EE611

Solution to EXAM II

1a) Check controllability: Completely controllable!!
Put in phase variable form:
Let w=-kx; Have: Want:

OR use Ackermann's formula:
b) Must assume same number of inputs as outputs we want to track!!

c) N_x is nxp, where p is the number of outputs we want to track = number of inputs

d)

e) N_u is mxp
f) We have 1 input and one output we want to track (y₂ = 5), so eliminate first row of C in calculations:

2a) r = rank(M), T_ccf. = [r linearly independent columns of M | n-r more columns such that T_ccf has full rank]
b) Controllable eigenvalues are {1,2,3}, uncontrollable are {1/2,1/2,1/2}
c) System is stabilizable!!
d)

3

Check observability: Completely observable!! Let's let A_o = A^T, b_o = c^T, and solve (A_o-B_oK_o^T)as the dual of the controller problem, :

c) The reduced-order observer is a better estimate AND it is estimating fewer states (1^st order as opposed to 2^nd order)

4a) We know Dw_k, so let , and let , and follow the rest of the design procedure for the reduced-order observer (see notes!!). Then A_RO=A₂₂-K_ROA₁₂, B_RO=B₂-K_ROB₁,C_RO=A_ROK_RO+A₂₁-K_ROA₁₁
b) Block diagram: