90% (or more) of control loops in industry are PID ... · • Simple control design model →...
Transcript of 90% (or more) of control loops in industry are PID ... · • Simple control design model →...
EE392m - Winter 2003 Control Engineering 4-1
Lecture 4 - PID Control
• 90% (or more) of control loops in industry are PID• Simple control design model → simple controller
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Example:Utilization control in a video server
P control• Integrator plant:
duy +=&
)( dP yyku −−=
• P controller:
admission rate
CPU
u(t)
completion rate
server utilization-u(t)
y(t)
d(t)
y(t)
yd(t)
-d(t)
Video stream i– processing time c[i], period p[i]– CPU utilization: U[i]=c[i]/p[i]
tUnew
∆∆=
tUdone
∆∆=
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P control• Closed-loop dynamics
dks
yks
kyP
dP
P
++
+= 1
dykyky dPP +=+&
• Steady-state (s = 0)
• Transient
• Frequency-domain (bandwidth)
SSP
dSS dk
yy 1+=
( )
P
TtSS
Pd
Tt
kT
edk
yeyty
/1
11)0()( //
=
−⋅
++= −−
( ) 1/
/)(ˆ)(ˆ)(ˆ
2 +
+=
P
Pd
k
kidiyiy
ω
ωωω
0.01 0.1 1 10-20-15-10-5
0
0 2 400.20.40.60.8
1
y(t)
|y(iω)|ti
tidd
eidtd
eiytyω
ω
ω
ω
)(ˆ)(
)(ˆ)(
=
=
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Example:• Servosystem command
• More:– stepper motor– flow through a valve– motor torque ...
I control
• Introduce integrator into control
• Closed-loop dynamics
)(,
dI yykvvu
−−==&
,dugy +⋅=
dgkssy
gksgky
Id
I
I
++
+=
y(t)
u(t)
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Sampled time I control
• Step to step update:
• Closed-loop dynamics
• Deadbeat control:
[ ]dI ytyktvtvtututdtugty
−=−+−=
+⋅=
)()()1()1()(
)()()(
[ ]dI yy
zku
dugy
−−
=
+⋅=
1
dgkz
zygkz
gkyI
dI
I
+−−+
+−=
11
1
sampled timeintegrator
1=Igk ( )dzyzy d11 1 −− −+=
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Run-to-run (R2R) control• Main APC (Advanced
Process Control) approach insemiconductor processes
• Modification of a productrecipe between tool "runs"
Tool
Run-to-run control
Cell controller
Process
Runtimecontroller Metrology
system
Tunable
recipe
parameters
• Processes:– vapor phase epitaxy– lithography– chemical mechanical
planarization (CMP)– plasma etch
[ ]dI ytyktututdtugty
−+−=+⋅=
)()1()()()()(
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ekvkuev
yye
PI
d
−−==
−=&
;
duyy ++−=&τ
PI control• First-order system:
• P control + integrator forcancelling steady state error
Example:• WDM laser-diode temperature control
• Other applications• ATE• EDFA optical amplifiers• Fiber optic laser modules• Fiber optic network equipment
duyy ++−=&τ
heat loss to environment
heat capacity
y(t) = temperature - ambient temperaturepumped
heatproduced
heat
TemperatureController
Peltier Heatpump
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Cascade loop interpretation:
PI control• P Control + Integrator for
cancelling steady state error
• Velocity form of the controller
{)(
;
vkekekvkuev
yye
P
Ikk
iPPI
d
−=−−==
−=&
Inner loop
Plant
kP
e(t)u(t)
ki∫
y-yd
PiI kkk ⋅=ekeku PI && −−=[ ])1()()()()1( −−−−=+ tetektektutu PI
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PI control• Closed-loop dynamics
• Steady state (s = 0): .No steady-state error!
