Mechatronic system design

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Mechatonic system design Mechatonic system design wb2414 2013/2014 Couse pat 5

Munnig Schmidt Mechatonic System Design 1 Lectue outline: What did you lean about

Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback

1 Mechatonic system design Mechatonic system design wb /2014 Couse pat 5 Motion contol Pof.i. R.H.Munnig Schmidt Mechatonic System Design 1 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 2 1

2 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 3 Motion Contol: Contol the tajectoy of a machine Position contol Velocity contol (e.g. scanning) Path planning Distubance ejection (vibations fom envionment, impefections of the guidance system, ) 4 2

3 A walk aound the contol loop Guidance signal Infomation Feedfowad Input distubance Enegy Output distubance Pocess distubance Pe filte - Feedback D/A Plant G(s) Senso distubance Output A/D senso 5 The plant is the pocess that needs to be contolled. The successive ode is fixed! 6 3

4 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 7 A walk aound the contol loop: Feedfowad contol equies no senso pedictable (if efeence is known) (faste than pue feedback) cannot intoduce instability no feeding back of senso noise plant has to be stable no compensation of model uncetainties can only compensate fo distubances that ae known (measued) 8 4

5 Feedfowad contol of a piezoelectic scanning unit Stefan Kuipe TUD/TNO 9 Model-based open-loop feedfowad contol of the piezo scanne Plant tansfe function: Gs () C C s s s s. 2 f f f f Poblem is low damped fist eigenmode 10 5

6 Model based open-loop feedfowad contol of the piezo scanne Plant tansfe function: Gs () C C s s s s. 2 f f f f Compensate dynamics by feedfowad contolle (pole zeo cancellation): s 2s C s G s C s C 2 2 f 1 1 ff () () 2 ff () f 1 Such a contolle is unfotunately not ealisable (infinite gain at infinite fequencies) 11 In pactice it can only be used to incease the damping Add a well damped ζ=1 second ode tansfe function: s 2 s s 2 s C () s C () s. ff f 1 1 f ff s 2 1 1s1 And with an additional fist ode low pass filte against noise: 2 2 s 2 f 1s1 Cff () s. 2 2 ( s1) s 21s1 Which combines into: C ( s 2 s) G s G s C s 2 2 f f 1 1 t,ff () () ff () ( s 2 f 1s1 ) ( s1) s 21s1 C ( s ) s 2 s f

7 Resulting esponse G () s G() s C () s f t,ff ff 2 2 ( s1) s 21s1 C 13 Tiangula scanning movement Without pole-zeo cancellation With pole-zeo cancellation 14 7

8 Second method: input shaping Without With 15 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 16 8

A walk aound the contol loop: Feedback contol can stabilize unstable systems can suppess distubances can handle uncetainties Input distubance Output distubance Pocess distubance Pe Filte F(s) f e _

9 A walk aound the contol loop: Feedback contol can stabilize unstable systems can suppess distubances can handle uncetainties Input distubance Output distubance Pocess distubance Pe Filte F(s) f e _ Feedback C fb (s) u v Plant G(s) x y senso equied slowe than FF (eo has to occu fist) (senso) noise is also fed back can intoduce instability (also of open loop stable systems) Senso distubance Measuement M(s) 17 Active stiffness by feedback in a CD playe, (fom Dynamics lectue) xˆ f f Hz , k f0 k 4 mf N/m 2 m 200 μm adial vibations at 25 Hz Mass lens: kg Max adial eo: 0,2 μm 18 9

) F k G G G G G p t m a p c Gives vitual sping stiffness k k G

10 Vitual stiffness Measue position Actuate with foce popotional and opposite to the deviation (feedback!) F k G G G G G p t m a p c Gives vitual sping stiffness k k G p t F 19 Sevo position contol Fo the example: m kg Gs () 2 2 ms s M() s

u Plant G(s) adding a P-gain (contolle) Feedback C fb (s) Output distubance y CD playe example: contol of lens position m = 10 g Measuement M(s) Is this system stable? e C () s C () s k 2.

