Control Systems 2
Lecture 6: Uncertainty and robustness in SISO systems
Roy Smith
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6.1
Robustness analysis
Nominal stability (NS) Is the closed-loop system stable when the plant is
known exactly?
Robust stability (RS) Is the closed-loop system stable when there is
uncertainty in our knowledge of the plant?
Nominal performance (NP) Does the closed-loop system meet the performance
specifications when the plant is known exactly?
Robust performance (RP) Does the closed-loop system meet the performance
specifications when there is uncertainty in our knowledge of the
plant?
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6.2
Uncertain models
Model sets
G(s) ∈ { Gnom (s) + ∆ | k∆k ≤ γ }
k∆k
Gnom (s)
Gnom (s)
=
Nominal plant
∆
=
unknown, but bounded,
perturbation (i/o operator)
Typically, ∆ is stable, causal and satisfies, k∆kH∞ ≤ γ.
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6.3
Uncertain models
Multiplicative perturbation model
d
Gd (s)
y
+
Wm (ω)
∆
+
y(s) = G(s)u(s) + Gd (s)d(s),
Gnom (s)
where
u
G(s) ∈ G,
G = { (1 + Wm (ω)∆)Gnom (s) | ∆ is causal, stable; k∆kH∞ ≤ 1 } .
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6.4
Uncertain models
Multiplicative perturbation model
d
Gd (s)
y
y
+
=
=
Wm (ω)
∆
+
Gnom (s)
u
(I + Wm (ω)∆)Gnom u + Gd d
Gnom u
| {z }
+
Wm (ω)∆Gnom u
|
{z
}
nominal
response
+
Gd d
|{z}
perturbation
response
disturbance
response
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6.5
Example: uncertain model (first order plus delay)
Ke−λs
G=
1 + τs
Nominal case
Magnitude
10
Gnom (jω)
0.1
1
10
0.1
1
10
log ω
(rad/sec)
1
K = 10
τ = 1.0
0.1
λ = 0.5
0
log ω
(rad/sec)
−90
Gnom (jω)
−180
−270
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Phase (deg.)
6.6
Example: uncertain model (first order plus delay)
imag
Ke−λs
G=
1 + τs
4
2
Nominal case
real
K = 10
−4
τ = 1.0
−2
2
4
6
8
10
−2
λ = 0.5
−4
Gnom (jω)
−6
−8
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6.7
Example: uncertain model (first order plus delay)
Random examples
Ke−λs
G=
1 + τs
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
G(jω)
Perturbed case
±15% uncertainty
K ∈ [8.5, 11.5]
τ ∈ [0.85, 1.15]
λ ∈ [0.425, 0.575]
also includes boundary
cases.
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0.1
0.1
1
10
0
log ω
(rad/sec)
−90
Gnom (jω)
−180
−270
G(jω)
Phase (deg.)
6.8
Example: uncertain model (first order plus delay)
Absolute error (random examples)
Error as a function of frequency: |Gnom (jω) − G(jω)|
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
|Gnom (jω) − G(jω)|
0.1
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6.9
Example: uncertain model (first order plus delay)
Relative error (random examples)
Error as a function of frequency:
|Gnom (jω) − G( jω)|
|Gnom (jω)|
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
|G(jω) − Gnom (jω)|
|Gnom (jω)|
0.1
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6.10
Example: uncertain model (first order plus delay)
Modeling problem: choose Gnom and Wm
Wm (ω)
y
∆
+
Gnom (s)
u
Upper bound on relative uncertainty: choosing Wm (s)
G(s) = (1 + Wm (ω)∆)Gnom (s) =⇒
so
G(jω) − Gnom (jω)
= Wm (ω)∆
Gnom (jω)
G(jω) − Gnom (jω) ≤ |Wm (ω)| =⇒ |∆| ≤ 1.
Gnom (jω)
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6.11
Example: uncertain model (first order plus delay)
Upper bound on relative uncertainty: choosing Wm (s)
G(jω) − Gnom (jω) ≤ |Wm (ω)|.
Gnom (jω)
(See Laughlin et al. for
the Wm (s) formula)
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
Wm (jω)
|G(jω) − Gnom (jω)|
|Gnom (jω)|
0.1
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6.12
Example: uncertain model (first order plus delay)
imag
Random examples
4
G=
−λs
Ke
1 + τs
2
real
Perturbations:
±15%
−4
−2
2
4
6
8
10
−2
on K, λ, τ .