• Transient dynamics: look at thecharacteristic equation
• Disturbance rejection
dSS yy =
0)1(2 =+++ IP kk λτλ
)(ˆ)()(ˆ ωωω idiHiy d ⋅=
dkskss
sykskss
kskyIP
dIP
IP
++++
++++=
)1()1( ττ
0
0.2
0.4y(t)
Frequency (rad/sec)
DISTURBANCE REJECTION
MA
GN
ITU
DE
(dB
)10 -2 100 102
-40
-30
-20
-10
0
)( ωiHd
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PLL Example• Phase-locked loop is arguably a
most prolific feedback system
PLL
VoltageControlledOscillator
Hi-freqLPF
Loop Filter(controller)
reference signal
signal phase-locked to reference
uK
ttAKettAK
vrKe
ooo
odom
oodm
m
=∆=
−+−≈+×+=
×=
ωθθθωω
θωθω
&
)sin()cos()sin(LPF2
LPF2e
u
EE392m - Winter 2003 Control Engineering 4-11
PLL Loop Model• Small-signal model:
PLL
VCO
Kd C(s)
reference error
phase
{o
uK
KKe
o
d
do
dd
θ
θωωθθθ
−+−=
≈=
43421&&
)sin(e
u
1<<−+−= odott θθωωθ
odott θθωωθ −+−=
sKo−
∫ ⋅+=
=−=
dtekeku
KeuKd
IP
d
o
θθ&
d
dkskKKs
sIPdo ++
= 2θ
• Loop dynamics:
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PD control• 2-nd order dynamics
• PD control
• Closed-loop dynamics
• Optimal gains (critical damping)
Example:• Disk read-write controlduy +=&&
ekekuyye
PD
d
−−=−=&
DISTURBVCM TTJ +=ϕ&&
VoiceCoilMotor
dekeke PD =++ &&&
dksks
ePD ++
= 21
2;2 ττ == PD kk
EE392m - Winter 2003 Control Engineering 4-13
es
eesse
D
D
DD 1/11
1 ++=
+≈
ττ
ττ&
PD control• Derivative (rate of e) can be obtained
– speed sensor (tachometer)– low-level estimation logic
• Signal differentiation– is noncausal– amplifies high-frequency noise
• Causal (low-pass filtered) estimate of the derivative
• Modified PD controller:eke
ssku P
DD −
+−=
1τ
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PD control performance• The performance seems to be infinitely improving for
• This was a simple design model, remember?• Performance is limited by
– system being different from the model• flexible modes, friction, VCM inductance
– sampling in a digital controller– rate estimation would amplify noise if too aggressive– actuator saturation– you might really find after you have tried to push the performance
• If high performance is really that important, carefulapplication of more advanced control approaches might help
∞→== τττ ;;2 2PD kk
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Plant Type
• Constant gain - I control• Integrator - P control• Double integrator - PD control• Generic second order dynamics - PID control
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PID Control• Generalization of P, PI, PD• Early motivation: control of first
order processes with deadtime
Example:•
us
geysTD
1+=
−
τ
Papermachinecontrol
DT
τ
g
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PID Control
• PID: three-term control
• Sampled-time PID
• Velocity form– bumpless transfer between
manual and automatic
∫ ⋅−−−=
−=
dtekekeku
yye
IPD
d
&
( ) ez
kekezku IPD 11
111 −
−
−++−−=
future present past
( )( ))2()1(2)(
)1()()()()1(−+−−−
−−−−=+tetetek
tetektektutu
D
PI
independent sensor or an estimate
1
2
1 −−=∆
−∆−∆−=∆
zekekeku IPD
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sim
Tuning PID Control
• Model-based tuning• Look at the closed-loop poles• Numerical optimization
– For given parameters run a sim,compute performance parametersand a performance index
– Optimize the performance indexover the three PID gains usinggrid search or Nelder method.
eskksku I
PD
++−=
Plant model
Optimizer
IPD kkk ,, Performance
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Zeigler-Nichols tuning rule
• Explore the plant:– set the plant under P control and start increasing the gain till the
loop oscillates– note the critical gain kC and oscillation period TC
• Tune the controller:
• Z and N used a Monte Carlo method to develop the rule• Z-N rule enables tuning if a model and a computer are both
unavailable, only the controller and the plant are.
kP kI kDP 0.5kC
______ ______
PI 0.45kC 1.2kP/TC______
PID 0.5kC 2kP/TC kPTC/8
EE392m - Winter 2003 Control Engineering 4-20
Integrator anti wind-up• In practice, control authority is
always limited:– uMIN ≤ u(t) ≤ uMAX
• Wind up of the integrator:– if the integral v
will keep growing while the controlis constant. This results in a heavyovershoot later
• Anti wind-up:– switch the integrator off if the
control has saturated
<≤≤
>=
−−==
MINcMIN
MAXcMINc
MAXcMAX
PIc
uuuuuuu
uuuu
ekvkuev
,,
,
&
MAXc uu >
<>≤≤
=MINcMAXc
MAXcMIN
uuuuuuue
vor if0,
for ,&
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Industrial PID Controller• A box, not an algorithm• Auto-tuning functionality:
– pre-tune– self-tune
• Manual/cascade mode switch• Bumpless transfer between
different modes, setpoint ramp• Loop alarms• Networked or serial port