1. Linearization:

Objective:

To provide a linear approximation of a process model that could predict local behavior around a chosen operating condition.

Procedure:

Given:

                                        (1)

To obtain a linearized model of equation (1) with respect to the point: x_o, u_o and d_o, first obtain the following constants:

then the linearized model is given by

2. Linear Ordinary Differential Equations with constant coefficients.

1. General Form:

      (2)

2. Solution:

where x_c is the complementary solution and x_p is the particular solution.

3. In solving for the complementary solution, the procedure involves obtaining the characteristic equation:

  

                                  (3)

whose n roots are the eigenvalues of the system described by (2).

4. The eigenvalues determine the inherent behavior of the system:

• the system is unstable if any of the eigenvalues has a positive real part.
• the system will have oscillatory behavior if any of eigenvalues has an imaginary part.
• the system will respond faster towards steady state if the real part is more negative.
• the frequency of oscillation is higher if the imaginary parts have larger magnitudes (absolute values).

 

 
 
 

3. Mulitiple Linear ODE System
 
 
• Standard Form

• Can be reduced into an nth order differential equation either by substitution or by using matrix operations. Then the characteristic equation is given by