Lecture 5 motion control for handouts - TU Delft OCW · 2016. 2. 23. · 5 Delft WB2414-Mechatronic System Design 2013-2014 9 University of Technology Feedforward control of a piezoelectric

1

WB2414-Mechatronic System Design 2013-2014 1DelftUniversity ofTechnology

Mechatronic system design

Mechatronic system design wb2414‐20132014

Course part 5

Profir RHMunnig SchmidtMechatronic System Design

Motion control


Lecture outline

bull What did you learn about PID ‐motion control so far

bull Introduction to motion control

bull Feedforward control of piezoelectric scanner

bull PD ndash feedback control of CD‐player

bull Sensitivity

bull Stability and Robustness

bull PID ‐ feedback control of mass‐spring system

bull PID ‐ feedback control of magnetic bearing

bull How to deal with difficult plants having multiple eigenmodes

2


Lecture outline





bull Sensitivity






Motion Control

bull Control the trajectory of a machine

bull Position control

bull Velocity control (eg scanning)

bull Path planning

bull Disturbance rejection (vibrations from environment imperfections of the guidance system hellip)

3


A walk around the control loop

Plant G(s)

sensorAD

DAFeedbackPre‐filter

Feedforward

Guidancesignal r

-

Input disturbance

Output disturbance

Processdisturbance

Sensor disturbance

Output

EnergyInformation


The plant is the process that needs to be controlled

The successive order is fixed

4


Lecture outline





bull Sensitivity






A walk around the control loop Feedforward control

bull requires no sensorbull predictable (if reference is known)

(faster than pure feedback)bull cannot introduce instabilitybull no feeding back of sensor noise

bull plant has to be stablebull no compensation of model uncertaintiesbull can only compensate for distur‐bances that are known (measured)

5


Feedforward control of a piezoelectric scanning unit

Stefan Kuiper TUDTNO


Model-based open-loop feedforward control of the piezo scanner

Plant transfer function

2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

Problem is low damped first eigenmode

6


Model based open-loop feedforward control of the piezo scanner

2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C


Compensate dynamics by feedforward controller (pole‐zero cancellation)

Such a controller is unfortunately not realisable (infinite gain at infinite frequencies)


In practice it can only be used to increase the damping

2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s

Add a well damped ζ=1 second order transfer function

And with an additional first order low pass filter against noise

Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s


Triangular scanning movement

Withoutpole-zerocancellation

Withpole-zerocancellation

8


Second method input shapingWithout

With


Lecture outline





bull Sensitivity





9


A walk around the control loop Feedback control

bull sensor requiredbull slower than FF (error has to occur first) bull (sensor) noise is also fed backbull can introduce instability

(also of open‐loop stable systems)

bull can stabilize unstable systemsbull can suppress disturbancesbull can handle uncertainties

Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v


Active stiffness by feedback in a CD player (from Dynamics lecture)

bull200 μm radial vibrations at 25 Hz

bullMass lens 10 10‐3 kg

bullMax radial error 02 μm

6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness

bull Measure position

bull Actuate with force proportional and opposite to the deviation (feedback)

bull Gives virtual spring stiffness

p r m a p ct r r F k G G G G G

r p tr

F

k k G


Servo position control

2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11


adding a P-gain (controller) hellip

CD‐player example control of lens positionm = 10 g

Is this system stable

5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k

L s G s C s M s kms s

Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


hellip closing the loop with P-controller

pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

Unity-gaincrossoverfrequency

12


We need some damping (PD-control)

fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


Look better at the graph

Increased unity-gain crossover frequency Reduced

slope at high frequencies

13


PD-control behaves like ldquotransmissibilityrdquo andcannot be made in reality (not proper)

Reduced reduction of high frequencies by continuous D‐actionInfinite gain at infinite frequency


The D-action must be terminated at high frequencies and kp must be reduced

dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s

C s k k s k s k s kk

pd dpdt 2

0 t

1( )

