Feedback linearization: Difference between revisions

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:<math>\begin{align}

\dot{y} = \frac{\mathrm{d}h(x)}mathord{\mathrmoperatorname{d}t} &=\frac{\mathrm{d}h(x)}{\mathrmmathord{\operatorname{d}x}\dot{xt}\\

&= \frac{\mathrm{d}partial h(x)}{\mathrm{d}partial x}f(x) + \fracdot{\mathrm{d}h(x)}{\mathrm{d}x}g(x)u\

&= \frac{\partial h(x)}{\partial x}f(x) + \frac{\partial h(x)}{\partial x}g(x)u

\end{align}</math>

Now we can define the [[Lie derivative]] of <math>h(x)</math> along <math>f(x)</math> as,

:<math>L_{f}h(x) =\triangleq \frac{\mathrm{d}partial h(x)}{\mathrm{d}partial x}f(x),</math>

and similarly, the Lie derivative of <math>h(x)</math> along <math>g(x)</math> as,

:<math>L_{g}h(x) =\triangleq \frac{\mathrm{d}partial h(x)}{\mathrm{d}partial x}g(x).</math>

With this new notation, we may express <math>\dot{y}</math> as,

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Note that the notation of Lie derivatives is convenient when we take multiple derivatives with respect to either the same vector field, or a different one. For example,

:<math>L_{f}^{2}h(x) = L_{f}L_{f}h(x) = \frac{\mathrm{d}partial (L_{f}h(x))}{\mathrm{d}partial x}f(x),</math>

and

:<math>L_{g}L_{f}h(x) = \frac{\mathrm{d}partial (L_{f}h(x))}{\mathrm{d}partial x}g(x).</math>

=== Relative degree ===

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\end{bmatrix}.</math>

So, with the appropriate choice of <math>kK</math>, we can arbitrarily place the closed-loop poles of the linearized system.

=== Unstable zero dynamics ===