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{{short description|Approach used in controlling nonlinear systems}}
{{More footnotes|date=May 2009}}
[[File:Feedback linearization.svg|thumb|Block diagram illustrating the feedback linearization of a nonlinear system]]
'''Feedback linearization''' is a common strategy employed in [[nonlinear control]] to control [[nonlinear systems]]. Feedback linearization techniques may be applied to nonlinear control systems of the form
{{NumBlk|:|<math>\dot{x}(t) = f(x(t)) + \sum_{i=1}^{m}\,g_i(x(t))\,u_i(t)</math><ref>{{cite book |last1=Isidori |first1=Alberto |title=Nonlinear Control Systems |date=1995 |publisher=Springer-Verlag London |isbn=978-1-4471-3909-6 |page=5 |edition=Third}}</ref>|{{EquationRef|1}}}}
where <math>x(t) \in \mathbb{R}^n</math> is the state, <math>u_1(t), \ldots, u_m(t) \in \mathbb{R}</math> are the inputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of variables and a suitable control input. In particular, one seeks a change of coordinates <math>z = \Phi(x)</math> and control input <math>u = a(x) + b(x)\,v,</math> so that the dynamics of <math>x(t)</math> in the coordinates <math>z(t)</math> take the form of a linear, controllable control system,
{{NumBlk|:|<math>\dot{z}(t) = A\,z(t) + \sum_{i=1}^{m} b_i\,v(t).</math><ref>H. Nijmeijer and A. van der Shaft, Nonlinear Dynamical Control Systems, Springer-Verlag, p. 163, 2016.</ref>|{{EquationRef|2}}}}
An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
== Feedback linearization of SISO systems ==
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
{{NumBlk|:|<math>u = a(x) + b(x)v\,</math><ref>{{cite book |last1=Isidori |first1=Alberto |title=Nonlinear Control Systems |date=1995 |publisher=Springer-Verlag London |isbn=978-1-4471-3909-6 |page=147 |edition=Third}}</ref>|{{EquationRef|3}}}}
that will render a linear input–output map from the new input <math>v \in \mathbb{R}</math> to the output <math>y</math>. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a [[diffeomorphism]]. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be [[smooth function|smooth]] so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic and the linearization results only hold in this smaller region.
Several tools are required to solve this problem.
=== Lie derivative ===
The goal of feedback linearization is to produce a transformed system whose states are the output <math>y</math> and its first <math>(n-1)</math> derivatives. To understand the structure of this target system, we use the [[Lie derivative]]. Consider the time derivative of (2), which
:<math>\begin{align}
\dot{y}
&= \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,
Line 39 ⟶ 46:
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 ===
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
:<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 \leq r-2</math>
:<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 explicitly. In an [[LTI system]], the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of [[pole (complex analysis)|pole]]s) and the degree of its numerator polynomial (i.e., number of [[zero (complex analysis)|zero]]s).
=== Linearization by feedback ===
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= \begin{bmatrix}y\\
\dot{y}\\
\vdots\\
y^{(n-1)}
\end{bmatrix}
Line 100 ⟶ 107:
renders a linear input–output map from <math>v</math> to <math>z_1 = y</math>. The resulting linearized system
:<math>\begin{cases}\dot{z}_1 &= z_2\\
\dot{z}_2 &= z_3\\
&\vdots\\
Line 123 ⟶ 130:
\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
== 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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* A. Isidori, ''Nonlinear Control Systems,'' third edition, Springer Verlag, London, 1995.
* H. K. Khalil, ''Nonlinear Systems,'' third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.
* M. Vidyasagar, ''Nonlinear Systems Analysis'', second edition, Prentice Hall, Englewood Cliffs, New Jersey, 1993.
* B. Friedland, ''Advanced Control System Design'',
{{refend}}
==References==
{{Reflist}}
== External links ==
* {{Scholarpedia|title=Feedback linearization|urlname=Feedback_linearization|date=2009|curator=Fabio Celani and Alberto Isidori|accessdate=31 December 2022}}
* [http://www2.ece.ohio-state.edu/~passino/lab5prelabnlc.pdf ECE 758: Modeling and Nonlinear Control of a Single-link Flexible Joint Manipulator] – Gives explanation and an application of feedback linearization.
* [http://www2.ece.ohio-state.edu/~passino/lab5_nonlinear_ball_tube_ex.pdf ECE 758: Ball-in-Tube Linearization Example] – Trivial application of linearization for a system already in normal form (i.e., no coordinate transformation necessary).
* [[Wolfram language]] functions to do [http://reference.wolfram.com/language/ref/FeedbackLinearize.html feedback linearization], compute [http://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html relative orders], and determine [http://wolfram.com/xid/0dbi23tfa58uxj73eu-ve3iy4 zero dynamics].
[[Category:Nonlinear control]]
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