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{{short description|Approach used in controlling nonlinear systems}}
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[[File:Feedback linearization.svg|thumb|Block diagram illustrating the feedback linearization of a nonlinear system]]
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An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
== Feedback
Here, consider the case of feedback linearization of a single-input single-output (SISO) system. Similar results can be extended to multiple-input multiple-output (MIMO) systems. In this case, <math>u \in \mathbb{R}</math> and <math>y \in \mathbb{R}</math>. The objective is to find a coordinate transformation <math>z = T(x)</math> that transforms the system (1) into the so-called [[Normal form (abstract rewriting)|normal form]] which will reveal a feedback law of the form
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:<math>\begin{align}
\dot{y} = \frac{\
&= \frac{\
&= \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)
and similarly, the Lie derivative of <math>h(x)</math> along <math>g(x)</math> as,
:<math>L_{g}h(x)
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{\
and
:<math>L_{g}L_{f}h(x) = \frac{\
=== Relative degree ===
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\end{bmatrix}.</math>
So, with the appropriate choice of <math>
=== Unstable zero dynamics ===
Feedback linearization can be accomplished with systems that have relative degree less than <math>n</math>. However, the normal form of the system will include [[zero dynamics]] (i.e., states that are not [[observable]] from the output of the system) that may be unstable. In practice, unstable dynamics may have deleterious effects on the system (e.g., it may be dangerous for internal states of the system to grow unbounded). These unobservable states may be controllable or at least stable, and so measures can be taken to ensure these states do not cause problems in practice. [[Minimum phase]] systems provide some insight on zero dynamics.
== Feedback linearization of MIMO systems ==
Although NDI is not necessarily restricted to this type of system, lets consider a nonlinear MIMO system that is affine in input <math>\mathbf{\mathbf{u}}</math>, as is shown below.
{{NumBlk|:|
<math>
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{f}(\mathbf{x}) + G(\mathbf{x})\mathbf{u}\\
\mathbf{y} &= \mathbf{h}(\mathbf{x})
\end{aligned}
</math>
|{{EquationRef|4}}}}
It is assumed that the amount of inputs is the same as the amount of outputs. Lets say there are <math>m</math> inputs and outputs. Then <math>G = [\mathbf{g}_1 \, \mathbf{g}_2 \, \cdots \, \mathbf{g}_m]</math> is an <math>n\times m</math> matrix, where <math>\mathbf{g}_j</math> are the vectors making up its columns. Furthermore, <math>\mathbf{u}\in \mathbb{R}^m</math> and <math>\mathbf{y}\in \mathbb{R}^m</math>. To use a similar derivation as for SISO, the system from Eq. 4 can be split up by isolating each <math>i</math>'th output <math>y_i</math>, as is shown in Eq. 5.
{{NumBlk|:|
<math>
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{f}(\mathbf{x}) + \mathbf{g}_1(\mathbf{x}) u_1 + \mathbf{g}_2(\mathbf{x}) u_2 + \cdots + \mathbf{g}_m(\mathbf{x}) u_m\\
y_i &= h_i(\mathbf{x})
\end{aligned}
</math>
|{{EquationRef|5}}}}
Similarly to SISO, it can be shown that up until the <math>(r_i-1)</math>’th derivative of <math>y_i</math>, the term <math>L_{g_j} h_i (\mathbf{x}) = 0</math>. Here <math>r_i</math> refers to the relative degree of the <math>i</math>'th output. Analogously, this gives
{{NumBlk|:|
<math>
\begin{aligned}
y_i =& h_i(\mathbf{x})\\
\dot{y}_i =& L_fh_i(\mathbf{x})\\
\ddot{y}_i =& L_f^2h_i(\mathbf{x})\\
&\vdots\\
y_i^{(r_i)} =& L_f^{r_i}h_i(\mathbf{x}) + \sum^m_{j=1} L_{g_j}L_f^{r_i-1}h_i(\mathbf{x})u_j\\
=& L_f^{r_i}h_i(\mathbf{x}) +
\begin{bmatrix}
L_{g_1}L_f^{r_i-1}h_i &
L_{g_2}L_f^{r_i-1}h_i&
\cdots&
L_{g_m}L_f^{r_i-1}h_i
\end{bmatrix} \mathbf{u}
\end{aligned}
</math>
|{{EquationRef|6}}}}
Working this out the same way as SISO, one finds that defining a virtual input <math>v_i</math> such that
{{NumBlk|:|
<math>
\begin{aligned}
v_i &= L_f^{r_i}h_i(\mathbf{x}) + \sum^m_{j=1} L_{g_j}L_f^{r_i-1}h_i(\mathbf{x})u_j\\
&= b_i(\mathbf{x}) +
\begin{bmatrix}
a_{i1} & a_{i1} & \cdots& a_{im}
\end{bmatrix} \mathbf{u}
\end{aligned}
</math>
|{{EquationRef|7}}}}
linearizes this <math>i</math>'th system. However, if <math>m>1</math>, <math>\mathbf{u}</math> can obviously not be solved given a value for <math>v_i</math>. However, setting up such an equation for all <math>m</math> outputs, <math>y_1,y_2,\ldots,y_m</math>, results in <math>m</math> equations of the form shown in Eq. 7. Combining these equation results in a matrix equation, which generally allows solving for the input <math>\mathbf{u}</math>, as is shown below.
{{NumBlk|:|
<math>
\begin{aligned}
\mathbf{v} &= \mathbf{b} + A \mathbf{u}\\
A^{-1} (\mathbf{v}-\mathbf{b}) &= \mathbf{u}
\end{aligned}
</math>
|{{EquationRef|8}}}}
== See also ==
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* B. Friedland, ''Advanced Control System Design'', facsimile edition, Prentice Hall, Upper Saddle river, New Jersey, 1996.
{{refend}}
==References==
{{Reflist}}
== External links ==
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