Disturbance Response
Dr. Kevin Craig
Professor of Mechanical Engineering
Rensselaer Polytechnic Institute
Disturbance Response
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Disturbance Response
• The focus up to now has been on command response.
Disturbance response is also important, and in some
applications, more important than command response.
• Disturbance response is more difficult to measure
because disturbances are more difficult to produce than
are commands.
• Both command response and disturbance response
improve with high loop gains.
– A high KP provides a higher bandwidth and better ability to reject
disturbances with high frequency content.
– A high KI helps the control system reject lower frequency
disturbances.
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– Setting KI high had minimal effect on the command-response
Bode plots. KI is aimed at improving response to disturbances,
not commands. In fact, the process of tuning KI is essentially to
raise it as high as possible without significant impact on the
command response. KI is not noticeable in the command
response until it is high enough to cause peaking and instability.
High KI provides unmistakable benefit in disturbance response.
• In addition, Disturbance-Compensated Feedforward
Control aids disturbance response by using measured or
estimated disturbances to improve disturbance response.
• Sometimes disturbance response is referred to by its
inverse – disturbance rejection or dynamic stiffness.
Control systems need to have high dynamic stiffness
(disturbance rejection) and low disturbance response.
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• Disturbances are undesirable inputs. We are concerned
about a response to an input other than a command.
• A properly placed integrator will totally reject DC (static)
disturbances. High tuning gains will help the system
reject dynamic disturbance inputs, but those inputs
cannot be rejected entirely.
Σ
Σ
The disturbance D(s) is applied just before the plant.
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• The control system cannot reject the disturbance
perfectly because the disturbance is detected only after it
moves the output; the controller cannot react until
system output has been disturbed.
• Disturbance response is defined as the response of the
system output C(s) to the disturbance D(s).
C(s)
G(s)
=
D(s) 1 + G c (s)G(s)H(s)
• One way to improve disturbance response is to use
slow-moving plants, e.g., large inertia, high capacitance,
to provide low plant gains. Reduce G(s). This is a timeproven technique. Large flywheels smooth motion; large
inductors and capacitors smooth voltage output.
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• A second way to improve disturbance response is to
increase the gains of the controller, GC(s). This is how
integral gains grant systems perfect response to DC inputs:
the gain of the ideal integrator at 0 Hz is infinite, driving up
the magnitude of the transfer function denominator and,
thus, driving down the disturbance response.
• At other frequencies, unbounded gain is impractical, so AC
disturbance response is improved with high gains but not
cured entirely.
• Dynamic Stiffness is the inverse of disturbance response. It
is a measure of how much force is required to move a
system as opposed to disturbance response, which is a
measure of how much the system moves in the presence of
a force. A system that is very stiff responds little to
disturbances.
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Proportional- Integral Control
R(s)
Σ
+-
1
G (s) =
Js
⎛ s + KI ⎞
GC (s ) = K P ⎜
⎟
s
⎝
⎠
H (s) = 1
E(s)
D(s)
+
Gc(s) + Σ
B(s)
G(s)
C(s)
H(s)
C(s)
G(s)
=
D(s) 1 + G c (s)G(s)H(s)
=
s
Js 2 + K P s + K P K I
High-Frequency Range
Low-Frequency Range
Mid-Frequency Range
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C
s
(s) = 2
D
Js + K Ps + K P K I
J =0.002
KP = 0.58
KI = 58
Increase J
Increase KP
Increase KI
What Happens?
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J = 0.02
KP =5.8
KI = 580
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• Some Observations
– Increasing the value of J reduces (improves) the disturbance
response in the higher frequencies. Disturbance response from
the inertia improves as frequency increases (1/Js term).
– In the medium frequency range the 1/KP term dominates. A
larger proportional gain helps in the medium frequencies.
– In the lowest frequency range, the s/KIKP term dominates.
Larger proportional gain improves the low-frequency disturbance
response, as does larger integral gain.
– Remember that larger proportional gain allows larger integral
gains. So increasing Kp improves medium- and low-frequency
disturbance response directly and also indirectly helps lowfrequency disturbance response by allowing larger KI.
– Raising J improves the high-frequency disturbance response
directly, but sometimes improves the rest of the frequency
spectrum indirectly by allowing a larger value of Kp (noise,
resolution, and resonance may limit improvement).
