18 - PoissonComplementarySensitivity18 filled

18 - PoissonComplementarySensitivity18 filled - ME 575...

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Unformatted text preview: ME 575 Handouts Poisson Integral on Complementary Sensitivity • Lemma 9.6 (Unstable Open-Loop Poles and Zeros) (Unstable OpenConsider the stable CL system with one-DOF controller configuration and open loop TF given by L(s) = G(s)C(s) = e − sτL(s), τ≥0 where L (s) is a rational TF. Assume that L(s) has open-loop unstable poles at p1,… p N , where pi = αi + jβi . Then, when L(s) has no unstable open-loop zeros , ∞ ∫ −∞ ln T ( jω) αi α + ( ω − βi ) 2 i 2 dω = πταi , ∀i = 1 …,N , When L(s) has unstable open-loop zeros at ∫ ∞ −∞ where ln T ( jω) Bz (s) αi z1,…,zM , dω = −π ln Bz ( pi ) + πταi 2 αi2 + ( ω − βi ) M s−z ∏ s + z k* , Blaschke product k =1 k ME575 Session 18 – Poisson Complementary Sensitivity Constraint School of Mechanical Engineering Purdue University Slide 1 Example of Design Implications • Poisson Complementary Integral for Unstable Poles ∫ ∞ −∞ ln T ( jω) W ( pi , ω) dω = −π ln Bz ( αi ) + πταi , pi = αi + jβi Lower bound for complementary sensitivity peak Tmax ≥ T( jω) : ln Tmax ≥ 1 ⎡ π ln B z ( α i ) + πτα i Ω ( pi , ωh ) − Ω ( pi , ωl ) ⎣ − ln (1 + ε S ) Ω ( pi , ωl ) + ( π − Ω ( pi , ωh ) ) ln ( ε T ) ⎤ ⎦ ME575 Session 18 – Poisson Complementary Sensitivity Constraint School of Mechanical Engineering Purdue University Slide 2 1 ME 575 Handouts Example of Design Implications Example • Observations – The lower bound on complementary sensitivity peak is larger for systems with pure delays and the influence of a delay increases for faster unstable poles (i.e., large αi ) – The lower bound grows unbounded when a NMP zero approaches an unstable pole, because then ln B z ( p i ) grows unbounded unbounded – When CL bandwidth is much smaller compared to unstable poles (i.e., ωh αi ), Ω pi , ωh will be very small, leading to very large complementary sensitivity peak. Therefore, to avoid large transient transient response, CL bandwidth should be chosen larger than unstable poles – Large sensitivity peaks leads to large deviations in transient response in time domain and small stability margins in frequency domain. Thus the design problem becomes more difficult when the system has fast unstable open-loop poles and slow NMP zeros. The fast openslow notions of fast and slow are relative to CL bandwidth fast slow ( ME575 Session 18 – Poisson Complementary Sensitivity Constraint ) School of Mechanical Engineering Purdue University Slide 3 Control of Inverted Pendulum • System Dynamics with Cart Position Feedback only Y(s) 2(s 2 − 10) G(s) = =22 F(s) s (s − 20) • Estimated Sensitivity Peaks Case 1: ωl = 10, ωh = 100, ε S = ε T = 0.1 ⇒ S max ≥ 432 ⇒ S max ≥ 16.7 ⇒ Tmax ≥ 3171 Case 2: ωl = 1, ωh = 100, ε S = ε T = 0.1 Case 3: ωl = 1, ωh = 20, ε S = ε T = 0.1 Case 4: ωl = 1, ωh = 3, ε S = ε T = 0.1 ME575 Session 18 – Poisson Complementary Sensitivity Constraint School of Mechanical Engineering Purdue University ⇒ Tmax ≥ 7.2 × 10 5 Slide 4 2 ME 575 Handouts Control of Inverted Pendulum • Desired CL Poles c c Case 1: p1d = p c d = p 3 d = − 5, p c 2 4,5,6,7 d = − 10 ± 10 j c c Case 2: p1d = p c d = p 3 d = − 0.5, p c 2 4,5,6,7 d = − 10 ± 3 j • Pole Placement Design MATLAB Command: numP=2*[1,0,-10]; denP=[1,0,-20,0,0]; denCLd=poly([-10+3j -10-3j -10+3j -10-3j -0.5 -0.5 0.5]); [denC,numC]=paq(denP,numP,denCLd) ⇒ C 2 (s) = C1(s) = 1691s 3 + 7213s 2 − 473s − 74 s 3 + 41.5s 2 − 2682s − 8279 ( 10 5 0.327s 3 + 1.14s 2 − 2s − 2.5 ) s + 55s − 64000s − 204200 3 ME575 Session 18 – Poisson Complementary Sensitivity Constraint 2 School of Mechanical Engineering Purdue University Slide 5 Control of Inverted Pendulum Note Case 1 has a much larger transient response as predicted ME575 Session 18 – Poisson Complementary Sensitivity Constraint School of Mechanical Engineering Purdue University Slide 6 3 ME 575 Handouts Control of Inverted Pendulum Note Case 1 has much larger sensitivity peaks as predicted ME575 Session 18 – Poisson Complementary Sensitivity Constraint School of Mechanical Engineering Purdue University Slide 7 4 ...
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