Elgersburg Workshop 2013
February 11–14, 2013
On the Kalman–Yakubovich–Popov Lemma
and its Application in Model Order Reduction
Peter Benner
Computational Methods in Systems and Control Theory
Max Planck Institute for Dynamics of Complex Technical Systems
Magdeburg, Germany
joint work (in parts) with Matthias Voigt and Xin Du, Guanghong Yang, and Dan Ye
MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
1/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Overview
1
Linear Systems Basics
2
Dissipativity and Structural Properties
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
5
Frequency-dependent KYP Lemma and Model Reduction
6
Numerical Examples
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
2/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Conclusions
References
Peter Benner, KYP and Balanced Truncation
3/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Linear Systems
LTI Systems
(
Σ:
ẋ(t) = Ax(t) + Bu(t),
y (t) = Cx(t) + Du(t),
x(0) = x0
with
A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m ,
state vector x(t) ∈ Rn ,
input vector u(t) ∈ Rm ,
output vector y (t) ∈ Rp .
u(t)
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ẋ(t) = Ax(t) + Bu(t)
y (t) = Cx(t) + Du(t)
y (t)
Peter Benner, KYP and Balanced Truncation
4/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Stability and Controllability
Definitions
The system Σ is called
(asymptotically) stable if limt→∞ x(t) = 0 for u ≡ 0;
controllable if for all x1 ∈ Rn there exist t1 > 0 and an input signal
u(t) such that x(t1 ) = x1 .
observable if y (t) ≡ 0 implies x(t) ≡ 0 (assuming u(t) ≡ 0).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
5/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Stability and Controllability
Definitions
The system Σ is called
(asymptotically) stable if limt→∞ x(t) = 0 for u ≡ 0;
controllable if for all x1 ∈ Rn there exist t1 > 0 and an input signal
u(t) such that x(t1 ) = x1 .
observable if y (t) ≡ 0 implies x(t) ≡ 0 (assuming u(t) ≡ 0).
Equivalent Conditions
The system Σ is
(asymptotically) stable ⇐⇒ all eigenvalues of A are in the open left
half-plane;
controllable ⇐⇒ rank λIn − A B = n for all λ ∈ C.
observable ⇐⇒ rank λIn − AT C T = n for all λ ∈ C.
minimal if it is controllable and observable.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
5/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Transfer function
Assume x(0) = 0. Then
(
L(Σ) :
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L{ẋ}(s) = AL{x}(s) + BL{u}(s),
L{y }(s) = C L{x}(s) + DL{u}(s),
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Transfer function
Assume x(0) = 0. Then
(
s(L{x}(s) − x(0)) = AL{x}(s) + BL{u}(s),
L(Σ) :
L{y }(s) = C L{x}(s) + DL{u}(s),
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Transfer function
Assume x(0) = 0. Then
(
L(Σ) :
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sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y }(s) = C L{x}(s) + DL{u}(s),
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Transfer function
Assume x(0) = 0. Then
(
L(Σ) :
sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y }(s) = C L{x}(s) + DL{u}(s),
Then
L{y }(s) = C (sIn − A)−1 B L{u}(s).
|
{z
}
=:G (s)
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency Domain Analysis
Laplace transform
Z
L{f }(s) :=
∞
e −st f (t)dt
0
Transfer function
Assume x(0) = 0. Then
(
L(Σ) :
sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y }(s) = C L{x}(s) + DL{u}(s),
Then
L{y }(s) = C (sIn − A)−1 B L{u}(s).
|
{z
}
=:G (s)
The transfer function G (s) maps inputs to outputs in the frequency
domain.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
6/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Conclusions
References
Peter Benner, KYP and Balanced Truncation
7/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Dissipative Systems
Definition
[Scherer, Weiland ’05]
A dynamical system Σ is called dissipative with respect to a supply
function s : Rp × Rm −→ R if there exists a storage function
V : Rn −→ R such that the dissipation inequality
Z t1
V (x(t1 )) ≤ V (x(0)) +
s(y (t), u(t))dt
0
is fulfilled for all 0 ≤ t1 .
