Breaking the ice - Get to know Lake Vostok

One of my favorite stories of the past couple of weeks has been the near completion (or completion by some accounts) of the Lake Vostok drilling project in Antarctica. Lake Vostok, for the unfamiliar, is a sub glacial lake in central Antarctica located more than 12,000 feet below the surface ice. Due to the extreme pressures underneath the ice, the lake consists of liquid water at about -3°C below freezing (remember your phase or PV diagrams!). Well, the reason this story is so exiting is the possibility of discovering life in an extremely isolated environment. The lake itself looks to have been isolated from the surface for around 15 to 25 million years, so any surviving organism would be extremophile to say the least. Discovery of life here though would have major implications for the prospects of life on other planetary bodies in our solar system. Europa (of Jupiter), for instance, is a classic example of a planet with icy surfaces with potential for sub-glacial oceans. 

Anyhow, back to the drilling project. The Russian have been drilling core sample at the Vostok station in Antarctica since the 1970’s, but only in 1998 did a joint venture penetrate to nearly 12,000 feet. At the time, the researchers called a halt to the drilling process out of fear of contamination of the lake due to drilling agents. As a result, there has been much controversy as to how to proceed to complete the drill into the lake. The Russian have been using Kerosene to prevent the bore hole from refreezing, but many environmentalists worry that the kerosene will flood into the lake and contaminate the water. It seems to me (after a couple of back of the envelope calculations) that the kerosene column pressure should not be greater than the lake pressure. Resulting in the lake forcing water up into the borehole, rather than the other way around. I can see the dilemma though… we don’t want to destroy something as pristine as Lake Vostok. It’ll be quite interesting to see how things proceed.

When prediction fails, and why.

In preparation for my dissertation proposal (hopefully sometime this semester), I’ve been diligently patching all potential logical flaws in my research assumptions. One of the overarching assumptions that I had been operating on was the inadequacy of existing energy storage prognostics techniques (batteries in particular) for non-constant loads. My brief original forays into the literature indicated this to be true, but I never independently confirmed this fact until Friday. Well, my treacherous spelunking in the literature allowed me to differentiate existing prediction algorithms, and I found that a shocking number or algorithms rely on an archaic empirical relationship, Peukert’s law. This law directly relates the run time of the battery to the current drawn. The peculiarity of battery systems, modeled by Peukert’s law, is the total available charge diminishes given higher currents (in other words, the efficiency of the battery drops). However, I ran some tests with real battery draw data and confirmed what a paper by Doerffel in 2006 illustrated… the invalidity of Peukert’s law in a vast majority of realistic cases.

The first tests that I ran used battery data with constant currents. In this case, Peukert’s law should apply. Accounting for the uncertainties in the parameters, we can generate the probability distribution of predictions. As shown in the figure, Peukert’s correlation is well founded. However, if we look at real battery data given a periodic load (on/off only), this correlation crumbles. As I mentioned above, a shocking number of published techniques still rely on this empirical correlation for prediction for realistic systems… completely discounting the inaccuracy. Dynamic battery models, on the other hand, directly account for variability in the load… given Occum’s razor, simpler prediction actually requires dynamic models.

Hybrid systems and mathematical oddities

While I was hoping to upload this post by the weekend, one thing (laziness) led to another (watching copious amounts of Battlestar) and my to-do list persists! So it goes. With a new semester upon us, I figure I should give just a hint of the fascinating mathematics that I’ve been delving into the past couple of weeks. Some of the most fascinating bits of hybrid control theory that I’ve been exposed to thus far have been illustrated with seemingly trivial problems. (Hybrid control, in a nutshell, is the control of continuous system which encounter discrete switched states. Hopefully this is clarified with the bouncing ball example). However, the results of the analysis prove to be profound. One simple example, a bouncing ball, illustrates how naïve modeling results in physically impossible implications. In this case, Zeno behavior. 

Zeno behavior arises in low fidelity hybrid system models in which the state trajectory gets “stuck” in a switched area in finite time. (This behavior is aptly named after the paradox suggested by the Greek philosopher, Zeno. He conjectured that one was to step halfway to a wall, then half way again, and so on ad infinitum… you would never actually reach the wall… well, his true example involved Achilles and a tortoise). In the bouncing ball example, with a simple model of the ball that attenuates velocity each bounce we encounter a point where the ball bounces an infinitely many times in a limited time. Physically, we know that this is impossible. However, if we’re not careful with our models, Zeno behavior can crop up and destroy both control design efforts and simulation ability.

My top 2011 albums

I know this discussion might seem out of place now with the new year nearly twenty days behind us now, but none the less, I’ve been wanting to mention my album choices of 2011. Thus far today has been heavily devoted to the daily standard grind, so I figured I would spend a little time mentioning music that has really helped to shape how I view 2011. The way I figure I’ll do this is mention the top five albums that I listened to this past year and then go into more detail about my favorite two (man did I listen to these albums relentlessly… 

And now, introducing the albums… in some particular order, beginning with my absolute favorite and moving to a more amorphous collection of albums that were constantly playing at work:

1. Yeasayer – Odd Blood
2. Beirut – The Rip Tide
3. Bon Iver – Bon Iver
4. Youth Lagoon – The Year of Hibernation
5. The Decemberists – The King is Dead 

What I love to find in music that I listen to on a regular basis is depth beyond a melody and a harmony. The Yeasayer’s album, Odd Blood, has intensely deep track layers that I find myself discovering a new sub-sub-sub melody after twenty listens. Of course, complexity alone isn’t what creates spectacular music, Yeasayer also manages to interlace all these melodies/harmonies/craziness into something almost pop-y. The upbeat melodies keep my focused intently on work, and the depth of the music keeps me coming back for more! 

