Consider the dynamics

$$x $=$ ax$2+$ cx $+$ bu

where a$,$ b$,$ c in R$>0$ are known constants. The control objective is to track a used$-$selected desired

trajectory xd : R$0$ R with known derivatives $$xd$,$xd$,...$ in R with known bounded derivatives such

that $|$xd$|$ xd$,|$xd$|$xd$,...$ x is the only measurable state.

$($a$)(15$ Points$)$ Describe the behavior of the system using a Lyapunov$-$based analysis.

$$ Be sure to:

Dene the error system$($s$).$

Dene the closed$-$loop error system$($s$).$

List the controller, update laws, and$/$or lter update policy all in one location

and box them.

Dene the candidate Lyapunov function.

List any gain conditions or assumptions if necessary.

Describe the behavior of the system, e$.$g$.,$ Local Asymptotic Stability.

Cite the theorem$/$denition that facilitates your stability result $($e$.$g$.,$ Theorem #$.$##$,$

Khalil or From Lecture ##$).$

Show that all conditions of that theorem$/$denition are satised