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Feedback linearization
Fabio Celani and Alberto Isidori (2009), Scholarpedia, 4(2):6517. | doi:10.4249/scholarpedia.6517 | revision #121144 [link to/cite this article] |
Feedback linearization is the process of determining a feedback law and a change of coordinates that transform a nonlinear system into a linear and controllable one.
Contents |
Problem formulation
Consider a single-input nonlinear system (without output)
\tag{1}
\dot x = f(x) + g(x)u
with internal state x \in \mathbb{R}^n and input u \in \mathbb{R}\ . Given a point x^\circ \in \mathbb{R}^n\ , the problem of feedback linearization consists in finding a neighborhood U of x^\circ\ , a coordinate transformation z=\phi(x) defined on U\ , and a feedback law u=\alpha(x)+\beta(x)v
Notations
Let \lambda be a real-valued function defined on a subset U of \mathbb{R}^n\ ; the differential of \lambda\ , denoted d\lambda(x)\ , is the row vector d\lambda(x) := \left( \frac{\partial \lambda}{\partial x_1}(x) \ \frac{\partial \lambda}{\partial x_2}(x) \ \cdots \ \frac{\partial \lambda}{\partial x_n}(x) \right):= {\partial \lambda \over \partial x}(x)\,.
Let g be another n-vector-valued function defined on U\ . The Lie product (or bracket) of f and g denoted by [f,g] is the n-vector-valued function defined by [f,g](x):=\frac{\partial g}{\partial x}f(x)-\frac{\partial f}{\partial x}g(x)\;.
Let f_i \ i=1,\ldots,d be n-vector-valued functions defined on U\ . The assignment, with each x \in U\ , of the subspace \Delta(x)=\mathrm{span} \{f_1(x), \ldots, f_d(x) \}
Relative degree and normal forms
The point of departure of the whole analysis is the notion of
relative degree of a system, which is formally described in the
following way. The single-input single-output nonlinear system
\tag{2}
\begin{array}{rcl}
\dot x&=& f(x)+g(x)u\\
y&=& h(x)
\end{array}
with internal state x \in \mathbb{R}^n\ , input u \in \mathbb{R}\ , and output y \in \mathbb{R} is said to have relative degree r at a point x^\circ if
(i) L_gL_f^kh(x)=0 for all x in a neighborhood of x^\circ and all k =0, \ldots, r-2
(ii) L_gL_f^{r-1}h(x^\circ )\ne 0\;.
It is possible to show that the functions h(x)\ , L_fh(x),\ldots,L_f^{r- 1}h(x) can be used in order to define, at least partially, a local coordinates transformation around x^\circ\ . This fact is based on the following property.
Lemma 1. If system (2) has relative degree r at x^\circ\ , then the row vectors dh(x^\circ ),dL_fh(x^\circ ),\ldots,dL_f^{r-1}h(x^\circ )
Lemma 1 shows that necessarily r \le n and that the r functions h(x), L_fh(x), \ldots, L_f^{r-1}h(x) qualify as a partial set of new coordinate functions around the point x^\circ\ . If r is strictly less than n\ , the set of new coordinates can be completed as described in the following.
Lemma 2. Suppose that system (2) has relative degree r at x^\circ\ . Then r\le n\ . If r is strictly less than n\ , it is always possible to find n-r more functions \psi_{1}(x),\ldots,\psi_{r-r}(x) such that the mapping \Phi(x)=\left(\begin{array}{c} \psi_1(x)\\ \ldots\\ \psi_{n-r}(x)\\ h(x)\\ L_fh(x)\\ \ldots\\ L_f^{r-1}h(x) \end{array} \right)
Setting z = \left( \begin{array}{c} \psi_{1}(x)\\ \psi_{2}(x) \\ \cdots \\ \psi_{n-r}(x) \end{array} \right), \quad \xi = \left( \begin{array}{c} h(x)\\ L_fh(x) \\ \cdots \\ L_f^{r-1}h(x) \end{array} \right)\;,
where \begin{array}{rcl} f_0(z,\xi) &=& \left( \begin{array}{c} L_f\psi_1(\Phi^{-1}(z,\xi))\\ \cdots\\ L_f\psi_{n-r}(\Phi^{-1}(z,\xi))\end{array} \right) \\ q(z,\xi)&=&L_f^rh(\Phi^{-1}(z,\xi))\\ b(z,\xi)&=&L_gL_f^{r-1}h(\Phi^{-1}(z,\xi))\;. \end{array}
Equations (3) are said to be in normal form.
Conditions for feedback linearization
In this section conditions and constructive procedures are given for finding a solution to the feedback linearization problem.
