Feedback linearizion for input-output linearizion - Lie Derivatives

Question

Short introduction to feedback linearizion: If we got a nonlinear system:

$$\dot x_1 = x_2$$ $$\dot x_2 = a x_1 ^2 + bx_1 + c x_2 + u$$

and we want to have state feedback by using feedback linearizion technique. We can simply say that the control law is:

$$u = -Kx - ax_1 ^2 + r$$

Where $x$ is a vector of both $x_1, x_2$ and $K$ is an arbitrary control law matrix and $r$ is the reference vector. Now we have cancel out the nonlinearities for the system due to the $-ax_1^2$

But then there is something called input-output linearizion. That technique requires Lie Derivative.

Assume that we have a nonlinear system:

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

For input-output linearizion, we say:

$$y^{(\rho)} = L^{\rho}_fh(x) + L_gL_f^{\rho -1}h(x)u$$

Where $$\frac{\partial h}{\partial x} f(x) = L_f h(x)$$

is called the Lie Derivative

So our control law will be:

$$u = \frac{1}{L_gL_f^{\rho -1}h(x)}[-L^{\rho}_fh(x) + r]$$

This control law reduce the system to:

$$y^{(\rho)} = r$$

Where $\rho$ is is called the relative degree of the system.

Question:

1. How should I find $\rho$ so $y^{(\rho)} = r$ ?
2. Why can't I use the control law $$u = \frac{1}{g(x)}[-f(x) + r]$$ instead?

Answer 1

1. You can find $\rho$ by repeatedly taking the derivative of $y$ with respect to time until that time derivative is a direct function of $u$. So the partial derivative of the $\rho$th time derivative of $y$ with respect to $u$ is none zero, however the partial derivative of the $\rho\!-\!1$th time derivative of $y$ with respect to $u$ is zero. Also note that $y^{(\rho)}$ is the $\rho$th time derivative of $y$.
2. You can only use the control law $$ u = g(x)^{-1}(r - f(x)) $$ when $g(x)$ is a matrix with full column rank for all $x$ (when $g(x)$ is not square but has full column rank, then you could use the right inverse). When it does not have full column rank, then the inverse is not defined. Unless you choose $r$ such that $r - f(x)$ always lies inside the column space of $g(x)$, but that puts constraints on $r$.

From this it can also be noted that for the input output linearization method the inverse of $L_g\,L^{\rho-1}_f\,h(x)$ might also not always be defined. For example when the relative degree $\rho$ changes as a function of $x$. However for a lot of common systems it is more often then not well defined compared to the inverse of $g(x)$. For example the inverse of $g(x)$ is never defined for SISO systems with the dimension of $x$ larger then one.