Consider the system is described in state variable form by

y(t) = Cx(r)
where

Assume that the input is a linear combination of the states, that is,
u(t) = -Kx(r) + r(f),
where r{t) is the reference input and the gain matrix is
K = [K1 K2]. Substituting u(t) into the state variable equation yields the closed-loop system

y(t) = Cx(t)
(a) Obtain the characteristic equation associated withA-BK.
(b) Design the gain matrix K to meet the following specifications: (i) the closed-loop system is stable; (ii) the system bandwidth ωb ‰¤ 1 rad/s; and (iii) the steady-state error to a unit step input R(s) = 1/s is zero.

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