SFWR ENG 4AA4 Assignment 7 Solutions

SFWR ENG 4AA4 Assignment 7 Solutions - Assignment 7 ,...

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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 z-transform equivalents using partial fractions and z-transform tables (you may use a z-transform 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 z-transfer 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 Z-plane 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 Z-plane 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 sampled-data 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.

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SFWR ENG 4AA4 Assignment 7 Solutions - Assignment 7 ,...

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