space Program.
this is the only answert to the question above and it's being downvoted? welcome to hacker news ...
I didn't downvote it, because it's not a bad attempt at an answer if you don't know the context, but the actual joke is completely different and about calculus and control systems. The right side of the plane is double meaning for the right half of the complex plane in dynamical systems analysis, which is the area where the real part of every point is positive. If your poles are on the right half of the complex plane, then the system is unstable and the output will tend towards exponential growth (going out of control) for any change in input.
The question is about the literal meaning of the joke, he knows what a pole is within the context of calculus. He didn't understand within the context of the joke what lead to the crash.
It's a joke. There is no cause within the literal context. The humor comes from word play.
After the plane crashed, one survivor was stranded in the wilderness. Miles from civilization, he cried and screamed until he got hoarse. Then he mounted the horse and rode back to civilization. Back at home, he found himself locked out of his house, since he lost his house keys with his luggage on the plane. He sat on the porch and sang various lamentations, until he found the right key and unlocked the door.
After the plane crashed, one survivor was stranded in the wilderness. Miles from civilization, he cried and screamed until he got hoarse. Then he mounted the horse and rode back to civilization. Back at home, he found himself locked out of his house, since he lost his house keys with his luggage on the plane. He sat on the porch and sang various lamentations, until he found the right key and unlocked the door.
The joke relies on the fact that keys unlock doors, this is the cause, just like the prior one relies on the assumption that people rushing to one side crashes the plane.
If there are 10 comments explaining what a key is in the context of music and I add another that says "I understand what the key refers to but how was the door opened", one can only assume I lack the knowledge that keys open doors.
Sometimes people understand the more complex behaviours but somehow miss the simpler explanations, it's happened to me before.
If there are 10 comments explaining what a key is in the context of music and I add another that says "I understand what the key refers to but how was the door opened", one can only assume I lack the knowledge that keys open doors.
Sometimes people understand the more complex behaviours but somehow miss the simpler explanations, it's happened to me before.
But the plane didn’t crash because of weight imbalance - if the Poles all went to the left half of the plane, it wouldn’t have crashed, because that would make the system stable!
There are enough actual LOT crashes he can form a model on his own…
That joke is at least 38 years old, as I heard it in my controls class in 1984 :)
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I think this article is trying to make a political statement, "Our leaders have just done something that is going to cause a future problem, a crash".
I have only read the linked article, I think the followup might be available, but I'm not going to read it before posting this -- because this article was meant to be clickbait, to get you hooked and possibly mis-informed.
The article as it stands (incomplete) contains some classic argument fallacies.
It is an "appeal to authority", by attempting to explain control theory. This person knows something.
The examples all result in bad things, airplanes crashing, hard drives (crashing - perhaps a stretch). Bicycle at the edge of a cliff. Eating too much ice cream. Chernobyl. This is a possible "appeal to emotion".
The final issue is an "incomplete comparison". The graph at the bottom of the article shows housing prices -- all of the previous examples have mathematical models which can explain the behavior. For the last graph no model (the equations) is presented. Without a model you cannot use control theory to analyze the zeros and poles.
The last graph wants you to believe in a continuing upward trend, the fallacious argument, implies that whatever the Canadian political change that has been made is going to make this trend to zero, despite the intention to keep it trending upwards.
Background: A long time ago, as a mathematics undergraduate, I did take a course in control theory, it was run out of the engineering department. It was somewhat awkward as quite a few mathematical terms were not-quite-the-same. I get the same unease when reading this for poles, zeros and right-half-planes. Any engineering textbooks that use j for sqrt(-1) instead of i (because that is taken) is an indicator. Yup, this could be construed as an "appeal to authority".
I have only read the linked article, I think the followup might be available, but I'm not going to read it before posting this -- because this article was meant to be clickbait, to get you hooked and possibly mis-informed.
The article as it stands (incomplete) contains some classic argument fallacies.
It is an "appeal to authority", by attempting to explain control theory. This person knows something.
The examples all result in bad things, airplanes crashing, hard drives (crashing - perhaps a stretch). Bicycle at the edge of a cliff. Eating too much ice cream. Chernobyl. This is a possible "appeal to emotion".
The final issue is an "incomplete comparison". The graph at the bottom of the article shows housing prices -- all of the previous examples have mathematical models which can explain the behavior. For the last graph no model (the equations) is presented. Without a model you cannot use control theory to analyze the zeros and poles.
The last graph wants you to believe in a continuing upward trend, the fallacious argument, implies that whatever the Canadian political change that has been made is going to make this trend to zero, despite the intention to keep it trending upwards.
Background: A long time ago, as a mathematics undergraduate, I did take a course in control theory, it was run out of the engineering department. It was somewhat awkward as quite a few mathematical terms were not-quite-the-same. I get the same unease when reading this for poles, zeros and right-half-planes. Any engineering textbooks that use j for sqrt(-1) instead of i (because that is taken) is an indicator. Yup, this could be construed as an "appeal to authority".
I have now read the followup article.
There are, to me some language red flags. The followup article now uses "we' -- suggesting this is not the work of an individual. This was not present in the initial post.
An attempt is made to build a model. Curiously a visual programming model is now used. No explicit differential equations are given. If the equations are not given there is no way to check the model. So "pole" and "zero" analysis is mute.
There are no tests to validate this model against past situations - an easy and free test, the past history is known. If the model that has been created and can predict the future, why would was it not run it against the past and show the predictive success?
The built model is only presented against the current situation -- is it circumstantial / opinion based.
My understanding of science is -- build a model that can predict the future / what happens accurately, i.e. matches the measurements. The consensus agrees that this is the (current) best model.
Prices are ultimately influenced by a sentiment factor, a value derived from a I have no idea what human feeling / property. Some power "enabled" humans have been endowed with a much higher influence on this parameter. Where is this value expressed in the predictive equations?
There are, to me some language red flags. The followup article now uses "we' -- suggesting this is not the work of an individual. This was not present in the initial post.
An attempt is made to build a model. Curiously a visual programming model is now used. No explicit differential equations are given. If the equations are not given there is no way to check the model. So "pole" and "zero" analysis is mute.
There are no tests to validate this model against past situations - an easy and free test, the past history is known. If the model that has been created and can predict the future, why would was it not run it against the past and show the predictive success?
The built model is only presented against the current situation -- is it circumstantial / opinion based.
My understanding of science is -- build a model that can predict the future / what happens accurately, i.e. matches the measurements. The consensus agrees that this is the (current) best model.
Prices are ultimately influenced by a sentiment factor, a value derived from a I have no idea what human feeling / property. Some power "enabled" humans have been endowed with a much higher influence on this parameter. Where is this value expressed in the predictive equations?
I agree. The first article was interesting when it was talking about control theory, but the hard pivot to economics and politics at the end turned the article from potentially informative to a disingenuous appeal to authority.
The author clearly knows a lot about control theory, but this seems to have led to an idea that he knows a lot about everything. This is a theme I see all too frequently in people who list “systems thinking” as one of their areas of expertise (like this author).
I have the benefit of the doubt and read the second article looking for perhaps some actual modeling and references to econ literature, but he merely refers to a couple papers and then admits that his model was home-grown and will be simpler. Some parts of it might be right, but this has too many elements of “I’m an expert in one thing therefore I’m qualified to talk about everything” for me.
The author clearly knows a lot about control theory, but this seems to have led to an idea that he knows a lot about everything. This is a theme I see all too frequently in people who list “systems thinking” as one of their areas of expertise (like this author).
I have the benefit of the doubt and read the second article looking for perhaps some actual modeling and references to econ literature, but he merely refers to a couple papers and then admits that his model was home-grown and will be simpler. Some parts of it might be right, but this has too many elements of “I’m an expert in one thing therefore I’m qualified to talk about everything” for me.
> There are, to me some language red flags. The followup article now uses "we' -- suggesting this is not the work of an individual. This was not present in the initial post.
First person plural ("we") is normal in technical and scientific writing.
First person plural ("we") is normal in technical and scientific writing.
Yes, agreed. It is a term of inclusivity that is applied to observations and models, the testable items. I don't think "we" it is acceptably used for individual opinions unless it reflects the opinions of multiple authors.
The followup definitely uses the "I" (for opinion) initially, but then switches to "we".
As a reader I am not a participant in "we" unless I agree with the statements being made. I think it is this is what made me frosty and say who is this other person, because it isn't me.
The followup definitely uses the "I" (for opinion) initially, but then switches to "we".
As a reader I am not a participant in "we" unless I agree with the statements being made. I think it is this is what made me frosty and say who is this other person, because it isn't me.
Canada does not have 30 year mortgages. Most home owners have mortgages with 3-5 year terms. Since 3-5 years ago rates were low, and home prices were low, people's mortgage expenses are still low. However that's about to end. As people either bought at much higher prices in the last 2 years, and/or the older mortgages are getting renewed at higher interest rates. This will result in drastic increase in what people need to pay to stay in their home.... possibly causing a market crush.
You are correct that in Canada we don't do 30-year fixed mortgages. We only lock in interest rates for a 5 year term.
However, we have what's called a stress-test. In order to get a mortgage at the previously low 5-year fixed rates of 2%, you still had to qualify as though the rates were >5%. This means that if rates go up to 5% everyone should still be able to afford their payments.
However, we have what's called a stress-test. In order to get a mortgage at the previously low 5-year fixed rates of 2%, you still had to qualify as though the rates were >5%. This means that if rates go up to 5% everyone should still be able to afford their payments.
This is all correct. It is worth nothing that other expenses are increasing quickly and consistently, so the stress test could be undermined slightly.
I don’t mean to be alarmist or anything here. It has just been on my mind as a recent home buyer. The stress test was performed on my finances under conditions which have already changed (for the worse).
I doubt the bottom will fall out of things, but I expect this to bite some people and to hear about it in the medias.
I don’t mean to be alarmist or anything here. It has just been on my mind as a recent home buyer. The stress test was performed on my finances under conditions which have already changed (for the worse).
I doubt the bottom will fall out of things, but I expect this to bite some people and to hear about it in the medias.
>This will result in drastic increase in what people need to pay to stay in their home
This will result in a drastic increase in what a relatively small number of homeowners need to pay to stay in their home. Specifically, those who bought a house to live in in the last 3-4 years.
>possibly causing a market crush.
Before losing their home, this relatively small number of people (plus a larger number of those who will experience much more moderate mortgage payment increases) will cut back on spending for literally everything else. This will have a very significant impact on the rest of our economy. Which, in fact, is the whole point of inflation targeting - cool down the economy when inflation is out of control.
This will result in a drastic increase in what a relatively small number of homeowners need to pay to stay in their home. Specifically, those who bought a house to live in in the last 3-4 years.
>possibly causing a market crush.
Before losing their home, this relatively small number of people (plus a larger number of those who will experience much more moderate mortgage payment increases) will cut back on spending for literally everything else. This will have a very significant impact on the rest of our economy. Which, in fact, is the whole point of inflation targeting - cool down the economy when inflation is out of control.
> Specifically, those who bought a house to live in in the last 3-4 years.
No, everyone in Canada has to renew their mortgage at least every 5 years, into whatever the prevailing interest rate market is at that time.
No, everyone in Canada has to renew their mortgage at least every 5 years, into whatever the prevailing interest rate market is at that time.
The vast majority of 28% of Canadian households who are currently carrying a mortgage did not buy their house at peak bubble prices, i.e. within the last 3-4 years. They carry a much more modest debt load, have paid off more of their mortgage, and this will significantly dampen the effect of rising rates on their finance.
Yes, my understanding is this tends to hit recent buyers and those who have had to remortgage. I don’t have numbers, but it seems reasonable to believe this makes up a small portion of homeowners.
Specifically, those who bought a house to live in in the last 3-4 years.
Also those who renewed in the last 3-4 years. Which realistically means most of those who bought in the last 20-25 years.
That's likely over half of all homeowners - not so small a fraction after all!
Also those who renewed in the last 3-4 years. Which realistically means most of those who bought in the last 20-25 years.
That's likely over half of all homeowners - not so small a fraction after all!
>Also those who renewed in the last 3-4 years.
No. Those who paid the exorbitant prices of the last 3-4 years. I've renewed recently, and borrowed the ~$100k remaining on my mortgage at a historically low rate. If I had to renew at twice that rate it would not make a huge impact on my finances, because I'm currently paying interest on a five-figure principal.
No. Those who paid the exorbitant prices of the last 3-4 years. I've renewed recently, and borrowed the ~$100k remaining on my mortgage at a historically low rate. If I had to renew at twice that rate it would not make a huge impact on my finances, because I'm currently paying interest on a five-figure principal.
But it still would impact you.
Across all Canadians, we're talking about $1.5 trillion (CAN) in mortgage debt. That's on par with the GDP. If you hike the interest rates to match your inflation, that's like sucking 5% of your GDP into interest payments. This will be a pretty dramatic event, no matter how it is distributed.
Across all Canadians, we're talking about $1.5 trillion (CAN) in mortgage debt. That's on par with the GDP. If you hike the interest rates to match your inflation, that's like sucking 5% of your GDP into interest payments. This will be a pretty dramatic event, no matter how it is distributed.
It will certainly be a dramatic event, but it still matters how it is distributed. To make up some numbers by way of example, if 4 million people have to pay $50/month more, I believe it has a significantly smaller effect on the economy than 1 million people having to pay $200/month more.
Yes, but interest is front-loaded on mortgages, so the impact is much lower for those 20-25 years in.
Governments are in a privileged position to get this data and they likely have it and have reasonable models.
The classic knob to influence inflation is interest rates. This can be also be modeled. Non-politicised civil servants have a good record of being able to do a reasonable job at this - only if the politics are removed. In my opinion, removing all political allegiance for civil servant agencies is probably the best way to get better long term stability and prosperity.
The classic knob to influence inflation is interest rates. This can be also be modeled. Non-politicised civil servants have a good record of being able to do a reasonable job at this - only if the politics are removed. In my opinion, removing all political allegiance for civil servant agencies is probably the best way to get better long term stability and prosperity.
> Most home owners have mortgages with 3-5 year terms.
