ABSTRACT
All our discussions up to now concerning stability, sensitivity, robustness, etc. are in principle very general. However, in all concrete calculations we have assumed that the system and its controller are linear, i.e., they are described by linear differential equations (or in the frequency domain by transfer functions). The reason for this is mainly that there is a well developed, comparatively simple, mathematical theory for linear systems. All real control systems contain some form of nonlinearity however. This can be seen by considering the consequences of exact linearity. If a linear system reacts with the output y to the input u, then it reacts with the output ay to the input au for all values of a. For, e.g., a DC motor this implies that the step response should have the same form, regardless of whether the motor current is in the //A range or in the kA range. In practice the former would mean a current, too weak to overcome friction, while the latter would burn up the motor. Similar considerations apply to other physical systems. In many cases the nonlinearity appears as an upper or lower bound of a variable. A valve, for instance, can be positioned in a range between completely open and completely closed. If the control signal from a regulator passes a D/A-converter there is a largest and smallest signal, corresponding to the largest and smallest integer which is accepted.
