ch03 - T H R E E Modeling in the Time Domain SOLUTIONS TO...

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T H R E E Modeling in the Time Domain SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: State-Space Representation For the power amplifier, E a (s) V p (s) = 150 s+150 . Taking the inverse Laplace transform, e a . +150e a = 150v p . Thus, the state equation is e a =− 150e a + 150v p For the motor and load, define the state variables as x 1 = θ m and x 2 = θ . m . Therefore, x . 1 = x 2 (1) Using the transfer function of the motor, cross multiplying, taking the inverse Laplace transform, and using the definitions for the state variables, x . 2 = - 1 J m (D m + K t K a R a ) x 2 + K t R a J m e a (2) Using the gear ratio, the output equation is y = 0.2x 1 (3) Also, J m = J a +5( 1 5 ) 2 = 0.05+0.2 = 0.25, D m = D a +3( 1 5 ) 2 = 0.01+0.12 = 0.13, K t R a J m = 1 (5)(0.25) = 0.8, and 1 J m (D m + K t K a R a ) = 1.32. Using Eqs. (1), (2), and (3) along with the previous values, the state and output equations are, x . = 01 0- 1 . 3 2 x + 0 0.8 e a ; y = 0.2 0 x
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48 Chapter 3: Modeling in the Time Domain Aquifer: State-Space Representation C 1 dh 1 dt = q i1 -q o1 +q 2 -q 1 +q 21 = q i1 -0+G 2 (h 2 -h 1 )-G 1 h 1 +G 21 (H 1 -h 1 ) = -(G 2 +G 1 +G 21 )h 1 +G 2 h 2 +q i1 +G 21 H 1 C 2 dh 2 dt = q i2 -q 02 +q 3 -q 2 +q 32 = q i2 -q o2 +G 3 (h 3 -h 2 )-G 2 (h 2 -h 1 )+0 = G 2 h 1 -[G 2 +G 3 ]h 2 +G 3 h 3 +q i2 -q o2 C 3 dh 3 dt = q i3 -q o3 +q 4 -q 3 +q 43 = q i3 -q o3 +0-G 3 (h 3 -h 2 )+0 = G 3 h 2- G 3 h 3 +q i3 -q o3 Dividing each equation by C i and defining the state vector as x = [h 1 h 2 h 3 ] T x . = ( G 1 + G 2 + G 3 ) C 1 G 2 C 1 0 G 2 C 2 ( G 2 + G 3 ) C 2 G 3 C 2 0 G 3 C 3 G 3 C 3 x + q i 1 + G 21 H 1 C 1 q i 2 q o 2 C 2 q i 3 q o 3 C 3 u ( t ) y = 100 010 001 x where u(t) = unit step function. ANSWERS TO REVIEW QUESTIONS 1. (1) Can model systems other than linear, constant coefficients; (2) Used for digital simulation 2. Yields qualitative insight 3. That smallest set of variables that completely describe the system 4. The value of the state variables 5. The vector whose components are the state variables 6. The n-dimensional space whose bases are the state variables 7. State equations, an output equation, and an initial state vector (initial conditions) 8. Eight 9. Forms linear combinations of the state variables and the input to form the desired output 10. No variable in the set can be written as a linear sum of the other variables in the set. 11. (1) They must be linearly independent; (2) The number of state variables must agree with the order of the differential equation describing the system; (3) The degree of difficulty in obtaining the state equations for a given set of state variables. 12. The variables that are being differentiated in each of the linearly independent energy storage elements
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Solutions to Problems 49 13. Yes, depending upon the choice of circuit variables and technique used to write the system equations. For example, a three -loop problem with three energy storage elements could yield three simultaneous second-order differential equations which would then be described by six, first-order differential equations.
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This note was uploaded on 04/04/2008 for the course MECH COntrol Sy taught by Professor Khurshid during the Spring '08 term at Michigan State University.

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ch03 - T H R E E Modeling in the Time Domain SOLUTIONS TO...

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