homework3_f10_sol

homework3_f10_sol - University of California, Davis...

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University of California, Davis Department of Chemical Engineering and Materials Science ECH 157 Process Dynamics and Control Ahmet Palazoglu HOMEWORK 3/SOLUTION Fall 2010 (40 points) 1. (10 pts) Problem II.6 from R&P. Solution: We start with the following expansion, ) ( ) ( ) ( ) , , ( ) , , ( 5 . 2 ) 4 ( ) ( ) ( ) ( ) , , ( ) , , ( ) ( 2 2 2 2 2 1 1 1 2 2 1 2 2 1 2 1 2 2 1 2 2 2 1 1 1 1 1 2 1 1 2 1 1 1 1 s ss s ss s ss s s s s ss s ss s ss s s s D D D f x x x f x x x f D x x f D x x f x D x dt dx D D D f x x x f x x x f D x x f D x x f x D dt dx + + + + + = = + + + + + = = μ The derivative terms can be calculated as follows, 0 12 . 0 53 . 0 2 2 1 1 11 = + = = ss ss D x x x f a 3866 . 0 ) 12 . 0 ( 53 . 0 ) 12 . 0 ( 53 . 0 2 2 2 1 2 1 2 1 12 = + + = = ss ss x x x x x x f a [] 4523 . 1 1 1 1 = = = ss ss x D f b 1 12 . 0 ) 53 . 0 ( 5 . 2 2 2 1 2 21 = + = = ss ss x x x f a 3649 . 1 ) 12 . 0 ( ) 53 . 0 ) 12 . 0 ( 53 . 0 ( 5 . 2 4 . 0 2 2 2 2 1 2 2 22 = + + + = = ss ss x x x x x f a 1
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[] 6308 . 3 4 2 2 2 = = = ss ss x D f b This yields the following state space form of the model, after defining the deviation variables: s s s D D u x x x x x x = = = , , 2 2 2 1 1 1 [] x c x x y u b x A u x x x x x = = + = ⎡− + = = 2 1 2 1 2 1 0 1 6308 . 3 4523 . 1 36 . 1 1 3866 . 0 0 To obtain the transfer function model, we need to take Laplace transform of the linear model, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 22 1 21 2 1 2 12 1 2 12 1 11 1 s u b s x a s x a s sx s u b s x a s u b s x a s x a s sx + + = + = + + = Solve the second equation for and replace in the second, ) s ( x 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 22 2 12 22 1 21 12 1 22 2 22 1 21 2 s u b a s s u b a a s s x a a s sx a s s u b a s s x a s x + + = + = Now, we have to collect the terms for the transfer function, and recognize that , ) ( ) ( 1 s x s y = 21 12 22 2 22 1 12 2 1 ) ( ) ( ) ( ) ( a a s a s a b a b s b s g s u s y + = = This is a second order system. 2. (10 pts) Problem II.10 from R&P. Solution: We start by linearizing the nonlinear equations, recognizing that we have two state variables and two inputs in the state equations. The first equation yields: 2
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() ) ( ) ( ) ( ) ( , , , ) , , , ( 2 2 , , , 2 1 1 1 , , , 1 1 2 2 , , , 2 1 1 1 , , , 1 1 2 1 2 1 1 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 s u u x x s u u x x s u u
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homework3_f10_sol - University of California, Davis...

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