The era after the Second World War can be called the classical period of control theory. It was characterized by the appearance of the first textbooks (MacColl, 1945; Lauer, Lesnick, & Matdon, 1947; Brown & Campbell, 1948; Chestnut & Mayer, 1951; Truxal, 1955), and by simple design tools that provided great perception and definite solutions to design problems. These tools were employed using hand calculations, or at most slide rules, with graphical techniques.
With the dawn of the space era, controls design in the United States prevented from the frequency-domain practices of classical control theory and back to the differential equation techniques of the late 1800's, which were inherent in the time domain. The reasons for this development are as follows.
The model of classical control theory was very fitting for controls design problems during and immediately after the World Wars. The frequency-domain approach was suitable for linear time-invariant systems. It is at its best when managing single-input/single-output systems, for the graphical techniques were problematic to use with numerous inputs and outputs.
Classical controls design had some successes with nonlinear systems. ...
Consequently, classical techniques can be applied on a linearized form of a nonlinear system, giving good results at an equilibrium position about which the system performance is more or less linear.
Frequency-domain methods can also be applied to systems with simple types of nonlinearities using the describing function approach, which relies on the Nyquist criterion. This method was first used by the Pole J. Groszkowski in radio transmitter design before the Second World War and complied with in 1964 by J. Kudrewicz.
Regrettably, it is not possible to design control systems for complex nonlinear multivariable systems, for example those arising in aerospace applications, using the assumption of linearity and treating the single-input/single-output transmission pairs individually.
Optimal Control and Estimation Theory
In view of the fact that naturally-occurring systems show optimality in their motion, it makes sense to design man-made control systems in a best possible fashion. A major gain is that this design may be realized in the time domain. In the context of modern controls design, it is common to reduce the time of transit, or a quadratic generalized energy functional or performance index, possibly with some constraints on the allowed controls.
R. Bellman (1957) employed dynamic programming to the optimal control of discrete-time systems, showing that the normal direction for solving optimal control problems is backwards in time. His modus operandi resulted in closed-loop, usually nonlinear, feedback schemes (Lewis, 1992).
PID & Robust and Optimal Controllers for Marine Engineering Systems: An Introduction
A Proportional-Integral-Derivative (or PID)