#control theory

[cyborgized]

Drawings by English mathematical physicist, mathematician, philosopher of science and Nobel Laureate in Physics, Roger Penrose

[mindblowingscience]

Fundamental engineering principles can help identify disease biomarkers more quickly

People often compare the genome to a computer's program, with the cell using its genetic code to process environmental inputs and produce ap

People often compare the genome to a computer's program, with the cell using its genetic code to process environmental inputs and produce appropriate responses. But the machine metaphor can be extended even further to any biological system, and applying established concepts of engineering to biology could revolutionize how scientists make their observations within biology, according to research from University of Michigan. In a paper published in Proceedings of the National Academy of Sciences, Indika Rajapakse, Ph.D., Joshua Pickard, Ph.D. (now an Eric and Wendy Schmidt Postdoctoral Fellow at the Broad Institute), and their team propose that fundamental principles of control theory and observability can be applied to study biological processes that change over time.

Continue Reading.

[cauchyfeelings]

Solving stability proofs are actually really easy.

All you have to do is just guess and hope. If it’s a particularly difficult one I heard praying might help.

[thursdayisbetterthanfriday]

The Unaccountability Machine

Information Theory and The Machine of the Economy

"things went off the rails starting in about 1970 when everyone decided that the way they were going to deal with complexity was basically to ignore it and to assume that the free market would solve all the problems because people you know from Hayek onwards in the economics profession have always regarded the market as being a magic all-purpose computer..."

"it's really weird that a profession that's based on the idea that there's no such thing as a free lunch is always so happy to help itself to an informational version of exactly the same that thing ... that the market is this giant information processing and decision making machine"

"the great tragedy of our friend Hayek is that he was thinking in a very sophisticated way about information in the economy but he was doing it in the 1920s and 30s and the concept of information Theory did not exist then"

"you know all of Hayek's works on information in the economy were before Alan Turing they were before Kurt Godel... they were before Shannon you know the concept of information was not on a sound kind of rigorous mathematic basis when he was doing these things"

"unfortunately all the economists who came after Hayek presumed that information in the economy was a solved problem. It was a done deal. " "It was the Socialist calculation problem everything that was interesting to say about it had been said in 1928 and so they never really got to grips with the idea that information is a quantity and as equally as important as balancing supply and demand is the ability to balance the information generated by the environment."

"the kind of chaos and variety and variability you know of the world with the capacity of the system that's meant to be regulating and managing that environment to process it because if those if your information balance sheet isn't balancing then the system has become unregulated and then not only is it going to go out of control but also because it's unregulated you cannot assume that your actions to try to control it are going to push it back to a steady state"

"it's quite often the case that you'll be pushing it out of equilibrium and you'll be causing you know what they call what you'd recognize as oscillation if it happened in an engineering framework because you're working on a broken model of reality..."

"and you know that's kind of my diagnosis of a lot of the state that we've got to which is that we're just working on a broken or insufficient model of reality and as a result very often we're making policy interventions that are just pushing things further away from our desired State without us really knowing that we're doing that" "The way in which the economics profession ignored most of the work done in information theory is striking. It ignored its own traditions relating to uncertainty and decision-making, instead ploughing ahead with a frighteningly simplistic view of the world in which everything could be reduced to a single goal of shareholder value maximisation."

- Dan Davies - The Unaccountability Machine

[phenomenologyofperception]

[alanshemper]

Some light reading in our lab library

[whitemagedelta]

Jesse REALLY does not like being addressed as Director. Marshall is the only one to try it twice and both times, Jesse tells her to just call her Jesse. Marshall compromises and says “I’ll call you Faden.”

In a previous post, I theorized that Jesse may just be uncomfortable with having that much authority, that it was the weight of the responsibility that led her to ask her people not to address her with the title. But after some consideration, I don’t think that’s it.

At one point, early on I believe, Jesse says “I’m nobody’s director.” She doesn’t want it. Who would? Who would willingly become a part (let alone chief) of the organization who took your baby brother away after a traumatizing supernatural event that made you feel like you were crazy during your formative years, which because of them you had to endure entirely alone, with no support at all?

Yeah. I think it’s animosity toward the FBC that leads to her aversion to being addressed as director.

[longestpathsearch]

Least Squares as a Natural Generalization of PID (Short Post)

I want to say this is a folk theorem (borrowing terminology from Game Theory) in that everyone who does optimal control theory knows about this stuff, probably [1] but I haven’t really seen it stated explicitly anywhere. Well, whatever, I thought it’d be interesting to state anyways. If anyone does indeed, work on optimal control, I’d love to know your thoughts (even if you don’t, I’d still like to know your thoughts on it)!

For context, currently, I’m leading a team on path planning for fixed-wing UAVs (I still don’t really know who put me in charge of this stuff, or why for that matter—overall, it seems pretty terrifying for them, but kinda fun for me), and I wondered why I hadn’t actually seen least squares in many papers on fixed-wing control. I still haven’t gotten an answer to the question, to be honest, but I did waste some potentially productive time showing that PID ⊂ LS. For context, let u(t) be our control input and allow ε(t) to be the error of the function, then PID is defined as

where each of the K variables are a gain or proportionality constant. Say Kp is the proportional constant (e.g. how much of u is proportional to the current error), Ki is the integral proportionality constant (e.g. how much of u is proportional to the integral of the error), and Kd is the derivative constant (e.g. ditto). For a more thorough explanation for what each of these means intuitively, see the PID wikipedia page.

Anyways, I’ll likely make a separate (more introductory post to LS) but, for now, I define an LS problem to be an optimization problem of the form, for arbitrary but given A, b, λ>0, C, d

where the norm is the usual ℓ-2 norm. It’s notable that this problem has an analytical solution (not that you’d necessarily want the analytical solution for most big-enough scenarios) and is extremely well-behaved for most optimization methods [2]. Now, consider the following objective function with trivial equality constraints (e.g. 0=0, for convenience, by setting C = (0,0,…,0)’, d = 0) and K being some proportionality constant (I’ll make the connection to the original K variables above, soon):

Minimizing this function by setting its gradient to zero (this is necessary and sufficient by differentiability, convexity, and coerciveness [that is E(u) → ∞, whenever ‖u‖ → ∞]) gives the solution

or, after rearranging

which allows the following correspondence between the original PID and the LS problem to be

So, now we’ve given the condition and we’re done!

Anyways, you may ask, why is this useful? I guess it kind of extends the framework to add constraints from your control surface, or secondary objectives. To be completely honest, though? I have no idea.

[1] I mostly know people who do hardware work, etc. on UAVs, so I don’t really have a representative sample of control people.

[2] PolitiFact: Mostly true. I mean the usual cases (e.g. first-order methods, second-order methods, or conjugate gradient/quasi-newton methods). It’s horribly behaved in conic program (SOCP) solvers.