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ps8sol

# ps8sol - in terms of a magnitude and a direction 1 G jω =...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Aeronautics and Astronautics 16.060: Principles of Automatic Control Fall 2003 PROBLEM SET 8 Solutions Problem 1 The transfer function G ( s ) can be written as: G ( s ) = K rl ( s + z 1 )( s + z 2 ) · · · ( s + p 1 )( s + p 2 ) · · · where K rl is the root locus gain of G ( s ), z 1 , z 2 , . . . are the zeros of G ( s ), and p 1 , p 2 , . . . are the poles of G ( s ). (Assume the system is stable, so all the poles lie in the left half-plane.) The input to the system in the Laplace domain is: R ( s ) = ( s + )( s ) So the output is: C ( s ) = G ( s ) R ( s ) = AK rl ( s + z 1 )( s + z 2 ) · · · ( s + )( s )( s + p 1 )( s + p 2 ) · · · Expanding into partial fractions: K 2 K 3 K 4 C ( s ) = K 1 + + + s + p 2 + · · · s + s s + p 1 c ( t ) = K 1 e jωt + K 2 e jωt + K 3 e p 1 t + K 4 e p 2 t + · · · Since the system is stable, the terms e p 1 t , e p 2 t , . . . all decay to zero as t . So the steady-state output is: c ( t ) = K 1 e jωt + K 2 e jωt c ss = lim t →∞ Now we need to find the residuals K 1 and K 2 . Using the coverup method, we get: ( s + ) K 1 = C ( s )( s + ) s = = ( s + )( s ) · G ( s ) = AG ( ) | s = 2 j ( s ) AG ( ) K 2 = C ( s )( s ) s = = ( s + )( s ) · G ( s ) = | s = 2 j Substituting back into the equation for c ss , we get the steady-state output is: AG ( ) e jωt AG ( ) e jωt c ss = + 2 j 2 j Now G ( ) is a complex number, that

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Unformatted text preview: in terms of a magnitude and a direction: 1 G ( jω ) = M ( ω ) e jφ ( ω ) where M ( ω ) = | G ( jω ) and φ ( ω ) = ∠ G ( jω ). Since all complex zeros and poles of G ( s ) come in | conjugate pairs (as they do for any real-life transfer function): G ( − jω ) = M ( ω ) e − jφ ( ω ) Now substitute again into the equation for c ss : A 2 j ( − M ( ω ) e − j ( ωt + φ ( ω )) + M ( ω ) e j ( ωt + φ ( ω ) ) c ss = But e jθ = cos θ + j sin θ and e − jθ = cos θ − j sin θ , so we get: A c ss = (2 jM ( ω ) sin( ωt + φ ( ω ))) 2 j = AM ( ω ) sin( ωt + φ ( ω )) A sin( ωt )--AM ( ω ) sin( ωt + φ ( ω )) G ( s ) 2...
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ps8sol - in terms of a magnitude and a direction 1 G jω =...

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