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Control Theory

Minimalist Description

Control is about achieving desirable outcomes. This could be strategic or it could be tactical. For instance, a football team wants to control the outcome of the game. Going into the game, they have a strategy: cover the outside receivers even if it means taking a bit of pressure off the quarterback; run to the inside and use the fake inside run to free up the receivers at the corners. But after the game starts, they emphasize tactics: the middle linebacker isn't falling for the fake inside run, and is instead peeling off to help the corners, so fake the inside run and then throw over the middle to the running back after the middle linebacker vacates to assist the cornerback.

Much of control theory deals with optimization, and in fact it is often called Optimal Control Theory. Optimization can take many forms. The two most basic are:

  • Continuous Optimization: These are problems described by smooth (or at least continuous) functions, such as the fuel burn schedule for a rocket engine or the watering schedule for a farm. 
  • Combinatorial Optimization: These are problems for which there is no smooth function, and the answers will be a finite set, such as which objects to take if the museum catches on fire (an unordered set), or what machines to process a job on (an ordered set). 

Both classes of problem come in constrained and unconstrained flavors. The rocket nozzle may restrict the fuel burn schedule to 50 pounds per minute even though if we could burn 75 pounds per minute at the very end, we could save on total fuel spent. There are also stochastic versions of both problems, where there is uncertainty in the schedule. We may not know, for instance, exactly how much time will be required to process a job on a particular machine. 

This uncertainty ma be random or non-random. A particularly interesting case non-random uncertainty arises in what are known as "partial information games". Chess is an example of a complete information game. Both players know exactly where they other player's pieces are. They don't know the other player's strategy but they know the other player's position. Battleship is a partial information game. They don't know the other player's position or strategy. So in Battleship, there is an incentive to use deception to fool the opposing player into thinking that your position is different from what it actually is. Deceptive strategies are also important in poker. If these were the only cases where deceptive strategies were important, partial information games would be interesting but not important. But deception also comes up in warfare and business. These deceptions take many forms including decoys, disguise, and stealth. In the first Gulf War, the Iraqis were effective enough at deception that it was nearly impossible to determine whether a SCUD kill destroyed an actual SCUD launcher or just a decoy and there were few confirmed SCUD launcher kills. Iraqi forces painted craters on their runways. In some cases these were sufficient to fool overhead photography, and required better sensors to uncover the deception and that diverted those sensors from other missions. In business, deception can occur in press releases, in patent applications, and in marketing. And certainly, stealth is a tactic in product release. These deceptions can have a dramatic impact on business and markets. For instance, in the mid-1980s when Lotus Software controlled spreadsheet market, they suffered many schedule delays in the release of their Jazz program, which industry experts expected to dominate the Apple Macintosh software market. Suddenly, with little advance warning, Microsoft released the Excel spreadsheet program for the Macintosh. Lotus faded from the spotlight and was never dominant in spreadsheets again. As for Microsoft, well, Excel is still the leader. No doubt, if Lotus had good industry intelligence on what Microsoft was doing, they would have re-targeted investment and certainly dealt with the industry press differently. Control in the face of deception is one area where we think we can bring world class expertise to bear. This expertise has been validated in DARPA programs dealing explicitly with deception, and through peer-reviewed publication process where the math has been checked.

S3 Data Science, copyright 2015