Fundamentals of Linear Control A Concise Approach

In Chapter 1 we contemplated solutions to our first control problem, a much simplified cruise controller for a car, without taking into account possible effects of time. System and controller models were static relations between the signals: the output signal, y, the input signal, u, the reference input, y, and then the disturbance, w. Signals in block-diagrams flow instantaneously, and closed-loop solutions derived from such block-diagrams were deemed reasonable if they could be implemented fast enough. We drew encouraging conclusions from simple analysis but no rationale was given to support the conclusions if time were to be taken into consideration.

Of course, it is perfectly fine to construct a static mathematical model relating a car's pedal excursion with its terminal velocity, as long as we understand the model setup. Clearly a car does not reach its terminal velocity instantaneously! If we expect to implement the feedback cruise controller in a real car, we have to be prepared to say what happens between the time at which a terminal target velocity is set and the time at which the car reaches its terminal velocity. Controllers have to understand that it takes time for the car to reach its terminal velocity. That is, we will have to incorporate time not only into models and tools but also into controllers. For this reason we need to learn how to work with dynamic systems.

In the present book, mathematical models for dynamic systems take the form of ordinary differential equations where signals evolve continuously in time. Bear in mind that this is not a course on differential equations, and previous exposure to the mathematical theory of differential equations helps. Familiarity with material covered in standard text books, e.g. [BD12], is enough. We make extensive use of the Laplace transform and provide a somewhat self-contained review of relevant facts in Chapter 3.

These days, when virtually all control systems are implemented in some form of digital computer, one is compelled to justify why not to discuss control systems directly from the point of view of discrete-time signals and systems. One reason is that continuous-time signals and systems have a long tradition in mathematics and physics that has established a language that most scientists and engineers are accustomed to.