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# ps10a - University of Maryland at College Park Dept of...

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University of Maryland at College Park Dept. of Aerospace Engineering ENAE 432: Aerospace Control Systems Problem Set #9 Issued: 23 Apr 2011 Due By: 29 Apr 2011 Question 1: Prove that: | S ( γ ) | = 1 2 sin( γ/ 2) Question 2: Prove that if the peak of the sensitivity diagram satisfies max ω 0 | S ( | ≤ 6 dB then the feedback system is guaranteed to have gain margin in the range 0 . 5 a 1 . 5 and the phase margin is guaranteed to satisfy | γ | > 29 . Hint: this is not so easy as applying the result in Question #1, since | S ( γ ) | is not generally the peak of the sensitivity diagram. We will show in class that | 1 + L ( ) | is the distance from the -1 point to the polar plot of L at each frequency ω . The statement above is equivalent to a minimum guaranteed value of this distance. Use this fact, and the definition of phase and gain margin, to establish the claimed properties. Question 3: A tracking antenna has azimuthal pointing dynamics given by ¨ θ ( t ) + 2 ˙ θ ( t ) = 5 u ( t ) where θ ( t ) is the azimuth angle, and u ( t ) is the current commanded to the driving servo- motor by the tracking computer. The quantity 5 u ( t ) is the actual torque developed by the servomotor in response to these commands. The system is required to perfectly track targets moving so that θ d ( t ) = A 0 + A 1 t where A 0 and A 1 can be of arbitrary sign and magnitude. The tracking bandwidth of the system should also be at least 2 rad/sec. The goal of this problem is to design a feedback

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ps10a - University of Maryland at College Park Dept of...

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