Lecture 3: How to Compute H2 and H∞ -Norms
Topics to discuss:
• Computing H2 -Norm for Stable Linear Systems
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 1/19
Lecture 3: How to Compute H2 and H∞ -Norms
Topics to discuss:
• Computing H2 -Norm for Stable Linear Systems
• Computing H∞ -Norm for Stable Linear Systems
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 1/19
Computing H2 -Norm:
H2 norm is the energy of the output of the system






w
C1
y1
 1 




Ẋ = A X + B1 , . . . , Bm  · · ·  ,  · · ·  =  · · ·  X
Cp
wm
yp
obtained in response to a vector impulse functions w, where
• y(t) is a p vector, y(t) = y1 (t), . . . , yp (t) T
T
• w is an m vector, w(t) = δ1 (t), . . . , δm (t)
T
−1
• G(s) = C1 , . . . , Cp (s In×n − A)
B1 , . . . , Bm is
the p × m transfer matrix
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 2/19
Computing H2 -Norm:
H2 norm is the energy of the output of the system






w
C1
y1
 1 




Ẋ = A X + B1 , . . . , Bm  · · ·  ,  · · ·  =  · · ·  X
Cp
wm
yp
obtained in response to a vector impulse functions w, where
• y(t) is a p vector, y(t) = y1 (t), . . . , yp (t) T
T
• w is an m vector, w(t) = δ1 (t), . . . , δm (t)
T
−1
• G(s) = C1 , . . . , Cp (s In×n − A)
B1 , . . . , Bm is
the p × m transfer matrix
The yi (t)-component to the impulse in wj -channel is
(
Ci eAt Bj , t > 0
yij (t) =
0,
t≤0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 2/19
Computing H2 -Norm:
H2 norm is the energy of the output of the system






w
C1
y1
 1 




Ẋ = A X + B1 , . . . , Bm  · · ·  ,  · · ·  =  · · ·  X
Cp
wm
yp
T
obtained in response to w(t) = δ1 (t), . . . , δm (t) .
The yi (t)-component to the impulse in wj -channel is

(
y11 (t) · · ·
At
C i e Bj , t > 0

yij (t) =
g(t) :=  · · ·
0,
t≤0
yp1 (t) · · ·
y1m (t)

··· 
ypm (t)
The H2 -norm of the system is the sum of the energies of yij (t)
v
v
u +∞
u +∞
Z X
p X
uZ
u
m
2 (t)dt = u
T
G(·) := u
t
t
trace
[g
(t)g(t)] dt
y
ij
2
0
c A. Shiriaev/L. Freidovich.
i=1 j=1
January 26, 2010.

0
Optimal Control for Linear Systems:
Lecture 3
– p. 2/19
Computing H2 -Norm:
Given a stable MIMO linear system
ẋ = A x + B w,
y =Cx
there are several ways to compute its H2 -norm:
+∞
R
1. Directly evaluate
trace [g T (t)g(t)] dt numerically!
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 3/19
Computing H2 -Norm:
Given a stable MIMO linear system
ẋ = A x + B w,
y =Cx
there are several ways to compute its H2 -norm:
+∞
R
1. Directly evaluate
trace [g T (t)g(t)] dt numerically!
0
2. Use Laplace transform and Parceval relation
Z +∞
• Lg(t) (s) =
g(t)e−st dt := G(s)
0
•
+∞
Z
trace [g (t)g(t)] dt =
T
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
1
2π
+∞
Z
trace [G∗ (jω)G(jω)] dω
−∞
Optimal Control for Linear Systems:
Lecture 3
– p. 3/19
Computing H2 -Norm:
Given a stable MIMO linear system
ẋ = A x + B w,
y =Cx
there are several ways to compute its H2 -norm:
+∞
R
1. Directly evaluate
trace [g T (t)g(t)] dt numerically!
0
2. Use Laplace transform and Parceval relation
Z +∞
• Lg(t) (s) =
g(t)e−st dt := G(s)
0
•
+∞
Z
trace [g (t)g(t)] dt =
T
0
1
2π
+∞
Z
trace [G∗ (jω)G(jω)] dω
−∞
3. Solve a Lyapunov equation without computing integrals!
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 3/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
v
u +∞
uZ
T
At
At
G(·) = u
t
trace Ce B Ce B dt
2
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
v
u +∞
uZ
Tt
T
A
T
At
G(·) = u
t
trace B e
C Ce B dt
2
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
v
u
u +∞
Z
u
AT t T
At
T
G(·) = u
trace
B
e
C
Ce
B dt
t
2
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
v


