Question

1.3. Assume a discrete-time system with input u(k) and output y(k). The system has a constant input disturbance d. Find the augmented state-space model with input Au(k) and output y(k) for the plant model given as below: Im(k +1) = AmIm (k) + Bmu(k) + Bad; y(k) = Cmlm (k) (1.82) [10.5 0 ] [ 0.51 where Am = 0.1 -0.1; Bm = 1 ; Cm = [101]; Ba= 100 0.8 -0.6 1. Calculate the plant transfer function that relates the input u(k) to the output y(k), and the transfer function of the augmented state-space model that relates the input Au(k) to the output y(k). 2. Compare the poles and zeros of these transfer functions.

Answer 1

lue 0 xml&+1) - Am dmck) + Bmuck) + B dd Y.CK) = Cm X mlk) e o can be written as [ack) Zrink) - lo.so (k) Tos xik) To 1-0.8 1 neurs + vekt loo 0.2 l n -0.6] By laplace Transform La ck) S X, (S) - Xics) 70-5X (S) +0.5USJ + d. X,(s) (5-1) = 0.5 X2 (8) +0.500) + Xi(s) - 0.5X₂ (5) + 0.5 usst da s-l SX200) = X3(s) - 0.Xges stoly X (S) (S-1) - -0.8 X3 (51 t ues) Te OT -0.8 x - X₃ (5) + US S7315) = X351) = 0.4 X3 159 -0.6uery -0.6 uus s-0.8

Xa SJ - -0.010 CO 0.48 - uest US) (S-1) (S-0-) 0.48 + 5-08 (SH) CS-08) S-0.32 wall (S-1)05-0.8) ② from ea X(S) = 0.5 's-1 I S-0.32 (SH) (5-0) wash + J 0 I US) .s- - ST-1.80-9.3 +0.64 +0.8d you X,) = 0.5(3-0:32) + (SV)(N-08) + CS-02)des 12 (S-132 (S-08) = 0.55 - 0:16 + 5%-0-85 -5 +0.3 + d(5-18870-3) - (SHU)?(8-0-8) XI(S) = ² (Hd) + S (-1.80-9.3 +0. Guinen (S-1) ² (3-0.8) from eq YiB) = TODJ[X10 tu Y S = X, (s) + Xg (1)

YS) - S21+ d) + (-1.80 1.3) + 0.64 to 8d wiss (54)? (5-08) + - 0.6 vs s-o.8 Saltd+S4.80.39 to 64 +0.80 -0.65 +1.25 (5+)2 (5-0-8) yers - 8²4 0.4+d) + s(Hidd-o. 1) to dd to ou (S-1)-(S-0.31 poles (4 (s. ond) =0 poles are 1, 1,02 2 eios - 18d 70-) Jl.204019 – 4 Cound 360-0u) 2co.utd]