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Unformatted text preview: 81 Chapter 8
Designing Feedback
Controllers for Motor Drives Exit 2001 by N. Mohan Print TOC " ! 82 Feedback Control Objectives
desired
(reference) + Σ
signal
 error error
amplifier P
P
! #"
"U $ Electric
Machine Mech
Load output ! #"
" $ Electrical
System Mechanical
System measured output signal ❏ Feedback control
x makes system insensitive to disturbances and
parameter variation
❏ Control Objectives
x Zero steadystate error
x Good dynamic response
 fast
 small overshoot
Exit 2001 by N. Mohan Controller
* X (s)
+ ∑
− E (s) Plant Gc ( s ) G p ( s) TOC X ( s) " ! 83 Definitions
100 ❏ Open loop
GOL ( s ) = Gc ( s )G p ( s ) 50 GOL
0 ❏ Closed loop 50 GCL ( s ) = GOL ( s ) /(1 + GOL ( s )) ❏ Crossover frequency fc , ωc 50 100 ∠GOL f c ,ω c phase
margin 150
−180 o 200 ❏ Gain Margin
❏ Phase Margin
> 45o for no oscillations
60o preferable
❏ Closed loop bandwidth ! fc 10 2 10 1 10 0 10 1 10 2 frequency GCL ( jω )
0 dB −3dB ω BW desired high for fast response
Exit 2001 by N. Mohan TOC " ! 84 Example
GOL ( s ) = K OL / s ; KOL = 2 × 10 3 GOL ( s )
x* (t ) ( dB ) ω c = 2 × 103 0 log ω 0.632 τ x(t ) GCL ( s ) ( dB )
0 −20 dB / decade log ω closed loop step response Exit 2001 by N. Mohan TOC " ! 85 Cascaded Control
speed*
position*
+
− Position
controller + − torque* Speed
controller + torque
Torque
controller − Electrical
System Mech
System speed
1
s position TOC " torque(current)
speed
position ❏ Torque loop : fastest
❏ Speed loop : slower
❏ Position loop : slowest Exit 2001 by N. Mohan ! 86 Steps in Designing the Controller
❏ Assume system is linear about the steady state
operating point ➝ design controller using Linear Control
Theory
❏ Simulate design under large signal conditions
and “tweak” controller as necessary System representation for small signal analysis
❏ Assume
x Steady state system operating point = 0
x Highest bandwidth at least an order of magnitude lower
than switching frequency ➝ neglect switching
frequency components
Exit 2001 by N. Mohan TOC " ! 87 Averaged Representation of the PPU
id (t ) id
idA idB
iA +
Vd − A +
B v
a
− iB − q A (t ) ia (t ) + + i A = ia
iB = − ia N
vcontrol (t ) ia
ea + Vd 1
− + d (t )
va (t ) ea
− − vcontrol (t ) 1
ˆ
Vtri d (t ) = d A( t) − d B )t
( qB ( t ) va (t ) = k PWM vc (t ) Va ( s )
Vc ( s ) k
PWM Va ( s ) = k PWM Vc ( s ) Exit 2001 by N. Mohan TOC " ! 88 Modeling of DC Machines and
Mechanical Load Combinations
d ia (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
TL JM V ( s ) − Ea ( s )
⇒ I a ( s) = a
( Ra + s La ) JL ; Ea ( s ) = k E ω m ( s )
TL ( s ) Tem ( s ) = kT I a ( s ) ωm (s) =
Exit 2001 by N. Mohan Tem ( s )
sJ eq Va ( s )
+
− I a (s)
1
Ra + sLa − + kT Tem ( s ) ωm (s ) 1
sJ eq kE TOC " ! 89 PI Controller
kp vc, p ( s )
G ( s) p
%""&""' X * (s)
+ E (s)
− ki
s vc,i ( s )
+ + X ( s) vcontrol ( s ) !"""
#"""
$
Gc ( s ) vc ( s )
ki ki s = k p + = 1+ E (s)
s s
ki / k p ❏ ProportionalIntegral (PI) Controller
x In the torque and speed loops, proportional control
without integral control input leads to steadystate error Exit 2001 by N. Mohan TOC " ! 810 Controller Design
❏ Procedure
x Design torque loop (fastest) first
x Design speed loop assuming torque loop to be ideal
x Design position loop (slowest) assuming speed loop to
be ideal Exit 2001 by N. Mohan TOC " ! 811 Design of the Torque (Current) Loop
Simplifying assumptions
*
I a (s)
+ Σ
− PI 1 / Ra
1 + sτ e V ( s)
k PWM a + Σ
− I a ( s) kT Tem −
Σ
+ *
I a ( s) ❏ Interleaved
Σ
+
loops redrawn −
as nested loops Ia (s)
❏ Assuming J
high enough,
inner loop can
be ignored ⇓
PI k PWM Va ( s )
+ Σ
− 1 / Ra
1 + sτ e I a ( s) kT Tem 1
sJ ωm 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) I a ( s) kiI s 1 / Ra
GI ,OL ( s ) = 1 + k PWM
#
s kiI / k p ! $ #"" PPU 1 + sτ e
!#$
!""