u
+∞
u
Z
u

 T
AT t T
At
G(·) = u
e
C
Ce
dt
B
B
trace


t
2
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
q
G(·) = trace B T Q B ,
2
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Q=
+∞
Z
e
AT t
C T CeAt dt
0
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Computing H2 -Norm:
v
u +∞
uZ
u
trace [g T (t)g(t)] dt
kG(·)k2 = t
0
q
G(·) = trace B T Q B ,
2
Q=
+∞
Z
e
AT t
C T CeAt dt
0
Important Observation:
The observability Grammian Q can be computed without
integration as a solution of the Lyapunov equation
QA + AT Q + C T C = 0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 4/19
Example 1
Consider the system
ẍ = −ẋ + u + w,
u is control variable, w is disturbance
To stabilize the position x = 0, we have chosen P-controller
u = −10 x
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 5/19
Example 1
Consider the system
ẍ = −ẋ + u + w,
u is control variable, w is disturbance
To stabilize the position x = 0, we have chosen P-controller
u = −10 x
Then the closed-loop system is
ẍ = −10x − ẋ + w
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 5/19
Example 1
Consider the system
ẍ = −ẋ + u + w,
u is control variable, w is disturbance
To stabilize the position x = 0, we have chosen P-controller
u = −10 x
Then the closed-loop system is
ẍ = −10x − ẋ + w
The state-space model of the system is
# " #
"
#
"
#"
d
x
0
x
0
1
w,
+
=
ẋ
1
−10 −1
dt ẋ
{z
}
| {z }
|
=A
=B
c A. Shiriaev/L. Freidovich.
January 26, 2010.
y = 1, 0
| {z }
=C
Optimal Control for Linear Systems:
Lecture 3
"
x
ẋ
#
– p. 5/19
Example 1
Consider the system
ẍ = −ẋ + u + w,
u is control variable, w is disturbance
To stabilize the position x = 0, we have chosen P-controller
u = −10 x
Then the closed-loop system is
ẍ = −10x − ẋ + w
The state-space model of the system is
# " #
"
#
"
#"
d
x
0
x
0
1
w,
+
=
ẋ
1
−10 −1
dt ẋ
{z
}
| {z }
|
=A
=B
y = 1, 0
| {z }
=C
"
x
ẋ
#
How to access the quality of the feedback design?
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 5/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
that is
"
# "
#T "
#"
#
q11 q12
0
1
0
1
q11 q12
=
+
q12 q22
q12 q22
−10 −1
−10 −1
= − 1, 0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
T Lecture 3
1, 0
– p. 6/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
that is
"
# "
#"
#"
#
#
"
q11 q12
0
1
0 −10
q11 q12
1 0
+
=−
q12 q22
q12 q22
−10 −1
1 −1
0 0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 6/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
that is
"
# "
#"
#"
#
#
"
q11 q12
0
1
0 −10
q11 q12
1 0
+
=−
q12 q22
q12 q22
−10 −1
1 −1
0 0
"
−10q12 q11 − q12
−10q22 q12 − q22
c A. Shiriaev/L. Freidovich.
# "
+
−10q12
−10q22
q11 − q12 q12 − q22
