Assessment of Achievable Performance
K. J. Åström
Department of Automatic Control, Lund University
K. J. Åström
Assessment of Achievable Performance
Congratulations
Seminar at MIT
Your monumental model reduction paper
Your tour de force on H ∞ theory and design methods
Great leadership in Cambridge
Great research
International network
Superb students
H ∞ loop shaping (elegant theory, lots of applications)
K. J. Åström
Assessment of Achievable Performance
Introduction
Goal: Capture the essence of a control design in a simple way
Useful for an teaching and an overall assessment
Useful for choosing the weights in H
∞
loopshaping
The idea
Assume that a controller is designed with a method like
H ∞ which guarantees robustness
Find a way to characterize essential trade-offs qualitatively
K. J. Åström
Assessment of Achievable Performance
Important Issues
Load disturbances
Measurement noise
Command signals
Process variations
Process dynamics, time delays, RHP poles and zeros
Actuator resolution and saturation
Sensor resolution and range
Results can be summarized in an assessment plot that can be
generated from the process transfer function
K. J. Åström
Assessment of Achievable Performance
Outline
1
Introduction
2
Preliminaries
3
Performance Trade-offs
4
Robustness Constraints for NMP systems
5
Assessment Plots
6
Summary
K. J. Åström
Assessment of Achievable Performance
A Basic Control System
n
d
r
Σ
F
e
C
u
Σ
v
P
x
Σ
y
−1
Controller
Process
Ingredinets
Controller: feedback C, feedforward F
Load disturbance d : Drives the system from desired state
Measurement noise n : Corrupts information about state x
Command signal r : Process state x should follow r
K. J. Åström
Assessment of Achievable Performance
Criteria for Control Design
n
d
r
Σ
F
e
C
u
Σ
v
P
x
Σ
−1
Controller
Process
Ingredients
Attenuate effects of load disturbance d
Do not feed in too much measurement noise n
Make the system insensitive to process variations
Make state x follow command r
K. J. Åström
Assessment of Achievable Performance
y
A Separation Principle for 2DOF
Design the feedback C to achieve
Low sensitivity to load disturbances d
Low injection of measurement noise n
High robustness to process variations
Then design the feedforward F to achieve the desired response
to command signals r
At least six transfer functions are required to characterize the
system (the Gang of Six)
Many books and papers show only the set point response
Interactive learning modules
K. J. Åström
Assessment of Achievable Performance
Process Control
The tuning debate: Should controllers be tuned for set-point
response or for load disturbance response?
Different tuning rules for PID controllers
Shinskey: Set-point disturbances are less common than
load changes.
Resolved by set-point weighting (poor mans 2DOF)
Z t
³ dr dy ´
¢
¡
¢
¡
f
r(τ )− y(τ ) dτ + kd γ
−
u(t) = k β r(t)− y(t) + ki
dt dt
0
Tune k, ki , and kd for load disturbances, filtering for
measurement noise and β , and γ for set-points
K. J. Åström
Assessment of Achievable Performance
PID Control with Set-Point Weighting
1.5
y
1
0.5
0
0
10
20
30
40
50
60
0
10
20
30
40
50
60
2
u
1.5
1
0.5
0
−0.5
K. J. Åström
Assessment of Achievable Performance
Outline
1
Introduction
2
Preliminaries
3
Performance Trade-offs
4
Robustness Constraints for NMP systems
5
Assessment Plots
6
Summary
K. J. Åström
Assessment of Achievable Performance
Disturbance Modeling gives Weights
n
d
r
F
Σ
e
Controller
C
−1
Y = PSD − SN,
u
Σ
v
P
x
Σ
y
Process
U = − PCSD − CSN
Stochastic modeling of d (drifting) and n (white noise) and
solution to the LQG problem
Z ∞
¡ 2
¢
y (τ ) + ρ u2 (τ ) dτ
J=E
0
gives a controller with integral action and high frequency roll-off
and weights for an H ∞ problem (mixed H 2 - H ∞ )
K. J. Åström
Assessment of Achievable Performance
Performance Assessment
Disturbance reduction by feedback
Ycl (s)
1
=
Yol (s)
1 + PC
Load disturbance attenuation (typically low frequencies)
X
Y
P
=
=
,
D
D
1 + PC
U
PC
=−
D
1 + PC
Measurement noise injection (typically high frequencies)
X
PC
=
,
N
1 + PC
U
C
=−
N
1 + PC
Command signal following
X
Y
PC F
=
=
,
R
R
1 + PC
K. J. Åström
U
CF
=
R
1 + PC
Assessment of Achievable Performance
Robustness
Robustness to process variations (large, additive, stable ∆ P)
¯ ∆ P ¯ p1 + PCp
1
¯
¯
=
¯
¯<
P
p PCp
pT p
Sensitivity of command signal response (small variations)
dG xr
dP
1
=
G xr
1 + PC P
Sensible design methods like H
good sensitivities.
