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Chapter 16: H∞Loop Shaping
•Robust Stabilization of Coprime factors
•Robust Stabilization of Normalized Coprime Factors
•H
∞Loop Shaping Design
•Weight ed H∞Control Interpretation
•Further Guidelines for Loop Shaping
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Lecture Notes on Loop Shaping - Optimal and Robust Control | ECE 7360, Study notes of Electrical and Electronics Engineering
Material Type: Notes; Class: Optimal and Robust Control; Subject: Electrical & Computer Engr; University: Utah State University; Term: Unknown 1989;
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Chapter 16: H ∞ Loop Shaping
- Robust Stabilization of Coprime factors
- Robust Stabilization of Normalized Coprime Factors
• H
∞ Loop Shaping Design
- Weighted H ∞ Control Interpretation
- Further Guidelines for Loop Shaping
Robust Stabilization of Coprime Factors
Robust Stabilization Condition:
Let P =
M
− 1 ˜ N be the nominal model and
P
∆
M +
M
− 1
(
N +
N
with
M,
N,
M
N
∈ RH
∞
and
∥ ∥ ∥ ∥ ∥
[
N
M
]∥ ∥ ∥ ∥ ∥
∞
f
f f
−
−
y
w
z 2 z 1
r
6
?
˜ ∆ M
˜ M
- − 1
˜ N
˜ ∆ N
K
The perturbed system is robustly stable iff
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1 ˜ M
− 1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
State Space Coprime Factorization:
Let
P =
A B
C D
and let L be such that A + LC is stable. Then
P =
M
− 1 ˜ N,
[
N
M
]
A + LC B + LD L
C D I
Denote
K = −K
Robust Stabilization of Normalized Coprime Factors
Suppose
M and
N are normalized coprime factors
M(jω)
M
∗
(jω) +
N(jω)
N
∗
(jω) = I
Then
M and
N can be obtained as
[
N
M
]
A − Y C
∗ C B −Y C
∗
C 0 I
where L = −Y C
∗ and Y ≥ 0 is the stabilizing solution to
AY + Y A
∗
− Y C
∗
CY + BB
∗
= 0
γ min
:= inf
K stabilizing
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1 ˜ M
− 1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
√
1 − λ max
(Y Q)
λ max
(Y Q) =
∥ ∥ ∥ ∥ ∥
[
N
M
]∥ ∥ ∥ ∥ ∥
2
H
where Q is the solution to
Q(A − Y C
∗
C) + (A − Y C
∗
C)
∗
Q + C
∗
C = 0.
Moreover, for any γ > γ min
a controller achieving
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1 ˜ M
− 1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
< γ
is given by
K(s) =
A − BB
∗ X ∞
− Y C
∗ C −Y C
∗
− B
∗ X ∞
where
X
∞
γ
2
γ
2 − 1
Q
I −
γ
2
γ
2 − 1
Y Q
− 1
- Let P =
M
− 1 ˜ N be a normalized left coprime factorization and
P
∆
M +
M
− 1
(
N +
N
with ∥ ∥ ∥ ∥ ∥
[
N
M
]∥ ∥ ∥ ∥ ∥ ∞
Then there is a robustly stabilizing controller for P ∆ if and only if
√
1 − λ max
(Y Q) =
√ √ √ √ √ 1 −
∥ ∥ ∥ ∥ ∥
[
N
M
]∥ ∥ ∥ ∥ ∥
2
H
- Let X ≥ 0 be the stabilizing solution to
XA + A
∗
X − XBB
∗
X + C
∗
C = 0
then
Q = (I + XY )
− 1
X
and
γ min
√
1 − λ max
(Y Q)
1 −
∥ ∥ ∥ ∥ ∥
[
N
M
]∥ ∥ ∥ ∥ ∥
2
H
− 1 / 2
√
1 + λ max
(XY ).
- Let P =
M
− 1 ˜ N be a normalized left coprime factorization. Then
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1 ˜ M
− 1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1
[
I P
]
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
I
K
(I + P K)
− 1
[
I P
]
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
I
P
(I + KP )
− 1
[
I K
]
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
- Let P =
M
− 1 ˜ N = NM
− 1
be respectively the normalized left and
right coprime factorizations. Then
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
K
I
(I + P K)
− 1 ˜ M
− 1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
∥ ∥ ∥ ∥ ∥
M
− 1
(I + KP )
− 1
[
I K
]∥ ∥ ∥ ∥ ∥ ∞
P
s
M
− 1
s
N
s
and
M
s
(jω)
M
∗
s
(jω) +
N
s
(jω)
N
∗
s
(jω) = I.
If max
1 return to (1) and adjust W 1
and W 2
b) Select ≤ max
, then synthesize a stabilizing controller K ∞
, which
satisfies ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
I
K
∞
(I + P
s
K
∞
− 1 ˜ M
− 1
s
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
− 1
.