January 26, 2010.
#
Optimal Control for Linear Systems:
=−
"
Lecture 3
1 0
0 0
#
– p. 6/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
that is
"
# "
#"
#"
#
#
"
q11 q12
0
1
0 −10
q11 q12
1 0
+
=−
q12 q22
q12 q22
−10 −1
1 −1
0 0
"
−10q12 q11 − q12
−10q22 q12 − q22
−20q12 = −1,
c A. Shiriaev/L. Freidovich.
# "
+
−10q12
−10q22
q11 − q12 q12 − q22
q11 − q12 − 10q22 = 0,
January 26, 2010.
#
=−
"
1 0
0 0
#
q12 − q22 = 0
Optimal Control for Linear Systems:
Lecture 3
– p. 6/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
that is
"
# "
#"
#"
#
#
"
q11 q12
0
1
0 −10
q11 q12
1 0
+
=−
q12 q22
q12 q22
−10 −1
1 −1
0 0
"
−10q12 q11 − q12
−10q22 q12 − q22
−20q12 = −1,
+
−10q12
−10q22
q11 − q12 q12 − q22
q11 − q12 − 10q22 = 0,
q12 =
c A. Shiriaev/L. Freidovich.
# "
1
20
,
q22 =
January 26, 2010.
1
20
,
q11 =
#
=−
"
1 0
0 0
#
q12 − q22 = 0
11
20
Optimal Control for Linear Systems:
Lecture 3
– p. 6/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
and the solution is
Q=
c A. Shiriaev/L. Freidovich.
1
20
January 26, 2010.
"
11 1
1 1
#
Optimal Control for Linear Systems:
Lecture 3
– p. 7/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
and the solution is
Q=
1
20
"
11 1
1 1
#
The H2 -norm of our linear system (squared) is then
"
#" #
2
1
0
G(·) = trace [B T QB] = 1 [0, 1] 11 1
=
= 0.05
2
1 1
1
20
20
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 7/19
Example 1 (Cont’d)
To compute H2 -norm, we need to solve the Lyapunov equation
QA + AT Q + C T C = 0
and the solution is
Q=
1
20
"
11 1
1 1
#
The H2 -norm of our linear system (squared) is then
"
#" #
2
1
0
G(·) = trace [B T QB] = 1 [0, 1] 11 1
=
= 0.05
2
1 1
1
20
20
A=[0, 1; -10, -1]; B=[0; 1]; C=[1, 0]; D=0;
plant=ss(A,B,C,D);
% computing H2-norm
H2=norm(plant,2), H2ˆ2
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 7/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
c A. Shiriaev/L. Freidovich.
Z
+∞
T
eAt BB T eA t dt
0
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
Z
+∞
0
+∞
g(t) g (t)dt =
T
0
c A. Shiriaev/L. Freidovich.
T
eAt BB T eA t dt
January 26, 2010.
Z
+∞
Ce
0
At
B Ce
At
B
Optimal Control for Linear Systems:
T
dt
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
Z
+∞
T
eAt BB T eA t dt
0
+∞
g(t) g (t)dt =
T
0
=
Z
+∞
0
Z +∞
Ce
At
Ce
At
B Ce
T
BB e
At
B
AT t
T
dt
C T dt
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
Z
+∞
T
eAt BB T eA t dt
0
+∞
g(t) g (t)dt =
T
0
=
Z
+∞
0
Z +∞
Ce
At
Ce
At
B Ce
T
BB e
At
B
AT t
T
dt
C T dt
0
= C
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Z
+∞
e
At
T
BB e
AT t
0
Optimal Control for Linear Systems:
dt C T
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
+∞
