Question

2. Olen)-11 27 +(6) 4. ). U skl= (^ 13. UG) Design an observer for the ② system with state feedback (with the poles o such that observer leads deatbeat response. Capply all methods) :) :) ii) (6 write the state equation of observer © write the state equation of overall systen.

Answer 1

The observability of the system can be found from the given state space equation and the observability of the system depends on the rank of the matrix which should be equal to the number of variables.

The observability (Q0) and the transfer function T(s) of the overall system is written below,

x(x+1)= C 8J x(k) +[Julie) x = AX+BU Y(K) = 1 D kulk) y = cu observability of a system is, 90 = [CT ACT CATCT.... (AT)"- et] where an is variables of the state equation. Here, n=2 9o = [CT ATC LI213 7 Here rank of the matrix is 2 , which is canal to the number of state varikly So, the System in oblervable

x(kt) = ( ºJxcx) + [!]ux) > 0 YCI) = [' Julie) - x = AX + BU y = cx+DU Comparing 0 & we get . - , & c = [1 ) DEO Transfer function T(s) = ({SI - AJETD. [SI-A] = s[! ;] - [8] = [- sz] IST-A1 = (S-1) (S-2) = 5-25-5+2=5435t2 rs-2 Š-35+2 S-1 S-35+2

Transfer function T(W= C[ST-A] BAD s-2 S-35+2 S-1 5-35tL 25-3 0 IT (s) = 5-35+2