1

k sL s

ms s

14



hellip closing the loop with tamed PD-controller

pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



Closed-loop step response of PD-controlled mass

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity




bull How to deal with difficult plants


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF

y d n rGC GC GCGC C CF

u d n rGC GC GC

Feedback Loop Sensitivity Functions

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _

Process disturbance d

Output disturbancen

e u v xr yrf

16


Response to reference Complementary sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC

Complementary sensitivity function T(s)

1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG

y d n rGC GC GCGC C FC

u d n rGC GC GC

Output disturbance rejection sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC

Sensitivity function T(s)

17


Relation of the two sensitivity functions

fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )

G s C sT s T s S s S s

G s C s G s C s

|T|

w

Complementary sensitivity

gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18


Stability and robustness

bull When is a (closed‐loop) system stable

bull A system is asymptotically stable if and only if all poles are in the left half of

the complex plane

bull When is a system robustbull Robustness of a feedback controlled system means that the closed‐loop

system can accept (some) model uncertainties (parameter variations)

bull Both properties are investigated with the Bode and Nyquist plot


Loop-Shaping design for stability and robustness

Shape the frequency response of the open‐loop system such thatbull high gain (gtgt1) at low frequencybull low gain (ltlt1) at high frequencybull less than 180 degree phase lag in crossover region (unity‐gain)

Robustnessbull Gain margin GMbull Phase margin PM

19


It happens around the unity-gain cross-over frequency

Stability Is the unity‐gain crossing of the magnitude plot at a frequency with less than 180ordm phase lag



Nyquist Plot of open-loop transfer L(s)

bull Stability The ‐1 point must be

passed at the left side when increasing the frequency

bull Robustnessbull Gain margin (GM) how much the gain can be increased until instabilitybull Phase margin (PM) how much the phase lag can be increased until instability

The circle that is just passed at the left gives the peak value of the resonance after closing the loop

20


Lecture outline





bull Sensitivity






The plant an undamped mass-spring system

21


Specifications of closed loop system

bull Bandwidth of 100 Hz (unity‐gain cross‐over frequency)

bull So far beyond the natural frequency on the mass line

bull Asymptotically stable all poles in the left half plane

bull Sufficiently damped (is requirement)

bull Specification max Q = 13 (+3dB)

bull Zero steady‐state error (0 Hz)


P-control

bull Lift the response to unity gain 100 Hz =gt kp~105

22


D-control for damping

bull reduce the phase lag 100Hz

bull Donrsquot forget to reduce kp


I-control for infinite gain at 0 Hz

bull Sensitivity =fb

1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )

t e tu t k e t k e k

t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e


The bode plot of the designed PID controller

24


Open-loop controller and plant


Nyquist plot

25


Closed loop total system


Increased sensitivity for disturbances above f0 on a linear scale

Waterbed effect due to Bode sensitivity integralEquals zero for system with 2 more poles than zeros 0

ln ( ) dS

26


Lecture outline





bull Sensitivity






Magnetic bearings are often based on the variable reluctance actuator

I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator

Force based on energy balanceEnergy stored in magnetic field

2

g 0

g 4

AnIF

27


Balancing and linearisation with two variable reluctance actuators

2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF

g1 g2 gIn the mid-position l l l

2 a 1 aWith and I I I I I I

2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1

What is the force toposition relation at a certain current level


Flux in a balanced reluctance actuator depends on the current direction

Sharing the flux paths by reversing the current in one of the coils

C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28


Inserting a permanent magnet in a double reluctance actuator

Creating the common mode flux with a permanent magnet

C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx


Hybrid actuator (right position) Permanent magnet biased reluctance actuator

movement

All PM flux passes through coil 1

Coil 1 Coil 2

29


Two ultimate positions

movement movement

Ratio of PM flux through the coils depends on the position of the mover


Superposition of the PM and current induced magnetic flux

Large dΦdxHigh forceShort stroke

Coil 1 Coil 2

F

Flux increase Flux decrease

I2

nn

I1

xAg

N

dx

g2g1

30


Hybrid actuator in Magnetic bearings

Discuss the 4 coordinatedirection bearing control


Control of a magnetic bearing

bull Bodeplot of negative stiffness

2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31


Shifting the pole to the left (root-locus)

