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# ps7 - EE 428 PROBLEM SET 7 DUE 12 NOV 2007 Reading...

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EE 428 PROBLEM SET 7 DUE: 12 NOV 2007 Reading assignment: Reading assignment: Ch 5, sections 5.1 through 5.5 Problem 29: (20 points) A control engineer is designing an attitude control system for a rocket. Figure 1 shows a block diagram of the attitude control system. The engineer finds that the input-output transfer function G p ( s ) of the dynamics of the rocket, from the input thrust U ( s ) to the output attitude θ ( s ), is G p ( s ) = Θ( s ) U ( s ) = K Js 2 where K [meters] is the aerodynamic gain of the rocket and J [kilogram-meter 2 ] is the moment of inertia of the rocket about its center of gravity. The control engineer specifies the use of a proportional-plus-derivative (PD) controller G c ( s ) = U ( s ) E ( s ) = K o + K d s, where the closed-loop error signal E ( s ) = Θ R ( s ) - Θ( s ) is the difference between the desired attitude Θ R ( s ) and the actual attitude Θ( s ). Figure 1: Rocket attitude control system. The control engineer subcontracts the following tasks to you: 1. (3 points) Assume K d is fixed and sketch the root locus of the closed-loop attitude control system as the proportional gain K o is increased from zero to infinity. 2. (3 points) Assume K o is fixed and sketch the root locus of the closed-loop attitude control system as the derivative gain K d is increased from zero to infinity. Construct the PD controller gain plane of K o versus K d (with K p on the ordinate and K d on the abscissa) and show the following regions in the plane: 3. (2 points) Stable region. 4. (2 points) Unstable region. 5. (2 points) Trajectory on which the closed-loop attitude control system is critically damped (that is, the trajec- tory on which the dimensionless damping ratio ζ of the closed-loop attitude control system is unity). 6. (2 points) Region in which the closed-loop attitude control system is overdamped (that is, the region in which ζ > 1). 7. (2 points) Region in which the closed-loop attitude control system is underdamped (that is, the region in which 0 < ζ < 1). 8. (2 points) Trajectory on which the underdamped natural frequency ω n of the closed-loop attitude control system is 40 radians/second.

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ps7 - EE 428 PROBLEM SET 7 DUE 12 NOV 2007 Reading...

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