Electronically controlled fusion reactor It seems that currently, only two types of fusion reactor styles are being used for large scale testing: the Stellarator and the Tokamak. Both of these and a whole bunch of other designs have been developed in the 1950s and '60s. They are based on the idea of confining very hot plasma in a magnetic field, shaped just weirdly enough to not let the plasma escape.
Shaping such a field seems to be extremely complicated and could apparently only recently be achieved using computational optimization on the coil shape. On a side note, these reactors are also massively huge and expensive, probably not unrelated to the design.
Why are we still relying on these half a century old designs? What is stopping us from creating a very simple coil arrangement and just using a control loop to keep the plasma confined? Surely, somebody already thought of this. The designs mentioned before rely on the magnet field to create an inherently stable confinement. But today we can use inherently unstable processes and artificially make them stable by using a control loop.
A viable coil arrangement could potentially consist of 6 coils, distributed around a cube shaped container. The plasma could be induced and heated using light or radiation. If necessary, the coils could be made superconductive. The plasma cloud's shape, size and orientation could be tracked and used to change the field's shape in a way to contain it. The plasma could be compressed by collectively increasing the field strength, further heating it up.
Why doesn't this work?
I'm posting this in the Physics StackExchange because, despite being somewhat of an engineering question, I'm quite sure the underlying physics is what makes the idea impractical.
 A: What do you mean by control loop? If you mean some sort of diagnostics to check when confinement gets lost and quickly reacting against it, then a) this is being done, b) confinement can fail in ways you can't even imagine without knowledge of plasma physics.
I suggest you check up on the history of the development of plasma reactors, that should clarify many of your questions and their follow-up questions. There are excellent youtube talks by high-ranking plasma people.
What you're describing in your question is a polywell reactor. Doesn't work due to massive radiative losses in the center of the polywell, if my memory serves.
These 'half-century-old' designs (of which only Tokamaks count, because Stellarators only existed as ideas) were the first ones to break the 1 million Kelvin bareer, and research effort into Tokamaks intensified.
Stellarators came later, as they are more complex to build, but they can overcome many plasma stability issues that Tokamaks have.
Those concepts are leading in a) stability b) simplicity c) the plasma triple-product. No other design can compete with those, although there's a bunch of secret military operations and private industries that try to develop competitors.
As a last word, the plasma-triple product is the main measure of energy effectivity of a plasma fusion reactor. Because it is simple to produce some fusion reactions in hand-held devices even, but it is hard to produce enough fusion reactions for the nuclear fire to be self-sustaining and economical.
A: 
It seems that currently, only two types of fusion reactor styles are
  being used for large scale testing: the Stellarator and the Tokamak.

Well right off, this is not true. The two approaches that are best studied are the "advanced tokamak" like ITER, and the laser-driven ICF approach like NIF. I'd wager that third place goes to the spherical toks, and fourth to the collection of maglif concepts. The stellarator is really not a mainstream approach these days, although the stream of news about X-7 might lead you to believe otherwise.

Shaping such a field seems to be extremely complicated and could
  apparently only recently be achieved using computational optimization
  on the coil shape.

That's not the problem. The problem is that when you squeeze a balloon, it tends to pop out through your fingers. We are squeezing the balloon HARD. Its just a hard problem no matter what solution you attempt.
The computer-aided shaping you refer to only applies to one particular style of the stellarator, the type X-7 uses, it is not needed for other types. Tokamaks are very simple to build, pinch devices are even easier.

A viable coil arrangement could potentially consist of 6 coils,
  distributed around a cube shaped container... Why doesn't this work?

Very simply because the magnetic lines aren't closed.
To start with, consider the fact that ions in a plasma at fusion temperatures are traveling at speeds around 1% of speed of light. If you have an active feedback system that can handle that, I want to buy it!
So in order to keep the plasma inside the reactor when it's going this fast, you want it to naturally stay inside. So in the tokamak, for instance, the magnets are arranged so the magnetic field is running around the inside of the donut. So a particle inside will just circle around and around the inside of the donut. In such an arrangement we call the field "closed", every line of force ends up on itself again at some point.
Now with the system you propose, the magnetic field would look like a star. You would have a high field in the center, but some of the magnetic field lines would run through the very center of each ring. And the ions would just follow those lines out of your reactor. Quickly. We call these arrangements open.
So whatever arrangement you make, the key is that the lines are closed. We've tried open lines in the magnetic mirror and the polywell (the latter is basically what you're proposing, see the field diagrams on that page), and they just don't work. With the mirror they kept adding more and more fields to bend the lines this way and that, but nothing worked, they leaked like a sieve.
A: The reason your arrangement isn't used is that it does not work for confining a plasma, because there's nothing that can be done to prevent leakage at the corners of your cube where the different magnetic fields come together. A toroid has no corners like that and hence stands a better chance of not leaking. 
This is a complex subject, as you note, and it has a long history. Most of your questions have answers in Charles Seife's book "Sun In A Bottle", which I recommend you have a look at.
A: 
Why are we still relying on these half a century old designs? What is
  stopping us from creating a very simple coil arrangement and just
  using a control loop to keep the plasma confined?

The thermal velocity of an ion at fusion temperatures is on the order of kilometers per second. Modern computers are fast. They aren't that fast. And then there's hysteresis.
The complex shaping seen in the stellarator and tokamak are designed, basically, to continually mix the fuel so that any drift in one direction gets evened out. It's the same basic concept as rifling in a gun, or missiles that are spin stabilized.
A: Active stabilization of a fusion plasma is possible.
This article describes a recently developed system for disrupting certain kinds of plasma instabilities in a Tokamak, by detecting an instability as it begins to form, then disrupting it using a "sabot" fired into the forming instability.  The "sabot" is a tiny pellet containing smaller pellets of low-z granules, and needs to be fired accurately within roughly 1 millisecond to a velocity of ~150 meters/sec. I think the result is that the plasma instability is rapidly locally cooled, draining its energy.
It's not exactly electronic control, but it is feedback control; and it- or an analogous technique- is likely to be important in future fusion reactors.