−4
Gnom (jω)
−6
−8
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6.13
Example: uncertain model (first order plus delay)
imag
Random examples
4
G=
−λs
Ke
1 + τs
|Gnom (jω)Wm (jω)|
2
real
Perturbations:
±15%
on K, λ, τ .
−4
−2
2
4
6
8
10
−2
−4
Gnom (jω)
−6
−8
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6.14
Example: uncertain model (first order plus delay)
imag
Random examples
4
G=
−λs
Ke
1 + τs
|Gnom (jω)Wm (jω)|
2
real
−4
Perturbations:
±15%
−2
2
4
6
8
10
−2
on K, λ, τ .
−4
Gnom (jω)
−6
−8
perturbation
regions
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6.15
Example: uncertain model (first order plus delay)
G(s) =
(
G(s) ∈
)
Ke
γ = 10, τ = 1, λ = 0.5, (all ± 15%)
1 + τs −λs
(1 + Wm (s)∆)Gnom (s) where here,
Wm (s) =
∆kH∞ ≤ 1 ,
1.15(1 + τ s) 0.15λs
e
−1
(1 + 0.85τ s)
(up to its maximum,
and then constant).
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
Wm (jω)
0.1
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6.16
Design example: Loopshaping design
Gnom (s) =
Ke−λs
,
1 + τs
K = 10,
τ = 1,
λ = 0.5
j
0.15(1 + τ s)
K(s) =
s
and
0.15Ke−λs
Lnom (s) =
s
imag
2
real
−2
2
4
6
8
−1
real
10
−2
−4
Gnom (jω)
−6
Lnom (jω)
Lnom (jω)
−8
−j
imag
−10
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6.17
Nominal stability
Nyquist criterion in more detail
0.15Ke−λs
Lnom (s) =
s
imag
-1
r −→ 0
real
R −→ ∞
real
Lnom (jω)
imag
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6.18
Nominal stability example: Loopshaping design
Ke−λs
Gnom (s) =
,
1 + τs
0.15(1 + τ s)
K(s) =
,
s
0.15Ke−λs
Lnom (s) =
s
Magnitude
10
Gnom (jω)
0.1
1
10
log ω
(rad/sec)
1
Lnom (jω)
K(jω)
0.1
0.1
1
10
0
K(jω)
−90
Gnom (jω)
−180
−270
log ω
(rad/sec)
Lnom (jω)
Phase (deg.)
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6.19
Nominal performance
|Wp (jω)Snom (jω)| < 1
for all ω.
imag
Lnom (jω)
|Wp (jω)|
−1
real
|1 + Lnom (jω)|
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6.20
Nominal performance
Nominal weighted sensitivity:
Nominal performance
⇐⇒
In the loopshaping example:
1
1 + Gnom (s)K(s)
Snom (s) =
|Snom (jω)| <
Wp( s) =
1
|Wp (jω)|
s + 2.5
.
5s
Magnitude
10
Wp −1 (jω)
0.1
1
log ω
(rad/sec)
10
1
Snom (jω)
0.1
0.01
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6.21
Robust stability
Closed-loop multiplicative perturbation model
Wm (ω)
y
+
∆
Gnom (s)
K(s)
+
r
−
Robust stability
Is this closed-loop stable for all stable, causal ∆, with k∆kH∞ ≤ 1?
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6.22
Robust stability
Wm (ω)
y
∆
+
Gnom (s)
K(s)
r
+
−
Robust stability
S(s) =
1
1
=
1 + G(s)K(s)
1 + (1 + Wm (s)∆)Gnom (s)K(s)
is stable for all ∆,
k∆kH∞ ≤ 1.
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6.23
Robust stability
Wm (ω)
y
∆
+
Gnom (s)
K(s)
r
+
−
−Wm (ω)Tnom (s)
Snom (s)Wm (ω)
y
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+
v
∆
z
+
Tnom (s)
r
6.24
Robust stability: an equivalent test
−Wm (ω)Tnom (s)
v
Snom (s)Wm (ω)
y
z
∆
+
+
Tnom (s)
r
∆
z
v
Wm (s)Tnom (s)
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6.25
Small gain theorem
u1
+
e1
y2
y1
M1 (s)
M2 (s)
e2
+
u2
Given e1 , e2 with ke1 k < ∞ and ke2 k < ∞, define,
u1 = e1 − M2 (s)e2
and
u2 = e2 − M1 (s)e1 .