$
PI controller 2001 by N. Mohan ωm 1
sJ kE I a (s) Exit TL motor TOC " ! 812 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 ❏ Select zero of PI to cancel motor pole ;
⇒ GI ,OL = k I ,OL
s kiI k k
; ki,OL = iI PWM
Ra =τ e ❏ Choose kiI to achieve desired crossover frequency
0
Magnitude (dB) Magnitude (dB) 60
40
20
0
20
40
0
10 10 1 3 2001 by N. Mohan 4 10 1 2 3 10
10
Frequency (Hz) 20 10 1 0 90 10 10 30
0
10 5 90.5 open loop
Exit 10 P has e (deg) P has e (deg) 89 2 10
10
Frequency (Hz) 89.5 91
0
10 k I ,OL = ωCI 10 4 10 5 2 3 2 3 10
10
Frequency (Hz) 10 4 10 5 50 100
0
10 10 1 10
10
Frequency (Hz) 10 4 10 5 closed loop
TOC " ! 813 Design of the Speed Loop
*
ω m (s)
+ *
I a (s) PI 1 I a ( s) Tem ( s ) kT − ω m (s) 1
Js ω m (s) ❏ Assume current loop to be ideal ➝ represent by unity
❏ Choose crossover frequency ωCω an order of magnitude
lower than ωCI
❏ Choose a reasonable phase margin φ PM ,ω
open loop closed loop
20
Magnitude (dB) Magnitude (dB) 150
100
50
0
50
1
10 0 10 1 10 2001 by N. Mohan 4 10 0 1 10 2 3 10
10
Frequency (Hz) 0 10 1 10 0 150 10 40
60
1
10 5 10 100 200
1
10 Exit 3 Phase (deg) Phase (deg) 50 2 10
10
Frequency (Hz) 0
20 4 10 5 10 2 3 2 3 10
10
Frequency (Hz) 4 10 5 10 50 100
1
10 0 10 1 10 10
10
Frequency (Hz) 4 10 5 10 TOC " ! 814 Design of the Position Loop
*
θ m (s)
+ *
ω m (s) k pθ 1 ωm (s) θ m (s) 1
s − ❏ Assume speed loop to be ideal
❏ 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 2 2001 by N. Mohan 3 10 0 1 2 10
10
Frequency (Hz) 40 10 0 0 90.5 10 20 60
1
10 4 90 open loop
Exit 10 P has e (deg) P has e (deg) 89 1 10
10
Frequency (Hz) 89.5 91
1
10 CP 10 3 10 4 1 2 1 2 10
10
Frequency (Hz) 10 3 10 4 50 100
1
10 10 0 10
10
Frequency (Hz) 10 3 10 4 closed loop
TOC " ! 815 Further Issues ❏ Feedforward: To improve dynamic response
Process computer
position* torque*
ff speed*
ff +
Position
controller + Σ
− + Speed
controller Torque
controller − torque* speed position 1
s torque(current) * Electrical
System Mech
System ˆ
+Vtri + Σ +Vd speed
position ❏ Effect of limits
 nonlinearity *
I a (s)
+ kiI
s − s
1+ kiI / k pI −Vd ˆ
−Vtri kp ❏ Antiwindup integration
 suspend integration
when output saturates
2001 by N. Mohan cp
max input co + co ' ki 0 Exit I a ( s) 1 / Ra
1 + sτ e k PWM min ci max −  co '  TOC " ! 816 Summary
❏ What are the various blocks of a motor drive?
❏ What is a cascaded control and what are its advantages?
❏ Draw the average models of a PWM controller and a
dcdc converter.
❏ Draw the dcmotor equivalent circuit and its representation
in Laplace domain. Is this representation linear?
❏ What is the transfer function of a proportionalintegral (PI)
controller?
❏ Draw the block diagram of the torque loop.
❏ What is the rationale for neglecting the feedback from speed
in the torque loop?
❏ Draw the simplified block diagram of the torque loop.
❏ Describe the procedure for designing the PI controller in the
torque loop.
Exit 2001 by N. Mohan TOC " ! 817 Summary
❏ How would we have designed the PI controller of the torque
loop if the effect of the speed were not ignored?
❏ What allows us to approximate the closed torque loop by
unity in the speed loop?
❏ What is the procedure for designing the PI controller in the
speed loop?
❏ How would we have designed the PI controller in the speed
loop if the closed torqueloop were not approximated by unity?
❏ Draw the positionloop block diagram.
❏ Why do we only need a P controller in the position loop?
❏ What allows us to approximate the closed speed loop by
unity in the position loop?
❏ Describe the design procedure for determining the controller
in the position loop
Exit 2001 by N. Mohan TOC " ! 818 Summary
❏ How would we have designed the position controller if the
closed speed loop were not approximated by unity?
❏ Draw the block diagram with feedforward. What are its
advantages?
❏ Why are limiters used and what are their effects?
❏ What is the integrator windup and how can it be avoided? Exit 2001 by N. Mohan TOC " ! ...
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This note was uploaded on 02/06/2012 for the course EE 4002 taught by Professor Scalzo during the Fall '06 term at LSU.
 Fall '06
 Scalzo

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