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e483h5s_07 - 1 ENSC 483 HW#5 Solutions by M Saif Simon...

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ENSC 483 HW#5 Solutions by M. Saif 1 Simon Fraser University School of Engineering Science ENSC 483 - Modern Control Systems 1. Electrical circuit– Obtain the state space description of the network shown in Figure 1. Use the voltages across the capacitors and the current through the inductor as the state variables. Note that v 1 is the input and v 2 is the output. Figure 1: An electrical network Solution: dv C 1 dt = 1 C 1 i C 1 dv C 2 dt = 1 C 2 i C 2 di L dt = 1 L v L R 1 i R 1 + v C 1 = v 1 R 2 i R 2 + R 3 i L + v L = v C 1 R 3 i L + v L = v C 2 v 2 = v C 2 i R 1 = i C 1 + i R 2 i R 2 = i L + i i = i C 2 Now if we take v C 1 ; v C 2 , and i L as our state variables, we get: ˙ v C 1 ˙ v C 2 ˙ i L = - 1 C 1 1 R 1 + 1 R 2 · 1 C 1 R 2 0 1 C 2 R 2 - 1 C 2 R 2 - 1 C 2 0 1 L - R 3 L v C 1 v C 2 i L + 1 R 1 C 1 0 0 v 1 v 2 = [ 0 1 0 ] v C 1 v C 2 i L 2. Mass-Spring Systems – Obtain the differential equations for the system shown in Figure 2. Draw a block diagram for this system and find the transfer function H ( s ) = X 2 ( s ) F ( s ) . Write the state variable formulation of this system using the state variables: z 1 = x 1 , z 2 = ˙ x 1 , z 3 = x 2 , z 4 = ˙ x 2 .
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ENSC 483 HW#5 Solutions by M. Saif 2 Figure 2: A mass-spring system. Figure 3: Electrical analogue of the mass-spring system
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ENSC 483 HW#5 Solutions by M. Saif 3 Solution: Draw the electrical analog of the mass-spring system as in Figure 3. From it we get, M 1 s 2 X 1 ( s ) + ( Ds + K 1 )( X 1 - X 2 ) = U ( s ) M 2 s 2 X 2 ( S ) + ( D 1 s + K 1 )( X 2 - X 1 ) + K 2 X 2 = 0 The block diagram representation of the system in terms of the input U ( s ) , and output X 2 ( s ) is shown in Figure 4. The transfer function of it is H ( s ) = Ds + K 1 M 1 M 2 s 4 + ( M 1 + M 2 ) Ds 3 + ( M 1 K 1 + M 2 K 1 + M 1 K 2 ) s 2 + K 2 Ds + K 1 K 2 Finally, the state space representation of the system is ˙ z ( t ) = d dt x 1 ˙ x 1 x 2 ˙ x 2 = 0 1 0 0 - K 1 M 1 - D M 1 K 1 M 1 D M 1 0 0 0 1 K 1 M 2 D M 2 - K 1 + K 2 M 2 - D M 2
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