$.$ Modeling: $(1)$ Derive the state space model with the duty cycle d$($t$)$ and the input voltage $()$ g v t as the inputs, and the output voltage $()$ v t as the output. $(2)$ Derive the linearized system by using the state space method $($not the circuit method$).(3)$ Derive the transfer function from d$($t$)$ to $()$ v t $.2.$ Control Design: $(1)$ Design c $()$ G s by using the frequency$-$domain method: Design a $$PI$$ controller, such that the closed$-$loop system has a phase margin at least $75$ degree. Discuss the tradeoff between bandwidth of the closed$-$loop system and phase margin. $(2)$ Design c $()$ G s by using the root locus method: Design a $$P$$ controller such that the the closed$-$loop system has a damping ratio larger than $0\mathrm{.}55.$ Discuss the tradeoff among the peak time, overshoot, settling time, and steady$-$state error. $(3)$ Simulate the overall closed$-$loop system, in its transfer function form, with your designed controllers. $(4)$ Simulate the overall closed$-$loop system by using the following forms: the converter is simulated in its circuit form $($including the PWM part$),$ the controllers by transfer functions. $(5)$ Discuss your design, performance, pros and cons.