As an American, this is easily misinterpreted to mean the mortgages are expected to be paid off in 3-5 years, which would be mind-blowing if true, because it would mean that either houses are super cheap, or only the rich are buying.
IMO, adjustable-rate mortgages are almost a scam. Are there any legal protections in place to disallow a bank from deciding "In the next period, we're going to raise your rate to 20% because fuck you the CEO wants another yacht"?
As an American, this is easily misinterpreted to mean the mortgages are expected to be paid off in 3-5 years, which would be mind-blowing if true, because it would mean that either houses are super cheap, or only the rich are buying.
IMO, adjustable-rate mortgages are almost a scam. Are there any legal protections in place to disallow a bank from deciding "In the next period, we're going to raise your rate to 20% because fuck you the CEO wants another yacht"?
Yes. Competitoon with other banks and other lenders. The market is your protection.
Just like one oil company could triple the gas price if they want. Doesn’t mean people will keep getting their gaz there however.
Just like one oil company could triple the gas price if they want. Doesn’t mean people will keep getting their gaz there however.
So you're able to change banks if the interest rate is not competitive?
Wait, so all mortgages in Canada are ARMs by another name?
3-5 year mortgage? Aren’t the monthly payments totally destructive to one’s financial health?
The renewal periods are typically 3-5 years. The standard mortgage amortization period in Canada is 25 years.
I'll be that guy. Please don't hate me.
>So "pole" and "zero" analysis is mute.
Moot.
>So "pole" and "zero" analysis is mute.
Moot.
I suppose if pole and zero analysis disagree with you, they probably aren't going to say anything.
Yup, it should have been moot.
I spotted that but thought your original worked nicely anwyay (it is silent on these things)
Thank you for taking the time to write these follow-ups. I was left with the same uneasy feeling at the end of the article, and you’ve articulated perfectly why.
[deleted]
That's not an appeal to authority. An appeal to authority is "P is a statement in the topic T. A is an expert in T. A says P is true. Therefore P is true." If I draw a parallel between two topics and proceed to explain one of them in order to argue something about the other one, that's not an appeal to authority.
It could be that the reader fails to understand the explanation and instead treats the writer as a trustworthy authority. That would not make the argument an appeal to authority or even fallacious.
It could be that the analogy is poorly justified, which would make the argument incomplete, or it could even be that the writer failed to understand some subtle aspect of the second topic that makes the analogy inappropriate. Neither of these would be appeals to authority.
It could be that the reader fails to understand the explanation and instead treats the writer as a trustworthy authority. That would not make the argument an appeal to authority or even fallacious.
It could be that the analogy is poorly justified, which would make the argument incomplete, or it could even be that the writer failed to understand some subtle aspect of the second topic that makes the analogy inappropriate. Neither of these would be appeals to authority.
In this case the author (A) is setting themselves up as an authority in (T). The statement (P) in this article has not been revealed, but because (and I suspected separated) is in the followup (which I have now read). The followup does refer back to the "now you know about poles" -- the appeal to authority.
Setting yourself up as an authority isn't a fallacy. It's a rhetorical device. (Though I am not sure what makes you say that anyway. The article didn't particularly strike me as the author setting themselves up as an authority, beyond what is inherent in writing an article that explains something.)
If they made the argument "I am an expert in control theory, therefore what I think about Canadian politics is correct", that would be an appeal to one's own authority. But I didn't see anything like that in the article.
If they made the argument "I am an expert in control theory, therefore what I think about Canadian politics is correct", that would be an appeal to one's own authority. But I didn't see anything like that in the article.
>The followup does refer back to the "now you know about poles" -- the appeal to authority.
How is that an appeal to authority?
How is that an appeal to authority?
In the classic social media sense.
You saw my post on X, now believe what I say on Y.
You saw my post on X, now believe what I say on Y.
So if a school teacher taught you addition and then later used it as the basis for teaching multiplication you'd reject it because they were referencing their own work on addition?
This is complicated. It involves identity, credibility (accreditation), reputation and authority.
For me to be taught, my parents ceded authority to the school to appoint credible teachers. Those teachers will have an identity, qualifications and perhaps a reputation. There are consequences for failing or abusing this position.
This is the way a lot of traditional things work.
To directly answer your question, no. A teacher that taught addition would likely be excellent for teaching multiplication.
But, I don't think this is how the internet now works, especially in the social media space. There is infinite identity, which creates trust problems for reputation and credibility. Applying traditional models breaks down. At the moment the dominant factor appears to be who can win a popularity content.
Hacker News is interesting in that karma is currency to build credibility and reputation. This model has flaws too. In my opinion, rewarding the most popular is not a long term winning strategy for a society or community. The demand for change only comes from dissenters. Dissenters and different opinions are the engines of change for any society / community, otherwise, without change, it would stagnate and fall into decline.
But, in the face of infinite identity what is the collective actually trying to say?
For me to be taught, my parents ceded authority to the school to appoint credible teachers. Those teachers will have an identity, qualifications and perhaps a reputation. There are consequences for failing or abusing this position.
This is the way a lot of traditional things work.
To directly answer your question, no. A teacher that taught addition would likely be excellent for teaching multiplication.
But, I don't think this is how the internet now works, especially in the social media space. There is infinite identity, which creates trust problems for reputation and credibility. Applying traditional models breaks down. At the moment the dominant factor appears to be who can win a popularity content.
Hacker News is interesting in that karma is currency to build credibility and reputation. This model has flaws too. In my opinion, rewarding the most popular is not a long term winning strategy for a society or community. The demand for change only comes from dissenters. Dissenters and different opinions are the engines of change for any society / community, otherwise, without change, it would stagnate and fall into decline.
But, in the face of infinite identity what is the collective actually trying to say?
I find it pretty funny that you're accepting one claim because it comes from what you think is an authority, while rejecting another because you think it relies on an appeal to authority.
An elementary school teacher doesn’t need to rely on appeal to authority to teach math. The student can quickly verify the method works themselves — they don’t have to take the teacher’s word for it alone.
In general, appeal to authority as a fallacy applies to subjects that are opinion-based or contentious.
In general, appeal to authority as a fallacy applies to subjects that are opinion-based or contentious.
No, but I wouldn't trust them if they referenced their math experience as a qualification to teach me SCUBA diving.
Would you mind pointing out in the second post the sentence where that line of reasoning is implied?
The person you're replying to already has answered this question in a post you've already replied to. But to be explicit, consider the first sentence of the follow up post [1]:
> Now that you know what right-half-plane zeros are, in this article I'm going to begin a deep-dive into a control-systems-based analysis of a certain feedback system that is drawing particular public attention today: inflation.
In my mind, this is explicitly drawing a connection between the author's engineering background, particularly in control theory, and the current topic, inflation. To be clear, it would be fine if the author wanted to make a _mathematical_ connection between control theory and inflation. However, since the author does not explain right-half-plane zeros with any meaningful technical detail, the rhetorical effect is an appeal: "just trust me on this." Then the author switches topics _and explicitly connects the two topics_. I cannot speak for the person you're replying to, but this rhetoric is what bothers me.
[1] https://jbconsulting.substack.com/p/on-shelter-futures-part-...
EDIT: Looks like VogonPoetry already replied while I was typing this up.
> Now that you know what right-half-plane zeros are, in this article I'm going to begin a deep-dive into a control-systems-based analysis of a certain feedback system that is drawing particular public attention today: inflation.
In my mind, this is explicitly drawing a connection between the author's engineering background, particularly in control theory, and the current topic, inflation. To be clear, it would be fine if the author wanted to make a _mathematical_ connection between control theory and inflation. However, since the author does not explain right-half-plane zeros with any meaningful technical detail, the rhetorical effect is an appeal: "just trust me on this." Then the author switches topics _and explicitly connects the two topics_. I cannot speak for the person you're replying to, but this rhetoric is what bothers me.
[1] https://jbconsulting.substack.com/p/on-shelter-futures-part-...
EDIT: Looks like VogonPoetry already replied while I was typing this up.
These:
"Now that you know what right-half-plane zeros are, in this article I'm going to begin a deep-dive into a control-systems-based analysis of a certain feedback system that is drawing particular public attention today: inflation."
And all of the statements made in the walkback "a bit of a preamble" -- basically issuing caveats or disclaiming all of what was said before. A classic "I got you here", but I'm going to disclaim all of that, you are a fool to read anything that follows.
"Now that you know what right-half-plane zeros are, in this article I'm going to begin a deep-dive into a control-systems-based analysis of a certain feedback system that is drawing particular public attention today: inflation."
And all of the statements made in the walkback "a bit of a preamble" -- basically issuing caveats or disclaiming all of what was said before. A classic "I got you here", but I'm going to disclaim all of that, you are a fool to read anything that follows.
I read that as "now that you know some of the concepts and terminology used in systems theory, I can use them to make an argument in economics, and you can use that knowledge to judge for yourself whether the argument is valid or not". It's literally the opposite of a appeal to authority. As I see it, the only way it could be dishonest would be if the systems theory explanation is incorrect or purposely leaves inconvenient things out (which it might).
It would be an (incompetent) appeal to authority if it said "now that you've seen that I know my stuff when it comes to systems theory, you know that this thing I say about economics is true".
It would be an (incompetent) appeal to authority if it said "now that you've seen that I know my stuff when it comes to systems theory, you know that this thing I say about economics is true".
> As I see it, the only way it could be dishonest would be if the systems theory explanation is incorrect or purposely leaves inconvenient things out (which it might).
That should be the default assumption, no? Today’s Control Theory has never been shown to be predictive of the economy or politics, wouldn’t you agree? The article framed the discussion as if control theory is useful, purposely implied it’s utility, and did not state it’s known lack of scientific validity, which is indeed leaving some very inconvenient facts out, right? Saying that they left it open with a ‘judge for yourself’ is not a great counter to the critique of the argument style. There is an intentional framing here that fails to list the alternatives, tries to establish itself as ‘correct’ via implication, and pivots to a separate topic that is unproven to be relevant. Pretending to be unbiased while presenting a one-sided set of “information” is a pretty common type of appeal to authority.
That should be the default assumption, no? Today’s Control Theory has never been shown to be predictive of the economy or politics, wouldn’t you agree? The article framed the discussion as if control theory is useful, purposely implied it’s utility, and did not state it’s known lack of scientific validity, which is indeed leaving some very inconvenient facts out, right? Saying that they left it open with a ‘judge for yourself’ is not a great counter to the critique of the argument style. There is an intentional framing here that fails to list the alternatives, tries to establish itself as ‘correct’ via implication, and pivots to a separate topic that is unproven to be relevant. Pretending to be unbiased while presenting a one-sided set of “information” is a pretty common type of appeal to authority.
I'm not particularly interested with whether the argument raised by the article is valid or not, which is why I left open several possible ways in which it could be fallacious. My point is solely that it's not an appeal to authority.
What makes appeal to authority fallacious is that one's supposed expertise has no bearing on the truth of the things one says. An appeal to authority is when someone argues something like, "I'm an expert, I think X and so should you" basing their opinion on X on their supposed expertise and not actual logic.
When someone gives reasons, their argument rests on those reasons, not on their supposed authority as an expert on the topic.
When someone gives reasons, their argument rests on those reasons, not on their supposed authority as an expert on the topic.
Actually, it seems more an effort to establish credibility. "I have a degree in such and such" or "I have worked in this field for over 20 years". That seems to be much different than appeal to authority.
It seems to only be an appeal to authority when it's a person or a position that one disagrees with. Otherwise it's "experience" or "consensus".
The author of the article has had their understanding of systems eat their understanding of humans. Every time they make a realization about people, it immediately gets turned to getting more understanding and control over the world. Which doesn't leave any room to grow for their understanding of humans.
This makes it really hard for them to write an article for non-technical people. To do so, they would have to connect the human side of things to the systems theory side of things. Which they can't do, because they don't understand the human side of things.
They aren't using bad-faith arguments to convince you that they are correct. They simply have no idea how to talk to you. They are in fact trying to be helpful, by presenting what they think is really useful information to understand and control the world with. Their help is useless to you, but they are trying.
This makes it really hard for them to write an article for non-technical people. To do so, they would have to connect the human side of things to the systems theory side of things. Which they can't do, because they don't understand the human side of things.
They aren't using bad-faith arguments to convince you that they are correct. They simply have no idea how to talk to you. They are in fact trying to be helpful, by presenting what they think is really useful information to understand and control the world with. Their help is useless to you, but they are trying.
I agree. The author uses technical jargon but is clearly intending the reader to be non-technical. Otherwise, footnote 1 should just be in the main text. The rhetorical effect, intended or otherwise, is an appeal to (the author's) authority. As you say, the author then switches to a topic he likely much less expertise in (housing prices, inflation, etc.) while relying on the reader to believe that this new problem shares some mathematical foundations with the original set of problems.
I call it the "Ian Malcolm syndrome", in reference to the character (a mathematician) in Michael Crichton's Jurassic Park who tries to apply mathematical concepts to the dinosaur attraction to try to show why it is doomed to collapse.
There are a lot of similar posts online about people trying to use chaos theory (or what they believe chaos theory is), "systems theory", and other obscure concepts to explain why, for example, climate change will be the end of us all, or the global economy will enter a depression, or any other extremely pessimistic macro scenario. I see three reasons why these people are doing it. (1) To draw attention to themselves as someone with a superior intellect (which they actually aren't) (2) To construct an analysis no one has constructed before, in a "all these professional economists are saying the same thing, but here's why they're all misleading themselves, thanks to my ace card of a theory" kind of way, and (3) A love for the hyperbole, for dramatic outcomes.
A classic economic prediction like "Inflation pressures will potentially slow down growth in the short term, but the behavior of the economy over the next 12 months remains unpredictable" is not enough. It MUST be "A catastrophic economic collapse is inevitable", or "We are going back to the Middle Ages by the end of the decade", or "Hundreds of millions will die in the next few years".
There are a lot of similar posts online about people trying to use chaos theory (or what they believe chaos theory is), "systems theory", and other obscure concepts to explain why, for example, climate change will be the end of us all, or the global economy will enter a depression, or any other extremely pessimistic macro scenario. I see three reasons why these people are doing it. (1) To draw attention to themselves as someone with a superior intellect (which they actually aren't) (2) To construct an analysis no one has constructed before, in a "all these professional economists are saying the same thing, but here's why they're all misleading themselves, thanks to my ace card of a theory" kind of way, and (3) A love for the hyperbole, for dramatic outcomes.
