Feedback linearization

Feedback linearization is a common approach used in controlling nonlinear systems. The approach involves coming up with a transformation of the nonlinear system into an equivalent linear system, through a change of variables and a suitable control input. Feedback linearization may be applied to nonlinear systems of the following form:

${\dot {x}}=f(x)+g(x)u\qquad (1)\,$

$y=h(x)\qquad \qquad \qquad (2)$

Where $x\in R^{n}$ is the state vector, $u\in R^{p}$ is the vector of inputs, and $y\in R^{m}$ is the vector of outputs. The goal, then, is to develop a control input $u$ that renders either the input-output map linear, or results in a linearization of the full state of the system.

Feedback Linearization of SISO Systems

We first consider the case of feedback linearization of a single-input single-ouput (SISO) system. In this case, $u\in R$ and $y\in R$. We wish to find a coordinate transformation $z=T(x)$ that transforms our system (1) into the so-called normal form. This transformation must be a diffeomorphism. Loosely speaking, a diffeomorphism is a function that is invertible, and both the function and its inverse are smooth.

We require several tools before we can solve this problem. The first tool is the Lie derivative. Consider the time derivative of (2), which we can compute using the chain rule,

${\dot {y}}={\frac {dh(x)}{dx}}{\dot {x}}$

$={\frac {dh(x)}{dx}}f(x)+{\frac {dh(x)}{dx}}g(x)u$

Now we can define the Lie derivative of $h(x)$ with respect to $f(x)$ as,

$L_{f}h(x)={\frac {dh(x)}{dx}}f(x)$

And similarly, the Lie derivative of $h(x)$ with respect to $g(x)$ as,

$L_{g}h(x)={\frac {dh(x)}{dx}}g(x)$

With this new notation, we may express ${\dot {y}}$ as,

${\dot {y}}=L_{f}h(x)+L_{g}h(x)u$

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,

$L_{f}^{2}h(x)=L_{f}L_{f}h(x)={\frac {d(L_{f}h(x))}{dx}}f(x)$

And,

$L_{g}L_{f}h(x)={\frac {d(L_{f}h(x))}{dx}}g(x)$

Before we proceed with developing the feedback linearizing control law, we must also introduce the notion of relative degree. Our system given by (1) and (2) is said to have relative degree $r$ at a point $x_{0}$ if,

$L_{g}L_{f}^{k}h(x)=0\qquad \forall x$ in a neighbourhood of $x_{0}$ and all $k<r-1$

$L_{g}L_{f}^{r-1}h(x_{0})\neq 0$

Considering this definition of relative degree in light of the expression of the time derivative of the output $y$, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output $y$ befor the input $u$ appears. For the discussion that follows, we will assume that the relative degree of the system is $n$. In this case, after differentiating the output $n$ times we have,

$y=h(x)\qquad$

${\dot {y}}=L_{f}h(x)$

${\ddot {y}}=L_{f}^{2}h(x)$

$\vdots$

$y^{(n)}=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u$

Then, the state transformation may be expressed as,

$T(x)={\begin{bmatrix}z_{1}(x)\\z_{2}(x)\\\vdots \\z_{n}(x)\end{bmatrix}}={\begin{bmatrix}h(x)\\L_{f}h(x)\\\vdots \\L_{f}^{n-1}h(x)\end{bmatrix}}$

Taking derivatives of the states of the transformed system, we have,

${\dot {z}}_{1}=L_{f}h(x)=z_{2}(x)$

${\dot {z}}_{2}=L_{f}^{2}h(x)=z_{3}(x)$

$\vdots$

${\dot {z}}_{n}=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u$

Now, choosing the control input $u$ as,

$u={\frac {1}{L_{g}L_{f}^{n-1}h(x)}}(-L_{f}^{n}h(x)+v)$

We end up with the following linearized system, which is effectively a bank of $n$ integrators,

${\dot {z}}_{1}=z_{2}$

${\dot {z}}_{2}=z_{3}$

$\vdots$

${\dot {z}}_{n}=v$

We can choose $v$ to be whatever we like, but note that choosing $v$ as,

$v=-Kx\qquad$

Where $K=[k_{1},k_{2},\ldots ,k_{n}]$ results in,

${\dot {z}}=Az$

With,

$A={\begin{bmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\ldots &1\\-k_{1}&-k_{2}&-k_{3}&\ldots &-k_{n}\end{bmatrix}}$

This means that, with the appropriate choice of $k$, we can place the closed-loop poles of the linearized system anywhere we like.

References

• 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.