# ch20 - 20-1 20-2 = =)13)115(sCsBsAsYm)13(115)91(21-ssss(2...

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Unformatted text preview: 20-1 20-2 =++++=)13()115()(sCsBsAsYm)13)(115()91(21++-ssss(2) where 2)13)(115()91(2=++-==ssssA60)13()91(2151-=+-=-=ssssB6)115()91(231=+-=-=ssssCSubstitute into (2) and take inverse Laplace transform: 3/15/242)(ttmeety--+-=(3) b)The new steady-state value is obtained from (3) to be ym(∞)=2 For t= t99, ym(t)=0.99ym(∞) = 1.98. Substitute into (3) 3/15/999924298.1ttee--+-=(4) Solving (4) for t99by trial and error gives t99≈79.5 min Thus, we specify that ∆t =79.5 min/40 ≈2 min Sample No SiSample No SiSample No Si1 -0.4739 16 1.5263 31 1.9359 2 -0.5365 17 1.5854 32 1.9439 3 -0.4106 18 1.6371 33 1.9509 4 -0.2076 19 1.6824 34 1.9570 5 0.0177 20 1.7221 35 1.9624 6 0.2393 21 1.7568 36 1.9671 7 0.4458 22 1.7871 37 1.9712 8 0.6330 23 1.8137 38 1.9748 9 0.8022 24 1.8370 39 1.9779 10 0.9482 25 1.8573 40 1.9807 11 1.0785 26 1.8751 12 1.1931 27 1.8907 13 1.2936 28 1.9043 14 1.3816 29 1.9163 15 1.4587 30 1.9267 Table S20.2. Step response coefficients20-3 20.3 From the definition of matrix S, given in Eq. 20-20, for P=5, M=1, with Siobtained from Exercise 20.1, S==2174.1344.06572.01811.54321SSSSSFrom Eq. 20-58Kc= (STS)-1STKc= [ ]1076.39206.19395.2589.= Kc1TBecause Kc1T is defined as the first row of Kc. Using the given analytical result, Kc1T= [ ]54321512)(1SSSSSSii∑== [ ]2174.1344.06572.01811.06995.1= [ ]1076.39206.19395.2589.which is the same as the answer obtained above using (20-58) 20.4 The step response is obtained from the analytical unit step response as in Example 20.1. The feedback matrix Kcis obtained using Eq. 20-57 as in Example 20.5. These results are not reported here for sake of brevity. The closed-loop response for set-point and disturbance changes are shown below for each case. MATLAB MPC Toolboxwas used for the simulations. 20-4 i)For this model horizon, the step response is over 99% complete as in Example 20.5; hence the model is good. The set-point and disturbance responses shown below are non-oscillatory and have long settling times 1020304050607080901000.511.5OutputsTimey1020304050607080901000.50....
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ch20 - 20-1 20-2 = =)13)115(sCsBsAsYm)13(115)91(21-ssss(2...

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