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Unformatted text preview: MAE143b Linear Control  Theory and Applications Lecture 5, Tuesday Aug. 18, 2:004:50pm Prof. R.A. de Callafon callafon@ucsd.edu CONTENTS OF THIS LECTURE Effect of pole location ( sec. 3.3, 3.5 ) Application to 2nd order system and lab demo model. Stability analysis for TF and SS models ( sec. 3.7 ) RouthHurwitz criterion Times for rise, settling & peak and overshoot ( sec. 3.4 ) Final Value Theorem Analysis of standard 2nd order system MAE143b, UCSD, Summer II 2009, R.A. de Callafon Lecture 5, Page 1 Dynamic Response Effect of Pole Locations Recap  Consider PFE of a TF model with real and complex poles: G ( s ) = k X i =1 C i s p i + n X k = i +1 C i s p i + C i s p i where p i for i = 1 , . . . , k are real poles with real valued coecients C i . For i = k + 1 , . . . , n , p i are complex conjugate pole pairs p i = i + j i with complex conjugate coecients C i = a i + jb i . Then the impulse response of the system is given by G ( t ) = k X i =1 C i e p i t + n X i = k +1 2 a i e i t cos i t 2 b i e i t sin i t Response is sum of exponents and exponential decaying sine and cosine functions From this response it can be seen that: System G with TF G ( s ) is stable if all poles p i satisfy Re { p i } < MAE143b, UCSD, Summer II 2009, R.A. de Callafon Lecture 5, Page 2 Dynamic Response Effect of pole locations For (complex conjugate pair of) pole locations p i = i + j i we can make a qualitative analysis of the response. Each pole location is indi cated by an . Poles in the Right Half Plane (RHP) indicate unstable dy namics. Poles in the Left Half Plane (LHP) indicate stable dynam ics. Poles on the imaginary axis indicate marginally stable dy namics. Each complex pole has a com plex conjugate. The larger the imaginary value i , the larger is the fre quency of oscillation. The larger the real value i , the faster is the response. MAE143b, UCSD, Summer II 2009, R.A. de Callafon Lecture 5, Page 3 Dynamic Response Application to 2nd order system Consider the TF model of a mass/damper/spring system: G ( s ) = 1 ms 2 + ds + k = 1 m 1 s 2 + d m s + k m By completing the squares on the denominator, this TF can be rewritten as 1 m 1 ( s + 1 2 d m ) 2 + k m 1 4 d 2 m 2 = 1 m 1 r k m 1 4 d 2 m 2 r k m 1 4 d 2 m 2 ( s + 1 2 d m ) 2 + k m 1 4 d 2 m 2 and application of inverse Laplace transform (table entry 20) yields the impulse response y ( t ) = 1 m 1 r k m 1 4 d 2 m 2 e 1 2 d m t sin s k m 1 4 d 2 m 2 t MAE143b, UCSD, Summer II 2009, R.A. de Callafon Lecture 5, Page 4 Dynamic Response Application to 2nd order system The formula for the impulse response can be simplified by realiz ing that in case d = 0 (no damping) we can define an undamped natural frequency n := s k m As a result we can define a damped frequency : d := s k m 1 4 d 2 m 2 =...
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This note was uploaded on 05/27/2010 for the course MAE 143B taught by Professor Paoc.chau during the Spring '06 term at UCSD.
 Spring '06
 PaoC.Chau

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