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linearization - ME 360 Modeling Analysis and Control of...

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ME 360: Modeling, Analysis, and Control of Dynamic Systems Professor Tilbury, University of Michigan, Winter 2009 Linearization Handout September 10, 2009 Basic idea: The local behavior of a nonlinear system can be approximated by its linearization about an operating point. 1 Taylor series approximation Let f ( x ) be a nonlinear function of x , and let x o be the operating point. Then x = x o + δx , where δx is the deviation (assumed to be small) from the operating point. Take the Taylor series expansion: f ( x o + δx ) = f ( x o ) + ∂f ∂x vextendsingle vextendsingle vextendsingle vextendsingle x = x o · δx + higher order terms f ( x o ) + kδx where k = ∂f ∂x vextendsingle vextendsingle vextendsingle x = x o is a constant. Note that k depends on the operating point x o . Also keep in mind that this is only an approximate relationship because we have kept only the constant and linear terms, and ignored the higher order terms. Nonlinear models can arise from experiments or empirical observations, or through first principles (e.g. geometry). Sometimes a nonlinear function is available; other times, there is only a data set or “look-up table” relating the variables of interest. 1.1 Nonlinear spring Assume that the force/displacement curve for the spring is obtained through experimentation, and plotted as shown in the figure.
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