The other album that I’ve consumed voraciously this past year is Beirut’s latest album, The Rip Tide. In a way, I want to claim similarities between it and the Yeasayer’s album… but that would be a trivial oversimplification. Beirut’s style lies completely in the folk domain, whereas Yeasayer’s could be considered something like psychedelic electronica. Again though, the depth of the melodies is what draws me to The Rip Tide. It is shocking what can be done with just vocals, horns, and the occasional synth.

Willpower boost with self-tracking awareness?

I remember when I first read about this Nicholas Felton character thinking he lost a couple of screws up top or something. Essentially each year, he distills his life down to a condensed annual report with various statistics about his year. Shear lunacy, I thought. More recently though, I’ve begun to experience the shear boost of willpower from knowledge of personal tracking. Whereas before I was only accountable to myself and my advisor (graduate school and all), I now feel like I hold accountability to anyone that follows any of my statistics.

Having not paid much attention to my personal tracking mechanisms till late, I didn’t realize the shear number of data that I already have on my life. More recently, I’ve started tracking (in detail) my exercise and sleep… all to my substantial benefit. In a sense, my Last.fm account was my first major foray into self-tracking, but my discover of Gmail year in review allowed even my ancient gmail account to become a massive source of personal information. My other older tracking account of course is my Good reads account with which I track all books that I read in a year.

Lately though, my obsession has been ratcheted up a notch with my discovery of endomondo and an android sleep tracker app. The past week or so, I've been tracking every single one of my workout with endomondo. This has proved to be of massive benefit for me personally. Before, my jogs were often short or my pace would slow down considerably as I ran. With this app, I feel constantly encouraged to push myself further and further. I should soon be breaking eight minute miles if I keep up my pace improvement! The sleep tracker on the other hand really aids with my knowledge of exactly how much sleep I’m getting. When I wake up in the morning, the last thing on my mind usually is to figure out how many hours I slept. To my surprise, I’m sleeping considerably less than I thought, so I’ve made it a prerogative to sleep at least seven hours a night. Willpower enhancement? Seems like I found something that works for me!

Guitar effects build!

Summing up the past week in just a single word is actually nearly a triviality… soldering! I spent the better part of last week constructing and attaching current sensors to our robot in the lab, but on top of this mandatory soldering madness I built a BYOC! For those unacquainted with BYOCs ( a quick google search would most likely ameliorate your knowledge, videos included!), you essentially build your own clone (hence the BYOC acronym) of guitar pedal effects. Stephanie surprised me for Christmas this year with one the most intricate (and downright epic!) pedals, the Digital echo/ping pong! I spend the better half of Friday soldering the thing together, and took a few pictures to document the effort… I figured I should share them with my readers (all one of you!). 

As any component heavy project should being, I ran through the parts and checked stock. The picture above shows all the components that actually go into this digital echo pedal… not trivial at all! Really though, at the heart of the electronics are two ping pong chips and an op amp… the rest of the pieces seem like voltage regulation and signal conditioning components (though I could be wrong, I haven’t inspected the schematic in detail). 

This next picture shows the shear (no pun intended) amount of circuitry here! Look at the size of that excess component wiring pile there, and this was only halfway through the build. 

After finishing the PCB, I had to construct and wire the pedal enclosure. The picture above shows the complete soldering of the PCB and pedal… but sadly I couldn’t finish everything! Even though the component list checked out with my stock, I was still missing a film capacitor (seems like there was a typo in the bill of materials…). I’m hopeful to hear back from the BYOC customer service to finish my pedal and start testing! I’ll update as soon as I can!.

On Model Predictive Control (My recent quest)

Well, winter break hardly began and it is already coming to a close. I’ll have to write a more journal-like post to document the trip up to Snoeshoe, WV, but today I want to discuss one technique of control theory that I recently came to know and love. Model Predictive Control (MPC) has always fascinated me and been something of an enigma to me since I knew little about it. However, the more I read about the subject matter, the more I wanted to share some of my findings with the interwebs. I’ll go over the basics of the theory and then I’ll wrap up with a little example that I coded in Matlab yesterday. 

MPC is, in a nutshell, an practical implementation of numerical optimal control. All good control graduate students are exposed to optimal control at one point or another, but usually to the analytic (archaic at this point) calculus of variations (COV) version. In the COV incarnation, optimal control is difficult and adding constraints (limits and room boundaries), makes analytic computations nearly impossible. Numerical implementations of optimal control can solve this however.

 I should mention, before I delve to deep into the theory, that optimal control at its core is a open loop control strategy… we find the optimal controller values and then run the system. If our system is not perfect or we hit a wall, optimal control is now confused and more or less screwed! So we run into the classic justification of closed loop control… where we watch what is going on and change accordingly. This, of course, is where MPC comes in the picture. With MPC, we implement optimal control at ever time step ( every second, say) and use only the next step of the resulting control… if something were to happen, we re-optimize accordingly! 

In the following example, we have a three state system with state constraints and control constraints. In other words, if we think of this as a robot, the robot has a maximum speed, and is contained in a room. This system attempts to follow a desired path, shown in the figure below. The optimization window for MPC is adjusted and we show that by having an extended window, the path of the MPC controller approaches the optimal path (as given by a purely optimal controller).