RMP Lecture Notes

Linearization

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"?

all variables present only to the first power
no product terms where variables are multiplied (constants are ok)
no square roots, exponentials, products, etc. involving variables

These can be understood by looking at some examples.

Example is linear as long as a is a constant and m(t) is linear.

Example is nonlinear because of the square root.

Example is nonlinear because of the cross-product x₁x₂ term, while Example is linear when m(t) is linear.

Linearity is useful, because of the mathematical theorems that state:

f(x) is a linear differential equation:

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.

The latter means that for a linear process, the result of two input changes is the sum of the results of the individual changes.

Making a Model Linear

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:

expand all nonlinear terms in a Taylor series, usually around the steady state value
truncate the expansion after the 1st order terms

This gives a general result for linearizing equations: Linearization

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.

EXAMPLE: Linearize Example

EXAMPLE: Linearize y=xz Example

Linearization in combination with perturbation variables has particular advantages. Recall that a deviation variable is defined as

so if we are at the steady state, the value of the deviation variable will be zero. All of the "pure constant" terms will then vanish from the linearized deviation variables. Moreover, when we switch to perturbation variables and then linearize around the steady-state, the initial conditions are zero. This means we can drop the x(0) terms as we take Laplace transforms.

EXAMPLE: Take the two variable case above into perturbation variables. Example

Notice how the terms containing only steady-state information vanish from the expression. If there were any constants (not functions of x, y, or z) they would also have vanished.

The last example showed that by combining linearizatio with perturbation variables, you effectively change the linearization equation to

References:

Coughanowr and Koppel, Process Systems Analysis and Control, McGraw-Hill, 1965, pp. 67-70.
Luyben, W.L. Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 171-76.
Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 74-77.
Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 166-72.
Seborg, D.E., T.F. Edgar, D.A. Mellichamp, Process Dynamics and Control, John Wiley, 1989, pp. 86-93.

R.M. Price
Original: 10/18/93
Modified: 1/24/97, 2/2/98; 5/20/2003, 8/12/2004