There is no such foundation of control theory because the goal is to make a dynamical system behave in a particular way. In that sense, anything goes. That is the reason why you have trazillion of methods to control the same system.
To give a necessarily brief account of how control theory become a discipline independent from the underlying problem feeder, control engineering, you have to consider a couple of different tracks simultaneously. James Watt's work on the steam governor, Stephen Black's feedback amplifier and Hugh Stoller's work on early mechanical televisions are always the same problem in disguise though many felt the connection inbetween these systems. During this time amplification was investigated under different names; regeneration theory was in particular facetious since they were trying to amplify the signal in open loop to transmit over long distances and this was causing the ringing due to noise amplification. Black actually had huge resistance to his feedback amplifier idea since negative feedback was reducing the gain greatly and people initially didn't buy his stability claims.
After Nyquist and Bode's work on Black's amplifier and making the problem stand on proper mathematical footing we need to include the works of AM Lyapunov over stability of motion. This is a completely separate branch of mathematics that came from Chebyshev school (Markov and Lyapunov were both students of him). Until 50s, these branches stayed exclusive.
Yet another branch of electrical engineering had similar problems in circuit network and microwave theory. This constitutes all the port formalism leaked into the control theory. Desoer/Vidyasagar gives a good account of an electrical engineering motivation. As an example, what has been called Raisbeck's conditions for 2-port network was rebranded/rediscovered as passivity theorem in control theory. Today we can combine all these frequency domain tools under the umbrella of the Integral Quadratic Constraints framework.
Kalman, Bucy and others also started a completely different track with state space methods that are relatively wide-known hence I'll skip that part.
During this time, one also has to consider the use of analog computers to solve differential equations by adjusting knobs. It was a messy process but once solved the equation was solved exactly. That is the reason why Laplace transforms dominate the field even though today they are merely an obstacle for undergrads and they don't even have the chance to learn it properly other than replace the number of dots with powers of $s$ fashion. In fact the only reason why people learn partial fraction expansions is mostly due to Laplace transforms.
At this point you have to realize that these are hard core engineering problems. Hence there is no foundations whatsoever. All things considered, they were trying to solve very real problems. For example, Bode was working on servomechanisms (ship gun stabilization and so on) during the WWII.
Anyways, I hope this gives an idea of why history is not even close to being a typical math subject storyline. The foundation you are seeking in my opinion does not exist because it doesn't have to have any. You can use abstractions and John Baez has been using category theory and uses morphisms to replace block diagram elements but they are too advanced for me to make a judgement, you can watch these videos to get an idea. http://math.ucr.edu/home/baez/networks_oxford/
In the meantime, another great figure, the late Jan C. Willems has created yet another abstraction, namely the behavioral approach, which I like the most and even though computationally has no real advantage, conceptually clears a lot of issues. As a personal view, I do think that it is the way control theory should be taught. Moreover I guess it is the closest thing to a foundation of control theory.
I hope this gives an idea why such article or book is hard to come by. There is a massive amount of details to be covered. To this day, each discipline investigated the elephant in a dark room from a different angle. And I didn't do any justice to any of the items above other than just mentioning them. For example there is still a lot to cover Pontryagin, Lur'e and many other domains such as fuzzy, adaptive, gain scheduling, nonlinear control etc.
A great anecdote I have from Jan Willems is this picture with Y.Yamamoto (they are students of Brockett and Kalman respectively):
What they are pointing to is a sailboat stabilizer fin and he used to say,
the only bunch who don't call this "a stabilizer" are control theorists.
I think it speaks volumes about control theory in general and its prospective foundations.
