Chapter 8 Designing Feedback Controllers for Motor Drives

© 2000 http://www.ece.umn.edu/groups/electricdrives

8-1

Feedback Control Objectives desired (reference) + Σ signal -

error

error amplifier

P P U 3 1424

Electric Machine

Mech Load

output

1 424 3

Electrical System

Mechanical System

measured output signal

o Feedback control u makes system insensitive to disturbances and parameter variation o Control Objectives u Zero steady-state error u Good dynamic response - fast - small overshoot © 2000 http://www.ece.umn.edu/groups/electricdrives

*

X ( s) + ∑ −

E ( s)

Controller

Plant

Gc (s )

G p ( s)

X (s )

8-2

Definitions 100

o Open loop GOL ( s ) = Gc (s )G p ( s )

50

GOL 0

o Closed loop

-50

GCL (s ) = GOL ( s ) /(1 + GOL ( s ))

o Crossover frequency

-100

∠GOL

fc ,ω c

phase margin

-150

−180 o -200

o Gain Margin o Phase Margin > 45o for no oscillations 60o preferable o Closed loop bandwidth ; fc desired high for fast response © 2000 http://www.ece.umn.edu/groups/electricdrives

f c , ωc

-50

-2

10

-1

10

0

10

1

2

10

10

frequency

GCL ( jω )

0 dB

−3dB BW

ω

8-3

Example GOL ( s ) = KOL / s ; KOL = 2 × 10 3

GOL ( s )

(dB)

x* (t ) ω c = 2 × 103

0

logω 0.632

τ

x (t )

GCL ( s )

(dB) 0

−20 dB / decade

logω

closed loop step response

© 2000 http://www.ece.umn.edu/groups/electricdrives

8-4

Cascaded Control speed* position* +

−

Position controller +

−

torque* Speed controller +

torque Torque controller

− torque(current)

speed position

o Torque loop : fastest o Speed loop : slower o Position loop : slowest

© 2000 http://www.ece.umn.edu/groups/electricdrives

Electrical System

Mech System

speed 1 s

position

8-5

Steps in Designing the Controller o Assume system is linear about the steady state operating point Ý design controller using Linear Control Theory o Simulate design under large signal conditions and “tweak” controller as necessary

System representation for small signal analysis o Assume u Steady state system operating point = 0 u Highest bandwidth at least an order of magnitude lower than switching frequency Ý neglect switching frequency components

© 2000 http://www.ece.umn.edu/groups/electricdrives

8-6

Averaged Representation of the PPU id (t )

id idA + Vd

−

idB iA

ia

+ B A va − iB i A = ia iB = − ia

N vcontrol ( t )

+ −

qA (t )

ia (t )

+ ea

+

Vd 1

−

+

d (t ) va ( t )

ea −

− vcontrol ( t )

1 Vˆtri

d (t ) = d A (t ) − d B ( t )

qB ( t )

va (t ) = k PWM vc (t ) Va ( s ) = k PWM Vc ( s )

© 2000 http://www.ece.umn.edu/groups/electricdrives

Va ( s ) Vc (s ) k PWM

8-7

Modeling of DC Machines and Mechanical Load Combinations dia (t ) dt ea (t ) = k E ω m (t )

va (t ) = ea (t ) + Ra ia (t ) + La

T ia = em kT

+

Ra

va

Va ( s ) = Ea ( s ) + ( Ra + s La ) I a ( s )

−

La

+

Tem

ea = k E ω m _

ωm TwL

JM

⇒ I a ( s) =

Va ( s ) − Ea ( s ) ( Ra + s La )

JL

; Ea ( s ) = k E ω m ( s ) TWL (s )

Tem ( s) = kT I a ( s) T (s) ω m ( s ) = em sJ eq © 2000 http://www.ece.umn.edu/groups/electricdrives

Va ( s ) + −

I a (s ) 1 Ra + sLa

−

+

kT

Tem (s )

kE

1 sJ eq

ωm ( s )

8-8

PI Controller kp

vc, p ( s ) G ( s)

p 6447448

X * (s ) +

E (s ) −

ki s

vc, i ( s ) +

+

X (s ) vcontrol ( s )

144424443 Gc ( s )

vc ( s ) ki ki  s  =kp + = 1+  E ( s) s s  ki / k p 

o Proportional-Integral (PI) Controller u In the torque and speed loops, proportional control without integral control input leads to steady-state error

© 2000 http://www.ece.umn.edu/groups/electricdrives

8-9

Controller Design o Procedure u Design torque loop (fastest) first u Design speed loop assuming torque loop to be ideal u Design position loop (slowest) assuming speed loop to be ideal

© 2000 http://www.ece.umn.edu/groups/electricdrives

8-10

Design of the Torque (Current) Loop Simplifing assumptions I a* (s ) + Σ −

PI

1 / Ra 1 + sτ e

V ( s) k PWM a + Σ −

I a (s )

kT

TWL

Tem − Σ +

ωm

1 sJ kE

I a (s ) I a* (s )

o Interleaved Σ + loops redrawn − as nested loops Ia (s) o Assuming J high enough, inner loop can be ignored

⇓ PI

k PWM

Va ( s ) + Σ −

1 / Ra 1 + sτ e

kT

Tem

1 sJ

k E kT sJ

⇓

kiI  s 1+ s  kiI / k pI

I a* ( s )

+ Σ −

   

k PWM

Va ( s )

1 / Ra 1 + sτ e

I a (s )

kiI  s  1 / Ra GI ,OL ( s ) =  1 + k PWM  s  kiI / k p  123 1 + sτ e 1442443 PPU 123 PI controller