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Open-Loop Input-Compensated Feedforward Control:
Plant Disturbance Input
Disturbance-Compensated
Disturbance
Com pensation
Flow of Energy
and/or Material
Control
Effector
Disturbance
Sensor
Controlled
Variable
Plant
Plant
Manipulated
Input
• Measure (or estimate) the disturbance
• Estimate the effect of the disturbance on the
controlled variable and compensate for it
Disturbance Response
Control
Director
Desired Value
of
Controlled Variable
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• Disturbance-Compensated Feedforward Control
• Basic Idea: Measure an input disturbance to the plant
and take corrective action (adjust the manipulated
variable) before it upsets the process (causes the
controlled variable to deviate from its set point). This
measurement provides an early warning that the
controlled variable will be upset some time in the future.
• In contrast, a feedback controller does not take
corrective action until after the disturbance has upset the
process and generated an error signal.
• This controller does not use feedback! However, it is
usually combined with feedback control so that the
important features of feedback are retained in the overall
strategy.
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• There are several disadvantages to disturbancecompensated feedforward control:
• The disturbance must be measured on line. In
many applications, this is not feasible.
• The quality of the feedforward control depends on
the accuracy of the process model; one needs to
know how the controlled variable responds to
changes in both the disturbance and manipulated
variables.
• Ideal feedforward controllers that are theoretically
capable of achieving perfect control may not be
physically realizable. Fortunately, practical
approximations of these ideal controllers often
provide very effective control.
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Feedback / Disturbance-Feedforward
Temperature Control Example
Objective:
Ensure that TC
remains at or near
the set point.
G i = dm i/dt
T U1
T U2
TC
V, M
TH
Disturbance Response
Q
T CM
G o = dm o /dt
Continuous
Stirred-Tank
Heater
Electrical
Heater
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• Physical Model Simplifying Assumptions
– No change of phase occurs in the tank fluid.
– Volume of fluid in the tank is constant.
– Perfect mixing is assumed in the tank, therefore
the temperature of the fluid in the tank is uniform
and equal to the exit temperature.
– Liquid in the tank has constant density, mass M ,
and specific heat C .
– Heater is an electrical resistance heater that
follows input voltage instantly with heater gain Kh.
– Heater mass, specific heat, and heat transfer area
are Mh, Ch, and Ah and the heat transfer coefficient
to liquid is Uh.
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– Tank wall is assumed to be pure thermal
resistance 1/UtAt (no energy storage)
– Assume an initial equilibrium operating point and
take all variables as perturbations.
– Temperature sensor (assumed negligible
dynamics) is located in the pipeline downstream of
the tank, due to tank vibration from stirrer and
electrical noise from heater, resulting in a dead
time τDT.
– Tank inlet temperature TU1 is uncontrolled and
thus is a disturbance input
– Ambient temperature TU2 is uncontrolled and thus
is a disturbance input.
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• Background on Mathematical Modeling:
Basic Equations
– Conservation of Mass: Tank
∂
0=
ρdV + ∫ ρvidA
∫
∂t CV
CS
•
•
d
Assume constant density.
( ρt Vt ) = ρt V ti − ρt V to
dt
d
( Vt ) = 0
Assume constant volume in the tank.
dt
•
•
V ti = V to
Disturbance Response
volume flow rate in = volume flow rate out
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– Conservation of Energy: Tank
•
•
∂
Q+ W =
eρdV + ∫ eρvidA
∫
∂t CV
CS
v2
e = u + + gz
2
• For most processes where there are thermal
effects, kinetic and potential energy terms can be
neglected because their contribution is generally at
least two orders of magnitude less than that of the
internal energy term.
