midterm04_sol

# midterm04_sol - ECE 147A FEEDBACK CONTROL SYSTEMS THEORY...

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ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY AND DESIGN F04 Midterm Exam Closed book and notes. No calculators. Show all work. Name: Solution Problem 1: /25 Problem 2: /25 Problem 3: /25 Total: /75 Miscellaneous: 1 . 8 ω n , 4 . 6 σ , exp - πζ p 1 - ζ 2 ! , atan([0.1 0.5 2/3 1 2 5 10])*180/pi = [5.7 26.6 33.7 45 63.4 78.7 84.3] - 1 dB 0 . 89, - 3 dB 0 . 71, - 5 dB 0 . 56 - 10 dB 0 . 32 - 15 dB 0 . 18 - 30 dB 0 . 0316.

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Name: Problem 1 Consider the nonlinear system ˙ z = ξ - z 2 ˙ ξ = 10 - ξ + pz with input p and output z . 1. What condition on the triple ( z * , ξ * , p * ) must be satisfed For it to qualiFy as a valid operating point? First note that we can take the states of the system to be x 1 = z , x 2 = ξ . Then we have f 1 ( x, u ) = x 2 - x 2 1 and f 2 ( x, u ) = 10 - x 2 + ux 1 . An operating point must satisfy f ( x * , u * ) = 0 . In terms of the original variables, we must have ξ * - ( z * ) 2 = 0 and 10 - ξ * + p * z * = 0 . In other words, we must have ξ * = ( z * ) 2 and ξ * = 10+ p * z * . 2. Using the defnition oF an operating point, explain graphically in the ( z, ξ )-plane whether For each z * > 0 it is possible to fnd p * and ξ * so that ( z * , ξ * , p * ) is a valid operating point. - 6 z ξ 10 0 Q Q Q Q Q Q Q Q Q Q Q Q ξ * z * ξ = z 2 ξ = 10 + p * z * , p * = - 1 On the diagram above, we have plotted the curve ξ = z 2 and the line ξ = 10 + pz for p = - 1 . Where the line and the curve intersect is an operating point ( z * , ξ * ) corresponding to p * = - 1 . It should be clear from this picture that for any desired z * > 0 , we can pick p * so that the line corresponding to p * intersects the curve at z * . The value of ξ at the intersection, denoted ξ * , gives the operating point ( z * , ξ * , p * ) . (continued on next page)
z * = 10. Hint: use the fact that " a b c d # - 1 = " d - b - c a # ad - bc . Using the definition of f in the first part and the definition of A , B , C and D in the notes, we have A = " - 2 z * 1 p * - 1 # , B = " 0 z * # , C = h 1 0 i , D = 0 . According to the analysis for the previous part, for z * = 10 , we must have ξ * = ( z * ) 2 = 100 and p * = z * - 10 /z * = 9 . We then have A = " - 20 1 9 - 1 # , B = " 0 10 # , C = h 1 0 i , D = 0 . Finally, the transfer function is given by P ( s ) = C ( sI - A ) - 1 B + D = h 1 0 i " s + 1 1 9 s + 20 # " 0 10 # ( s + 20)( s + 1) - 9 = 10 s 2 + 21 s + 11 . 4. Suppose a controller C has been designed to control the system above near the operating point given above, using the negative unity feedback con±guration with the reference pre±lter F . Draw the corresponding block diagram implementation of the control system.

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midterm04_sol - ECE 147A FEEDBACK CONTROL SYSTEMS THEORY...

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