11 u Plant G(s) adding a P-gain (contolle) Feedback C fb (s) Output distubance y CD playe example: contol of lens position m = 10 g Measuement M(s) Is this system stable? e C () s C () s k fb p p Open loop gain: Lp() s G() s Cfb() s M() s kp ms 2 s ms s u Plant G(s) closing the loop with P- contolle Feedback C fb (s) Output distubance y kp m 1 f Hz 2 e Measuement M(s) L () s 1 1 T ( s) 1 1 p p 2 Lp () s 1 m 2 s 2 kp 0 Unity-gain cossove fequency 22 11

We need some damping (PD-contol) Feedback C fb (s) e u

d 0 d pd p d 0 p 0 0 23 Look bette at the gaph Inceased

12 We need some damping (PD-contol) Feedback C fb (s) e u Output distubance Plant G(s) Cfb() s Cpd () s kp kds kp kds Lpd () s G() s Cpd () s 2 ms 2j 1 L () s k k s 2 k Tpd ( s) 2 2 L () s 1 ms k k s k 2j 1 2 y Measuement M(s) pd p d 0 d pd p d 0 p Look bette at the gaph Inceased unity-gain cossove fequency Reduced slope at high fequencies 24 12

13 PD-contol behaves like tansmissibility and cannot be made in eality (not pope) Reduced eduction of high fequencies by continuous D action. Infinite gain at infinite fequency! 25 The D-action must be teminated at high fequencies and k p must be educed k s C ( s) k k s k (1 s) k (1 s) k (1 ) d pd p d p p d p kp d L () s k 1 s s d p d pdt 2 0 ms 1 t 26 13

position m = 10 g 27 Closed-loop step esponse of PD-contolled mass Feedback C fb (s) e u

14 u closing the loop with tamed PDcontolle Feedback C fb (s) Output distubance e L () s k 1 s s d p d pdt 2 0 ms 1 t Plant G(s) Measuement M(s) y CD playe example: contol of lens position m = 10 g 27 Closed-loop step esponse of PD-contolled mass Feedback C fb (s) e u Output distubance Plant G(s) y CD playe example: contol of lens position m = 10 g Measuement M(s) 28 14

15 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants 29 Feedback Loop Sensitivity Functions Pocess distubance d Output distubance n Pe Filte F(s) f e Feedback u v Plant x _ C fb (s) G(s) y G GC GCF x d n 1GC 1GC 1GC G 1 GCF y d n 1GC 1GC 1GC GC C CF u d n 1GC 1GC 1GC 30 15

16 Response to efeence, Complementay sensitivity Pocess distubance d Output distubance n Pe Filte F(s) f e Feedback u v Plant x _ C fb (s) G(s) y G GC GCF x d n 1GC 1GC 1GC G 1 GCF y d n 1GC 1GC 1GC GC C CF u d n 1GC 1GC 1GC If F(s) = 1 u x x y GC d n 1 GC Complementay sensitivity function T(s) 31 Output distubance ejection, sensitivity Pocess distubance d Output distubance n Pe Filte F(s) f e Feedback u v Plant x _ C fb (s) G(s) y G GC FCG x d n 1GC 1GC 1GC G 1 FCG y d n 1GC 1GC 1GC GC C FC u d n 1GC 1GC 1GC If F(s) = 1 y n 1 1 GC Sensitivity function T(s) 32 16

17 Relation of the two sensitivity functions Pocess distubance d Output distubance n Pe Filte F(s) f e Feedback u v Plant x _ C fb (s) G(s) y Complementay sensitivity Sensitivity GsC () fb() s 1 T( s) T( s) S( s) 1 S( s) 1 G() s C () s 1 G() s C () s fb fb T gain=1 S w w 33 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 34 17

Stability and obustness When is a (closed loop) system stable? A system is asymptotically stable if and only if all poles ae in the left half of the complex plane. When is a system obust?

18 Stability and obustness When is a (closed loop) system stable? A system is asymptotically stable if and only if all poles ae in the left half of the complex plane. When is a system obust? Robustness of a feedback contolled system means that the closed loop system can accept (some) model uncetainties (paamete vaiations). Both popeties ae investigated with the Bode and Nyquist plot. 35 Loop-Shaping design fo stability and obustness Shape the fequency esponse of the openloop system such that: high gain (>>1) at low fequency low gain (<<1) at high fequency less than 180 degee phase lag in cossove egion (unity gain). Robustness: Gain magin GM Phase magin PM 36 18

19 It happens aound the unity-gain coss-ove fequency Stability: Is the unity gain cossing of the magnitude plot at a fequency with less than 180º phase lag? Robustness: Gain magin GM Phase magin PM 37 Nyquist Plot of open-loop tansfe L(s). Stability: The 1 point must be passed at the left side when inceasing the fequency. Robustness: Gain magin (GM), how much the gain can be inceased until instability Phase magin (PM), how much the phase lag can be inceased until instability The cicle that is just passed at the left gives the peak value of the esonance afte closing the loop 38 19

20 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 39 The plant, an undamped masssping system 40 20

21 Specifications of closed loop system Bandwidth of 100 Hz (unity gain coss ove fequency) So fa beyond the natual fequency on the mass line Asymptotically stable, all poles in the left half plane Sufficiently damped (is equiement!) Specification, max Q = 1.3 (+3dB) Zeo steady state eo (0 Hz) 41 P-contol Lift the esponse to unity 100 Hz => k p ~