(3) The final controller K
K = W
1
K
∞
W
2
A typical design works as follows: the designer inspects the open-loop
singular values of the nominal plant, and shapes these by pre- and/or
postcompensation until nominal performance (and possibly robust stabil-
ity) specifications are met. (Recall that the open-loop shape is related to
closed-loop objectives.) A feedback controller K ∞
with associated stabil-
ity margin (for the shaped plant) ≤ max
, is then synthesized. If max
is
small, then the specified loop shape is incompatible with robust stability
requirements, and should be adjusted accordingly, then K ∞
is reevaluated.
Weighted H ∞ Control Interpretation
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
∥
I
K
∞
(I + P
s
K
∞
− 1 ˜ M
− 1
s
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
∥
∞
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
∥
I
K
∞
(I + P
s
K
∞
− 1
[
I P
s
]
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
∥
∞
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
W
2
W
− 1
1
I
K
(I + P K)
− 1
[
I P
]
W
− 1
2
W
1
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞ =
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
I
P
s
(I + K
∞
P
s
− 1
[
I K
∞
]
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥
W
− 1
1
W
2
I
P
(I + KP )
− 1
[
I P
]
W
1
W
− 1
2
∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∞
This shows how all the closed-loop objective are incorporated.
z 1
z 2
W
2
W
− 1
1
I
K
(I + P K)
− 1
[
I P
]
W
− 1
2
W
1
w 1
w 2
h h h K
W
− 1
1
W
1
P W^
− 1
2
W
2
?
??
?
6
6
6
−
z 2
w 2
w 1
z 1
from which the gain margin result follows. Similarly, at frequencies where
P (jω)K(jω) = −e
jθ
,
b P,K
| 1 − e
jθ |
√ √ √ √ √ (1 + |P |
2 )(1 +
|P |
2
|2 sin
θ
2
√ √ √ √ √ √min
P
(1 + |P |
2
)(1 +
|P |
2
|2 sin
θ
2
which implies θ ≥ 2 arcsin b P,K
For example, b P,K = 1/2 guarantees a gain margin of 3 and a phase
margin of 60
o
.
b p,k = emargin(P, K); % given P and K, compute b P,K
[K
opt
, bp ,k
] = ncfsyn(P, 1 ); % find the optimal controller K opt
[K
sub
, bp ,k
] = ncfsyn(P, 2 ); % find a suboptimal controller K sub
Further Guidelines for Loop Shaping
P = NM
− 1 : normalized right coprime factorization.
b opt (P ) ≤ λ(P ) := inf
0
σ
M(s)
N(s)
small λ(P ) =⇒ small b opt
(P ).
open right-half plane zeros and poles of P :
z 1
, z 2
,... , z m
, p 1
, p 2
,... , p k
Define
N
z
(s) =
z 1
− s
z 1
- s
z 2
− s
z 2
- s
z m
− s
z m
- s
, N
p
(s) =
p 1
− s
p 1
- s
p 2
− s
p 2
- s
p k
− s
p k
- s
Then
P (s) = P 0 (s)N z (s)/N p (s)
where P 0
(s) has no open right-half plane poles or zeros.
Let N 0
(s) and M 0
(s) be stable and minimum phase spectral factors:
N
0 (s)N
∼
0
(s) =
1 +
P (s)P
∼ (s)
− 1
, M
0 (s)M
∼
0
(s) = (1+P (s)P
∼
(s))
− 1
.
Then P 0
= N
0
/M
0 is a normalized coprime factorization and (N 0
N
z ) and
(M
0
N
p ) form a pair of normalized coprime factorizations of P. Thus
b opt
(P ) ≤
√
|N
0
(s)N z
(s)|
2
- |M 0
(s)N p
(s)|
2 , ∀<(s) > 0.
ln |N 0 (re
jθ
)| =
∫ ∞
−∞
ln
√
1 + 1/|P (jω)|
2
K
θ (ω/r) d(ln ω)
ln |M 0
(re
jθ
)| =
∫ ∞
−∞
ln
√
1 + |P (jω)|
2
K
θ
(ω/r) d(ln ω)
When can b opt (P ) be small
Let s = re
jθ and note that N z
(z i
) = 0 and N p
(p j
) = 0. Then the
bound can be small if
|N
z (s)| and |N p (s)| are both small for some s. That is, |N z (s)| ≈ 0
(i.e., s is close to a right-half plane zero of P ) and |N p (s)| ≈ 0 (i.e.,
s is close to a right-half plane pole of P ). This is only possible if
P (s) has a right-half plane zero near a right-half plane pole. (See
Example 0.1.)
|N
z
(s)| and |M 0
(s)| are both small for some s. That is, |N z
(s)| ≈ 0
(i.e., s is close to a right-half plane zero of P ) and |M 0
(s)| ≈ 0 (i.e.,
|P (jω)| is large around ω = |s| = r). This is only possible if |P (jω)|
is large around ω = r, where r is the modulus of a right-half plane
zero of P. (See Example 0.2.)
|N
p (s)| and |N 0 (s)| are both small for some s. That is, |N p (s)| ≈ 0
(i.e., s is close to a right-half plane pole of P ) and |N 0 (s)| ≈ 0 (i.e.,
|P (jω)| is small around ω = |s| = r). This is only possible if |P (jω)|
is small around ω = r, where r is the modulus of a right-half plane
pole of P. (See Example 0.3.)
|N
0
(s)| and |M 0
(s)| are both small for some s. That is, |N 0
(s)| ≈ 0
(i.e., |P (jω)| is small around ω = |s| = r) and |M 0
(s)| ≈ 0 (i.e.,
|P (jω)| is large around ω = |s| = r). The only way in which |P (jω)|
can be both small and large at frequencies near ω = r is that |P (jω)|
is approximately equal to 1 and the absolute value of the slope of
|P (jω)| is large. (See Example 0.4.)
