Although automatic control has been around in simple forms for thousands of years, it was the industrial revolution in the late 18th and early nineteenth centuries that really triggered the development of more complex systems.
Prior to the industrial revolution, human labour was the main mechanism of industry and every human came complete with its own biological control system! As machines started to replace humans it became clear that automatic mechanisms were required to regulate them. The most famous of the industrial revolution control devices is the Watt Flyball Governor, invented by James Watt to control the speed of steam engines.
Machines and their accompanying control systems developed into the twentieth century, but they tended to be designed on a relatively ad-hoc basis with little unifying theory. This changed completely during the Second World War when control systems were pressed into military service. The theory of automatic control developed rapidly during the war and led to the development of highly advanced weapon control systems (Anti-aircraft gunsights, Ballistic missile control systems, Bomb sights, etc).
Along with control systems, process plant design also developed during the early part of the twentieth century. At the start of the century chemical production largely occurred in batch processes, most of which were simply large tanks were reactants were added and mixed together. These simple processes were relatively simple to control (most didn't have much that could be controlled - it was simply a case of getting the proportions right and hoping for the best!). As time moved on, however, processes became more sophisticated and more and more continuous process plants came into service. After the second world war almost all process plants had some form of automatic control fitted (based around pneumatic systems - the controller was a mechanical device which used air pressure as a communication and calculation medium).
In the late fifties the transistor started to come into industrial use, followed by integrated circuit technology. This led to the move away from pneumatic controllers to electronic systems (except in applications in highly flammable areas). This continued through the 1960's and 70's. By the early 1980's computer systems became cheap and powerful enough to use as process control systems (computers were tried, unsuccessfully, in the 1960's) and discrete electronic controllers began to be partially replaced.
The situation today is one where computers largely dominate the process control scene. These computers are either at a higher, supervisory, level and look a bit like a normal PC (i.e. they have a keyboard and screen), or they can be embedded inside a low level controller.
Some Web resources on the history of automatic control. The American Institution of Electrical and Electronic Engineers. A handy chart with dates. Three interesting, but kind of weird, pages: page 1, page 2, page 3