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Internal dynamics —— In feedback linearization

17/04/2011

While reviewing the feedback linearization part, I learned a lot, especially on internal dynamics.Here I just want to talk how to interpret the internal dynamics.

1. Difference between input-state and input-output feedback linearization

For input-state feedback linearization, the control goal is to let states track some references. For input-output feedback linearization, the control goal is to let output to track certain inference. This is not the most important difference. What’s most important, there are internal dynamics in input-output feedback linearization problem, which does not exist in input-state feedback linearization.

2. Where does internal dynamics come from?

Consider an nth order nonlinear system:

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

In order to control y to track reference y_{re}, we may use input-output feedback linearization. Differentiate y until control u appear:

y^{(r)}=\phi (u)

r is known as the relative degree. After linearization, we usually get y^{(r)}=v where v is a new control variable. It is easy to prove we can control y,\dot{y}, ... ,y^{r-1} to track y_{re}, \dot{y_{re}}, ..., y_{re}^{(r-1)}, respectively. Now problem appears:

In an nth system, the system is uniquely determined by n state variables. In other words, if only some (not all) of the states are determined, then the system is not uniquely determined and there are still some DOFs. Note the input-output feedback linearization determines r states (y,\dot{y}, ... ,y^{(r-1)}) which are r new states of the system. Now n-r DOFs are still undetermined. These n-r states are called internal dynamics. We must know whether the internal dynamics are bounded.

3. Which states represent internal dynamics?

The original states of the system are x_1,...,x_n. In input-output linearization, the new states are y, \dot{y},...,y^{(r-1)}. In fact, this is a state transformation. We only need to choose certain n-r states combining with y, \dot{y},...,y^{(r-1)} to form a diffeomorphism.

4. Highlight something

a) As mentioned, y, \dot{y},...,y^{(r-1)} are new sys states, which do contain n-r DOFs.
b) We must check whether the internal dynamics are bounded. This is actually a BIBO problem. An SISO LTI system is BIBO stable if and only if all its poles has negative real parts or \lambda (A) have negative real parts.

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