Suppose there exists γ1 > 0 and γ2 > 0 such that,
kM1 (s)e1 k ≤ γ1 ke1 k
and
kM2 (s)e2 k ≤ γ2 ke2 k.
If γ1 γ2 < 1 then,
ky1 k ≤
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γ1
(ku1 k + γ2 ku2 k).
1 − γ1 γ2
6.26
Robust stability: an equivalent test
∆
z
v
Wm (s)Tnom (s)
By applying the small-gain theorem:
If
k∆kkWm (s)Tnom (s)k < 1
then the perturbed closed-loop system is stable.
Or:
If
kWm (s)Tnom (s)kH∞ < 1
then the perturbed closed-loop system is stable (RS).
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6.27
Robust stability
Robust stability
⇐⇒
1
1 + G(s)K(s)
is stable for all G(s) ∈ G.
⇐⇒
1
1 + (1 + Wm (s)∆)Gnom (s)K(s)
is stable for all k∆kH∞ ≤ 1
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⇐⇒
kWm (s)Tnom (s)kH∞ < 1
⇐⇒
|Wm (jω)Tnom (jω)| < 1
⇐⇒
|Tnom (jω)| <
1
|Wm (jω)|
for all ω
for all ω.
6.28
Robust stability
|1 + Gnom (jω)K(jω)| > |Gnom (jω)Wm (jω)K(jω)|
⇐⇒ |Tnom (jω)| <
1
|Wm (jω)|
for all ω.
imag
Lnom (jω)
−1
real
|1 + Lnom (jω)|
|Lnom (jω)Wm (jω)|
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6.29
Robust stability: Loopshaping design example
Robust stability ⇐⇒ |Tnom (jω)| <
1
|Wm (jω)|
for all ω.
Magnitude
10
Wm −1 (jω)
0.1
1
10
log ω
(rad/sec)
1
Tnom (jω)
0.1
0.01
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6.30
Example: uncertain model (first order plus delay)
Nyquist criteria: perturbed loopshape regions
imag
−2
−1
real
|Lnom (jω)Wm (jω)|
loopshape
perturbation
regions
−2
Lnom (jω)
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6.31
Loopshaping approximations
Nominal performance and robust stability
If Lnom (jω) 1 (high freq.)
Magnitude
100
T (jω) < 1/|Wm (jω)|
is approximately
Lnom (jω)
|Lnom (jω)| < |Wm−1 (jω)|
10
Wp (jω)
0.1
1
log ω
(rad/sec)
10
1
Wm−1 (jω)
0.1
If Lnom (jω) 1 (low freq.)
S(jω) < 1/|Wp (jω)|
0.01
is approximately
|Lnom (jω)| > |Wp (jω)|
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6.32
Robust performance
Robust performance
⇐⇒
⇐⇒
W
(s)
p
1 + G(s)K(s) H∞
< 1, for all G(s) ∈ G.
W
(s)
p
1 + (1 + Wm (s)∆)Gnom (s)K(s) < 1,
H∞
for all k∆kH∞ ≤ 1
⇐⇒
|Wp (jω)|
< 1,
|1 + (1 + Wm (jω)∆)Gnom (jω)K(jω)|
for all ω and all |∆| ≤ 1
which is equivalent to,
|Wp (jω)Snom (jω)| + |Wm (jω)Tnom (jω)| < 1
for all ω.
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6.33
Robust performance
|Wp (jω)Snom (jω)| + |Wm (jω)Tnom (jω)| < 1
for all ω.
imag
Lnom (jω)
|Wp (jω)|
−1
real
|1 + Lnom (jω)|
|Lnom (jω)Wm (jω)|
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6.34
Robust performance example
|Wp (jω)Snom (jω)| + |Wm (jω)Tnom (jω)| < 1
for all ω.
1.25
1.0
0.75
Robust performance
0.5
Nominal performance
0.25
Robust stability
0
0.1
1
10
log ω
(rad/sec)
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6.35
Notes and references
Skogestad & Postlethwaite (2nd Ed.)
Pertubation models: sections 7.1, 7.2, 7.3, 7.4
SISO robust stability: section 7.5
SISO robust performance: section 7.6
Perturbation bound, Wm (s), for the RS example
“Smith predictor design for robust performance,”DL Laughlin, DE Rivera, M
Morari, Int. J. Control, v. 46, no. 2, pp. 477–504, 1987.
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6.36