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
8/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Dissipative Systems
Definition
[Scherer, Weiland ’05]
A dynamical system Σ is called dissipative with respect to a supply
function s : Rp × Rm −→ R if there exists a storage function
V : Rn −→ R such that the dissipation inequality
Z t1
V (x(t1 )) ≤ V (x(0)) +
s(y (t), u(t))dt
0
is fulfilled for all 0 ≤ t1 .
Interpretation
Z
t1
s(y (t), u(t))dt can be seen as the energy supplied to the system
0
in the time interval [0, t1 ].
s(y (t), u(t)) is a measure for the power at time t.
V (x(t)) is the internal energy at time t.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
8/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Quadratic Supply Functions
Often, we consider
T W
y (t)
s(y (t), u(t)) =
u(t)
ST
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S
R
y (t)
u(t)
with W = W T , R = R T
Peter Benner, KYP and Balanced Truncation
9/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Quadratic Supply Functions
Often, we consider
T W S y (t)
y (t)
s(y (t), u(t)) =
with W = W T , R = R T
u(t)
S T R u(t)
T W S Cx(t) + Du(t)
Cx(t) + Du(t)
=
u(t)
u(t)
ST R
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Peter Benner, KYP and Balanced Truncation
9/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Quadratic Supply Functions
Often, we consider
T W S y (t)
y (t)
s(y (t), u(t)) =
with W = W T , R = R T
u(t)
S T R u(t)
T W S Cx(t) + Du(t)
Cx(t) + Du(t)
=
u(t)
u(t)
ST R
T T
x(t)
C WC
C T WD + C T S
x(t)
=
u(t)
D T WC + S T C D T WD + D T S + S T D + R u(t)
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
9/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Quadratic Supply Functions
Often, we consider
T W S y (t)
y (t)
s(y (t), u(t)) =
with W = W T , R = R T
u(t)
S T R u(t)
T W S Cx(t) + Du(t)
Cx(t) + Du(t)
=
u(t)
u(t)
ST R
T T
x(t)
C WC
C T WD + C T S
x(t)
=
u(t)
D T WC + S T C D T WD + D T S + S T D + R u(t)
T x(t)
W̃ S̃ x(t)
=:
u(t)
S̃ T R̃ u(t)
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
9/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Quadratic Supply Functions
Often, we consider
T W S y (t)
y (t)
s(y (t), u(t)) =
with W = W T , R = R T
u(t)
S T R u(t)
T W S Cx(t) + Du(t)
Cx(t) + Du(t)
=
u(t)
u(t)
ST R
T T
x(t)
C WC
C T WD + C T S
x(t)
=
u(t)
D T WC + S T C D T WD + D T S + S T D + R u(t)
T x(t)
W̃ S̃ x(t)
=:
u(t)
S̃ T R̃ u(t)
=: s̃(x(t), u(t)).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
9/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Passivity
T y (t)
0 Im y (t)
s(y (t), u(t)) =
,
u(t)
Im 0 u(t)
T x(t)
x(t)
0
CT
.
s̃(x(t), u(t)) =
u(t)
C D + D T u(t)
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Peter Benner, KYP and Balanced Truncation
10/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Passivity
T y (t)
0 Im y (t)
s(y (t), u(t)) =
,
u(t)
Im 0 u(t)
T x(t)
x(t)
0
CT
.
s̃(x(t), u(t)) =
u(t)
C D + D T u(t)
Contractivity
T y (t)
−Ip 0 y (t)
,
u(t)
0
Im u(t)
T x(t)
−C T C
−C T D
x(t)
s̃(x(t), u(t)) =
.
