Beautiful Mathematics in Control Theory

March 04, 2010

I became nostalgic as an angry Boman Irani asks Sharman Joshi in 3 Idiots,”how does an induction motor start?” Sharman Joshi answers “broom, brooooom......” imitating the noise that a motor makes as its starts accelerating. Many years back, I spent many days over a couple of months to understand how exactly the rotating magnetic field is produced by the coils in the stator of an induction motor and how this magnetic field cuts the squirrel cage rotor making the rotor to rotate to oppose the cause of electricity.

Most students of electrical engineering in those days of course loved the great bible by B.L. Theraja for mugging up all the concepts and mathematics just for exams. One day my Iranian friend showed me the Iranian (translated) version of this bible.

However, as the guys of these Engineering disciplines graduate to final semesters, they encounter a serious course called “Control Theory”. This course throws mathematical bouncers from day one and there is no one who can mug up this subject. The Laplace transforms, differential equations, the stability criteria, and the observability and controllability conditions make their life miserable. Most guys in those days used to feel greatly relieved once they escaped with minimum required credits to clear the course.

However, those few who can spare time and persist long enough during and after the course, can hope to tame the wild dragon and its mathematical equations and land riding on it at a beautiful place, which unravels many secrets to working of machines and complex systems all around.

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The process of designing a system to control behaviour of a dynamical system is called “control philosophy”.

Control Systems are not about anything specific. It is all about information flow and feedbacks. It is all about massive chains of feedbacks that run inside machines, humans, societies and systems all around. In a way, it connects the researcher to “Unified Systems Theory”. The researchers practice to observe only at the information flow, the non-linear elements in systems and the feedbacks that take place. Soon they realise that the feedbacks can be altered and the behaviour of the system can be changed.

At an undergraduate, graduate course, project or dissertation work, it mostly involves the machines. Most students start practicing to observe the behaviour of machines and systems around, model and control them. Then they work to improvise the mathematical algorithms for optimisation.

For example, a student can find a crankshaft of vehicle behaving same way as an electrical network of inductors, capacitors and resistors. A human being walking can be visualised as an inverted pendulum balancing itself. It is all about behaviour of dynamical systems, their stability and their control.

Inverted Pendulum Video:

Robot balancing itself:

Controlling a massive inverted pendulum in sky:

The students learn that corrective actions to reach a desired state in a dynamical system not only depend on the error, but also the integration of errors over a period of time. It is also important that overshoots due to a badly tuned control systems can lead to unwanted oscillations with the states of the system badly swinging across a desired state creating havoc.

The students know mathematically, how deadly the time delays between actions and results (inherent in systems) coupled with bad control strategy can make a system unstable, instead of improving its behaviour. For example, imagine how difficult it would be to drive a car, when there is a 2 seconds delay in the behaviour of steering wheel and/or the accelerator pedal. You will not be able to drive such a car. However, huge delays exist in systems around us on which we intervene so often. For example, the way a child responds to positive or negative feedback, is known only after a certain delay, for which many people never have any patience. As a result, there are enough small children who manipulate and control their parents instead of getting controlled.

The feedback and interventions play an important role in global markets. Bad sentiments in market makes people withdraw money and remain in cash and this positive feedback (or vicious circle) causes further crash to happen. Similarly, the higher liquidity due to government stimulus or funds from across the world can boost markets and make the feedback loop run in the opposite direction boosting the sentiments. Now, this entire phenomenon can be modelled using an autoregressive with moving averages (ARMA) models or by difference equations.

The future states of any dynamical system are a function of its current states and also a function of its series of part states and series ofpast inputs (or interventions). When a dynamical system has many actors, many couplings, times delays, non-linear behaviour, then that system is intelligent enough to fool any number of best brains in the world. So, most often sincere and highly popular interventions further screw up a badly behaving system, making the people demand further increase of interventions, which leads to a completely screwed up system. In fact, this entire phenomenon can be understood when one mathematically simulates it in a program in Matlab.

Sumanth is an Indian Men's Rights Activist.
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