$(25$ Points$)$ Consider the dynamical system

$\backslash [$

$\backslash$begin$\{$aligned$\}$

a $\backslash$dot$\{$x$\}$ & $=$x$+$y$-$b $\backslash$cos $($t$)\backslash \backslash$

$\backslash$dot$\{$y$\}$ & $=$c $\backslash$tanh $($t$)+\backslash$tau

$\backslash$end$\{$aligned$\}$

$\backslash ]$

where only $\backslash ($ x$,$ y $\backslash$in $\backslash$mathbb$\{$R$\}\backslash )$ are measurable, $\backslash ($ a$,$ b$,$ c $\backslash$in $\backslash$mathbb$\{$R$\}\backslash )$ are unknown positive constants. Design a continuous controller $\backslash (\backslash$tau $\backslash$in $\backslash$mathbb$\{$R$\}\backslash )$ such that at least $\backslash ($ x$($t$)\backslash )$ converges to $0.$

$($a$)(17$ Points$)$ Describe the behavior of the system using a Lyapunov$-$based analysis.

$-$ Be sure to:

$-$ Define the error system$($s$).$

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

$-$ List the controller, update laws, and$/$or filter update policy all in one location and box them.

$-$ Define the candidate Ly apunov function.

$-$ List any gain conditions or assumptions if necessary.

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

$-$ Cite the theorem$/$definition that facilitates your stability result $($e$.$g$.,$ "Theorem $\backslash$#$.\backslash$#$\backslash$#$,$ Khalil" or "From Lecture $\backslash$#$\backslash$#$").$

$-$ Show that all conditions of that theorem$/$definition are satisfied.

$($b$)(3$ Points$)$ Prove the controller is bounded and implementable.

$-$ You must perform signal chasing to show that the controller is bounded, and that it is composed of known signals. For example, if you need do design an output feedback controller, then you would need to show all signals are implementable $($e$.$g$.,$ "design a p$-$filter"$).$

$($c$)(5$ Points$)$ Describe the individual components $($i$.$e$.,$ each term$)$ of your controller. List at least one benefit and one drawback of one term in your controller $($e$.$g$.,$ "sliding mode feedback is $($un$)$ desir able because...," or "high gain controllers are prone to$...").$