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# Therefore the conversion of a to b increases however

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Therefore, the conversion of A to B increases. However, the control objective is not satisfied since the reactor temperature exceeds its allowable limit. (3 pts) (f) State variables - x = [ c a , c b , T r , T c ] Input variables - u = [ q r , q c , T ri , T ci ] We now linearize equations (1) to (4) by calculating the A and B matrices. A 11 = ∂f 1 ∂c a = - ( q rs V rs + k 10 e - E 1 Trs + k 20 e - E 2 Trs ) A 12 = ∂f 1 ∂c b = 0, and so on. The A matrix is given by: - ( q rs V rs + k 10 e - E 1 Trs + k 20 e - E 2 Trs ) 0 - ( k 10 E 1 e - E 1 Trs + k 10 E 1 e - E 1 Trs ) c as T 2 rs 0 k 10 e - E 1 Trs - q rs V rs k 10 E 1 e - E 1 Trs c as T 2 rs 0 - h 1 k 1 ( T rs )+ h 2 k 2 ( T rs ) ρ r c pr 0 - h 1 k 1 ( T rs ) E 1 + h 2 k 2 ( T rs ) E 2 ρ r c pr c as T 2 rs - q rs V rs - UA V rs ρ r c pr UA V rs ρ r c pr 0 0 UA V cs ρ c c pc - q cs V cs - UA V cs ρ c c pc Similarly, the B matrix is given by: c ais - c as V rs 0 0 0 - c bs V rs 0 0 0 - T ris - T rs V rs 0 q rs V rs 0 0 T cis - T cs V cs 0 q cs V cs Substitute the steady state values to get the numerical values of these matrices. (Scoring: Expressions for A and B: 10 pts; numerical values: 5 pts) 2

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