K. J. Åström
∞
loop shaping guarantees
Assessment of Achievable Performance
Simple Performance Assessment
Ycl (s)
1
=
Yol (s)
1 + PC
PC
P
D−
N
X =
1 + PC
1 + PC
C
PC
D−
N
U =−
1 + PC
1 + PC
Load disturbances typically have low frequencies, and
measurement noise typically has high frequencies → integral
action and high frequency roll-off
K. J. Åström
Assessment of Achievable Performance
Minimum Phase Systems
Any transfer function can be realized. No limitations because of
system dynamics. High bandwidth attenuates disturbances
effectively but measurement noise is also amplified. Gain
crossover frequency ω ˆc captures
Disturbance attenuation
Ycl = SYol
Noise injection to state
X = −T N
ω ˆc
How about noise
injection to u?
ω sc
U = − CSN
K. J. Åström
Assessment of Achievable Performance
Effect of Noise on Control Signal
Loop shaping design
Determine desired crossover frequency ω ˆc
Required phase lead at crossover frequency
ϕ l = − arg P(iω ˆc ) − π + ϕ m
Add phase lead to give desired phase margin
Adjust gain to make loop gain 1 at ω ˆc
Phase lead is requires gain.
K. J. Åström
Assessment of Achievable Performance
Gain of a Simple Lead Networks
G n (s) =
Phase lead ϕ = n arctan
√
n
³
s + a ´n
√
.
n
s/ K + a
K −1
√
.
2n
2 K
q
³
´n
Gain K n = 1 + 2 tan2 ϕn + 2 tan ϕn 1 + tan2 ϕn
Phase lead
90○
180○
225○
n=2
34
-
n=4
25
1150
14000
n=6
24
730
4800
n=8
24
630
3300
n=∞
23
540
2600
As n goes to infinity K n → K ∞ = e2ϕ , exponential increase
K. J. Åström
Assessment of Achievable Performance
Lead Networks of 2nd 3rd and 10th Order
3
p G (iω )p
10
2
10
1
10
0
10
−2
10
−1
0
10
10
1
10
2
10
140
arg G (iω )
120
100
80
60
40
20
0
−2
10
−1
0
10
10
ω
K. J. Åström
1
10
2
10
Assessment of Achievable Performance
Bode’s Phase Area Formula
Let G (s) be a transfer function with no poles and zeros in the
right half plane. Assume that lims→∞ G (s) = G∞ . Then
Z
Z
G (∞)
dω
2 ∞
2 ∞
log
=
=
arg G (iω )
arg Ḡ (iu)du
G (0)
π 0
ω
π −∞
The gain K required to obtain a given phase lead ϕ is an
exponential function of the area under the phase curve
( )
K = e4cϕ 0 /π = e2γ ϕ 0
2c
γ =
ϕo
π
c
K. J. Åström
c
c
Assessment of Achievable Performance
Estimate of Controller Gain
log p Cp
− log K c
− log p P(iω ˆc )p
log
p
Kϕ
log
p
Kϕ
ω ˆc
K c = max p C(iω )p =
ω ≥ω ˆc
p
Kϕ
p P(iω ˆc )p
=
log ω
eγ ϕ l
eγ (−π +ϕ m −arg P(iω ˆc ))
=
.
p P(iω ˆc )p
p P(iω ˆc )p
Right hand side only depends on the process!
K. J. Åström
Assessment of Achievable Performance
Estimating Controller Gain
This largest high frequency gain of the controller is
approximately given by (γ ( 1)
K c = max p C(iω )p =
ω ≥ω ˆc
eγ ϕ l
eγ (−π +ϕ m −arg P(iω ˆc ))
=
p P(iω ˆc )p
p P(iω ˆc )p
Notice that K c only depends on the process
Compensation for process gain 1/p P(iω ˆc )p
Gain required for phase lead: eγ (−π +ϕ m −arg P(iω ˆc ))
The largest allowable gain is determined by sensor noise and
resolution and saturation levels of the actuator. Results also
hold for NMP systems but there are other limitations for such
systems.
K. J. Åström
Assessment of Achievable Performance
A Classic Problem
For linear systems it follows Bode’s phase area formula
that phase advance requires gain
An observation: higher order compensator gives lower gain
A key question: Can we get a given phase advance with
less gain by using a nonlinear systems?
The Clegg integrator
A problem worth revisiting?
K. J. Åström
Assessment of Achievable Performance
Outline
1
Introduction
2
Preliminaries
3
Performance Trade-offs
4
Robustness Constraints for NMP systems
5
Assessment Plots
6
Summary
K. J. Åström
Assessment of Achievable Performance
Robustness and Gain Crossover Frequency
Factor process transfer function as P(s) = Pmp(s) Pnmp(s) such
that p Pnmp(iω )p = 1 and Pnmp has negative phase. Requiring a
phase margin ϕ m we get
arg L(iω ˆc ) = arg Pnmp(iω ˆc ) + arg Pmp(iω ˆc ) + arg C(iω ˆc )
≥ −π + ϕ m
But arg Pmp C ( nπ /2, where n is the slope at the crossover
frequency. (Exact for Bodes ideal loop transfer function
Pmp(s) C(s) = (s/ω ˆc )n ). Hence
arg Pnmp(iω ˆc ) ≥ −π + ϕ m − n
π
2
The phase crossover inequality implies that robustness
constraints for NMP systems can be expressed in terms of ω ˆc .
