The world around us is full of complex systems, both natural and anthropogenic. Many times, it is desirable to shape the dynamics of these systems to achieve stable operation, robustness to unpredicted disturbances, or other useful performance characteristics. I speak abstractly here intentionally... these words are equally applied to biological feedback mechanisms, airplanes and spacecraft, industrial systems, and human infrastructure like water or power distribution. The study of control aims to develop the right mathematical and computational tools for characterizing these systems and shaping their dynamics to achieve a desired behavior.
Here, I summarize many concepts in control. Those experienced in control will notice that my summary has a distinctly modern flavor, focused on optimal control in high-dimensional systems. This is not motivated by a disregard for the existing classical techniques (in the frequency domain), but rather by a desire to unify content across Linotype under the lingua franca of nonlinear dynamical systems and optimization. This common thread is enhanced by frequent nods to ideas, both old and new, which have reshaped our understanding of control, such as the calculus of variations and convex analysis.
The linear quadratic regulator is the quintessential optimal controller.
Increasingly common in the Age of Information are methods for numerical optimal control. A common paradigm for numerical optimal control is model-predictive control, in which the controller solves an optimization problem at each step to determine the next control effort.
Frequently, methods for model-predictive control require writing a continuous-time problem in terms of a countable number of decision variables. The process of constructing these formulations is frequently known as transcription. Numerous transcription methods exist, with various benefits and limitations.