DP9.10 Consider the system is described in state vari

able form by

(b) Design the gain matrix K to meet the following specifications: (i) the closed-loop system is sta

ble; (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.

x(t) = Ax(t) + Bu(t) y(t) = Cx(1) where 01 A 2 3 ]B= | | C = [10] Assume that the input is a linear combination of the states, that is, u(t) = -Kx(t)+r(t), where r(t) is the reference input and the gain matrix is K = K K2]. Substituting u(t) into the state variable equation yields the closed-loop system x(t) = [A - BK]x(t) + Br(1) y(t) = Cx(t). (a) Obtain the characteristic equation associated with A-BK.