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m →Relative degree: Wikified poles and zeros. Changed < r-1 too leq r-2. Made it clear that r is a whole number. |
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=== Relative degree ===
In our feedback linearized system made up of a state vector of the output <math>y</math> and its first <math>(n-1)</math> derivatives, we must understand how the input <math>u</math> enters the system. To do this, we introduce the notion of [[relative degree]]. Our system given by (1) and (2) is said to have relative degree <math>r \in \mathbb{W}</math> at a point <math>x_0</math> if,
:<math>L_{g}L_{f}^{k}h(x) = 0 \qquad \forall x</math> in a [[neighbourhood (mathematics)|neighbourhood]] of <math>x_0</math> and all <math>k
:<math>L_{g}L_{f}^{r-1}h(x_0) \neq 0</math>
Considering this definition of relative degree in light of the expression of the time derivative of the output <math>y</math>, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output <math>y</math> before the input <math>u</math> appears. In an [[LTI system]], the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of
=== Linearization by feedback ===
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