g(t) g (t)dt =
T
0
=
Z
+∞
T
eAt BB T eA t dt
0
Z
+∞
0
Z +∞
Ce
At
Ce
At
B Ce
T
BB e
At
B
AT t
T
dt
C T dt
0
= C
c A. Shiriaev/L. Freidovich.
Z
+∞
e
0
January 26, 2010.
At
T
BB e
AT t
dt C T = CP C T
Optimal Control for Linear Systems:
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
2
G(·) =
2
Z
Z
+∞
T
eAt BB T eA t dt
0
+∞
g(t) g T (t)dt = CP C T
0
+∞
trace [g T (t) g(t)] dt =
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Z
+∞
trace [g(t) g T (t)] dt
0
Optimal Control for Linear Systems:
Lecture 3
– p. 8/19
Alternative Way of Computing H2 -Norm:
Important Observation: A solution of the linear equation
AP + P AT + BB T = 0
i.e. the controllability Grammian, has the form
P =
Z
G(·)2 =
2
Z
+∞
T
eAt BB T eA t dt
0
+∞
g(t) g T (t)dt = CP C T
0
Z
+∞
trace [g(t) g T (t)] dt = trace [CP C T ]
0
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 8/19
Lecture 3: How to Compute H2 and H∞ -Norms
• Computing H2 -Norm for Stable Linear Systems
• Computing H∞ -Norm for Stable Linear Systems
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 9/19
H∞ -norm for SISO stable systems
The steady-state output of the SISO stable system
Ẋ = A X + B w,
y =CX
to the input function
w(t) = a sin(ω t + φ)
with unknown a, ω, φ ∈ R1 , a 6= 0, is
yss (t) = G(jω) · a · sin ω t + φ + arg G(jω)
where G(s) = C(sI − A)−1 B ⇐ Y (s) = G(s) W (s).
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 10/19
H∞ -norm for SISO stable systems
The steady-state output of the SISO stable system
Ẋ = A X + B w,
y =CX
to the input function
w(t) = a sin(ω t + φ)
with unknown a, ω, φ ∈ R1 , a 6= 0, is
yss (t) = G(jω) · a · sin ω t + φ + arg G(jω)
where G(s) = C(sI − A)−1 B ⇐ Y (s) = G(s) W (s).
The H∞ -norm is the maximal possible amplification, i.e.
sup |Y (jω)|
1
ω∈R
G(jω)
G(·) =
=
sup
∞
sup |W (jω)|
ω∈R1
ω∈R1
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 10/19
Example 1 (Cont’d)
Our closed-loop system is
# " #
"
#
"
#"
d
x
0
x
0
1
+
w,
=
ẋ
1
−10 −1
dt ẋ
{z
}
| {z }
|
=A
=B
c A. Shiriaev/L. Freidovich.
January 26, 2010.
y = 1, 0
| {z }
=C
Optimal Control for Linear Systems:
Lecture 3
"
x
ẋ
#
– p. 11/19
Example 1 (Cont’d)
Our closed-loop system is
# " #
"
#
"
#"
d
x
0
x
0
1
+
w,
=
ẋ
1
−10 −1
dt ẋ
{z
}
| {z }
|
=A
=B
y = 1, 0
| {z }
=C
"
x
ẋ
#
% computing Hinf-norm
Hinf=norm(plant,inf)
ga=Hinf;
[mag,phase,w] = bode(plant,{0.01,10}), grid;
magnitudes=mag(:);
figure(1),
ga line=ga*ones(size(w));
plot(w,magnitudes,w,ga line,’r--’), grid
xlabel(’frequency’), ylabel(’magnitude’)
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 11/19
H∞ -norm for MIMO stable systems
The steady-state k-th output of the general stable system




