Robustness. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo. Outline of Today’s Lecture. Review Important transfer functions Gang of Six Gang of Four Disturbance Rejection Noise Rejection Limitations Robustness
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Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
N
D
Measurement
Noise
Disturbances
u
h
Y
U
R
E
+
+
Y
1
Controller
Process
+
+
+
+
N
D
u
h
R
Y
U
E
+
+
Y
1
Controller
Process
“Gang of Six”
Complementary
Sensitivity
Function
Load
Sensitivity
Function
+
+
+
“Gang of Four”
+
Noise
Sensitivity
Function
Sensitivity
Function
N
D
u
h
Y
U
R
E
+
+
Y
1
Controller
Process
+
+
+
+
P(s)
P(s)
P(s)
Dm
d
D
Additive
Multiplicative
Feedback
The smaller the value of T the more robust the system
C
P
+

sm
CD
Del Positive
Unstable
Region
Del Negative
Sensing
Compute
Actuate
Actuate
Sense
Controller
Plant
Compute
Sensor
Disturbance
+ or 
+ or 
Output
Input
+ or 
+ or 
REAL WORLD
OBSERVATIONS
SENSE
FORMULATE
TEST
EXPLANATION/
PREDICTION
MATHEMATICAL
MODEL
INTERPRET
f(t)
f(ti)
ti+1
ti
Problems occur if the slope reverses
sign such as in an oscillation or
becomes very flat
t
Often called the system path, trajectory or time series
{
Overshoot
Mp
Steady State
Rise time, tr
Transient period=settling time, ts
Phase
Shift, DT
Amplitude
Ay
Amplitude
Au
Input Sin(t)
Period,T
Transient Response
is called the “Convolution Equation”
Expresses the effect of an input on the system
and assuming that the eigenvalues A do not equal s
}
}
Steady
State
Transient
0
Such a structure can be represented by blocks as
…
y
S
S
S
S
D
c1
c2
cn1
cn
…
z1
z2
zn
zn1
u
S
1
a1
a2
an1
an
…
S
S
S
Given a system with the dynamics and the output
Design a linear controller with a single input which is
stable at an equilibrium point that we define as
Disturbance
Controller
u
Plant/Process
Input
r
Output
y
S
S
kr
State Controller
Prefilter
x
K
State Feedback
wn=1
z=0.6
Im(l)
Im(l)
x
wn=4
1
1
z=0.1
x
x
x
wn=2
z=0.4
x
z=0
x
x
wn=1
z=0.6
Re(l)
Re(l)
z=1
z=1
x
x
wn
1
1
z
z=0.6
wn=1
x
x
z=0
x
wn=2
x
z=0.4
x
x
1
1
z=0.1
x
wn=4
We say the system is observable if for any time T>0 it is possible to determine the state vector, x(T), through the measurements of the output, y(t), as the result of input, u(t), over the period between t=0 and t=T.
to have full rank, thus also being invertible, the common test
Where ai are the
coefficients of the
characteristic equation
…
u
bn
bn1
b2
b1
D
y
z2
zn
zn1
…
z1
S
S
S
S
S
an
an1
a2
a1
1
…
Output y(t)
Input u(t)
y
+
+
B
L
A
B
C
C
A
u
Noise
_
+
+
+
+
+
Observer/Estimator
State
}
}
Steady
State
Transient
Transfer function is defined as
General form of linear time invariant (LTI) system is expressed:
For an input of u(t)=estsuch that the output is y(t)=y(0)est
Note that the transfer function for a simple ODE can be written as the ratio
of the coefficients between the left and right sides multiplied by powers of s
The order of the system is the highest exponent of s in the denominator.
Gx
x
G(xz)
Gxz
(GH)x
Gx
Hx
x
x
x
x
Gx
GH
G
G
G
G
G
G
G
H
G
Gx
z
Gxz
(GH)x
x
x
Gx
Gx
z
x
+
+
+
+
+





Gx
G(xz)
z
Gz
z
G
Negative Feedback
Y(s)
R(s)
E(s)
Positive Feedback
H(s)
Y(s)
R(s)
E(s)
H(s)
B(s)
G(s)
B(s)
+
+
+

Amplifies
Attenuates
Input
Response
The magnitude is in decibels
decade
also, cycle
Note: The scale for w is logarithmic
(IEEE Standard 100 Dictionary of IEEE Standards Terms, Seventh Edition, The Institute of Electrical and Electronics Engineering, New York, 2000; ISBN 0738126012; page 288)
Because decibels is traditionally used measure of power, the decibel value of a magnitude, M, is expressed as 20 Log10(M)
General form of linear time invariant (LTI) system was previously expressed as
We now want to examine the case where the input is sinusoidal. The
response of the system is termed its frequency response.
negative if the factor is in the denominator
Fortunately, we rarely have to use these integrals as there are other methods
Now, you should see the advantage
of having zero initial conditions
Disturbance/Noise
Reference
Input
R(s)
Error
signal
E(s)
Output
y(s)
Controller
C(s)
Plant
G(s)
Prefilter
F(s)
Open Loop
Signal
B(s)
Sensor
H(s)
+
+


The plant is that which is to be controlled with transfer function G(s)
The prefilter and the controller define the control laws of the system.
The open loop signal is the signal that results from the actions of the
prefilter, the controller, the plant and the sensor and has the transfer function
F(s)C(s)G(s)H(s)
The closed loop signal is the output of the system and has the transfer function
Error
signal
E(s)
Output
y(s)
Input
r(s)
Controller
C(s)
Plant
P(s)
Open Loop
Signal
B(s)
Imaginary
B(iw)
Plane of the Open Loop
Transfer Function
Sensor
1
1
B(0)
Real
+
+
B(iw)
1 is called the
critical point
Error
signal
E(s)
Output
y(s)
Input
r(s)
Imaginary
Controller
C(s)
Plant
P(s)
B(iw)
Plane of the Open Loop
Transfer Function
Open Loop
Signal
B(s)
1 is called the
critical point
1
B(0)
Sensor
1
Real
Stable
B(iw)
Unstable
+
+
Simple Nyquist Theorem:
For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point 1.
1
Magnitude, dB
0
Positive Gain Margin
w
Phase, deg
180
Phase Margin
w
Phase Crossover Frequency
+
+
+
+
+
1
Rise
Time
T
Lag L
Rise
Time
T
Lag L
Error
signal
E(s)
Output
y(s)
Input
r(s)
Controller
C(s)
Plant
P(s)
Open Loop
Signal
B(s)
Sensor
1
+
+
Error
signal
E(s)
Output
y(s)
Input
r(s)
Controller
C(s)
Plant
P(s)
Open Loop
Signal
B(s)
Sensor
1
+
+
We achieved the specifications once the pole of the compensator was moved
out to 9 and we adjusted the gain for the 0.6 damping.
N
D
u
h
R
Y
U
E
+
+
Y
1
Controller
Process
“Gang of Six”
Complementary
Sensitivity
Function
Load
Sensitivity
Function
+
+
+
“Gang of Four”
+
Noise
Sensitivity
Function
Sensitivity
Function
N
D
u
h
Y
U
R
E
+
+
Y
1
Controller
Process
+
+
+
+