K. J. Åström
Assessment of Achievable Performance
Bode’s Ideal Cut-off Characteristics
The repeater problem. Large gain variations in vacuum tube amplifiers. What
should a loop transfer function look like
to make the properties independent of
open-loop gain?
³ s ´n
L(s) =
ω ˆc
Phase margin invariant with loop gain. For this transfer function
we have arg L(iω ) = nπ /2.
The slope n = −1.5 gives the phase margin ϕ m = 45○ .
Horowitz extended Bodes ideas to deal with arbitrary plant
variations not just gain variations in the QFT method.
K. J. Åström
Assessment of Achievable Performance
The Crossover Frequency Inequality
The inequality
arg Pnmp(iω ˆc ) ≥ −π + ϕ m − nˆc
π
2
implies that robustness requires that the phase lag of the
non-minimum phase component Pnmp at the crossover
frequency is not too large!
Simple rule of thumb:
ϕ m = 45○ , nˆc = −1/2 [ arg Pnmp(iω ˆc ) ≥ −
π
2
π
ϕ m = 60○ , nˆc = −2/3 [ arg Pnmp(iω ˆc ) ≥ −
3
π
ϕ m = 45○ , nˆc = −1 [ arg Pnmp(iω ˆc ) ≥ −
4
K. J. Åström
Assessment of Achievable Performance
Useful to Plot the Phase of Pnmp
Example from Doyle, Francis and Tannenbaum 1992 and the
Bhattacharya fragility debate.
P(s) =
s2
s−1
,
+ 0.5s − 0.5
Pnmp =
(1 − s)(s + 0.5)
(1 + s)(s − 0.5)
arg P(iω )
0
−90
−180
−1
10
0
10
ω
K. J. Åström
Assessment of Achievable Performance
1
10
Bicycle with Rear Wheel Steering
Transfer function
RHP zero at V0 /a
0
arg Pnmp
V0
am{ V0 −s + a
P(s) =
mˆ{
bJ
s2 −
J
p
RHP pole at mˆ{/ J
V0 = 1, 2, 3, 4, 5 m/s
−90
−180
0
10
K. J. Åström
ω ˆc
1
10
Assessment of Achievable Performance
2
10
Outline
1
Introduction
2
Preliminaries
3
Performance Trade-offs
4
Robustness Constraints for NMP systems
5
Assessment Plots
6
Summary
K. J. Åström
Assessment of Achievable Performance
The Assessment Plot
The assessment plot has a gain curve K c (ω ˆc ) and two phase
curves arg P(iω ) and arg Pnmp(iω )
Attenuation of disturbance captured by ω ˆc
Injection of measurement noise captured by the high
frequency gain of the controller K c (ω ˆc )
Robustness limitations due to time delays and RHP poles
and zeros captured by arg Pnmp(ω ˆc )
Controller complexity is captured by arg P(iω ˆc )
K. J. Åström
Assessment of Achievable Performance
Assessment Plot for P(s) = 1/(s + 1)4
4
10
3
Kc
10
2
10
1
10
0
10
−1
10
0
10
1
10
0
arg P
−90
−180
P
−270
−360
−1
10
0
ω10ˆc
K. J. Åström
1
10
Assessment of Achievable Performance
Assessment Plot for P(s) = e−
√
s
4
10
3
Kc
10
2
10
1
10
0
10
0
10
1
2
10
10
arg P
0
−90
−180
P
−270
0
10
1
ω10ˆc
K. J. Åström
2
10
Assessment of Achievable Performance
Assessment Plot for P(s) = e−0.01s /(s2 − 100)
4
10
3
Kc
10
2
10
1
10
0
arg P, arg Pnmp
10
0
10
1
2
10
10
3
10
0
−90
−180
−270
−360
0
10
D
1
10
K. J. Åström
ω ˆc
2
10
3
10
Assessment of Achievable Performance
Outline
1
Introduction
2
Preliminaries
3
Performance Trade-offs
4
Robustness Constraints for NMP systems
5
Assessment Plots
6
Summary
K. J. Åström
Assessment of Achievable Performance
Summary
Issues in control system design
Load disturbances, measurement noise, command signals
Robustness to rocess variations
Process dynamics, time delays, RHP poles and zeros
Resolution and range of actuators and sensors
The assessment plot captures many issues qualitatively.
Trading off attenuation of load disturbance against injection
of measurement noise
Robustness for NMP systems
Assessment of controller complexity
Weights can be chosen based on gain crossover frequency
K. J. Åström
Assessment of Achievable Performance