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lec05_print - MAE143b Linear Control Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 5, Tuesday Aug. 18, 2:00-4:50pm Prof. R.A. de Callafon [email protected] 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 ) Routh-Hurwitz 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 i =1 C i s p i + n 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 coefficients C i . For i = k + 1 , . . . , n , p i are complex conjugate pole pairs p i = σ i + i with complex conjugate coefficients C i = a i + jb i . Then the impulse response of the system is given by G ( t ) = k i =1 C i e p i t + n i = k +1 2 a i e σ i t cos ω i t 2 b i e σ i t sin ω 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 } < 0 MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 5, Page 2
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Dynamic Response - Effect of pole locations For (complex conjugate pair of) pole locations p i = σ i + 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 ω i , the larger is the fre- quency of oscillation. The larger the real value σ 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 k m 1 4 d 2 m 2 · 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 k m 1 4 d 2 m 2 · e 1 2 d m t sin k m 1 4 d 2 m 2 t MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 5, Page 4
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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 ω n := k m As a result we can define a damped frequency : ω d := k m 1 4 d 2 m 2 = ω n 1 β 2 , with β := 1 2 d mk as the damping ratio . This also makes βω n = 1 2 d m and reduces the impulse response formula to y ( t ) = 1 d · e βω n sin( ω d t ) The definition of these variables also rewrites the TF model into: G ( s ) = 1 ms 2 + ds + k = 1 k · ω 2 n s
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