© 2000 http://www.ece.umn.edu/groups/electricdrives

I a (s )

motor

I a (s )

ωm

8-11

Design of the Torque (Current) Loop Selecting Parameters I a* ( s ) + Σ −

kiI s

 s 1+  kiI / k pI 

   

k PWM

Va ( s )

I a (s )

1 / Ra 1 + sτ e

I a (s )

k pI

o Select zero of PI to cancel motor pole ; ⇒ GI ,OL =

k I ,OL s

kiI

k k ; ki,OL = iI PWM Ra

=τ e

o Choose kiI to achieve desired cross-over frequency 0

40

Magnitude (dB)

Magnitude (dB)

60

20 0 -20 -40 0 10

10

1

10

2

10

3

10

4

10

-10

-20

-30 0 10

5

10

1

Frequency (Hz) -89

2

10

3

10

4

10

5

0

Phase (deg)

Phase (deg)

10

Frequency (Hz)

-89.5 -90 -90.5 -91 0 10

k I ,OL = ωCI

10

1

10

2

10

3

10

Frequency (Hz)

open loop © 2000 http://www.ece.umn.edu/groups/electricdrives

4

10

5

-50

-100 0 10

10

1

10

2

10

3

10

4

Frequency (Hz)

closed loop

10

5

8-12

Design of the Speed Loop * ωm (s ) +

I a* (s )

PI

1

I a (s )

Tem (s )

kT

−

ωm (s )

1 Js

ωm (s )

o Assume current loop to be ideal Ý represent by unity o Choose crossover frequency ωCω an order of magnitude lower than ωCI o Choose a reasonable phase margin φPM ,ω open loop

closed loop 20 Magnitude (dB)

Magnitude (dB)

150 100 50 0 -50 -1 10

0

10

1

10

2

3

10 10 Frequency (Hz)

4

10

-40 -60 -1 10

5

10

0

10

10

1

2

3

2

3

10 10 Frequency (Hz)

4

10

5

10

0 Phase (deg)

-50 Phase (deg)

0 -20

-100

-150

-50

-100

-200 -1

10

0

10

1

10

2

3

10 10 Frequency (Hz)

© 2000 http://www.ece.umn.edu/groups/electricdrives

4

10

5

10

10

-1

0

10

10

1

10 10 Frequency (Hz)

4

10

5

10

8-13

Design of the Position Loop * θm ( s) +

* ωm (s )

k pθ

1

ωm (s )

θ m ( s)

1 s

−

o Assume speed loop to be ideal o Proportional gain ( k Pθ ) alone is adequate due to presence of k Pθ pure integrator G = ⇒ k =ω θ ,OL

Pθ

s

0 Magnitude (dB)

Magnitude (dB)

50

0

-50 -1 10

10

0

1

2

10 10 Frequency (Hz)

10

3

10

Phase (deg)

Phase (deg)

-90 -90.5 -91 10

0

1

2

10 10 Frequency (Hz)

open loop © 2000 http://www.ece.umn.edu/groups/electricdrives

-40

10

0

10

3

10

4

10

1

10

2

10

3

10

4

Frequency (Hz)

0

-89

-1

-20

-60 -1 10

4

-89.5

10

CP

-50

-100 -1 10

10

0

10

1

10

2

10

3

Frequency (Hz)

closed loop

10

4

8-14

Further Issues o Feed-forward: To improve dynamic response Process computer position* +

Σ

−

torque*ff

speed*ff

+ Position controller + Σ −

Torque controller

torque*

torque(current)

*

speed

+

Speed controller

Electrical System

Mech System

+Vˆtri

+Vd

1 s

position

speed

position

o Effect of limits - nonlinearity

I a* (s ) +

kiI s

−

  s 1+    / k k iI pI  

cp

kp

max

input

co

+ co ' ki

0

© 2000 http://www.ece.umn.edu/groups/electricdrives

I a (s )

−Vd

−Vˆtri

o Antiwindup integration - suspend integration when output saturates

1 / Ra 1 + sτ e

k PWM

cl

max − | co ' |

min

8-15

Summary o What are the various blocks of a motor drive? o What is a cascaded control and what are its advantages? o Draw the average models of a PWM controller and a dc-dc converter. o Draw the dc-motor equivalent circuit and its representation in Laplace domain. Is this representation linear? o What is the transfer function of a proportional-integral (PI) controller? o Draw the block diagram of the torque loop. o What is the rationale for neglecting the feedback from speed in the torque loop? o Draw the simplified block diagram of the torque loop. o Describe the procedure for designing the PI controller in the torque loop. © 2000 http://www.ece.umn.edu/groups/electricdrives

8-16

Summary o How would we have designed the PI controller of the torque loop if the effect of the speed were not ignored? o What allows us to approximate the closed torque loop by unity in the speed loop? o What is the procedure for designing the PI controller in the speed loop? o How would we have designed the PI controller in the speed loop if the closed torque-loop were not approximated by unity? o Draw the position-loop block diagram. o Why do we only need a P controller in the position loop? o What allows us to approximate the closed speed loop by unity in the position loop? o Describe the design procedure for determining the controller in the position loop © 2000 http://www.ece.umn.edu/groups/electricdrives

8-17

Summary o How would we have designed the position controller if the closed speed loop were not approximated by unity? o Draw the block diagram with feed-forward. What are its advantages? o Why are limiters used and what are their effects? o What is the integrator windup and how can it be avoided?

© 2000 http://www.ece.umn.edu/groups/electricdrives