C1 X
y1
m
X




Ẋ = A X +
=
Bi wi ,
·
·
·
·
·
·




i=1
Cp X
yp
to the input functions wi (t) = ai sin(ω t + φi ),
m
X
yk (t) =
Gki (jω) · ai · sin ω t + φi + arg Gki (jω)
i=1
where
Gki (s) = Ck (sI − A)−1 Bi ⇐ Yk (s) = Gki (s) Wi (s).
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 12/19
H∞ -norm for MIMO stable systems
The steady-state k-th output of the general stable system




C1 X
y1
m
X




Ẋ = A X +
=
Bi wi ,
·
·
·
·
·
·




i=1
Cp X
yp
to the input functions wi (t) = ai sin(ω t + φi ),
m
X
yk (t) =
Gki (jω) · ai · sin ω t + φi + arg Gki (jω)
i=1
where
Gki (s) = Ck (sI − A)−1 Bi ⇐ Yk (s) = Gki (s) Wi (s).
The H∞ -norm is the maximal
possiblevector amplification
p
P
|Yk (jω)|2
sup
1
G(·) = ω∈R k=1
= sup σ̄ G(jω)
m
∞
P
ω∈R1
2
|Wi (jω)|
sup
ω∈R1
c A. Shiriaev/L. Freidovich.
i=1
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 12/19
Computing an Approximation for H∞ -Norm
Given a stable linear system
ẋ = Ax+Bw,
y = Cx+Dw,
G(s) = D + C(sIn − A)
−1
B ,
its H∞ -norm equals the largest singular value
G(·)
∞
= sup σ̄ G(jω) = sup
ω∈R1
c A. Shiriaev/L. Freidovich.
January 26, 2010.
ω∈R1
r
max λi {G∗ (jω) G(jω)}
1≤i≤p
Optimal Control for Linear Systems:
Lecture 3
– p. 13/19
Computing an Approximation for H∞ -Norm
Given a stable linear system
ẋ = Ax+Bw,
y = Cx+Dw,
G(s) = D + C(sIn − A)
−1
B ,
its H∞ -norm equals the largest singular value
G(·)
∞
= sup σ̄ G(jω) = sup
ω∈R1
To approximate G(·)
∞
ω∈R1
r
max λi {G∗ (jω) G(jω)}
1≤i≤p
we can choose a set of frequencies
{ω1 , ω2 , . . . , ωN }
and search for
max σ̄ {G(jωk )} ≈ kG(·)k∞
1≤k≤N
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 13/19
Computing a Bound on H∞ -Norm if D
=0
Given a number γ > 0 and a stable linear system
−1
ẋ = A x + B w, y = C x,
G(s) = C(sIn − A) B
Then G(·)∞ < γ if and only if
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 14/19
Computing a Bound on H∞ -Norm if D
=0
Given a number γ > 0 and a stable linear system
−1
ẋ = A x + B w, y = C x,
G(s) = C(sIn − A) B
Then G(·)∞ < γ if and only if the following 2n × 2n matrix

H(A, B, C, γ) = 
A
−C T C
1
γ2
B BT
−AT
has no imaginary eigenvalues.
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:


Lecture 3
– p. 14/19
Computing a Bound on H∞ -Norm if D
=0
Given a number γ > 0 and a stable linear system
−1
ẋ = A x + B w, y = C x,
G(s) = C(sIn − A) B
Then G(·)∞ < γ if and only if the following 2n × 2n matrix

H(A, B, C, γ) = 
A
−C T C
1
γ2

B BT

−AT
has no imaginary eigenvalues.
The proof is based on defining
Φ(s) :=
γ2

I − G (−s) G(s),
T
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Φ−1
A


=  −C T C

0
1
γ2
BB
T
1
γ2
−AT
1
γ2
BT
Optimal Control for Linear Systems:
B
0
1
γ2
Lecture 3
I





– p. 14/19
Computing a Bound on H∞ -Norm if D
=0
Given a number γ > 0 and a stable linear system
−1
ẋ = A x + B w, y = C x,
G(s) = C(sIn − A) B
Then G(·)∞ < γ if and only if the following 2n × 2n matrix

H(A, B, C, γ) = 
A
−C T C
1
γ2
B BT
−AT
has no imaginary eigenvalues.


The proof is based on defining
Φ(s) := γ 2 I − GT (−s) G(s)
and using the fact that G(·)∞ < γ ⇔ Φ(jω) > 0 (positive
definite) for ω ∈ R1 ⇔ Φ−1 (jω) has no imaginary poles.
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 14/19
Computing a Lower Bound on H∞ -Norm
Given a number γ > 0 and a stable linear system
−1
ẋ = Ax+Bw, y = Cx+Dw,
G(s) = D + C(sIn − A) B
Then |G(·)|∞ < γ if and only if
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 15/19
Computing a Lower Bound on H∞ -Norm
Given a number γ > 0 and a stable linear system
−1
ẋ = Ax+Bw, y = Cx+Dw,
G(s) = D + C(sIn − A) B
Then |G(·)|∞ < γ if and only if
• σ̄ D < γ ;
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 15/19
Computing a Lower Bound on H∞ -Norm
Given a number γ > 0 and a stable linear system
−1
ẋ = Ax+Bw, y = Cx+Dw,
G(s) = D + C(sIn − A) B
Then |G(·)|∞ < γ if and only if
• σ̄ D < γ ;
• The following 2n × 2n matrix H = H(A, B, C, D, γ) has
no imaginary eigenvalues
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 15/19
Computing a Lower Bound on H∞ -Norm
Given a number γ > 0 and a stable linear system
−1
ẋ = Ax+Bw, y = Cx+Dw,
G(s) = D + C(sIn − A) B
Then |G(·)|∞ < γ if and only if
• σ̄ D < γ ;
• The following 2n × 2n matrix H = H(A, B, C, D, γ) has
no imaginary eigenvalues