A classic economic prediction like "Inflation pressures will potentially slow down growth in the short term, but the behavior of the economy over the next 12 months remains unpredictable" is not enough. It MUST be "A catastrophic economic collapse is inevitable", or "We are going back to the Middle Ages by the end of the decade", or "Hundreds of millions will die in the next few years".
Er, Ian Malcolm, the character who was introduced so that he could arrogantly predict that the park would collapse, only to get his satisfying come-uppance when... he turns out to be completely correct?
Why doesn't all the same analysis apply to your making of "Ian Malcolm syndrome" itself?
This seems uncharitable. The author is trying to explain how knowing a bit of control theory could be helpful for understanding economics. Unfortunately they’re having trouble writing for a general audience, perhaps due to being too comfortable with the math. This doesn’t mean anything nefarious, just that they could use an editor.
Textbooks that use j instead of i are an indicator of what? Why is slightly different notation awkward? It's not as if notation is always consistent within the field of mathematics, right?
The context of the writer. Apologies if I implied that the notation semantics were important -- letter exchange is free; concept change is not. What I indented to mean was the use of the terms poles and zeros is a bit wonky. Attempting to be cute with a physicist's hat, a pole is a "black hole", an entity you must not cross because the calculus will not work. It is a discontinuity, a singular value. There are worse things in mathematical generalizations; a cut / line in a manifold. But tools to cope.
[deleted]
I’m curious what the right half plane zero in housing is. In general, we correct inflation through interest which we raise gradually waiting a month or two to observe the effects of a 0.5% raise. I think that should be sufficient to avoid the control feedback effects he talks about in this article.
The RHP0 in (Canadian) housing comes from including mortgage interest payments -- a term that increases in proportion to interest rates -- in the measurement of the CPI.
I admit I was a bit disappointed by the end of the article. But I don't think it was strictly an appeal to authority. The author is evidently a very good writer, but he might have done a better job at summing up a conclusion. It's very difficult to tell just what point he was trying to make. He never even said just what his inputs and outputs are in his housing model. Even if he is saving that for a later article, he should have at least mentioned that.
Comment: j is used instead of i in certain contexts -- e.g. circuits -- when it is possible to confuse it with the current (same letter used). As e.g. an Electrical and Computer Engineer you just move back and forth on sqrt(-1) notation appropriately. Not sure beyond that about the not quite the same terminology -- not my experience but who knows.
The article is kind of a "layman's article" to put it politely. I am erring towards the side of hocus pocus -- if economics were that well behaving and linear and composable as the author models, it would be a solved field years now. And control theory wouldn't exist as a field for 50 years, as it would have been solved. Do the components diverge? Where in the complex plane are they defined to begin with? Anyways.
Also: if you should learn something: Central Banks have a terrible track record, and despite what they keep saying are political. The joke is that they are the blind driver with a gas and break pedal that have a 2 years delay. So good luck to us all.
The article is kind of a "layman's article" to put it politely. I am erring towards the side of hocus pocus -- if economics were that well behaving and linear and composable as the author models, it would be a solved field years now. And control theory wouldn't exist as a field for 50 years, as it would have been solved. Do the components diverge? Where in the complex plane are they defined to begin with? Anyways.
Also: if you should learn something: Central Banks have a terrible track record, and despite what they keep saying are political. The joke is that they are the blind driver with a gas and break pedal that have a 2 years delay. So good luck to us all.
The follow-up article tries to argue there is an over five year lag to interest correcting inflation based on undisclosed principles of a model they don’t provide equations for. Their argument is in effect that the growth of roughly 30% of the inflation basket of goods (housing) dominates the shrinking of the other 70% of the inflation basket of goods. The analysis is suspect as they don’t seem to model how much the 70% shrinks or slows in growth at all. They also aren’t very numerically precise on the interest effects on housing prices and misanalyze how mortgages work in Canada.
Their analysis argues that roughly 4% of mortgages are replaced every year while 96% retain the same rates. This makes it hard to argue interest rates have the desired effect size but that doesn’t stop them from trying.
Their analysis argues that roughly 4% of mortgages are replaced every year while 96% retain the same rates. This makes it hard to argue interest rates have the desired effect size but that doesn’t stop them from trying.
> The analysis is suspect as they don’t seem to model how much the 70% shrinks or slows in growth at all.
It doesn't model it at all. The trajectory of ex-shelter CPI is taken as an input. For the provided scenarios, all non-shelter components are assumed (as a scenario premise, not a prediction) to instantly return to a 2% growth rate and stay there.
It doesn't model it at all. The trajectory of ex-shelter CPI is taken as an input. For the provided scenarios, all non-shelter components are assumed (as a scenario premise, not a prediction) to instantly return to a 2% growth rate and stay there.
“the math that I know is THE math to explain inflation, political instability and more”
I am not sure what to take from this. The title certainly makes sense from a engineers perspective. But from a layman's? I don't know.
It seems, that he tries to explain an important concept in control theory but does not explain the meaning of "plane" at all. Even worse: He uses airplanes as an example.
Isn't it terribly confusing for non-control-theory people? If I don't know control theory, then how on earth would I know that he means a 2D-plane? And what's this "minimum-phase-system" he mentions once? Is a pole a number? And what about RHP poles?
I would be interested in how readers without a background in control theory and higher maths understood that article and what questions arose.
It seems, that he tries to explain an important concept in control theory but does not explain the meaning of "plane" at all. Even worse: He uses airplanes as an example.
Isn't it terribly confusing for non-control-theory people? If I don't know control theory, then how on earth would I know that he means a 2D-plane? And what's this "minimum-phase-system" he mentions once? Is a pole a number? And what about RHP poles?
I would be interested in how readers without a background in control theory and higher maths understood that article and what questions arose.
As someone with a rapidly (exponentially?) decaying background in engineering, all the article did was create an insufferable itch of mathematical fuzziness about why these were called zeros until I got to the footnote and the actual article began.
Thanks for the tip, the rest made way more sense after reading that. I don't think I'm particularly poorly read or mathematically versed but I've never heard a pole called a "right-half-plane zero" before, so maybe the author could have lead with that (or even just called it a pole, which I have heard of so at least I'd have been less lost.)
The article wasn’t doing that, and specifically distinguished poles from zeroes:
>> Control theorists like to classify the behaviour of dynamical systems based on what we call poles and zeros. … Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
>> Control theorists like to classify the behaviour of dynamical systems based on what we call poles and zeros. … Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
Whoops, adding that to my list of things to read again... >.>
Seemed a decent if over-long article to explain something I needed to know about feedback theory. Summary: to do x you may have to do, or briefly get, anti-x which you need to be aware of and account for. Key sentence where the importance clicked for me was:
"Let’s say our airplane is running in auto-pilot. We’ve sent a request to gain altitude, so the flight controller tilts the elevators to initiate a climb. But suddenly the airplane is losing altitude, moving farther away from our target? Do we pull up even harder?"
It's ok.
"Let’s say our airplane is running in auto-pilot. We’ve sent a request to gain altitude, so the flight controller tilts the elevators to initiate a climb. But suddenly the airplane is losing altitude, moving farther away from our target? Do we pull up even harder?"
It's ok.
Talk about a borked example.
First of all, I'd call airplane/plane an "aircraft" instead, not because that's what aviators and Wikipedia editors do, but because it's super confusing in an article about some other thing called "plane".
And, as an aviator would probably tell you, in order to climb you should care more about the throttle (i.e. increase power) than about them elevators! So, in many cases there is absolutely no such dip.
And, I know of no aircraft capable of achieving that loopy flight path using just elevators. That would require quite a bit of kinetic energy, to say the least.
The bike counter-steering example clicked instantly with me, but I guess not many people have an intuition about that. I rode for decades before learning this and I think most other cyclists are not aware as well. (Most motorbike instructors teach that however.)
First of all, I'd call airplane/plane an "aircraft" instead, not because that's what aviators and Wikipedia editors do, but because it's super confusing in an article about some other thing called "plane".
And, as an aviator would probably tell you, in order to climb you should care more about the throttle (i.e. increase power) than about them elevators! So, in many cases there is absolutely no such dip.
And, I know of no aircraft capable of achieving that loopy flight path using just elevators. That would require quite a bit of kinetic energy, to say the least.
The bike counter-steering example clicked instantly with me, but I guess not many people have an intuition about that. I rode for decades before learning this and I think most other cyclists are not aware as well. (Most motorbike instructors teach that however.)
> The bike counter-steering example clicked instantly with me, but I guess not many people have an intuition about that. I rode for decades before learning this and I think most other cyclists are not aware as well.
Veritasium did a video on this:
https://youtu.be/9cNmUNHSBac
Veritasium did a video on this:
https://youtu.be/9cNmUNHSBac
> And, as an aviator would probably tell you, in order to climb you should care more about the throttle (i.e. increase power) than about them elevators! So, in many cases there is absolutely no such dip.
An autopilot responds the same way, where an altitude error input affects the throttle control. (wincing... when implemented with classical SISO controls...).
An autopilot responds the same way, where an altitude error input affects the throttle control. (wincing... when implemented with classical SISO controls...).
I mean, to a layman like myself, the takeaway is something about stateful, irreversible systems where x is on both sides of the equation in unequal or inverse measures, and how those systems (maybe? sometimes?) progress toward equilibrium by going through unexpected phases that seem to contradict what you're trying to input. I read the whole thing as a metaphor for Fed policies or how we elect Democrats every time the economy is about to crash, and Republicans every time it's just recovered.
To the degree that some field of math can be best exemplified by various types of aircraft stalls, this didn't do a great job of explaining either the types of stalls or the feedbacks (what you might call negative REPL loops) leading to them.
I agree it's ok.
To the degree that some field of math can be best exemplified by various types of aircraft stalls, this didn't do a great job of explaining either the types of stalls or the feedbacks (what you might call negative REPL loops) leading to them.
I agree it's ok.
So, to me, this is just calibrating against a continuum. Right? I think of this as binary-searching a kinda normal'ish distribution? I'm not good at math so I had a hard time once the OP article got into differential equations
Fair enough, but then he wouldn't have needed to use control theory and just stuck with the examples.
Edit: removed ambiguity
Edit: removed ambiguity
As a mathematician it is just a lot of fuzzy rambling, until they have the actual definition. I really don’t know what an average person is supposed to take away, you can end up in some feedback loop, lol okay. Communicating maths to a wider audience you can still be correct but there is zero reason to make it fuzzy. It’s okay to use an analogy if there is something people can latch onto, carry it with them mentally, and then you slowly deform it into the correct statement, but you have to be careful to not be fuzzy.
If you are writing something just minimise the number of non informative sentences. They probably could have explained poles, planes, etc all in the same space instead of rambling. Cutting to the examples and then giving the definition would have been better.
If you are writing something just minimise the number of non informative sentences. They probably could have explained poles, planes, etc all in the same space instead of rambling. Cutting to the examples and then giving the definition would have been better.
Yeah, as someone with a math degree, I already know what the right half-plane refers to, I already know what zeroes and poles are (in a complex analysis way, not a control theory way), and I am just befuddled. The article doesn't really seem to have much concrete content at all, and could be cut down by a lot. Which is an achievement considering how short it already is.
Just cutting out this junk:
> Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
and similar waffle would be good. It's so vague as to be counter-productive.
The conclusions in the article aren't easily drawn from the vague tools we're given, in my opinion. Maybe I'm just not the target audience, but I don't really understand who that is.
Just cutting out this junk:
> Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.
and similar waffle would be good. It's so vague as to be counter-productive.
The conclusions in the article aren't easily drawn from the vague tools we're given, in my opinion. Maybe I'm just not the target audience, but I don't really understand who that is.
At the beginning I didn't understand what the post was about. At the ending, it felt like I was at the beginning.
I never studied control theory, but I was fine with this. My pure maths education stopped at say first year undergrad. I didn't even consider that there were two kinds of plane. I thought the example was a good example of the subject.
To answer the confusion about poles...I think the teaching method of, 'here are some terms you won't understand until later' is very common, isn't it. I bet it even has a name
To answer the confusion about poles...I think the teaching method of, 'here are some terms you won't understand until later' is very common, isn't it. I bet it even has a name
It's a nice explanation of the phenomenon of "getting tilted" in games:
Tilt originated from Poker and it's usually a state of emotional frustration and confusion.
It's most commonly used if you're going on a losing streak and then you become so frustrated that you start playing worse because you cannot focus anymore.
Part of it, as I see it, is that you are using that frustration to fuel further efforts, which ends up in a downward spiral feedback loop
https://gaming.stackexchange.com/questions/190507/what-is-th...
Tilt originated from Poker and it's usually a state of emotional frustration and confusion.
It's most commonly used if you're going on a losing streak and then you become so frustrated that you start playing worse because you cannot focus anymore.
Part of it, as I see it, is that you are using that frustration to fuel further efforts, which ends up in a downward spiral feedback loop
https://gaming.stackexchange.com/questions/190507/what-is-th...
[deleted]
> I am not sure what to take from this
I think the author is using a lot of words to say "things often need to get a bit worse in order to start getting better"
I think the author is using a lot of words to say "things often need to get a bit worse in order to start getting better"
And, critically, you potentially need to ignore the signal of "things getting worse". Otherwise you might creature a vicious cycle of more corrective action -> more things getting worse (at first) -> even more corrective action, until your system crashes and you never got into the phase where most of the desired result of the corrective action materialized.
And a second important point: You mustn't ever get into a state where the "things getting worse at first" already pushes you over a red line (see the example of the plane that loses even more altitude before it rises again. If you hit the ground in between you don't care that you theoretically would have risen later on).
And a second important point: You mustn't ever get into a state where the "things getting worse at first" already pushes you over a red line (see the example of the plane that loses even more altitude before it rises again. If you hit the ground in between you don't care that you theoretically would have risen later on).
That's only half of the story. The second half of the story is "if the getting worse temporarily cancels out the getting better, stop trying to correct, or you're going to crash!"
