Many of an engineer's tools for analyzing dynamic systems apply only to linear systems. The Laplace transform, for instance, only works if the equations to be transformed are linear.
What makes an equation "linear"?
is linear as long as a is a constant and m(t) is linear.
is nonlinear because of the square root.
is nonlinear because of the cross-product x1x2 term, while is linear when m(t) is linear.
Linearity is useful, because of the mathematical theorems that state:
f(x) is a linear differential equation:The latter means that for a linear process, the result of two input changes is the sum of the results of the individual changes.
- If x1 is a solution to the equation and c1 a constant, then c1x1 is also a solution
- If x1 and x2 are solutions to the equation, then x1+x2 is also a solution.
Many chemical engineering systems are highly nonlinear and general methods for working with nonlinear models are few, so it is important to know how to approximate nonlinear equations with linear ones.
The approach is really pretty straightforward:
Notice that when you linearize, you do so around a specific point. Choice of this point is important. If the linear version of your model is to work, you must be operating close to the chosen point, so that you remain within the region where the linear approximation is valid. The steady state value is the usual choice since control systems are most often used to reject disturbances moving the plant away from steady state.
Linearization in combination with perturbation variables has particular advantages. Recall that a deviation variable is defined as
The last example showed that by combining linearizatio with perturbation variables, you effectively change the linearization equation to
Modified: 1/24/97, 2/2/98; 5/20/2003, 8/12/2004
Copyright 1998, 2003, 2004 by R.M. Price -- All Rights Reserved