Fundamentals of Linear Control A Concise Approach

In controls we make use of the abstract concept of a system: we identify a phenomenon or a process, the system, and two classes of signals, which we label as inputs and outputs. A signal is something that can be measured or quantified. In this book we use real numbers to quantify signals. The classification of a particular signal as an input means that it can be identified as the cause of a particular system behavior, whereas an output signal is seen as the product or consequence of the behavior. Of course the classification of a phenomenon as a system and the labeling of input and output signals is an abstract construction. A mathematical description of a system and its signals is what constitutes a model. The entire abstract construction, and not only the equations that we will later associate with particular signals and systems, is the model.

We often represent the relationship between a system and its input and output signals in the form of a block-diagram, such as the ones in Fig. 1.1 through Fig. 1.3. The diagram in Fig. 1.1 indicates that a system, G, produces an output signal, y, in the presence of the input signal, u. Block-diagrams will be used to represent the interconnection of systems and even algorithms. For example, Fig. 1.2 depicts the components and signals in a familiar controlled system, a water heater; the block-diagram in Fig. 1.3 depicts an algorithm for converting temperature in degrees Fahrenheit to degrees Celsius, in which the output of the circle in Fig. 1.3 is the algebraic sum of the incoming signals with signs as indicated near the incoming arrows.

Models and Experiments

Systems, signals, and models are often associated with concrete or abstract experiments. A model reflects a particular setup in which the outputs appear correlated with a prescribed set of inputs. For example, we might attempt to model a car by performing the following experiment: on an unobstructed and level road, we depress the accelerator pedal and let the car travel in a straight line. We keep the pedal excursion constant and let the car reach constant velocity. We record the amount the pedal has been depressed and the car's terminal velocity.