{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec05_print

lec05_print - MAE143b Linear Control Theory and...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}