• Rate of work done on Control Volume by
surroundings: •
•
•
W = W shaft + W normal
•
W normal =
∫σ
nn
vidA
σ nn ≈ − p
CS
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•
•
⎛
p⎞
∂
uρdV + ∫ ⎜ u + ⎟ ρvidA
=
∫
∂t CV
ρ⎠
CS ⎝
•
•
•
•
dU •
=
+ V t ρ t ( h to − h ti )
dt
d ( H − pV ) •
=
+ V t ρ t ( h to − h ti )
dt
Q+ W shaft
Q+ W shaft
Q+ W shaft
• But since the volume of the tank is constant and the
mean pressure change is negligible (good
assumption for liquids provided the pressure
change is not too large), we have:
d ( pV )
dt
Disturbance Response
dp
dV
=V +p
=0
dt
dt
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•
•
dH •
= V t ρ t ( h ti − h to ) + Q+ W shaft
dt
– Neglect the work done by the mixing impeller and
assume single phase and a constant heat
capacity:
h = c p ΔT
•
dTt •
ρt Vt c pt
= V t ρ t c pt ( Tti − Tto ) + Q
dt
•
•
dTt V t
Q
=
( Tti − Tto ) +
ρt Vt c pt
dt
Vt
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• Mathematical Model
– Conservation of Mass
• Mass flow rate in = Mass flow rate out
– Conservation of Energy applied to the Heater
dTh
Q − U h A h ( Th − TC ) = M h C h
dt
M h C h dTh
Q
+ Th =
+ TC
U h A h dt
Uh Ah
– Conservation of Energy applied to the Tank
dTC
U h A h ( Th − TC ) + GC TU1 − GC TC − U t A t ( TC − TU 2 ) = M C
dt
dTC
MC
+ ( U h A h + GC + U t A t ) TC = U h A h Th + GC TU1 + U t A t TU 2
dt
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T U1
Open-Loop System
Block Diagram
Q
K4
+
Σ
+
1
Th
( τ as + 1)
K1
+
Σ
+
K2
+
K3
+
T U2
Σ
TC
1
( τ bs + 1)
TC
GC
= 0.0694
K1 =
U h A h + GC + U t A t
GC = 0.0154
Ut At
= 0.2775
K2 =
U h A h + GC + U t A t
U h A h = 0.145
M h C h = 13.05
U t A t = 0.0616
Uh Ah
= 0.6532
K3 =
U h A h + GC + U t A t
M h Ch
τa =
= 90
Uh Ah
1
= 6.90
K4 =
Uh Ah
MC
τb =
= 138.7
U h A h + GC + U t A t
Disturbance Response
M C = 30.8
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– Elimination of Th leads to:
( τ1D + 1)( τ2 D + 1) TC = K Q Q + K U1 ( τ3D + 1) TU1 + K U 2 ( τ3D + 1) TU 2
M C τ3
τ1τ2 =
GC + U t A t
τ1 + τ2 =
M h Ch
τ3 =
Uh Ah
τ3 ( U h A h + GC + U t A t ) + M C
GC + U t A t
1
KQ =
GC + U t A t
K U2
Disturbance Response
GC
K U1 =
GC + U t A t
Ut At
=
GC + U t A t
GC = 0.0154
M h C h = 13.05
U h A h = 0.145
U t A t = 0.0616
M C = 30.8
τ1 = 600, τ2 = 60
τ3 = 90, K Q = 12.99
K U1 = 0.2, K U 2 = 0.8
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T U1
K U1
+
Σ
+
τ 3s + 1
Q
+
KQ
+
Σ
K U2
T U2
Open-Loop System
Block Diagram
TC
1
( τ 1s + 1 )( τ 2 s + 1 )
( τ1D + 1)( τ2 D + 1) TC = K Q Q + K U1 ( τ3D + 1) TU1 + K U 2 ( τ3D + 1) TU 2
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T U1
K U1
+
Σ
+
K U2
T U2
τ 3s + 1
TV
eR
Ka
+
eE
Σ
Kh
-
+
Q
KQ
+
Σ
TC
1
( τ 1s + 1 )( τ 2 s + 1 )
eB
Ka
e
− τ DT s
Proportional Control of Temperature: Block Diagram
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• Dead-Time Approximations
– The simplest dead-time approximation can be obtained
graphically or by taking the first two terms of the Taylor series
expansion of the Laplace transfer function of a dead-time element,
τDT.
Qo
( s ) = e−τDTs ≈ 1 − τDTs
Qi
q o ( t ) ≈ q i ( t ) − τDT
dq i
dt
– The accuracy of this approximation depends on the dead time
being sufficiently small relative to the rate of change of the slope
of qi(t). If qi(t) were a ramp (constant slope), the approximation
would be perfect for any value of τDT. When the slope of qi(t)
varies rapidly, only small τDT's will give a good approximation.
– A frequency-response viewpoint gives a more general accuracy
criterion; if the amplitude ratio and the phase of the approximation
are sufficiently close to the exact frequency response curves of
for the range of frequencies present in qi(t), then the
approximation is valid.
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Dead-Time Graphical Approximation
qi
τDT
q o = q i ( t − τDT )
tangent line
qi(t)
q o = q i ( t ) − τDT
dq i
dt
t
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– Use this simple dead-time approximation to
simplify the closed-loop system equation and
facilitate gain setting: −τDTs
e
≈ −τDT D + 1
– The closed-loop system equation is:
⎛ D 2 2ζ D ⎞
K U1
K
+ 1⎟ TC =
TV +
( τ3D + 1) TU1
⎜ 2 +
ωn
K +1
K +1
⎝ ωn
⎠
K U2
+
( τ3D + 1) TU 2
K +1
K +1
ωn =
τ1τ2
Disturbance Response
ζ=
τ1 + τ2 − KτDT
2 τ1τ2 ( K + 1)
K = Ka Kh KQ
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• Parameter
Values
–
–
–
–
–
–
–
–
–
Disturbance Response
Ka = 0.05
Kh = 10.0
KQ = 12.99
KU1 = 0.2
KU2 = 0.8
τ3 = 90
τ1 = 600
τ2 = 60
τDT = 5, 50
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• Disturbance Feedforward Compensation
– Since both disturbances TU1 and TU2 are
measurable, the feedback system can be
augmented with a feedforward system.