For robustnesskp gt 2 kn

After closing the loop


Bode plot of P(I)D-controlledmagnetic bearing

p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32


Nyquist explains what happens


Lecture outline





bull Sensitivity





33


4 types of responses


Type I -2 slope zero pair pole pair -2 slope

34


Type II -2 slope pole pair zero pair -2 slope


Type III -2 slope pole pair -4 slope

35


Type IV -2 slope pole pair -2 slope


non-collocated measuring at non-rigid body

36


Bode and Nyquist with PID control open loop of sheet 48


Pole-zero cancellation (notching) But donrsquot overdo it

Risk of frequency drift

37


Adding a low-pass filter (a pole)


Only Nyquist gives the real answer on stability

38


Optimisation with LPF corner frequency

2


Lecture outline





bull Sensitivity






Motion Control

bull Control the trajectory of a machine

bull Position control

bull Velocity control (eg scanning)

bull Path planning

bull Disturbance rejection (vibrations from environment imperfections of the guidance system hellip)

3



Plant G(s)

sensorAD


Feedforward

Guidancesignal r

-

Input disturbance

Output disturbance

Processdisturbance

Sensor disturbance

Output

EnergyInformation




4


Lecture outline





bull Sensitivity










5







2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s


6



2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C






2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s



Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



3



Plant G(s)

sensorAD


Feedforward

Guidancesignal r

-

Input disturbance

Output disturbance

Processdisturbance

Sensor disturbance

Output

EnergyInformation




4


Lecture outline





bull Sensitivity










5







2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s


6



2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C






2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s



Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



4


Lecture outline





bull Sensitivity










5







2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s


6



2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C






2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s



Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



5







2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s


6



2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C






2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s



Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



6



2f f 1

2 2 2f 1 1

f21 1

( ) 2

2 1

C CG s

s s s s

2 2f 1 1

ff ff f21

2( ) ( ) ( )

s sC s G s C s C






2 2 2 2f 1 1 f 1 1

ff ff2 2 21 1 1

2 2( ) ( )

2middot1middot

s s s sC s C s

s s



Which combines into

2 2

f 1 1ff 2 2

1 1 1

2( )

( ) 2

s sC s

s s s

2 2f f 1 1

tff ff 2 2 2 2f 1 1 1 1 1

f2 2

1 1 1

( 2 )( ) ( ) ( )

( 2 ) ( ) 2

( ) 2

C s sG s G s C s

s s s s s

C

s s s

7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



7


Resulting response

f

t ff ff 2 21 1 1

( ) ( ) ( )( ) 2

CG s G s C s

s s s





8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



8



With


Lecture outline





bull Sensitivity





9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



9






Plant

G(s)Feedback

Cfb (s)Pre‐Filter

F(s) _

e ur y

Measurement

M(s)

rf

Input disturbance

Output disturbance

Process disturbance

x

Sensor disturbance

v






6r

0 6r

r0

2 2 5r 0

1middot10

ˆ 200 10 25 800

0

2

2 10

4 25 Nm

xf f Hz

kf k mf

m

10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



10


Virtual stiffness





r p tr

F

k k G



2

2 2

1 10 kg

1 100( )

( ) 1

m

G sms s

M s

For the example

11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



11





5fb p

7

p fb p 2 2

( ) ( ) 25 10

1 25 10( ) ( ) ( ) ( )

pC s C s k


Open loop gain

2 2

1 100

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y



pp 2

2p2

p 0

( ) 1 1( )