H = 
A + BR−1 D T C
−C T I + DR−1 D
R = γIn − D T D
c A. Shiriaev/L. Freidovich.
January 26, 2010.
T
BR−1 B T
C − A + BR−1 D T C
Optimal Control for Linear Systems:

T 
Lecture 3
– p. 15/19
Example 2
m1 ẍ1 = −k1 (x1 − x2 ) − b1 (ẋ1 − ẋ2 ) + u1
m2 ẍ2 = k1 (x1 − x2 ) + b1 (ẋ1 − ẋ2 ) − k2 x2 − b2 ẋ2 + u2
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 16/19
Example 2 (Cont’d)
The dynamical system
m1 ẍ1 = −k1 (x1 − x2 ) − b1 (ẋ1 − ẋ2 ) + u1
m2 ẍ2 = k1 (x1 − x2 ) + b1 (ẋ1 − ẋ2 ) − k2 x2 − b2 ẋ2 + u2
can be written in the state-space from as follows



0
z1

 0
d 
z

 2
  =  k1
 − m1
dt  z3 
k1
z4
m2
0
0
1
0
b1
−m
1
k1
m1
+k2
− k1m
2
"
c A. Shiriaev/L. Freidovich.
y1
y2
#
b1
m2

0
z1
 z   0
  2 
   + 1
b1
  z3   m1
m1
+b2
z4
0
− b1m
2
0
1


# z1

1 0 0 0 
 z2 
=
 
0 1 0 0  z3 
z4
"
January 26, 2010.
 
Optimal Control for Linear Systems:
Lecture 3
0
0
0
1
m2

" #
 u

1

 u2
– p. 17/19
Example 2 (Cont’d)
k1=1; k2=4; b1=0.2; b2=0.1; m1=1; m2=2;
A=[0, 0, 1, 0; 0, 0, 0, 1;...
-k1/m1, k1/m1, -b1/m1, b1/m1;
k1/m2, -(k2+k1)/m2, b1/m2, -(b1+b2)/m2];
B=[0, 0; 0, 0; 1/m1, 0; 0, 1/m2];
C=[1, 0, 0, 0; 0, 1, 0, 0];
D=zeros(2);
plant=ss(A,B,C,D);
% computing H2-norm
H2=norm(plant,2), h2norm(pck(A,B,C,D)),
% computing Hinf-norm
Hinf=norm(plant,inf), hinfnorm(pck(A,B,C,D),0.00001),
figure(2), sigma(plant,0.1,10); grid on
disp(20*log10(Hinf)),
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 18/19
Next Lecture / Assignments:
Next meeting: January 27, 13:15-15:00, in A205,
Next lecture (January 29, 10:15-12:00, in A206):
“Well-Posedness and Internal Stability”.
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 19/19
Next Lecture / Assignments:
Next meeting: January 27, 13:15-15:00, in A205,
Next lecture (January 29, 10:15-12:00, in A206):
“Well-Posedness and Internal Stability”.
Problem: Let G(s) =
1
(s2
.
+ 2 ζ s + 1)(s + 1)
Compute G(·)2 and G(·)∞ using Bode plots and
state-space algorithms, respectively for ζ ∈ 1, 0.1, 0.0001
and compare the results.
Verify your solutions using Matlab. Show all your work.
The assignment is due at 13:15 on February 2, 2010.
c A. Shiriaev/L. Freidovich.
January 26, 2010.
Optimal Control for Linear Systems:
Lecture 3
– p. 19/19