Linear Approximations of Linear Systems

 

      There are various nonlinearities in practice such as friction, deadzone, saturation, hysterisis etc. None of the practical systems is linear in nature. But most of the important mathematical methods are available for the analysis of linear systems only. Hence practically a compromise is done between ease of analysis and accuracy of analysis and a nonlinear system is treated as linear under linear approximations. This technique of making nonlinear system approximations, This technique of making nonlinear system linear under linear approximations is called linearization technique.

      Thus whenever situation justifies, the linearization technique is used to analyse the nonlinear systems.

      The common situation where linearization can be used is a smooth curve which differs very little from the point of tangency. Hence in the narrow region near the point of tangency a nonlinear curve can be approximated as a straight line for the analysis purpose. This situation is very common in control systems in which the purpose of the systems is to keep controlled variable very close to its desired value. Hence the range of variation of a controlled variable is narrow and close to a set point and hence can be assumed linear. But if the system is essentially following a curve then we can analyze the system by linearizing it at several points along the curve.

      Consider a general control system with x(t) as the input while y(t) as the output. The variation of an output variable is given by a function f(x) as shown in the Fig. 1. The relationship is nonlinear but is continuous.

      Expansion of the equation y = f(x) into a Taylor's series about the normal operating point (x_n, y_n) is given by,

     It is assumed that from the point of tangency the variation is very small hence (x - x_n) about the normal operating point is very small. Hence higher order terms of (x - x_n) can be easily neglected which gives,

     The equation represents a straight line of the form y = mx + c which is perfectly linear.

      If the output variable y depends on multiple input variables x₁, x₂, x₃, .....x_m i.e.y= f(x₁, x₂, x₃........x_m) then the linear approximation for y can be achieved by this equation into Taylor's series about the operating point (x_1n, x_2n, x_3n, ..........x_mn, y_n) and neglecting all higher order derivative terms. Such a linearization gives,