( ) 1 1 1

L sT s

mL s sk

p 30

0 0

5 10

1800 Hz

2

k

m

f

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y


12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



12



fb pd p d

p dpd pd 2

( ) ( )

( ) ( ) ( )

C s C s k k s

k k sL s G s C s

ms

pd p d 0 dpd 22

pd p d 0 p20 0

2 1( ) 2

( ) ( ) 1

2 1

jL s k k s k

T sL s ms k k s k

j

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y





13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



13






dpd p d p p d p

p d

( ) (1 ) (1 ) (1 )k s


pd dpdt 2

0 t

1( )

1

k sL s

ms s

14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



14




pd dpdt 2

0 t

1( )

1

k sL s

ms s

Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y




Plant

G(s)

Feedback

Cfb (s)

Measurement

M(s)

e

u Output

disturbance

r y

15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



15


Lecture outline





bull Sensitivity






1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



16



Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

u x x y GC

d n r r GC


1 1 11

1 1 1

1 1 1

G GC GCFx d n r

GC GC GCG GCF


u d n rGC GC GC


1 1 11

1 1 1

1 1 1

G GC FCGx d n r

GC GC GCG FCG


u d n rGC GC GC


Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf

If F(s) = 1

1

1

y

n GC


17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



17



fb

fb fb

( ) ( ) 1( ) ( ) ( ) 1 ( )

1 ( ) ( ) 1 ( ) ( )


G s C s G s C s

|T|

w


gain=1|S|

w

Sensitivity

Plant G(s)

Feedback Cfb (s)

Pre‐FilterF(s) _


Output disturbancen

e u v xr yrf


Lecture outline





bull Sensitivity





18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



18





the complex plane








19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



19











20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



20


Lecture outline





bull Sensitivity







21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



21










P-control


22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



22








1( )

1 ( ) ( )S s

G s C s

23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



23


PID-control in math

p i d

ip d

ipid p d

ipid p d

0

d ( ) ( ) ( ) ( )d

d

( ) ( )

( )( )

( )

( )( )

( )


t

ku s e s k k s

s

ku sC s k k s

e s s

kuC k j jk

e



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



24




Nyquist plot

25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



25






ln ( ) dS

26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



26


Lecture outline





bull Sensitivity







I

F

n

xdx

g

ys

wF

gA gA

ymmover

stator


2

g 0

g 4

AnIF

27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



27



2 1

2 2 2 2 2g 0 g 0 g 02 1 2 1

2 2g2 g1 g2 g14 4 4

F F F

A A n AnI nI I I

2 2 2g 0 2 1

2g4

n A I IF



2 22 2g 0 g 0a a a

2 2g g

2a g 0

2g

4

4 4

n A n AI I I I I IF

I n AF

I

C2C1

I1

x

F

F1 F2I2

dx

g2g1





C2C1

I1

C2C1

I1

I2

I2

F F

dx dx

28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



28




C2C1

I1PM

I2

C2C1

F

PMI2

Elastic hinge

I1

x xF

N N

dx dx



movement


Coil 1 Coil 2

29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



29



movement movement





Coil 1 Coil 2

F


I2

nn

I1

xAg

N

dx

g2g1

30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



30







2 2n

n

2 4

1 1

10000 Nm

01

( )

( )1

k

middot10

g

1

01

t

t

Cms ms

k

m

C

xs

F k k

x

F

2n

n 316

0 ms

ks

m

k

Poles

31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



31







p 2n

pn

2 2

n n

plant+P

1

12

1

)

1

(

Cms

s kk

kk

k km m

s s

32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



32




Lecture outline





bull Sensitivity





33





34





35





36






37





38



33





34





35





36






37





38



34





35





36






37





38



35





36






37





38



36






37





38



37





38



38



Documents

Lecture 5 motion control for handouts - TU Delft OCW · 2016. 2. 23. · 5 Delft WB2414-Mechatronic System Design 2013-2014 9 University of Technology Feedforward control of a piezoelectric