Consider the nonlinear system with output (2) and suppose that at some point x^\circ the system has relative degree equal to the dimension of the state, i.e. r=n\ . In this case, the change of coordinates that puts the system in normal form is given by
\tag{4}
\Phi(x)= \left(
\begin{array}{c}
h(x)\\
L_f h(x)\\
\cdots\\
L_f^{n-1}h(x)
\end{array}
\right) \;.
Performing the change of coordinates \xi=\Phi(x) and neglecting the output, the system is described by \begin{array}{rcl} \dot \xi_1&=& \xi_2\\ \dot \xi_2&=& \xi_3\\ &&\cdots\\ \dot \xi_{n-1}&=&\xi_n\\ \dot \xi_n&=& q(\xi)+b(\xi)u \end{array}
Apply now the following feedback control law u=\frac{1}{b(\xi)}(-q(\xi)+v)
In general, the two transformations used in order to obtain a linear and controllable system can be interchanged: one can first apply a feedback and then change the coordinates without modifying the result. Note that the feedback just used expressed in the x coordinates is given by u=\frac{1}{b(\Phi(x))}(-q(\Phi(x))+v)=\frac{1}{L_gL_f^{n-1}h(x)}(-L_f^nh(x)+v)\;.
Of course, the basic feature that made feedback linearization possible was that system (2) had relative degree n (at x^{\circ}). Thus, the system without output (1) can be feedback linearized if it is possible to find an "output" function h(x) such that the corresponding system with output (2) has relative degree n (at x^{\circ}). The above condition turns out to be also necessary for solving the feedback linearization problem as stated in the following.
Lemma 3. The feedback linearization problem is solvable if and only of there exists a neighborhood U of x^{\circ} and a real-valued function h(x) defined on U\ , such that system (2) has relative degree n at x^{\circ}\ .
The question now arises of when, given a system of the form (1) (namely a pair of n-vector-valued functions f(x) and g(x)) there exists an "output function" h(x) such that the resulting system (2) has relative degree n at x^{\circ}\ . This is answered by the following result.
Theorem 1. Let f(x)\ , g(x)\ , and x^{\circ} be given. The feedback linearization problem is solvable (i.e. there exists an "output" function h(x) for which system (2) has relative degree n at x^{\circ}) if and only if the following conditions are satisfied
(i) the matrix \left( g(x^{\circ}) \ ad_f g(x^{\circ}) \cdots \ ad_f^{n-2} g(x^{\circ}) \ ad_f^{n-1} g(x^{\circ}) \right) is nonsingular
(ii) the distribution \mathrm{span} \{ g,\ ad_f g, \cdots, ad_f^{n-2}g \} is involutive near x^{\circ}\ .
It can be shown that the "output" function h(x) for which the system has relative degree n at x^{\circ} is a solution of the following system of first order partial differential equations L_gh(x)=L_{ad_{f}g}h(x)= \ldots = L_{ad^{n-2}_{f}g}h(x)=0\;.
Example
Consider the system \dot x = \left(\begin{array}{c} x_3(1+x_2) \\x_1 \\x_2(1+x_1) \end{array}\right) +\left(\begin{array}{c}0 \\1+x_2 \\-x_3 \end{array}\right) u\;.
In order to check whether or not this system is feedback linearizable around x=0\ , we compute the functions ad_f g(x) and ad^2_f g(x) and test the conditions of Theorem 1.
Appropriate calculations show that ad_f g(x)=\left(\begin{array}{c}0 \\x_1 \\-(1+x_1)(1+2 x_2) \end{array}\right)
In the present case, it is easily seen that a function h(x) that solves the system of equations L_g h(x)=L_{ad_{f}g}h(x)= 0
From our previous discussion, we know that considering this as "output" will yield a system having relative degree 3 at the point x=0\ . Then, locally around x=0\ , the system will be transformed into a linear and controllable one by means of the state feedback u=\frac{-L_f^3h(x)+v}{L_g L_f^2 h(x)}= \frac{-x_3^2(1+x_2)-x_2 x_3(1+x_2^2)-x_1(1+x_1)(1+2x_2)-x_1x_2(1+x_1)+v} {(1+x_1)(1+x_2)(1+2x_2)-x_1x_3}
References
- Isidori, A (1995). Nonlinear Control Systems. Springer Verlag, London.
- Jakubczyk(1980). On linearization of control systems. Bulletin de l'Academie Polonaise des Sciences. Serie des Sciences Mathematiques, Astronomiques et Physiques 28: 517-522.
- Khalil, H K (2002). Nonlinear Systems. Prentice Hall, Upper Saddle River.
- Sastry, S (1999). Nonlinear Systems: Analysis, Stability, and Control. Springer Verlag, New York.