RHP Poles/Zeros are close
Example 0.
P
1 (s) =
K(s − r)
(s + 1)(s − 1)
b opt
(P
1 ) will be very small for all K whenever r is close to 1 (i.e.,
whenever there is an unstable pole close to an unstable zero).
r 0. 5 0. 7 0. 9 1. 1 1. 3 1. 5
K = 0. 1 b opt
(P
1
r 0. 5 0. 7 0. 9 1. 1 1. 3 1. 5
K = 1 b opt
(P
1
r 0. 5 0. 7 0. 9 1. 1 1. 3 1. 5
K = 10 b opt
(P
1
10
− 10
− 10
0 10
1 10
2
10
−
10
−
10
0
10
1
K=0.
K=
K=
Figure 0.30: Frequency responses of P 1 for r = 0.9 and K = 0. 1 , 1, and 10
Complex Nonminimum Phase Zeros
P
3
(s) =
K[(s − cos θ)
2
- sin
2 θ]
s[(s + cos θ)
2
- sin
2
θ]
θ (degree) 0 45 60 80 85
K = 0. 1 b opt
(P
3
θ (degree) 0 45 60 80 85
K = 1 b opt
(P
3
θ (degree) 0 45 60 80 85
K = 10 b opt
(P
3
- b opt
(P
3 ) will be small if |P 3 (jω)| is large around the frequency of ω = 1
(the modulus of the right-half plane zero).
- for zeros with the same modulus, b opt
(P
3 ) will be smaller for a plant
with relatively larger real part zeros than for a plant with relatively
larger imaginary part zeros (i.e., a pair of real right-half plane zeros
has a much worse effect on the performance than a pair of almost pure
imaginary axis right-half plane zeros of the same modulus).
Unstable Poles
Example 0.
P
4
(s) =
K(s + 1)
s(s − 1)
b opt
(P
4
) will be small if |P 4
(jω)| is small around ω = 1 (the modulus of
the right-half plane pole).
K 0. 01 0. 1 1 10 100
b opt
(P
4
Note that b opt
(P
4
) −→ 0 .707 as K −→ ∞. This is because |P 4
(jω)|
is very large around the frequency of the modulus of the right-half plane
pole as K −→ ∞.
P
5
(s) =
K[(s + cos θ)
2
- sin
2
θ]
s[(s − cos θ)
2
- sin
2
θ]
The optimal b opt
(P
5
) for various θ’s are listed in the following table:
θ (degree) 0 45 60 80 85
K = 0. 1 b opt
(P
5
θ (degree) 0 45 60 80 85
K = 1 b opt
(P
5
θ (degree) 0 45 60 80 85
K = 10 b opt
(P
5
- b opt
(P
5 ) will be small if |P 5 (jω)| is small around the frequency of the
modulus of the right-half plane pole.
Large Slope near Crossover
Example 0.
P
6
(s) =
K(0. 2 s + 1)
4
s(s + 1)
4
10
− 10
− 10
− 10
0 10
1 10
2 10
3
10
−
10
−
10
−
10
0
10
5
10
10
10
15
K=0.
K=0.
K=
Figure 0.32: Frequency response of P 6 for K = 10
− 5 , 10
− 1 and 10
4
• K = 10
− 5 : slope near crossover is not too large =⇒ b opt
(P
6
) not too
small.
• K = 10
4 : Similar.
- K = 0.1: slope near crossover is quite large =⇒ b opt
(P
6
) quite small.
K 10
− 5 10
− 3
- 1 1 10 10
2 10
4
b opt
(P
6
Guidelines
Based on the preceding discussion, we can give some guidelines for the
loop-shaping design.
♥ The loop transfer function should be shaped in such a way that it has
low gain around the frequency of the modulus of any right-half plane
zero z. Typically, it requires that the crossover frequency be much
smaller than the modulus of the right-half plane zero; say, ω c < |z|/ 2
for any real zero and ω c < |z| for any complex zero with a much larger
imaginary part than the real part (see Figure 0.29).
♥ The loop transfer function should have a large gain around the fre-
quency of the modulus of any right-half plane pole.
♥ The loop transfer function should not have a large slope near the
crossover frequencies.
These guidelines are consistent with the rules used in classical control
theory (see Bode [1945] and Horowitz [1963]).