Internal dynamics —— In feedback linearization
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 th order nonlinear system:
In order to control to track reference , we may use input-output feedback linearization. Differentiate until control appear:
is known as the relative degree. After linearization, we usually get where is a new control variable. It is easy to prove we can control to track , respectively. Now problem appears:
In an th system, the system is uniquely determined by 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 states () which are new states of the system. Now DOFs are still undetermined. These 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 . In input-output linearization, the new states are . In fact, this is a state transformation. We only need to choose certain states combining with to form a diffeomorphism.
4. Highlight something
a) As mentioned, are new sys states, which do contain 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 have negative real parts.