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Unformatted text preview: Assignment 7 , Mechtron/Sfwr Eng 4AA4 Sample Solution Q1. [2, 2] For the transfer functions given below, obtain the ztransform equivalents using partial fractions and ztransform tables (you may use a ztransform equivalent to a Laplace transform from tables). (a) 1 s 2 ( s + 1) 1 s 2 ( s + 1) = A s + B s 2 + C s + 1 = 1 s + 1 s 2 + 1 s + 1 Z [ 1 s 2 ( s + 1) ] = Z [ 1 s ] + Z [ 1 s 2 ] + Z [ 1 s + 1 ] = z z 1 + T z ( z 1) 2 + z z e T (b) 1 s ( s 2 + 1) 1 s ( s 2 + 1) = A s + Bs + C s 2 + 1 = 1 s s s 2 + 1 Z [ 1 s ] = z z 1 Z [ s s 2 + 1 ] = z ( z cosT ) z 2 (2 cosT ) z + 1 Z [ 1 s ( s 2 + 1) ] = z z 1 z ( z cosT ) z 2 (2 cosT ) z + 1 = z ( z + 1)(1 cosT ) ( z 1)( z 2 (2 cosT ) z + 1 1 Q2. [4] Find f ( kT ), for the F ( z ) given below: z ( z + 2)( z + 5) ( z . 4)( z . 6)( z . 8) F ( z ) z = ( z + 2)( z + 5) ( z . 4)( z . 6)( z . 8) Calculate constants: A = ( z + 2)( z + 5) ( z . 6)( z . 8)  z =0 . 4 = 162 B = ( z + 2)( z + 5) ( z . 4)( z . 8)  z =0 . 6 = 364 C = ( z + 2)( z + 5) ( z . 6)( z . 4)  z =0 . 8 = 203 = 162 1 z . 4 364 1 z . 6 + 203 1 z . 8 F ( z ) = 162 z z . 4 364 z z . 6 + 203 z z . 8 f ( kT ) = 162(0 . 4) k 364(0 . 6) k + 203(0 . 8) k Q3. [2 + 3 +1 + 3 +1] a) A closed loop proportional law is used to control the continuous time plant shown in Figure 1 . Sketch the root locus for K 0.[2] Figure 1: Analog Control System . 2 Figure 2: Analog Control System Root Locus b) A digital controller is used to implement the proportional gain K, as shown in Figure 3. Derive a ztransfer function for only the open loop plant preceded by a ZOH.[3] The discrete transfer function is given by: Figure 3: Digital Control System G ( z ) = (1 z 1 ) Z [ G ( s ) s ] = (1 z 1 ) Z [ 8 s ( s + 4) ] Partial fractions 8 s ( s + 4) = A s + B s + 4 3 A = 2 , B = 2 G ( z ) = (1 z 1 ) Z [ 2 s 2 s + 4 ] G ( z ) = 2(1 z 1 ) Z [ 1 s 1 s + 4 ] From Tables = 2(1 z 1 )[ z z 1 z z e 4 T = 2 ( z 1) z [ z ( z e 4 T z + 1 ( z 1)( z 4 T e ) = 2 1 e 4 T z e 4 T c) Sketch a Zplane root locus for the system. [3] The closed loop TF is: T ( z ) = KG ( z ) 1 + KG ( z ) Which gives characteristic equation: 1 + 2 K 1 e 4 T z e 4 T = 0 z e 4 T + 2 K (1 e 4 T ) = 0 z + [2 K 2 Ke 4 T e 4 T ] = 0 Gives a pole at: z = [2 K (1 e 4 T ) e 4 T ] K = 0 z = e 4 T Let e 4 T = . The Zplane root locus is shown below: 4 Figure 4: Root Locus for Digital Control System 5 d) As the gain increases, what happens in the discrete case which is not a characteristic of the continuous case?[1] Answered in the figure 4. Q4. [5] The transfer function of a system preceded by a ZOH is given by: G ( s ) = 1 e T s s ( s + 2) ( s + 1) Use Matlab to find the sampleddata transfer function, G(z), if the sampling period is 0 . 5 second....
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This note was uploaded on 12/13/2011 for the course SOFTWARE E SFWR ENG 4 taught by Professor Bokhari during the Fall '11 term at McMaster University.
 Fall '11
 Bokhari

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