I think "when things get worse we tend to double down on what caused things to get worse, until everything around us crash and burn" would be more accurate, given the current state of affairs.
Not sure I share the pessimism. It seems to me that we're adjusting to situations pretty rapidly as a species. We're definitely at least attempting to find out what inputs lead to which short- and long-term outputs, which is a unique behavior among mammals.
To me, that immediately sounds like someone making excuses for their side and the midterm elections. I don’t care how much math there is.
It really all depends on how low to the ground you are when things begin to get worse.
For example, a virus that takes 10 IQ points off people who've been infected with it might make China more apt to consume Disney movies, while it might plow America straight into the ground.
For example, a virus that takes 10 IQ points off people who've been infected with it might make China more apt to consume Disney movies, while it might plow America straight into the ground.
> He uses airplanes as an example.
Oh wow, i didn't notice that. It's either pretty clever, or unfortunate wording.
Oh wow, i didn't notice that. It's either pretty clever, or unfortunate wording.
Same. I ran to the comments to check my understanding.
The greatest sin of undergraduate engineering education is sequestering signals and systems into electrical engineering curricula. I understand why it's done that way (I even had to fight to take the course early, sidestepping some prereqs for reasons).
But it's really so foundational to understanding concepts of stability, resonance, information/energy flow (from the conceptual perspective), and the simple analytical tools for building a solid conceptual base. It takes a semester to hammer home that step response matters, positive feedback bad, negative feedback usually good, and topologies are useful.
But it's really so foundational to understanding concepts of stability, resonance, information/energy flow (from the conceptual perspective), and the simple analytical tools for building a solid conceptual base. It takes a semester to hammer home that step response matters, positive feedback bad, negative feedback usually good, and topologies are useful.
I took some electrical engineering courses but mainly software engineering with a focus on low-level details and graphics.
The more I specialize in graphics the more I realize I need much more knowledge about signal processing than we were taught at the university.
The more I specialize in graphics the more I realize I need much more knowledge about signal processing than we were taught at the university.
Could you demonstrate with an example?
I can give a few. In general, the theory behind rendering generally treats what's on the screen as a discretization of a continuous visual signal. Post processing especially is largely about signal processing.
Anti-aliasing (AA) is a very clear example where the lack of it leads to moiré patterns and jaggies. Understanding a lot of AA techniques is simplified by seeing the frame not as a discrete set of pixels but as a continuous signal being sampled (and can be sampled at multiple sub-pixel points per pixel).
A lot of other screen effects are essentially filters applied to the graphical signal (sobel, gaussian blur, ...) and understanding them from a signal processing view helps understanding how to modify and optimize them. A good example here is identifying whether your effect is a separable filter which can be split into a horizontal and vertical pass.
Seeing the image as a continuous signal/field being sampled is also the theoretical basis for a lot of visual effects used in physically-based rendering and things like screen-space ambient occlusion.
Finally, if you want to write your own ray-tracer it really helps to be able to take this view of things once you get past the basics.
Anti-aliasing (AA) is a very clear example where the lack of it leads to moiré patterns and jaggies. Understanding a lot of AA techniques is simplified by seeing the frame not as a discrete set of pixels but as a continuous signal being sampled (and can be sampled at multiple sub-pixel points per pixel).
A lot of other screen effects are essentially filters applied to the graphical signal (sobel, gaussian blur, ...) and understanding them from a signal processing view helps understanding how to modify and optimize them. A good example here is identifying whether your effect is a separable filter which can be split into a horizontal and vertical pass.
Seeing the image as a continuous signal/field being sampled is also the theoretical basis for a lot of visual effects used in physically-based rendering and things like screen-space ambient occlusion.
Finally, if you want to write your own ray-tracer it really helps to be able to take this view of things once you get past the basics.
I went to school for mechanical engineering (though I now work in software). We were required to take signals and systems, but if I remember right it was a weed-out course for most MechEs (it certainly was a challenge for me, though I think that had more to do with the curriculum than the topic).
Those lessons might have been hard-won on my part, but I definitely still use them. The general concepts (feedback loops etc) are applicable basically everywhere in life, and I still find uses for literal actual math (like using a convolution kernel to do rolling window sampling in numpy).
Those lessons might have been hard-won on my part, but I definitely still use them. The general concepts (feedback loops etc) are applicable basically everywhere in life, and I still find uses for literal actual math (like using a convolution kernel to do rolling window sampling in numpy).
I can imagine it's used for car suspension?
That's not considered signals and systems, at least not where I went to school (unless you're talking about fully-active suspensions, but those are very rare and highly specialized). Rather, that would be down the "dynamics" course progression -- which is bread and butter for MechEs in those lines of work. That's also extremely useful, but it's generally a different subject matter, at least until you get into graduate-level dynamics combined with upper-undergrad-to-graduate-level numerical methods.
Vibration isolation can definitely be thought of within a systems framework, although it's perhaps overkill for passive filtering as you won't be doing much in the way of feedback.
Wasn't control theory invented to understand the dynamic behavior of steam engine governors?
"signals and systems" - covering signals processing, systems and control theory is very generic subject.
You need it in economics, biology, chemistry, physics, computer science, statistics, electrical engineering, robotics , automation, logistics, ...
It should be taught like calculus or linear algebra, so that everyone gets gist of the basics before learning to apply it into their specific field.
You need it in economics, biology, chemistry, physics, computer science, statistics, electrical engineering, robotics , automation, logistics, ...
It should be taught like calculus or linear algebra, so that everyone gets gist of the basics before learning to apply it into their specific field.
I always thought my college education was backwards, with the exception that differential equations (and laplace transforms, which can help lay the groundwork for other transforms) came early enough that I could get by -- though it'd have been better if they were earlier, like high school, and if I hadn't been able to skip two semesters of calculus thanks to my HS calculus then differential equations would have come even later in college. But as a CE student at a school nominally more about video game making education I ended up first taking as CS electives an audio processing course, and then an image processing course, before the CE side reached control systems, which I had to retake after taking the next digital signal processing course, after which control systems made a lot more sense. It was only after graduating that I felt like I had reached a level of sophistication to go back and really grok all the related theory and go deep into applications. Maybe that's how graduate students are meant to feel? But I just went into full time enterprise work and 'retired' after 6 years of that, so I've since forgotten a lot... Still, the concept of feedback has proved useful in systems analysis from time to time, and it's a framework that I think could yield many low hanging fruit in other disciplines. (The book Behavior: The control of perception applies control theory to psychology in a convincing way but it's understandably been neglected by psychologists who aren't often very sophisticated mathematically.)
Can you recommend any good introductory books on the topics you mentioned for someone who studied (theoretical/mathematical) physics but never electrical engineering or signal processing? (Read this as: I am more or less familiar with Nyquist-Shannon's theorem but that's about it.)
Not a book, but this series on Youtube is excellent.
https://www.youtube.com/watch?v=Pi7l8mMjYVE&list=PLMrJAkhIeN...
> familiar with Nyquist-Shannon's theorem
BTW: In control theory there is also https://en.wikipedia.org/wiki/Nyquist_stability_criterion
https://www.youtube.com/watch?v=Pi7l8mMjYVE&list=PLMrJAkhIeN...
> familiar with Nyquist-Shannon's theorem
BTW: In control theory there is also https://en.wikipedia.org/wiki/Nyquist_stability_criterion
ime the books are very loaded on theory and it's hard to connect that to practice and concepts. I don't know of any book that touches on what poles/zeros mean or why they're useful outside what are (seemingly) contrived properties of formulae like stability. It's one of those things that's just useful to go through in a university course, even just auditing it. The practical problems and lectures are extremely useful, particularly due to the overhead of notation (the practice is relatively young and derived from a different tradition of education, and a lot of contemporary notation can seem a little foreign, or is filled with shorthand).
Applied Digital Signal Processing by Manolakis & Ingle is the book I always turn to for reference and the code examples don't suck. Oppenheim & Schafer is a classic but frankly only useful as a reference, that tome is a bit dated otherwise. The Scientists and Engineer's guide to DSP is also not bad as a practical text.
Applied Digital Signal Processing by Manolakis & Ingle is the book I always turn to for reference and the code examples don't suck. Oppenheim & Schafer is a classic but frankly only useful as a reference, that tome is a bit dated otherwise. The Scientists and Engineer's guide to DSP is also not bad as a practical text.
Signals & Systems by Oppenheim is a classic introductory textbook.
Does anyone know of any courses which could explain concepts of stability, resonance, information/energy flow and help build a solid conceptual base for managers or entrepreneurs? These concepts are crucial when making business decisions. I've been building up an understanding of this through experience. If there was a way to shortcut this and provide such education deliberately, it would allow people to become better decision makers quicker.
I was already thinking of taking it this summer, but you convinced me to do it for sure.
Thank you!
Thank you!
The classic paper, Maxwell's "On governors". (1869) [1] This is the first theoretical analysis of feedback.
It will be seen that the motion of a machine with its governor consists in general of a uniform motion, combined with a disturbance which may be expressed as the sum of several component motions. These components may be of four different kinds:-
(1) The disturbance may continually increase.
(2) It may continually diminish.
(3) It may be an oscillation of continually increasing amplitude.
(4) It may be an oscillation of continually decreasing amplitude.
The first and third cases are evidently inconsistent with the stability of the motion; and the second and fourth alone are admissible in a good governor. This condition is mathematically equivalent to the condition that all the possible roots, and all the possible parts of the impossible roots, of a certain equation shall be negative.
That is, in the left half-plane.
(Terminology has changed. Maxwell says "disturbance" where today, the term "error" would be used. Today, "disturbance" means an input which disturbs stability, while error is an output.)
Maxwell got so much right in that paper, and it was a long time before anybody picked up on that result.
Now, where it looks like the author is going is into economic territory. Basic economics talks about "economic equilibrium". The concept is that restoring forces will bring supply and demand into equilibrium. But basic control theory tells us that may not happen. Any system with delay in it can potentially be unstable. Too much delay, and even simple systems will not stabilize.
[1] https://en.wikisource.org/wiki/On_Governors
It will be seen that the motion of a machine with its governor consists in general of a uniform motion, combined with a disturbance which may be expressed as the sum of several component motions. These components may be of four different kinds:-
(1) The disturbance may continually increase.
(2) It may continually diminish.
(3) It may be an oscillation of continually increasing amplitude.
(4) It may be an oscillation of continually decreasing amplitude.
The first and third cases are evidently inconsistent with the stability of the motion; and the second and fourth alone are admissible in a good governor. This condition is mathematically equivalent to the condition that all the possible roots, and all the possible parts of the impossible roots, of a certain equation shall be negative.
That is, in the left half-plane.
(Terminology has changed. Maxwell says "disturbance" where today, the term "error" would be used. Today, "disturbance" means an input which disturbs stability, while error is an output.)
Maxwell got so much right in that paper, and it was a long time before anybody picked up on that result.
Now, where it looks like the author is going is into economic territory. Basic economics talks about "economic equilibrium". The concept is that restoring forces will bring supply and demand into equilibrium. But basic control theory tells us that may not happen. Any system with delay in it can potentially be unstable. Too much delay, and even simple systems will not stabilize.
[1] https://en.wikisource.org/wiki/On_Governors
In the real world all exponentials are sigmoids eventually, so what we actually get is is a recession with a drastic reallocation of resources (creative destruction).
Not really, there are plenty that are sinusoids which are just complex exponentials.
The real trouble with sigmoids is when the saturation point is beyond physically meaningful quantities of the system. See the tacoma narrows bridge collapse.
The real trouble with sigmoids is when the saturation point is beyond physically meaningful quantities of the system. See the tacoma narrows bridge collapse.
Isn't it true to say that a theoretical exponential becomes a practical sigmoid precisely because some property of the system has become saturated and gone nonlinear?
(Just trying to get this clear in my own head by writing it down.)
When a real world system goes non linear (like the Tacoma Narrows case) you don’t get a sigmoid but something catastrophic.
(If you graph the amplitude of the vibrations they increase and increase — and then — if it was a sigmoid they’d level out and stay at the max amplitude… but in reality they go to zero as there is no bridge left to vibrate.)
When a company is “growing exponentially” it may saturate the market and then the growth slows in a nice sigmoid function. That’s common. But if, for example, the investors insist that the company must maintain the growth at all costs… it breaks laws, gets destroyed and there’s no company left to grow. No exponential curve, no sigmoid, no signal at all.
Both the sigmoid and the total collapse are typical real world results of what a simple model would expect to be an unbounded exponential curve.
When a real world system goes non linear (like the Tacoma Narrows case) you don’t get a sigmoid but something catastrophic.
(If you graph the amplitude of the vibrations they increase and increase — and then — if it was a sigmoid they’d level out and stay at the max amplitude… but in reality they go to zero as there is no bridge left to vibrate.)
When a company is “growing exponentially” it may saturate the market and then the growth slows in a nice sigmoid function. That’s common. But if, for example, the investors insist that the company must maintain the growth at all costs… it breaks laws, gets destroyed and there’s no company left to grow. No exponential curve, no sigmoid, no signal at all.
Both the sigmoid and the total collapse are typical real world results of what a simple model would expect to be an unbounded exponential curve.
Control theory assumes the system is linear. As soon as you get saturation the system stops being linear.
So in trad undergrad control theory instability implies "And then the system blows up" - numerically, literally, or sometimes both.
But depending on the system you can end up in regions of recursive instability which are better modelled by logistic/chaos theory:
https://en.wikipedia.org/wiki/Period-doubling_bifurcation
So in trad undergrad control theory instability implies "And then the system blows up" - numerically, literally, or sometimes both.
But depending on the system you can end up in regions of recursive instability which are better modelled by logistic/chaos theory:
https://en.wikipedia.org/wiki/Period-doubling_bifurcation
[deleted]
Shouldn't there be a diagram of the complex plane so that people can see what it's the right half plane of? On top of it, there's a picture of a plane which is confusing.
Fascinating subject though, in engineering class it was quite surprising how this bunch of functions tracing lines and dots on the complex plane would be relevant to just about everything. Perhaps the first lesson is that even if you know how a system works, you can't just take the inverse function to control what comes out.
