This appendix illustrates techniques that are available for determining the system accuracy. These are referred to several times in this book.
A method for determining the steady-state performance of any control system is to apply the final-value theorem of the Laplace transform. Let us consider the unity-feedback system shown in Figure C.1. The relation between the resulting system error, E(s), for a given input R(s) is given by
The steady-state error can be expressed as
The control engineer is usually interested in test inputs of position, velocity, and acceleration. A step, ramp, and paraboloid are simple mathematical expressions which represent these physical quantities, respectively, and is illustrated in Figure C.2. They are defined in Eqs. (C.3)–(C.5), where the notation U(t) means a unit step for t 0: