Consider a normalized inverted pendulum with a rate sensor de- scribed by d == dt (2) = ( )) (2) y = (01) (21) = + A u = Ax + Bu, = Cx (a) Is this system reachable? Is it observable? (b) Suppose we wish to design a controller for disturbance rejection (r = 0) using output feedback with an observer. Find K and L such that the matrices A- BK and A-LC have the characteristic polynomials s+a1s+ a2 and s + bs+b2 (with all coefficients positive). (c) In state space, the controller for disturbance rejection (r = 0) is given by u = -K, d dt A+ Bu+L(y- C) In the frequency domain, we can represent the output feedback con- troller for disturbance rejection as C(s) in the following diagram: y C(s) P(s) What is the controller transfer function, C(s)? (d) Using the K and L found in part b, Show that the controller is unstable.