Fascinating subject though, in engineering class it was quite surprising how this bunch of functions tracing lines and dots on the complex plane would be relevant to just about everything. Perhaps the first lesson is that even if you know how a system works, you can't just take the inverse function to control what comes out.
> Shouldn't there be a diagram of the complex plane so that people can see what it's the right half plane of?
Author here. Yes! Very fair criticism. I was trying to strike a balance between making the concept approachable for those who don't have a background involving complex numbers, but that certainly leaves the name of the concept more confusing. I should add it in a footnote at least.
And I did honestly not think about the potential for confusion between plane // airplane. An airplane was the most familiar example system I could think of to explain the concept. Oops!
> Perhaps the first lesson is that even if you know how a system works, you can't just take the inverse function to control what comes out.
That's a great point too. It probably even deserves its own article.
Author here. Yes! Very fair criticism. I was trying to strike a balance between making the concept approachable for those who don't have a background involving complex numbers, but that certainly leaves the name of the concept more confusing. I should add it in a footnote at least.
And I did honestly not think about the potential for confusion between plane // airplane. An airplane was the most familiar example system I could think of to explain the concept. Oops!
> Perhaps the first lesson is that even if you know how a system works, you can't just take the inverse function to control what comes out.
That's a great point too. It probably even deserves its own article.
I'd definitely read a series about control theory. The first time round was very badly presented at my university. A lot of math, motivation hidden.
Stuff like Nyquist criterion just sort of appears out of nowhere as functions.
I guess the big one is feedback being the magic, and then the complex plane tells you where that blows up.
Stuff like Nyquist criterion just sort of appears out of nowhere as functions.
I guess the big one is feedback being the magic, and then the complex plane tells you where that blows up.
> Stuff like Nyquist criterion just sort of appears out of nowhere as functions.
Black's canonical 1934 paper[1] Stabilized Feedback Amplifiers, which had an outsized influence on EE classical control theory, may have something to do with that:
Results of experiments, however, seemed to indicate something more was involved and these matters were described to Mr. H. Nyquist, who developed a more general criterion for freedom from instability applicable to an amplifier having linear positive constants.
...which directly cites Nyquist's canonical 1932 paper[2] Regeneration Theory.
[1] http://archive.org/details/bstj13-1-1
[2] http://archive.org/details/bstj11-1-126
Black's canonical 1934 paper[1] Stabilized Feedback Amplifiers, which had an outsized influence on EE classical control theory, may have something to do with that:
Results of experiments, however, seemed to indicate something more was involved and these matters were described to Mr. H. Nyquist, who developed a more general criterion for freedom from instability applicable to an amplifier having linear positive constants.
...which directly cites Nyquist's canonical 1932 paper[2] Regeneration Theory.
[1] http://archive.org/details/bstj13-1-1
[2] http://archive.org/details/bstj11-1-126
I went down this rabbit hole in grad and my opinion is that Control Theory is good for writing academic papers, but has few applications besides the classical ones (mostly in mechanics).
Techniques often need very strong assumptions about the systems being modeled, which severely limits their usefulness.
In fact, CT is sort of the antitheses of the currently most hyped way of modeling systems: Machine Learning.
Techniques often need very strong assumptions about the systems being modeled, which severely limits their usefulness.
In fact, CT is sort of the antitheses of the currently most hyped way of modeling systems: Machine Learning.
Interesting username for such an opinion.
Also systems modeling is not the same as control theory. You could indeed utilize machine learning to model a system, which you could then control by classical controllers. On the other hand, control algorithms that use machine learning are a thing.
Also systems modeling is not the same as control theory. You could indeed utilize machine learning to model a system, which you could then control by classical controllers. On the other hand, control algorithms that use machine learning are a thing.
> Interesting username for such an opinion.
Perhaps just a coincidental namespace collision: Process ID, not Proportional-Integral-Derivative.
Perhaps just a coincidental namespace collision: Process ID, not Proportional-Integral-Derivative.
Yes, probably.
I think you're sweeping a ton of valuable applications under the rug by saying "few applications besides the classical ones" ?
I hope I will be able to change your mind!
Really good article. Make changes and corrections to it, sure, but be careful not to over react to the criticism and make too many changes such that you then get even more critical feedback and make more corrections and get worse feedback and bigger corrections and oh god help me worse feedback and bigger changes and stop me please worse feedback and help bigger changes seriously kill me bigger feedback worse changes arggggh
> the potential for confusion between plane // airplane
This reminds me of the famous (possibly apocryphal) story of the algebraic geometer of middle eastern descent who was brought aside by Air Marshals for talking about how a particular problem could be solved by “blowing up points on a plane”
This reminds me of the famous (possibly apocryphal) story of the algebraic geometer of middle eastern descent who was brought aside by Air Marshals for talking about how a particular problem could be solved by “blowing up points on a plane”
At Berkeley the legend is that there was a class on blowing up with special application to local problems.
The main thing missing for me: presumably if there's a plane, you're graphing something vs something. What are the X and Y? (or, uh, the x and y in the x+iy) Talk of a plane remains rather vague without knowing that. Maybe I missed it, but I looked twice, and read the comments on this page, and couldn't see it mentioned at all. There are a couple of graphs/planes, but they seem to be a different kind of thing.
It is the complex plane, x is real, y is imaginary.
the author should have clarified this, you are correct, as the author also makes clear he is writing for a non-technical audience.
He probably forgot to mention it because the concept is so fundamental to signal processing that he assumed it was common knowledge.
the author should have clarified this, you are correct, as the author also makes clear he is writing for a non-technical audience.
He probably forgot to mention it because the concept is so fundamental to signal processing that he assumed it was common knowledge.
Wikipedia has it highlighted on the graph: https://en.wikipedia.org/wiki/Right_half-plane
Also see page 4 here: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
Also see page 4 here: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
Uh that first link is just a picture of a right complex half-plane... I know what that is, having done quite a bit of complex maths, but the kind that uses i, not the kind that uses j, e.g. I love Visual Complex Analysis. I just don't know what things are being graphed in the article! Sorry I didn't explain better. Ok thanks for the 13pp article, I will have a look sometime soon. I was hoping someone could just tell me the answer.
edit: I looked at the first few pages of the paper but I feel none the wiser, at all.
edit2: Ah... "the poles and zeros of a transfer function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex s-plane". The transfer function (whatever that is) is a rational function of the complex variable s, i.e. (in my words) it's a fraction with complex polynomials for numerator and denominator. The zeros are the roots of the numerator and the poles are the roots of the denominator.
Ok, I still don't know what the transfer function is or means or comes from, but am much less in the dark, thank you! :-)
edit: I looked at the first few pages of the paper but I feel none the wiser, at all.
edit2: Ah... "the poles and zeros of a transfer function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex s-plane". The transfer function (whatever that is) is a rational function of the complex variable s, i.e. (in my words) it's a fraction with complex polynomials for numerator and denominator. The zeros are the roots of the numerator and the poles are the roots of the denominator.
Ok, I still don't know what the transfer function is or means or comes from, but am much less in the dark, thank you! :-)
> I know what that is, having done quite a bit of complex maths, but the kind that uses i, not the kind that uses j
Some things in life leave a lasting impression[1]. :eye_roll:
It sounds like what you're looking for is an explanation of root locus analysis[2].
In the simplest control case, a transfer function is nothing more than the expression of a continuous closed-loop LTI system's output Y(s) over its input X(s) in the Laplace domain, conveniently abstracted as its forward path G(s) and negative feedback path H(s).
From there, Routh-Hurwitz method[3] can be used to determine stability of the system.
...and I will continue to use j, thanks.
[1] https://youtu.be/1rqJl7Rs6ps?t=1828
[2] https://en.wikipedia.org/wiki/Root_locus
[3] https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stabilit...
Some things in life leave a lasting impression[1]. :eye_roll:
It sounds like what you're looking for is an explanation of root locus analysis[2].
In the simplest control case, a transfer function is nothing more than the expression of a continuous closed-loop LTI system's output Y(s) over its input X(s) in the Laplace domain, conveniently abstracted as its forward path G(s) and negative feedback path H(s).
From there, Routh-Hurwitz method[3] can be used to determine stability of the system.
...and I will continue to use j, thanks.
[1] https://youtu.be/1rqJl7Rs6ps?t=1828
[2] https://en.wikipedia.org/wiki/Root_locus
[3] https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stabilit...
hehehe Thank you very much! :-) Funny video. I'll check out that course. Thanks for the links.
edit: Took a lil while to work out that LTI system is Linear time-invariant system.
https://en.wikipedia.org/wiki/Linear_time-invariant_system
edit: Took a lil while to work out that LTI system is Linear time-invariant system.
https://en.wikipedia.org/wiki/Linear_time-invariant_system
You were asking "What are the X and Y?" so the graph plot seemed useful in that it has the axes labelled.
I see that /u/metaphor has given you some formal references.
I'd like to chime in with a more intuition-based explanation of what transfer functions are, from my recollections of college control theory classes in both electrical signals and a more general "systems engineering" application:
Basically, the transfer function is a different perspective on modelling/representing a system's output as a function of its input. Classically, when modelling and/or reasoning about a system in physics, the perspective we adopt is that of "input" being the forward advance of time (and sometimes initial conditions) and "output" being the amplitude of the physical quantity(ies) or dimension(s) of the system that interest(s) us. The transfer function, then, is when we switch perspectives to consider the "input" to be a sinusoidal signal (characterized by amplitude and phase over time), and the "output" is the new amplitude and phase of that signal [after "traversing" the system]. Of course, you're actually working with a closed-loop, but most input/output systems can be modeled as a closed-loop if you sufficiently broaden the system's boundaries.
This turns out to be useful for/in several reasons/contexts:
- many physical phenomena are sine waves (or, thanks to Fourier, a sum of sometimes many different sine waves), and often times a system's purpose (to us humans) is to control such a phenomena precisely along the lines of "do this to the amplitude, and/or adjust the phase like so" - dampening, feedback loops, more sophisticated processes like hysteresis, maintaining a steady state given incoming perturbations, etc. In these cases the transfer function ends up being the mathematical expression of that system's function in the "domain language" of that problem, so to speak.
- It turns out that often, when working with systems whose "classical" representation involve components like exponentials or sine and cosine of time (which are "just" complex exponentials of those quantities), the corresponding transfer functions are "simple" fractions of polynomials. More precisely, passing into the Langrange domain allows transforming a differential equation problem into a complex polynomial fractions problem - often much easier to crunch/solve. Furthermore, in the Lagrange domain, de-phasing a signal by pi/2 is equivalent to simply adding 1/(j * signal's frequency) to that signal (if I recall correctly). This makes much of the math more accessible to human intuition, and especially on more complex systems that have several "moving parts" the linear quality of polynomials becomes invaluable.
Personally, I remember quickly adopting, once I'd grokked it, the transfer function perspective when trying to reason about the effect of introducing a capacitor into an existing circuit - analog or DC[0] - as well as things like how the material properties of a door contribute to its behavior as a low-pass filter on sound waves. Sitting down and doing the math, the formulas that I would arrive at spoke much more clearly to me. Also, you are sort of adopting a "time-agnostic" (or perhaps time-invariant) perspective, where the system itself does not change over time. Instead, its' input is characterized by how it behaves over time, and the transfer function (especially when plotted) gives you a clear, direct sense of what the output's "behavior over time" will accordingly be. Notably, it's here that the zeroes of the OP become so meaningful.
[0]: part of what initially started making things "tick" for me was when a professor explained that an impulse on an input signal (i.e. a quasi-instant variation, then back to the preceding "steady state" value of it - i.e. a DC current "turning on"), to a transfer function, "looks like" a sine wave signal with a constant amplitude but monotonously increasing phase offset - again I forget if the rate is constant, polynomial, exponential or what.
I'd like to chime in with a more intuition-based explanation of what transfer functions are, from my recollections of college control theory classes in both electrical signals and a more general "systems engineering" application:
Basically, the transfer function is a different perspective on modelling/representing a system's output as a function of its input. Classically, when modelling and/or reasoning about a system in physics, the perspective we adopt is that of "input" being the forward advance of time (and sometimes initial conditions) and "output" being the amplitude of the physical quantity(ies) or dimension(s) of the system that interest(s) us. The transfer function, then, is when we switch perspectives to consider the "input" to be a sinusoidal signal (characterized by amplitude and phase over time), and the "output" is the new amplitude and phase of that signal [after "traversing" the system]. Of course, you're actually working with a closed-loop, but most input/output systems can be modeled as a closed-loop if you sufficiently broaden the system's boundaries.
This turns out to be useful for/in several reasons/contexts:
- many physical phenomena are sine waves (or, thanks to Fourier, a sum of sometimes many different sine waves), and often times a system's purpose (to us humans) is to control such a phenomena precisely along the lines of "do this to the amplitude, and/or adjust the phase like so" - dampening, feedback loops, more sophisticated processes like hysteresis, maintaining a steady state given incoming perturbations, etc. In these cases the transfer function ends up being the mathematical expression of that system's function in the "domain language" of that problem, so to speak.
- It turns out that often, when working with systems whose "classical" representation involve components like exponentials or sine and cosine of time (which are "just" complex exponentials of those quantities), the corresponding transfer functions are "simple" fractions of polynomials. More precisely, passing into the Langrange domain allows transforming a differential equation problem into a complex polynomial fractions problem - often much easier to crunch/solve. Furthermore, in the Lagrange domain, de-phasing a signal by pi/2 is equivalent to simply adding 1/(j * signal's frequency) to that signal (if I recall correctly). This makes much of the math more accessible to human intuition, and especially on more complex systems that have several "moving parts" the linear quality of polynomials becomes invaluable.
Personally, I remember quickly adopting, once I'd grokked it, the transfer function perspective when trying to reason about the effect of introducing a capacitor into an existing circuit - analog or DC[0] - as well as things like how the material properties of a door contribute to its behavior as a low-pass filter on sound waves. Sitting down and doing the math, the formulas that I would arrive at spoke much more clearly to me. Also, you are sort of adopting a "time-agnostic" (or perhaps time-invariant) perspective, where the system itself does not change over time. Instead, its' input is characterized by how it behaves over time, and the transfer function (especially when plotted) gives you a clear, direct sense of what the output's "behavior over time" will accordingly be. Notably, it's here that the zeroes of the OP become so meaningful.
