Summary of Week 2

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: xo, uo and do, first obtain the following constants:

then the linearized model is given by

2. Linear Ordinary Differential Equations with constant coefficients.
1. General Form:
(2)
1. Solution:
x = xc+x p

where xc is the complementary solution and xp is the particular solution.
1. 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).
1. 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).

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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