KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-sol-hw-6

KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-sol-hw-6

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Unformatted text preview: 6.1 a) By using MATLAB, the poles and zeros are: Zeros: (-1 +1i) , (-1 -1i) Poles: -4.3446 (-1.0834 +0.5853i) (-1.0834 0.5853i) (+0.7557 +0.5830i) (+0.7557 -0.5830i) These results are shown in Fig E6.1 Figure S6.1. Poles and zeros of G(s) plotted in the complex s plane. b) c) Process output will be unbounded because some poles lie in the right half plane. By using Simulink-MATLAB 2 x 10 8 0 -2 -4 -6 Output -8 -10 -12 -14 -16 0 5 10 15 Time 20 25 30 Figure E6.1b. Response of the output of this process to a unit step input. As shown in Fig. S6.1b, the right half plane pole pair makes the process unstable. 6.6 Y (s) = K1 K2 K2 K U (s) + U ( s) = 1 + U ( s ) s s + 1 s s + 1 Y ( s ) K 1 s + K 1 + K 2 s ( K 1 + K 2 ) s + K 1 = = U ( s) s (s + 1) s (s + 1) Put in standard K/ form for analysis: K K 1 + 2 s + 1 K1 Y (s) G( s) = = U ( s) s (s + 1) a) b) c) Order of G(s) is 2 (maximum exponent on s in denominator is 2) Gain of G(s) is K1. Gain is negative if K1 < 0. Poles of G(s) are: s1 = 0 and s2 = 1/ s1 is on imaginary axis; s2 is in LHP. d) Zero of G(s) is: sa = - K1 -1 = K K1 + K 2 + 2 K1 If K1 < 0 , the zero is in RHP. K1 + K 2 Two possibilities: 1. K1<0 and K1 + K2 >0 2. K1 > 0 and K1 + K2 < 0 e) Gain is negative if K1 < 0 Then zero is RHP if K1 + K2 > 0 This is the only possibility. f) g) Constant term and e-t/ term. If input is M/s, the output will contain a t term, that is, it is not bounded. 6.9 a), b) Represent processes that are (approximately) critically damped. A step response or frequency response in each case can be fit graphically or numerically. c) d) e) f) g) h) = 2, = 10 Exhibits strong overshoot. Can't approximate it well. = 0.5, = 10 = 1, = 10 Underdamped (oscillatory). Can't approximate it well. = 2, = 0 By using Simulink-MATLAB, models for parts c), e), f) and h) are compared: (Suppose K = 1) Part c) 1 0.9 0.8 0.7 0.6 Output 0.5 0.4 0.3 0.2 0.1 Exact model Approximate model 0 0 5 10 15 20 25 time 30 35 40 45 50 Figure S6.9a. Unit step responses for exact and approximate model in part c) 6-10 Part e) 1 0.9 0.8 0.7 0.6 Output 0.5 0.4 0.3 0.2 0.1 Exact model Approximate model 0 0 5 10 15 20 25 Time 30 35 40 45 50 Figure S6.9b. Unit step responses for exact and approximate model in part e) Part f) 1 0.9 0.8 0.7 0.6 Output 0.5 0.4 0.3 0.2 0.1 Exact model Approximate model 0 0 5 10 15 20 25 Time 30 35 40 45 50 Figure S6.9c. Unit step responses for exact and approximate model in part f) 6-11 Part h) 1.5 1 0.5 Output 0 -0.5 Exact model Approximate model -1 0 5 10 15 20 25 Time 30 35 40 45 50 Figure S6.9d. Unit step responses for exact and approximate model in part h) ...
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