[0]: part of what initially started making things "tick" for me was when a professor explained that an impulse on an input signal (i.e. a quasi-instant variation, then back to the preceding "steady state" value of it - i.e. a DC current "turning on"), to a transfer function, "looks like" a sine wave signal with a constant amplitude but monotonously increasing phase offset - again I forget if the rate is constant, polynomial, exponential or what.
You had me at "a more intuition-based explanation of what transfer functions are". :-) Thank you so much for this.
edit: By "Lagrange" did you possibly mean to write "Laplace"? I confuse those two gentlemen too. p.s. I just learnt Lagrange was Italian! born Giuseppe Luigi Lagrangia.
edit: By "Lagrange" did you possibly mean to write "Laplace"? I confuse those two gentlemen too. p.s. I just learnt Lagrange was Italian! born Giuseppe Luigi Lagrangia.
Thank you for this article and the follow up. The information is fascinating, and I eagerly await your next installments.
If you haven’t read it, I recommend: https://www.gutenberg.org/files/6949/6949-h/6949-h.htm
I felt that the article ended just where it was ripe to dig into the maths.
Not to mention, it's not just a complex plane. It's the s-domain, reached through the laplace transform.
I lost interest before I got to his political point (about inflation), but the comments made me go back and while it’s an interesting point, it ignores the fact that the current bout of inflation is almost certainly a consequence of the Covid stimulus plus the supply chain disruptions with a dose of war in Ukraine. And it turns out that inflationary pressures seem to be declining outside of the volatile food and energy sectors (see https://jabberwocking.com/inflationary-pressure-seems-to-be-...). I have a bit more faith in domain-specific analysis than abstract mathematical approaches.
It's like "I know math so take me seriously" but then the posts don't use the math in any meaningful way.
Not that I think they should use the math. Just skip the systems lectures and the "there is math behind this" and go straight to the analysis.
Not that I think they should use the math. Just skip the systems lectures and the "there is math behind this" and go straight to the analysis.
Really should've run this by someone who knows how airplanes work, because they don't do what these hypothetical airplanes do. First off, "elevator flaps" is a howler. Elevators and flaps are two very different things. More importantly, you don't climb with the elevator; you climb with excess thrust. You can kinda sorta climb a little bit with just elevator, but unless you're making a very minor altitude correction, you'll slow way down and your climb rate will be really anemic, if you climb at all. But most importantly, an airplane won't just keep climbing and do a loop like that (unless you've got a thrust-to-weight ratio greater than one, which, to put it simply, you don't); the wing will stall way before you get anywhere near vertical, and the airplane will just stop flying until the angle of attack is reduced.
> More importantly, you don't climb with the elevator; you climb with excess thrust. You can kinda sorta climb a little bit with just elevator, but unless you're making a very minor altitude correction, you'll slow way down and your climb rate will be really anemic, if you climb at all.
So what happens to something like a jet fighter that is in level flight with the engines at maximum power if the pilot uses the elevators to raise the nose?
So what happens to something like a jet fighter that is in level flight with the engines at maximum power if the pilot uses the elevators to raise the nose?
Powered airplanes hide the energy-management nature of climb/descent.
Essentially, you have potential energy (altitude) and kinetic energy (speed), and you trade one for the other. For safe range of inputs, your lift is function of pitch and airspeed, with drag as result. Increasing pitch increases lift (most of the time) at expense of increased drag. Lift gets you higher raising your potential energy, which you can spend back on glide, exchanging it for kinetic energy necessary for airflow. With powered plane, you provide extra kinetic energy that can be spent on higher lift. You use your stick to manipulate energy/speed, and in fact it's common to descend while pulling the stick on purpose.
A fighter jet with TWR above 1 has enough extra energy that it's going to be able to maneuver rapidly (afterburner/reheat exists for it, even, as jet engines are slow to spool up). Plane with lower TWR, or with significant mass, is going to behave closer to glider (heavy planes like airliners make energy management a big issue)
Essentially, you have potential energy (altitude) and kinetic energy (speed), and you trade one for the other. For safe range of inputs, your lift is function of pitch and airspeed, with drag as result. Increasing pitch increases lift (most of the time) at expense of increased drag. Lift gets you higher raising your potential energy, which you can spend back on glide, exchanging it for kinetic energy necessary for airflow. With powered plane, you provide extra kinetic energy that can be spent on higher lift. You use your stick to manipulate energy/speed, and in fact it's common to descend while pulling the stick on purpose.
A fighter jet with TWR above 1 has enough extra energy that it's going to be able to maneuver rapidly (afterburner/reheat exists for it, even, as jet engines are slow to spool up). Plane with lower TWR, or with significant mass, is going to behave closer to glider (heavy planes like airliners make energy management a big issue)
It will climb (and slow down), but again, it's the thrust that's producing the climb, and this is an extreme example that isn't illustrative. Fighter jets produce enormous thrust for their weight with many having thrust-to-weight ratios greater than one. If they weren't air-breathing, they could literally fly into space. A normal airplane, even if it weren't air-breathing, couldn't do that.
Depending on the configuration an F-16 has a T/W a little under 1.1, while a transport category aircraft (an airliner) will have a T/W somewhere between 0.20 and 0.35. Totally different performance characteristics.
Depending on the configuration an F-16 has a T/W a little under 1.1, while a transport category aircraft (an airliner) will have a T/W somewhere between 0.20 and 0.35. Totally different performance characteristics.
Always appreciate control theory articles. But this one needs some editing. Take ice cream example;that is not zero dynamics that's still pretty much convolution. Now zero dynamics examples must have no effect on the system. Thus it is not a zero if you continuously chugging ice cream. It is indeed a step input that you equilibriate at a constant ice cream input and some happiness comes out constantly. The moment you stop happiness goes away. That is not related to zero.
Also a pure zero action is supposed to cancel the input completely. Not at first but completely (restricting the discussion to linear systems).
Zeros effects are not so trivial to untangle as the article suggests unfortunately but fun read anyways and very nice flow.
Also a pure zero action is supposed to cancel the input completely. Not at first but completely (restricting the discussion to linear systems).
Zeros effects are not so trivial to untangle as the article suggests unfortunately but fun read anyways and very nice flow.
> Now zero dynamics examples must have no effect on the system. Thus it is not a zero if you continuously chugging ice cream
Thank you for your input! I wonder if you might have misread that example. In this system there's indeed a RHP0 in the transfer function from ice cream consumption happiness. A continuously increasing rate of ice cream intake results in exactly no effect on the output.
Thank you for your input! I wonder if you might have misread that example. In this system there's indeed a RHP0 in the transfer function from ice cream consumption happiness. A continuously increasing rate of ice cream intake results in exactly no effect on the output.
Your happiness shouldn't change if ice cream input has a zero. Also it should be independent from the amount of ice cream. In other words if we have an ODE say
ddot y + 2 dot y + y = dot u - 2 u
As long as my input is pure C exp(2t) independent of C I see no happiness and it's not working on my mood. In your example input u effect is cancelled by decay of y cancelling the guilt. Making it not a zero.
In my personal case eating celery is a zero i see absolute no point eating it :) no harm and no benefit just pointless chewing
ddot y + 2 dot y + y = dot u - 2 u
As long as my input is pure C exp(2t) independent of C I see no happiness and it's not working on my mood. In your example input u effect is cancelled by decay of y cancelling the guilt. Making it not a zero.
In my personal case eating celery is a zero i see absolute no point eating it :) no harm and no benefit just pointless chewing
It’s a bit disappointing that an article purporting to educate us about something we should know but may be unaware of, is completely wrong about the example of how an airplane climbs - you don’t climb by pitching up, you have to increase power.
Just pitching up will have you climb, but at the expense of losing velocity right?
Right up to the stall speed limit. You can't just keep pitching up, you'd have to apply some power too if you don't eventually want to run out of forward (air)speed.
In a glider you can't do that so there when in level flight you have a limited amount of forward momentum available to help you climb if the air itself isn't moving up, you are continuously trading altitude for speed and vv (easy to see in a dive: everybody expects you to gain speed in a dive because can all relate so something falling, it's obvious the reverse has to happen when you climb and the stall speed is a design parameter of the aircraft at a given altitude combined with a bunch of other factors).
https://aviation.stackexchange.com/questions/27693/how-does-...
In a glider you can't do that so there when in level flight you have a limited amount of forward momentum available to help you climb if the air itself isn't moving up, you are continuously trading altitude for speed and vv (easy to see in a dive: everybody expects you to gain speed in a dive because can all relate so something falling, it's obvious the reverse has to happen when you climb and the stall speed is a design parameter of the aircraft at a given altitude combined with a bunch of other factors).
https://aviation.stackexchange.com/questions/27693/how-does-...
Briefly.
Fair. Perhaps a glider would have been a better example.
How? A glider climbs when the air it is in rises faster than the airplane sinks in that air to maintain its forward speed.
If you pitch up you can only exchange speed for a little bit of altitude, briefly.
If you pitch up you can only exchange speed for a little bit of altitude, briefly.
Exactly. A glider in air that doesn't rise is like a yo yo on a string, you need to add energy to the system (rising air, pulling up on the string at the right moment) to be able to overcome the eventual return to the ground state.
For those wondering: this article is an introduction into inflation of house prices in Canada. You can easily find the next article (it's already online) by going to the substack's index.
Very good article, and well worth your time to read.
Very good article, and well worth your time to read.
It’s funny as soon as I read the ice cream but I knew where he was going with it.
The interesting thing is that his modelling concludes that raising interest rates is inflationary which is what Warren Mosler also says.
Anyone interested in what to do about inflation would enjoy the recent Macro n Cheese podcast episode featuring Randall Wray
The interesting thing is that his modelling concludes that raising interest rates is inflationary which is what Warren Mosler also says.
Anyone interested in what to do about inflation would enjoy the recent Macro n Cheese podcast episode featuring Randall Wray
Let’s hope Erdogan doesn’t read this
Link, because I assumed it wasn't out yet since the author hadn't provided it: https://jbconsulting.substack.com/p/on-shelter-futures-part-...
This is very well explained, but there should be a simpler name for this effect than right-half-plane zeros if the author wants to spread the concept beyond control theory.
> The danger of the right-half-plane zero is that it lures you into reacting to it, but that is precisely the wrong thing to do. Attempting to apply a new control input to cancel the inverse response only sets off an even worse chain of events, where the resulting secondary inverse reaction becomes even more severe, requiring even more corrective action, until finally you’ve slammed into the ground.
> In this situation, the flight controller’s only option is to ignore the initial misdirection and wait patiently until the airplane eventually begins to climb as intended.
Of course that is easier said than done; adding a sleep() to your control loop to ignore the initial misdirection is also very bad. The right way to solve this is to not just tell the control loop to "go up", but to plan a realistic trajectory that the control loop can execute. That way, the error between the desired trajectory and the actual trajectory will be much smaller, and the closer the error is to zero, the less chance of a control loop to go wild.
Of course that is easier said than done; adding a sleep() to your control loop to ignore the initial misdirection is also very bad. The right way to solve this is to not just tell the control loop to "go up", but to plan a realistic trajectory that the control loop can execute. That way, the error between the desired trajectory and the actual trajectory will be much smaller, and the closer the error is to zero, the less chance of a control loop to go wild.
> the first important lesson of inverse response is: don’t ride your bicycle on the edge of a cliff
I don't know if many people often ride their bicycle on cliff edges, but many plane (as in airplane) accidents occur because it's difficult / impossible to recover from a stall near the ground.
I don't know if many people often ride their bicycle on cliff edges, but many plane (as in airplane) accidents occur because it's difficult / impossible to recover from a stall near the ground.
You can also get your weight to the inside of a turn for a bicycle by leaning while riding straight.
Every time this comes up there is a big debate with a bunch of people saying counter steering is required. Please, go out and try it. Drive perfectly straight along a painted line. Lean left, turn left. Lean right, turn right. There is no requirement for counter steering.
Every time this comes up there is a big debate with a bunch of people saying counter steering is required. Please, go out and try it. Drive perfectly straight along a painted line. Lean left, turn left. Lean right, turn right. There is no requirement for counter steering.
I think it's somewhat speed-dependent. I ride motorcycles and bikes. On a motorcycle at high speed it's impossible (in my experience) to take a turn without counter steering. You sometimes have to push really hard on the handle on the side where you want to go.
Also, leaning is very similar to counter steering; counter steering makes the bike "fall" on the side where you want to go.
Interestingly, you can also steer a horse without doing anything on the reins; the horse will usually go where you put your weight; I think it's because it needs to compensate for the weight differential; or maybe it takes it as a hint about which way you're looking. In any case it works.
Also, leaning is very similar to counter steering; counter steering makes the bike "fall" on the side where you want to go.
Interestingly, you can also steer a horse without doing anything on the reins; the horse will usually go where you put your weight; I think it's because it needs to compensate for the weight differential; or maybe it takes it as a hint about which way you're looking. In any case it works.
This is some of the best non-rigorous writing about math that I've encountered. Far superior to the awful quanta articles that sometimes get posted here.
Great article and blog!
I would also mention, that we essentially design controllers to shift the poles and zeroes of the total system (which consists of the plant system and the controller system) to more desirable positions, than those of the plant system alone.
I would also mention, that we essentially design controllers to shift the poles and zeroes of the total system (which consists of the plant system and the controller system) to more desirable positions, than those of the plant system alone.
OP: dy/dt = dx/dt + Ax
Comments: The OP is a troubling covert political statement. The real issue here seems to be US midterm elections / climate change / COVID / appeal to authority.
Comments: The OP is a troubling covert political statement. The real issue here seems to be US midterm elections / climate change / COVID / appeal to authority.
> Countersteering on a bicycle is another example: To turn right, a cyclist will first steer slightly to the left.
It's very visible and pronounced on heavier bikes, like motorcycles. Especially if you try riding a very heavy cruiser bike -- you'll immediately notice that countersteering is the only way to turn it. No matter how you try to lean it, it won't respond and will just go straight, but it'll respond very easly to handlebar inputs.
