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Unformatted text preview: ECE 147A FEEDBACK CONTROL SYSTEMS  THEORY AND DESIGN F06 Midterm Exam Closed book and notes. No calculators. Show all work. Name: Solution Problem 1: /25 Problem 2: /35 Problem 3: /25 Total: /85 Miscellaneous: b b 2 4 ac 2 a , 1 . 8 n , 4 . 6 , exp  p 1 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] acos([0.5 0.7071 0.866])*180/pi =[60 45 30] asin([0.5 0.7071 0.866])*180/pi =[30 45 60] 1 dB . 89, 3 dB . 71, 5 dB . 56 10 dB . 32 15 dB . 18 30 dB . 0316. &quot; m 11 m 12 m 21 m 22 # 1 = &quot; m 22 m 12 m 21 m 11 # m 11 m 22 m 21 m 12 Name: Problem 1 The dynamical equations for the height of a bead on a rotating hoop are given by = 2 sin( ) cos( ) m  g R sin( ) e y = R [1 cos( )] where the control input is the angular velocity of the hoop , the variable denotes the angle the bead makes relative to the vertical down position (see the figure), the coefficient is used to characterize friction, R is the radius of the hoop, m is the mass of the bead, g is the gravitational constant, and e y is the height of the bead above the floor. y ~ R Figure 1: The bead on a hoop 1. For each possible value of the height y * , find all of the possible input values * and state values that correspond to an operating point with the given height. (Im not looking for numerical values here...) We take x 1 = , x 2 = , e u = and we get x 1 = x 2 =: f 1 ( x, e u ) x 2 = e u 2 cos( x 1 ) sin( x 1 ) m x 2 g R sin( x 1 ) = m x 2 sin( x 1 ) g R e u 2 cos( x 1 ) =: f 2 ( x, e u ) e y = R [1 cos( x 1 )] =: h ( x, e u ) . Operating points, which enforce f ( x * , * ) = 0 , must satisfy x * 2 = 0 and sin( x * 1 ) = 0 or g = R ( * ) 2 cos( x * 1 ) . Such points either have the form ( k, , * ) where k is an arbitrary integer and * is an arbitrary real number, or else, as long as x * 1 is such that cos( x * 1 ) &gt; , x * 1 , , s g R cos( x * 1 ) ! . It follows that the operating height y * must be in the interval [0 , R ) or else equal to 2 R . For y * = 0 or y * = 2 R , this corresponds to x * 1 = k for some integer k and * can be anything. For y * (0 , R ) , x * 1 and * must satisfy cos( x * 1 ) = 1 y * R , * = s g R (1 y * /R ) = r g R y * . Name: 2. Compute the transfer function from  * to e y y * for any * that goes with y * = 0 . 5 R . Note that the formula for the inverse of a 2...
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This note was uploaded on 08/03/2010 for the course ECE PROF. VOLK taught by Professor Volkanrodoplu during the Spring '10 term at UCSB.
 Spring '10
 VolkanRodoplu

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