– Ideally the feedforward dynamics should be: τ3s + 1
– However, the derivative signal might accentuate
noise, so we include a low-pass filter, even though
this prevents perfect dynamic compensation:
1
1
=
τ4s + 1 0.1τ3s + 1
– KU1FF = 0.2
– KU2FF = 0.8
– τ4 = 0.1τ3
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K U1FF
τ 3s + 1
T U1
K h K Q ( τ 4s + 1)
Σ
+
K U1
+
Σ
K U 2 FF
+
+
K U2
T U2
τ 3s + 1
TV
eR
Ka
+
Feedback
+
Feedforward
Control
-
eE
Σ
Kh
-
+
Q
KQ
+
Σ
TC
1
( τ 1s + 1 )( τ 2 s + 1 )
eB
Disturbance Response
Ka
e
− τ DT s
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• Important Issues in Feedforward Control
– The feedforward controller must be physically
realizable and stable.
– The disturbance time delay must be greater than the
process time delay for perfect feedforward
compensation. If the process time delay is greater
than the disturbance time delay, the feedforward
controller will have no time-delay compensation, and
perfect control cannot be achieved.
– If the process has a RHP zero, it must be factored out
before designing the feedforward controller.
Otherwise, the feedforward controller will be unstable.
– If the process is higher order than the disturbance, fast
time constants probably must be neglected in
designing the feedforward controller.
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– Often a static feedforward controller will have good
performance, particularly if the disturbance dynamics
are the same time scale as the process dynamics.
– A feedforward controller does not change the closedloop stability of the feedback system, assuming that
the feedforward controller is stable. Also, a
feedforward controller does not change the set point
response of a closed-loop system.
– Feedforward control can be implemented with either
classical feedback (PID-type) or Internal Model Control
structure.
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• Numerical Values
τ1 = 600 sec
τ2 = 60 sec
τDT = 5 sec
– If we design for ζ = 0.6, we get K = 6.48 and ωn =
0.0144 rad/sec (period is 435 sec/cycle).
• Simulations
– Run numerical simulations, using both
approximate and exact dead-time models, to
check these results and get additional information
about system behavior.
– Ka = 0.05
– Kh = 10.0
– KQ = 12.96
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–
–
–
–
–
–
–
–
KU1 = 0.2
KU2 = 0.8
TU1 = 0, initially, then a unit step
TU2 = 0, initially, then a unit step
τ3 = 90
τ1 = 600
τ2 = 60
τDT = 5 (then increase to 50 sec, to see decrease
in accuracy of approximation)
– Command Input: Unit Step, initially, then zero
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– Since both disturbances TU1 and TU2 are
measurable, the feedback system can be
augmented with a feedforward system.
– Ideally the feedforward dynamics should be: τ3s + 1
– However, the derivative signal might accentuate
noise, so we include a low-pass filter, even though
this prevents perfect dynamic compensation:
1
1
=
τ4s + 1 0.1τ3s + 1
– KU1FF = 0.2
– KU2FF = 0.8
– τ4 = 0.1τ3
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K U1FF
τ 3s + 1
K h K Q ( τ 4s + 1)
T U1
Σ
+
K U1
+
Σ
K U 2 FF
+
+
K U2
T U2
τ 3s + 1
TV
eR
Ka
+
Feedback
+
Feedforward
Control
- e
E
Σ
Kh
-
+
Q
KQ
+
Σ
TC
1
( τ 1s + 1 )( τ 2 s + 1 )
eB
Disturbance Response
Ka
e
− τ DT s
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– Investigate both momentary and sustained
disturbances:
T U1
8.0
250
T U2
350
tim e
2.0
50
Disturbance Response
tim e
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• Compare the feedback with the feedback +
feedforward system.
– What do you observe about transient errors?
– What do you observe about steady-state errors?
– The total elimination of error depends on a perfect
match of numerical parameter values in the
feedforward and main system signal paths, a
condition we can only approximate in practice.
– The transient errors remaining in the simulation
can be reduced by making τ4 smaller. Let τ4 = 1.0
sec and observe the results. Whether a practical
system could use such a small time constant
value depends on the noisiness of the signal to
the feedforward controller.
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