In order to analyse the behaviour of engineering systems mathematical models are required for the various components. It is common practice to try and obtain linear models as a rich mathematical theory exists for linear systems. These models will always be approximate, although possibly quite accurate for defined ranges of system variables, but inevitably nonlinear effects will eventually be found for large excursions of system variables. Linear systems have the important property that they satisfy the superposition principle. This leads to many important advantages in methods for their analysis. For example, in circuit theory when an RLC circuit has both d.c and a.c input voltages, the voltage or current elsewhere in the circuit can be found by summing the results of separate analyses for the d.c. and a.c. inputs taken individually, also if the magnitude of the a.c. voltage is doubled then the a.c. voltages and currents elsewhere in the circuit will be doubled. Thus, mathematically a linear system with input x t() and output y t() satisfies the property that the output for an input ax1 2 ()t b + x t() is ay1 2 ()t b + y t(), if y t1() and y t2 () are the outputs in response to the inputs x t1() and x t 2 (), respectively, and a and b are constants. A further point is that when the a.c. voltage is sinusoidal, which is normally assumed in using the term a.c., then all the other voltages and currents in the circuit are sinusoidal with the same frequency as the input.
A nonlinear system is defined as one which does not satisfy the superposition property. The simplest form of nonlinear system is the static nonlinearity where the output depends only on the current value of input but in a nonlinear manner, for example
y t() cx()t dx t() 3 = + (1.1)
where the output is the summation of a linear and a nonlinear, cubic, term. This description of nonlinearity is given by a simple mathematical expression. Practical nonlinearities which occur in control engineering, however, can often not be so easily described; so that approximating them may require more complex mathematical models, such as a high order power series or a linear segmented approximation.
More commonly a nonlinear differential equation, for example
d y()t d/ (t c ) (dy()t d/ )t y()t x(()t 2 2 3 + + = (1.2)
will describe the behavior. From an engineering viewpoint it may be desirable to think of this equation in terms of a block diagram, as shown in Figure 1.1, consisting of linear dynamic elements and a static nonlinearity, in this case a cubic, which with input dy()t d/ t gives an output c d( (y t)/dt) 3. Unfortunately there is no general approach to solving nonlinear, unlike linear, differential equations. The major point about nonlinear systems, however, is that their response is amplitude dependent so that if a particular form of response, or some measure of it, occurs for one input magnitude it may not result for some other input magnitude.