Lie derivative

The goal of feedback linearization is to produce a transformed system whose states are the output ${\displaystyle y}$ and its first ${\displaystyle (n-1)}$ derivatives. To understand the structure of this target system, we use the Lie derivative. Consider the time derivative of (2), which can be computed using the chain rule,

${\displaystyle {\begin{aligned}{\dot {y}}={\frac {\mathrm {d} h(x)}{\mathrm {d} t}}&={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}{\dot {x}}\\&={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}f(x)+{\frac {\mathrm {d} h(x)}{\mathrm {d} x}}g(x)u\end{aligned}}}$

Now we can define the Lie derivative of ${\displaystyle h(x)}$ along ${\displaystyle f(x)}$ as,

${\displaystyle L_{f}h(x)={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}f(x),}$

and similarly, the Lie derivative of ${\displaystyle h(x)}$ along ${\displaystyle g(x)}$ as,

${\displaystyle L_{g}h(x)={\frac {\mathrm {d} h(x)}{\mathrm {d} x}}g(x).}$

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

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

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,

${\displaystyle L_{f}^{2}h(x)=L_{f}L_{f}h(x)={\frac {\mathrm {d} (L_{f}h(x))}{\mathrm {d} x}}f(x),}$

and

${\displaystyle L_{g}L_{f}h(x)={\frac {\mathrm {d} (L_{f}h(x))}{\mathrm {d} x}}g(x).}$

Relative degree

In our feedback linearized system made up of a state vector of the output ${\displaystyle y}$ and its first ${\displaystyle (n-1)}$ derivatives, we must understand how the input ${\displaystyle u}$ enters the system. To do this, we introduce the notion of relative degree. Our system given by (1) and (2) is said to have relative degree ${\displaystyle r\in \mathbb {W} }$ at a point ${\displaystyle x_{0}}$ if,

${\displaystyle L_{g}L_{f}^{k}h(x)=0\qquad \forall x}$ in a neighbourhood of ${\displaystyle x_{0}}$ and all ${\displaystyle k\leq r-2}$

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

Considering this definition of relative degree in light of the expression of the time derivative of the output ${\displaystyle y}$, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output ${\displaystyle y}$ before the input ${\displaystyle u}$ 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 poles) and the degree of its numerator polynomial (i.e., number of zeros).

Linearization by feedback

For the discussion that follows, we will assume that the relative degree of the system is ${\displaystyle n}$. In this case, after differentiating the output ${\displaystyle n}$ times we have,

${\displaystyle {\begin{aligned}y&=h(x)\\{\dot {y}}&=L_{f}h(x)\\{\ddot {y}}&=L_{f}^{2}h(x)\\&\vdots \\y^{(n-1)}&=L_{f}^{n-1}h(x)\\y^{(n)}&=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u\end{aligned}}}$

where the notation ${\displaystyle y^{(n)}}$ indicates the ${\displaystyle n}$th derivative of ${\displaystyle y}$. Because we assumed the relative degree of the system is ${\displaystyle n}$, the Lie derivatives of the form ${\displaystyle L_{g}L_{f}^{i}h(x)}$ for ${\displaystyle i=1,\dots ,n-2}$ are all zero. That is, the input ${\displaystyle u}$ has no direct contribution to any of the first ${\displaystyle (n-1)}$th derivatives.

The coordinate transformation ${\displaystyle T(x)}$ that puts the system into normal form comes from the first ${\displaystyle (n-1)}$ derivatives. In particular,

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

transforms trajectories from the original ${\displaystyle x}$ coordinate system into the new ${\displaystyle z}$ coordinate system. So long as this transformation is a diffeomorphism, smooth trajectories in the original coordinate system will have unique counterparts in the ${\displaystyle z}$ coordinate system that are also smooth. Those ${\displaystyle z}$ trajectories will be described by the new system,

${\displaystyle {\begin{cases}{\dot {z}}_{1}&=L_{f}h(x)=z_{2}(x)\\{\dot {z}}_{2}&=L_{f}^{2}h(x)=z_{3}(x)\\&\vdots \\{\dot {z}}_{n}&=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u\end{cases}}.}$

Hence, the feedback control law

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

renders a linear input–output map from ${\displaystyle v}$ to ${\displaystyle z_{1}=y}$. The resulting linearized system

${\displaystyle {\begin{cases}{\dot {z}}_{1}&=z_{2}\\{\dot {z}}_{2}&=z_{3}\\&\vdots \\{\dot {z}}_{n}&=v\end{cases}}}$

is a cascade of ${\displaystyle n}$ integrators, and an outer-loop control ${\displaystyle v}$ may be chosen using standard linear system methodology. In particular, a state-feedback control law of

${\displaystyle v=-Kz\qquad ,}$

where the state vector ${\displaystyle z}$ is the output ${\displaystyle y}$ and its first ${\displaystyle (n-1)}$ derivatives, results in the LTI system

${\displaystyle {\dot {z}}=Az}$

with,

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

So, with the appropriate choice of ${\displaystyle k}$, we can arbitrarily place the closed-loop poles of the linearized system.

Unstable zero dynamics

Feedback linearization can be accomplished with systems that have relative degree less than ${\displaystyle n}$. 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.