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Linearizing first or discretizing first?

09/01/2012

Given a nonlinear model: \dot{x}=f(x)

In practice, we need to both discretize and linearize the nonlinear model. The question is which procedure should be first?

(1) Method 1: Linearizing first, then discretizing

first linearizing: \dot{x}=\frac{\partial f}{\partial x}x

then discretizing: x(t+1)=(I+\Delta t \frac{\partial f}{\partial x})x

(2) Method 2: Discretizing first, then  linearizing

first discretizing: x(t+1)=x(t)+\Delta t f[x(t)]

then linearizing: x(t+1)=(I+\Delta t \frac{\partial f}{\partial x})x

Conclusions: the two methods are equivalent.

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