22 D-contol fo damping educe the phase Don t foget to educe k p 43 I-contol fo infinite gain at 0 Hz Sensitivity = 1 Ss () 1 GsC ( ) ( s ) fb 44 22

23 PID-contol in math. ut ( ) ket ( ) k e( )d k p i d 0 d() et dt ki us () es () kp ks d s us () ki Cpid () s kp kds es () s u( ) ki Cpid ( ) kp j jkd e( ) t 45 The bode plot of the designed PID contolle 46 23

24 Open-loop contolle and plant 47 Nyquist plot 48 24

25 Closed loop total system 49 Inceased sensitivity fo distubances above f 0 on a linea scale. Watebed effect due to Bode sensitivity integal! Equals zeo fo system with 2 moe poles than zeos 0 ln S( ) d 50 25

26 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 51 Magnetic beaings ae often based on the vaiable eluctance actuato y,m move dx x A g F A g g Foce based on enegy balance Enegy stoed in magnetic field I n F w y,s stato F ni 2 A g 0 g

27 I 1 Balancing and lineaisation with two vaiable eluctance actuatos C 1 x F 1 F 2 I 2 g,1 dx g,2 F C 2 F F2 F ni A 2 g 0 ni A 1 g0 n Ag 0 I2 I g,2 4 g,1 4 4 g,2 g,1 In the mid-position: l l l : g,1 g,2 g nag 0 I2 I 1 F 2 4 g With: I I I and I I I 2 a 1 a na I I I I I I F F I g0 na a a g0 4 a g 4 g 2 InA a g0 2 g What is the foce to position elation at a cetain cuent level? 53 Flux in a balanced eluctance actuato depends on the cuent diection dx dx I 1 I 1 I 2 C 1 C 2 I 2 C 1 C 2 F F Shaing the flux paths by evesing the cuent in one of the coils 54 27

28 Inseting a pemanent magnet in a double eluctance actuato x dx F x dx F N N I 1 PM I 2 I 1 PM I 2 C 1 C 2 C 1 C 2 Elastic hinge Ceating the common mode flux with a pemanent magnet 55 Hybid actuato (ight position) Pemanent magnet biased eluctance actuato movement Coil 1 Coil 2 All PM flux passes though coil

29 Two ultimate positions movement movement Ratio of PM flux though the coils depends on the position of the move 57 Supeposition of the PM and cuent induced magnetic flux x Flux incease A g dx F Flux decease N I 1 g,1 g,2 n n Coil 1 Coil 2 I 2 Lage dφ/dx High foce Shot stoke 58 29

Hybid actuato in Magnetic beaings Discuss

30 Hybid actuato in Magnetic beaings Discuss the 4 coodinate diection beaing contol 59 Contol of a magnetic beaing Bodeplot of negative stiffness x 1 1 Ct () s kn N/m m 0.1 kg x 1 Ct ( ) 2 4 F Poles: 2 ms kn 0 kn s 316 m 2 2 F ms k ms kn 60 30

31 Shifting the pole to the left (oot-locus) Afte closing the loop! Fo obustness k p > 2 k n 61 Bode plot of P(I)D-contolled magnetic beaing C plant+p n ( s) k 1 ms k p 2 1 kp k n 2 m 2 m 2 s 1 s 1 k k n n 62 31

32 Nyquist explains what happens. 63 Lectue outline: What did you lean about PID motion contol so fa? Intoduction to motion contol Feedfowad contol of piezoelectic scanne PD feedback contol of CD playe Sensitivity Stability and Robustness PID feedback contol of mass sping system PID feedback contol of magnetic beaing How to deal with difficult plants having multiple eigenmodes 64 32

33 4 types of esponses 65 Type I: -2 slope, zeo pai, pole pai, -2 slope 66 33

34 Type II: -2 slope, pole pai, zeo pai, -2 slope 67 Type III: -2 slope, pole pai, -4 slope 68 34

35 Type IV: -2 slope, pole pai, -2 slope 69 non-collocated measuing at nonigid body 70 35

36 Bode and Nyquist with PID contol open loop of sheet Pole-zeo cancellation (notching) But don t ovedo it! Risk of fequency dift! 72 36

37 Adding a low-pass filte (a pole) 73 Only Nyquist gives the eal answe on stability 74 37

38 Optimisation with LPF cone fequency 75 38

ME 3600 Control Systems Frequency Domain Analysis

ME 3600 Control Systems Frequency Domain Analysis ME 3600 Contol Systems Fequency Domain Analysis The fequency esponse of a system is defined as the steady-state esponse of the system to a sinusoidal (hamonic) input. Fo linea systems, the esulting steady-state