It's very visible and pronounced on heavier bikes, like motorcycles. Especially if you try riding a very heavy cruiser bike -- you'll immediately notice that countersteering is the only way to turn it. No matter how you try to lean it, it won't respond and will just go straight, but it'll respond very easly to handlebar inputs.
As a layman, I had to think of drugs or other dopamine hits first - actions which increase well-being in the short term but are harmful in the long term.
But what he really talks about seems to be the opposite: Actions which case some mild harm in the short term but increase well-being in the long term. So I guess something like working out or going on a diet or making a downpayment for a house?
Except the failure mode is also counterintuitive: Normally, we tend to overvalue the short-term downsides of those actions and therefore shy away from them, missing out on the long-term benefits. But he talks about a situation where we overvalue the long-term benefits but ignore the short-term and overdo the action until the short-term harm becomes critical.
So, e.g. someone working out, getting muscle-ache - and then working out more to counter the ache - which will only lead to more of it until the workout actually starts to become detrimental to their health.
It's easy to see how this would trip up automated control loops, but I don't really see how this has practical application outside of control theory.
But what he really talks about seems to be the opposite: Actions which case some mild harm in the short term but increase well-being in the long term. So I guess something like working out or going on a diet or making a downpayment for a house?
Except the failure mode is also counterintuitive: Normally, we tend to overvalue the short-term downsides of those actions and therefore shy away from them, missing out on the long-term benefits. But he talks about a situation where we overvalue the long-term benefits but ignore the short-term and overdo the action until the short-term harm becomes critical.
So, e.g. someone working out, getting muscle-ache - and then working out more to counter the ache - which will only lead to more of it until the workout actually starts to become detrimental to their health.
It's easy to see how this would trip up automated control loops, but I don't really see how this has practical application outside of control theory.
This stuff is fun. Surprised I haven't seen direct mention of state space representation yet:
https://en.wikipedia.org/wiki/State-space_representation
I recall going through this at university and it being a bit of a struggle. Looking back at it now, this stuff makes a lot more sense. A decade+ of practical experience probably helps.
https://en.wikipedia.org/wiki/State-space_representation
I recall going through this at university and it being a bit of a struggle. Looking back at it now, this stuff makes a lot more sense. A decade+ of practical experience probably helps.
Something I don't understand: on the one hand, people on HN seems to enjoy this kind of articles. On the other hand, they upvote any article who decries interviews where candidates are asked things that involves knowing some theory.
So, if we think knowing theory is useful in cracking hard problems, why is it wrong to asses its knowledge in an interview?
So, if we think knowing theory is useful in cracking hard problems, why is it wrong to asses its knowledge in an interview?
HN's readership isn't homogeneous. There's room for both points of view.
I've done a lot of useful work in feedback systems without ever really grokking Laplacian notation and the notion of complex frequency in general. A lot of the actual numerical methods used in real life boil down to a few canned formulas. But I know enough about the underlying theory to appreciate where the canned formulas come from, and fully intend to sit down some day and go through the whole process. Articles like this are interesting if only for the occasional gems in the comments, such as John N.'s pointer to Maxwell's 'On Governors' paper that I'd never run across before.
At the same time, I don't see much upside in making hiring decisions on the basis of whether someone can regurgitate a bunch of textbook math. I'd rather spend the interview talking about control problems the candidate has dealt with personally, how they were handled, and what the candidate learned from them.
I've done a lot of useful work in feedback systems without ever really grokking Laplacian notation and the notion of complex frequency in general. A lot of the actual numerical methods used in real life boil down to a few canned formulas. But I know enough about the underlying theory to appreciate where the canned formulas come from, and fully intend to sit down some day and go through the whole process. Articles like this are interesting if only for the occasional gems in the comments, such as John N.'s pointer to Maxwell's 'On Governors' paper that I'd never run across before.
At the same time, I don't see much upside in making hiring decisions on the basis of whether someone can regurgitate a bunch of textbook math. I'd rather spend the interview talking about control problems the candidate has dealt with personally, how they were handled, and what the candidate learned from them.
While I don't claim to have the answer to your question, I think there's a jump you're making that may not be so straightforward and somewhat explain things.
You say "people on HN seems to enjoy this kind of articles" which seems reasonable, given the comments here. But then you jump to "we think knowing theory is useful in cracking hard problems".
Going from the first to the second is not quite so clear. That is, someone may enjoy such an article and even learning some theory, but not necessarily because they think they will directly apply it. People sometimes just enjoy learning stuff or reading about it and then forgetting it.
You also make a second jump, because other factors may be involved. Maybe the theory asked in the interviews is completely unrelated to the things involved in the job. The job may not even require cracking hard problems. These are frequent occurrences -in my experience, at least-, and clearly seem compatible with thinking that knowing theory is good in general.
You say "people on HN seems to enjoy this kind of articles" which seems reasonable, given the comments here. But then you jump to "we think knowing theory is useful in cracking hard problems".
Going from the first to the second is not quite so clear. That is, someone may enjoy such an article and even learning some theory, but not necessarily because they think they will directly apply it. People sometimes just enjoy learning stuff or reading about it and then forgetting it.
You also make a second jump, because other factors may be involved. Maybe the theory asked in the interviews is completely unrelated to the things involved in the job. The job may not even require cracking hard problems. These are frequent occurrences -in my experience, at least-, and clearly seem compatible with thinking that knowing theory is good in general.
Because programming is rarely about cracking hard problems. That's actually why the people HN enjoy the kind of article, it takes them away from the drudgery of their job.
[deleted]
Programming and engineering are neighbors, so it is interesting to see what's going on over there. But many engineering analysis tools are not really useful for normal programming jobs (unless you happen to be working on engineering software, control systems, or something like that).
[deleted]
Odd that OP would give a grim warning without an equation.
I graduated in 1991 with BSEE and the curriculum was a rush to get us to diffeq and linear systems because 90% of the remaining three years of classes were taught using S and Z plane analysis. My engineering professors were incapable of introducing a concept without starting with a differential equation.[1]
For the author to spend so much time explaining control theory, then sort of give up on it, was disappointing. Ironically, knowledge of control systems and warnings about them should pervade the thought process of the thinker and prevent them from making grandiose claims. Perhaps OP only has a topical knowledge but can spin a good yarn.
[1] I'm a career programmer, but I always start my thoughts with a d/dt: whether it's an RTOS project, a graph GraphQL endpoint, or USB driver. I guess "if all you have is a hammer..."
I graduated in 1991 with BSEE and the curriculum was a rush to get us to diffeq and linear systems because 90% of the remaining three years of classes were taught using S and Z plane analysis. My engineering professors were incapable of introducing a concept without starting with a differential equation.[1]
For the author to spend so much time explaining control theory, then sort of give up on it, was disappointing. Ironically, knowledge of control systems and warnings about them should pervade the thought process of the thinker and prevent them from making grandiose claims. Perhaps OP only has a topical knowledge but can spin a good yarn.
[1] I'm a career programmer, but I always start my thoughts with a d/dt: whether it's an RTOS project, a graph GraphQL endpoint, or USB driver. I guess "if all you have is a hammer..."
Counterpoint: no you don’t.
By the time you have a perfectly reasonable model of a system that is good enough such that computing the transfer function’s poles actually tell you something interesting about the system, there’s way more you can say about the system than “it is stable.”
There are maybe some lessons to be drawn from basic “classical” control theory, but many are better stated by just analyzing the system directly.
(As a side note, I’m not saying there’s Zero value in analyzing transfer functions, just that it’s a long way to the top from there.)
By the time you have a perfectly reasonable model of a system that is good enough such that computing the transfer function’s poles actually tell you something interesting about the system, there’s way more you can say about the system than “it is stable.”
There are maybe some lessons to be drawn from basic “classical” control theory, but many are better stated by just analyzing the system directly.
(As a side note, I’m not saying there’s Zero value in analyzing transfer functions, just that it’s a long way to the top from there.)
The article claims to be for a broad audience, and contains insinuations of political relevance. But right-half-plane zeros are a purely technical concept, and they're not explained with enough technical detail to be air-dropped into the middle of a broad audience piece.
It's almost good. It would be good if it dropped the pretensions to technicality.
It's almost good. It would be good if it dropped the pretensions to technicality.
I watched this video in preparation for a Drone license test.
Really interesting stuff.
Human Factors for Drone Pilots
https://www.youtube.com/watch?v=UHYjl0u2UMA
Based on what real pilates need to know about human physiology as it relates to steering airplanes.
Human Factors for Drone Pilots
https://www.youtube.com/watch?v=UHYjl0u2UMA
Based on what real pilates need to know about human physiology as it relates to steering airplanes.
If you are going slow on a bicycle, and want to turn right, you might have to turn your front wheel left and then right, just so you don't fall over and maintain upright balance. Which would be a problem if you are riding along a cliff's edge.
TLDR; If you want to make something go up (or down, or left or right) sometimes it goes down (or whatever the opposite is) a bit first. If you were to try harder to make it go up at that point, the problem gets worse, repeat until your system crashes. Times like these are called right-half-plane zeros and any control system you build has to be aware of them and designed to avoid causing a feedback loop.
So sometimes it is better to cut losses rather than double down because planes exist.
Are they still important if you use something like model predictive control ?
They are certainly something that your MPC controller will need to account for, and they will constrain the theoretical maximum performance you can get out of the system no matter how optimal your control algorithm is.
I have a feeling this is going to be about interest rates and housing.
I don't think they should use the term right-half-plane zero if they don't explain laplace transform and the s-domain. Just call it an unstable system, or unbounded growth.
The point of the article, perhaps not stressed enough, is that there are stable systems that can be very difficult to control. Zeros in the right half-plane don't indicate instability. They indicate nulls in the transfer function for unbounded inputs.
Poles in the right half-plane indicate instability. Zeros in the right half-plane indicate something a little more subtle. I believe the article is about the latter.
Poles in the right half-plane indicate instability. Zeros in the right half-plane indicate something a little more subtle. I believe the article is about the latter.
Good point, I mixed up the two. However, the right-half plane still refers to the s-domain so it doesn't make sense to keep referring to the a right half plane without explaining what plane it is
That would be a right-half-plane pole.
An explanation of the Laplace transform is coming! This is an article, not a book.
An explanation of the Laplace transform is coming! This is an article, not a book.
The follow-up article on inflation of shelter costs in response to mortgage interest rates is also fascinating and worth a read, imo [0]. The tl;dr is that in the medium and longer term, higher interest rates will decrease inflation, but there's a right-half-plane zero effect that will cause higher inflation rates in the short term in response to interest rate hikes (with a lot of simplifying assumptions).
[0] https://jbconsulting.substack.com/p/on-shelter-futures-part-...
[0] https://jbconsulting.substack.com/p/on-shelter-futures-part-...
It's a little more complex than that because of the potential feedback loop via the main price channel.
In other words if you can have a wage/price spiral you can have a profit/price spiral and an interest/price spiral for the same reasons.
[0]:https://new-wayland.com/blog/interest-price-spiral/
Once you abandon the concepts of general equilibrium and a fixed amount of money for what happens in reality, then the functional control mechanisms get a whole lot more interesting.
In other words if you can have a wage/price spiral you can have a profit/price spiral and an interest/price spiral for the same reasons.
[0]:https://new-wayland.com/blog/interest-price-spiral/
Once you abandon the concepts of general equilibrium and a fixed amount of money for what happens in reality, then the functional control mechanisms get a whole lot more interesting.
The danger of "systems thinking" is that it misleads otherwise clever people into believing they've found a cheat sheet for subjects they don't have deep knowledge in. This article is a fine example of that.
I didn't know what right-half-plane zeroes are. I knew some of the examples it gave (Veritasium has a fun video on the bicycle steering phenomenon), but not that there was a category they all fit into. Neat.
But I got squinty when the author said the intended audience was everybody, and got a hunch it was going to wander into the current economy ... which it did, sort of, except that in this case, "wandering" into the subject meant, "wrap it up with a graph and then point at it and go, see! See! I can predict the economy now!"
The follow-up at https://jbconsulting.substack.com/p/on-shelter-futures-part-... goes into more detail and concludes that interest rate hikes will increase inflation over the next 5-ish years, but there's no "part 2" in the series to be found (paywalled?).
There are lots of graphs and the author tries to build a case out of several arguments, buuuuut in the end it feels like that IASIP conspiracy board meme. I smell a faint whiff of gish galloping here and there, but one thing that stands out to me is that the author chooses to normalize housing costs against inflation, but there's no mention of wage stagnation anywhere.
Rents and housing costs can't rise a whole lot more, certainly not to the extent the author seems to be predicting, because people can't afford them. This is already a conversation happening in every housing market in at least several countries. In the US, pick literally any local subreddit and ask if anyone knows of an affordable place to rent, I dare you. There's already a huge epidemic of unhoused people and van-dwelling is more popular than it has ever been, and consumers are currently getting squeezed in a lot of directions. Here, one of the things in my browser history before this article was this thread: https://old.reddit.com/r/news/comments/v8knl5/gas_prices_hit...
So, if wages don't rise to meet these costs, then something big is going to break way before the cost increases the article is doomsaying.
I didn't know what right-half-plane zeroes are. I knew some of the examples it gave (Veritasium has a fun video on the bicycle steering phenomenon), but not that there was a category they all fit into. Neat.
But I got squinty when the author said the intended audience was everybody, and got a hunch it was going to wander into the current economy ... which it did, sort of, except that in this case, "wandering" into the subject meant, "wrap it up with a graph and then point at it and go, see! See! I can predict the economy now!"
The follow-up at https://jbconsulting.substack.com/p/on-shelter-futures-part-... goes into more detail and concludes that interest rate hikes will increase inflation over the next 5-ish years, but there's no "part 2" in the series to be found (paywalled?).
There are lots of graphs and the author tries to build a case out of several arguments, buuuuut in the end it feels like that IASIP conspiracy board meme. I smell a faint whiff of gish galloping here and there, but one thing that stands out to me is that the author chooses to normalize housing costs against inflation, but there's no mention of wage stagnation anywhere.
