(Solved): 2. Consider the inverted pendulum system given in Take-home Exam 2, Question 3 . (a) (10 pts) Using ...

2. Consider the inverted pendulum system given in Take-home Exam 2, Question 3 . (a) (10 pts) Using the linearized model of the system, design a state feedback controller such that the dominant poles have a damping ratio of $$?=0.707$$ and the system has a settling time of $$t_s=1\mathrm{s}$$. (b) ( $$10\mathrm{pts})$$ If the initial state is given as $${\mathrm{x}}_0={\left[\begin{array}{llll}0.1& 0& 0& 0\end{array}\right]}^T$$, plot the initial condition response of the closed loop system. You can use the initial() command of MATLAB. (c) (15 pts) Note that to apply state feedback, you have to measure all the state variables. Suppose it is not possible. You can only afford to buy a single encoder. You can use that encoder to measure $$?$$ only, or $$?$$ only. Which one would you measure? Why? Explain by considering the observability features of the system. (d) (10 pts) Design a linear quadratic regulator for this system to minimize $$PI={?}_0^{?}{?}^2+2{?}^2+T^{2}dt$$