Points$)$ Consider the dynamical systemax $=$ x $+$ y b cos $($t$)$y $=$ c tanh $($t$)+,$where only x$,$ y R are measurable, a$,$ b$,$ c R are unknown positive constants. Design a contin$-$uous controller R such that at least x $($t$)$ converges to $0.($a$)(17$ 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 locationand 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 #$.$##$,$ Khalilor From Lecture ##$).$ Show that all conditions of that theorem$/$denition are satised.$($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 iscomposed 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$-$lter"$).($c$)(5$ Points$)$ Describe the individual components $($i$.$e$.,$ each term$)$ of your controller. List at leastone benet and one drawback of one term in your controller $($e$.$g$.,$ sliding mode feedback is