Rents and housing costs can't rise a whole lot more, certainly not to the extent the author seems to be predicting, because people can't afford them. This is already a conversation happening in every housing market in at least several countries. In the US, pick literally any local subreddit and ask if anyone knows of an affordable place to rent, I dare you. There's already a huge epidemic of unhoused people and van-dwelling is more popular than it has ever been, and consumers are currently getting squeezed in a lot of directions. Here, one of the things in my browser history before this article was this thread: https://old.reddit.com/r/news/comments/v8knl5/gas_prices_hit...
So, if wages don't rise to meet these costs, then something big is going to break way before the cost increases the article is doomsaying.
> Rents and housing costs can't rise a whole lot more, certainly not to the extent the author seems to be predicting, because people can't afford them.
House prices are passively controlled because there's only so much money to spend on housing?
House prices are passively controlled because there's only so much money to spend on housing?
The footnote is nonsense. Needs revision, author is currently arguing that poles are indistinguishable from zeros.
The footnote currently appears to be correct.
I'm not aware of what you mean.
This is weird to say, but I've never had this happen before: as someone who's never heard of control theory, the initial illustrations and the leading icecream example seemed highly confusing and put me off from reading the rest.
Not to mention poles in the unit circle.
My dynamical systems prof loved that xD
Economics is not a physical theory.
Full stop.
Full stop.
I'm nX 1 too<p>;)\n
can anybody tell me what's f in this scenario?
dy/dt = dx/dt + Ax
is it the case that y = f(x,t)?
thanks
dy/dt = dx/dt + Ax
is it the case that y = f(x,t)?
thanks
Can someone provide a TL;DR? I don't have time to read the whole article, and I don't even understand what's the topic based on title and reading a few paragraphs, nor from googling "right-half-plane zeros".
The plane refers to the plane of complex numbers (a+b*i), the right half is where the real part (a) is positive. If the zeros of your feedback loop are on the right hand side that means it is a unstable feedback loop, if they are on the left side it is stable feedback loop (this happens because the zeros eventually end up in exponents which either tend to infinity or to zero with time).
If you squint a little bit most systems looks like a feedback loop and my not very confident interpretation is, judging from the last picture, that the author thinks we have a runaway inflation problem (zeros on the right hand side of the plane) and we might or might not have the controls to move them to the left hand side to tame it.
If you squint a little bit most systems looks like a feedback loop and my not very confident interpretation is, judging from the last picture, that the author thinks we have a runaway inflation problem (zeros on the right hand side of the plane) and we might or might not have the controls to move them to the left hand side to tame it.
I thought this would be something about positive zero in a 2D coordinate system, but seeing drawings of airplanes in the article I concluded “bummer, it’s not” and immediately hit backspace. Thanks for the summary. :)
This is fantastic.
The author lost all credibility when he said I have to countersteer to turn my bike. The author is a "professional engineer and consultant, specializing in feedback systems and motors" too. There are >50 million cyclists in the US who would presumably immediately recognize this as a false statement and he didn't ask a single one of them to proofread.
Edit since some people are confused by this apparently: You can steer your bike away from the edge of a cliff by leaning to initiate the turn instead of countersteering. This makes the author's advice of "Don’t ride your bicycle on the edge of a cliff" much less critical.
Edit since some people are confused by this apparently: You can steer your bike away from the edge of a cliff by leaning to initiate the turn instead of countersteering. This makes the author's advice of "Don’t ride your bicycle on the edge of a cliff" much less critical.
In the interest of defending my credibility, consider the challenge of explaining some concept to an audience using concrete examples, when any example has subtle associated details and exceptions that, while fun to explore on their own, do not further the goal of explaining the original concept.
If you'd prefer, I can explain the concept of the right-half-plane zero using only abstract systems described by transfer functions. But I suspect that will be less effective from an education standpoint.
I maintain that countersteering a bicycle is an excellent example of a RHP0, and it's no coincidence that a translating inverted pendulum also has an RHP0 in its control solution.
I also am one of those cyclists and, at least prior to the pandemic, commuted by bicycle every day.
If you'd prefer, I can explain the concept of the right-half-plane zero using only abstract systems described by transfer functions. But I suspect that will be less effective from an education standpoint.
I maintain that countersteering a bicycle is an excellent example of a RHP0, and it's no coincidence that a translating inverted pendulum also has an RHP0 in its control solution.
I also am one of those cyclists and, at least prior to the pandemic, commuted by bicycle every day.
One can always test the hypothesis via an experiment:
https://youtu.be/9cNmUNHSBac
https://youtu.be/9cNmUNHSBac
Heh was about to reply with the same video. It’s really counterintuitive even for people who ride a lot
It's counterintuitive to cyclists because we don't countersteer. We lean instead, generally subconsciously. If we didn't lean, it's true that we'd have to countersteer... but nobody rides that way, making it a flawed analogy at best.
You always countersteer; you're just not doing it consciously. Because of that, you're doing it very slightly. Because you're doing it slightly, you enter into turns slowly. You're not able to take sharp corners at a decent speed.
You cannot lean without countersteering first; the lean happens because you induced a fall in the desired direction, through a tiny countersteering move. With the tiny, subconscious counter-steers, you induce only small tilts. I suspect it takes multiple small counter-steering maneuvers to "build up" a decent tilt for a sharp turn, which takes time.
Once I learned to deliberately countersteer, I then started doing it all the time. I hardly take any turn (big or small) on a bicycle without pushing forward the handle-bar on that side.
If you don't make deliberate countersteering your main steering method, so that it becomes second nature, you will not be able to count on yourself to use it in an emergency.
You cannot lean without countersteering first; the lean happens because you induced a fall in the desired direction, through a tiny countersteering move. With the tiny, subconscious counter-steers, you induce only small tilts. I suspect it takes multiple small counter-steering maneuvers to "build up" a decent tilt for a sharp turn, which takes time.
Once I learned to deliberately countersteer, I then started doing it all the time. I hardly take any turn (big or small) on a bicycle without pushing forward the handle-bar on that side.
If you don't make deliberate countersteering your main steering method, so that it becomes second nature, you will not be able to count on yourself to use it in an emergency.
Maybe you countersteer. I don't. I checked by riding through a puddle, turning right, and then checking the resulting trail left by the tires. No overshoot whatsoever.
Your interpretation of the trail must have missed something. We know that all turning of a bike is via countersteering. You wouldn't be able to turn right on a bike whose steering is locked out from turning left. (See video upthread.)
You can ride a bicycle that is locked out from steering at all. (You can ride one-footed on anti-rocker (2 wheel) inline skates, with some practice).
I’m not good downhill mtb rider but my understanding it’s really common technique there. Even on road i do that not to take a spill in sharp-ish downhill turn going at speed
Indeed. I like this clip where they show bike steering to the left, without its front wheel ever going to the right side of its original path, by only leaning and no countersteering. https://youtu.be/9cNmUNHSBac?t=445
That's with no rider on the bike. Earlier in the video when he attempts to turn with the device that prevents countersteering, he falls off every time. It seems unlikely that 50M cyclists could easily do this but he could not.
The bicycle and riding style used in the video do not provide a good illustration, I have no idea what he was trying to do as he crashed repeatedly, and he did not seem (or the frame prevented him) to use the ability of his body to lean the bike into the turn, riding naively (which didn't prevent them from talking down as if there is a clear consensus in the field of bicycle dynamics).
At low speeds on e.g. a light MTB stop pedaling, lift your butt off the saddle, lean bike far into desired side balancing your body and handlebars to keep going straight. You have most of what's needed to turn. You may countersteer but your line does not veer into the direction of countersteer, counter to illustrations.
Riding edge of a cliff is not a great idea anyway because balancing a bicycle tends to require countless small adjustments even due to pedaling alone.
At low speeds on e.g. a light MTB stop pedaling, lift your butt off the saddle, lean bike far into desired side balancing your body and handlebars to keep going straight. You have most of what's needed to turn. You may countersteer but your line does not veer into the direction of countersteer, counter to illustrations.
Riding edge of a cliff is not a great idea anyway because balancing a bicycle tends to require countless small adjustments even due to pedaling alone.
> The bicycle and riding style used in the video do not provide a good illustration
It's good enough if you already understand it and know what to look for; just not the best for convincing someone who doesn't believe the physics.
It's good enough if you already understand it and know what to look for; just not the best for convincing someone who doesn't believe the physics.
You don't just believe the physics. If there is consensus in physics that says you cannot turn a bicycle without measurably turning the opposite way first, feel free to link. It's a ridiculous claim that assumes certain speeds, bicycle type and riding style. I've seen a guy ride a narrow rocky ridge, even jumping from stone to stone where needed (without stepping off the bicycle), was he violating laws of physics or is the world more complex than simplified models of it? Either way, it is just bikeshedding.
> That's with no rider on the bike.
That's irrelevant; bicycles countersteer to keep stable, without a rider present (above a certain speed).
If the riderless experiment were repeated with a bike whose steering is locked out, it would fall down rather quick.
That's irrelevant; bicycles countersteer to keep stable, without a rider present (above a certain speed).
If the riderless experiment were repeated with a bike whose steering is locked out, it would fall down rather quick.
Now just imagine the possibilities with a rider on the bike. Being able to dynamically shift weight to either the left or right side of the bike at will! You can even do turns while riding with no hands.
When you shift your weight to the left, the bicycle tilts right to keep the center of gravity above the track line.
If this were the case, you wouldn't be able to turn while riding with no hands. Yet reality has us out here riding bikes with no hands making turns. https://youtu.be/_SWdCutqSWQ?t=73
That is false, because you can induce a motion of the steering column without using your hands. If you shift your center of gravity around without your hands on the bars, the steering column will do things on its own and you can learn to predict and control that. Not just simply 0shifting the center of gravity in the seat; you can do things if you seize the top bar between your knees.
A steering column which somehow turns right without your hands will induce a countersteer to the left the same way as if you had used your hands.
A steering column which somehow turns right without your hands will induce a countersteer to the left the same way as if you had used your hands.
The video shows the exact opposite of your claim. The bicycle is given an initial push by the experimenter. Its steering is initially pointing a bit to the left. Because of this, it begins falling to the right, and recovers by steering in that direction, away from the guardrail.
The bike seems to be initially leaning to the left, because it was induced into motion that way by the experimenter. Yet in spite of this, it recovers by tilting to the right.
The bike seems to be initially leaning to the left, because it was induced into motion that way by the experimenter. Yet in spite of this, it recovers by tilting to the right.
Looks to me like 1. The bike is initially rolling straight with no handlebar angle, and then starts falling over to the left. 2. Due to the mechanic explained in the video, this causes the handlebar to turn left.
If there was a human on that bike wanting to turn, here they would continue leaning, turning, and pedaling in the correct amounts.
But because there's no human to keep their weight leaned to the left, then 3. The left turn of the handlebars causes the road contact points to move to the left of the center of gravity, so that 4. It starts turning right.
If there was a human on that bike wanting to turn, here they would continue leaning, turning, and pedaling in the correct amounts.
But because there's no human to keep their weight leaned to the left, then 3. The left turn of the handlebars causes the road contact points to move to the left of the center of gravity, so that 4. It starts turning right.
I'm curious how you're so confident about this. I think if you ask most people (even most cyclists) what happens when you turn the handlebars to the right, they'd say you turn to the right. I cycle almost every day and I thought this for a long time. Have you actually tried it? When I tried it, I found that it forces me into a left turn.
Have you tried leaning to the left while biking in a straight line? It results in you turning left without your front wheel ever having gone to the right (i.e., turning without countersteer).
> Have you tried leaning to the left while biking in a straight line?
That is impossible. If you lean your body left while biking in a straight line, the bike will lean right to keep your center of gravity above the track.
For bike and body to be leaning left, the bike must already be in a turn. Or else, it must be in the middle of a fall.
Two-track vehicles always begin a turn with a countersteering move which induces a fall in the opposite direction. Then the steering immediately switches in the falling direction, to convert the fall into a turn.
(In the absence of wind anyway. A left turn could plausibly start without a countersteer if a gust of wind blows over the bike into a left lean; the subsequent left steer and turn will supply the compensating acceleration to prevent a fall.)
A bicycle is always countersteering. A bicycle whose steering column is locked out, prevented from turning, cannot be ridden. It will fall over.
A bicycle doesn't require a rider in order to maintain balance, either. Above a certain speed, a bicycle can correct itself. This is due to countersteering. Whenever the bicycle accidentally steers slightly to the right, it begins to fall to the left, and this provokes a left steer which prevents a fall.
A rider who doesn't understand countersteering nevertheless intuitively "bootstraps" the turn out of these small wobbles, gradually increasing the tilt and and degree of turn, through a sequence of small counter-steering maneuvers.
That is impossible. If you lean your body left while biking in a straight line, the bike will lean right to keep your center of gravity above the track.
For bike and body to be leaning left, the bike must already be in a turn. Or else, it must be in the middle of a fall.
Two-track vehicles always begin a turn with a countersteering move which induces a fall in the opposite direction. Then the steering immediately switches in the falling direction, to convert the fall into a turn.
(In the absence of wind anyway. A left turn could plausibly start without a countersteer if a gust of wind blows over the bike into a left lean; the subsequent left steer and turn will supply the compensating acceleration to prevent a fall.)
A bicycle is always countersteering. A bicycle whose steering column is locked out, prevented from turning, cannot be ridden. It will fall over.
A bicycle doesn't require a rider in order to maintain balance, either. Above a certain speed, a bicycle can correct itself. This is due to countersteering. Whenever the bicycle accidentally steers slightly to the right, it begins to fall to the left, and this provokes a left steer which prevents a fall.
A rider who doesn't understand countersteering nevertheless intuitively "bootstraps" the turn out of these small wobbles, gradually increasing the tilt and and degree of turn, through a sequence of small counter-steering maneuvers.
"A LOT (Polish airline) airplane is about to land in New York City; as they align for final approach, the first officer notifies the passengers that those seated on the right can now see the Statue of Liberty. A number of passengers get up from their seats left of the aisle and lean over the people seated on the right to get a glimpse of the statue. Plane promptly crashes.
Why? There were too many Poles in the right half of the plane.
I'll lead myself out.