u(t)
−D T C Im − D T D u(t)
s(y (t), u(t)) =
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
10/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Dissipativity in the Frequency Domain
Definition: Popov function
Φ(s) =
Max Planck Institute Magdeburg
H W
(sIn − A)−1 B
Im
ST
S
R
(sIn − A)−1 B
Im
Peter Benner, KYP and Balanced Truncation
11/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Dissipativity in the Frequency Domain
Definition: Popov function
Φ(s) =
H W
(sIn − A)−1 B
Im
ST
S
R
(sIn − A)−1 B
Im
Theorem
Let Σ be controllable. Then, Σ is dissipative with respect to
T W S x(t)
x(t)
s̃(x(t), u(t)) =
if and only if Φ(iω) < 0 holds
u(t)
S T R u(t)
for all iω ∈ iR\Λ(A).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
11/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Passivity and positive realness
A dynamical system is passive if and only its transfer function G is
positive real, i.e.,
G (s) + G H (s) < 0
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∀s ∈ C+ .
Peter Benner, KYP and Balanced Truncation
12/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Passivity and positive realness
A dynamical system is passive if and only its transfer function G is
positive real, i.e.,
G (s) + G H (s) < 0
∀s ∈ C+ .
Contractivity and bounded realness
A dynamical system is contractive if and only its transfer function G is
bounded real, i.e.,
Im − G H (s)G (s) < 0
Max Planck Institute Magdeburg
∀s ∈ C+ .
Peter Benner, KYP and Balanced Truncation
12/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Passivity and positive realness
A dynamical system is passive if and only its transfer function G is
positive real, i.e.,
G (s) + G H (s) < 0
∀s ∈ C+ .
Contractivity and bounded realness
A dynamical system is contractive if and only its transfer function G is
bounded real, i.e.,
Im − G H (s)G (s) < 0
∀s ∈ C+ .
Remark
In contrast to general dissipativity, positive and bounded realness are
properties of Φ(s) in the whole open right half-plane. It can be shown
that for these cases V (x(t)) = x(t)T Xx(t) for an X = X T < 0.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
12/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Relations to H∞ Optimal Control
Problem setting
[Green, Limebeer ’95]
Plant P, dynamic compensator K,
w
z
u
P
K
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y
noise w , estimation error z.
Goal: Find K that stabilizes the system
and minimizes the influence of w on z!
( = minimizing the H∞ -norm of
closed-loop transfer function)
Peter Benner, KYP and Balanced Truncation
13/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Relations to H∞ Optimal Control
Problem setting
[Green, Limebeer ’95]
Plant P, dynamic compensator K,
w
z
u
P
K
y
noise w , estimation error z.
Goal: Find K that stabilizes the system
and minimizes the influence of w on z!
( = minimizing the H∞ -norm of
closed-loop transfer function)
H∞ -spaces
p×m
H∞
(iω) = Banach space of p × m matrix-valued functions which are
analytic and bounded in the open right half-plane.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
13/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Relations to H∞ Optimal Control
Problem setting
[Green, Limebeer ’95]
Plant P, dynamic compensator K,
w
z
u
P
y
noise w , estimation error z.
Goal: Find K that stabilizes the system
and minimizes the influence of w on z!
( = minimizing the H∞ -norm of
closed-loop transfer function)
K
H∞ -spaces
p×m
H∞
(iω) = Banach space of p × m matrix-valued functions which are
analytic and bounded in the open right half-plane.
H∞ -norm (in this setting)
kG kH∞ = sup σmax (G (s)) = sup σmax (G (iω))
s∈C+
ω∈R
= inf γ 2 Im − G H (iω)G (iω) < 0 ∀ω ∈ R .
γ≥0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
13/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Conclusions
References
Peter Benner, KYP and Balanced Truncation
14/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Algebraic Characterizations
Dissipativity can be characterized by properties of various algebraic
structures such as
linear matrix inequalities,
quadratic matrix inequalities,
algebraic matrix equations (Riccati equations, Lur’e equations),
(structured matrices and matrix pencils).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
15/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Consider again the dissipation inequality (in differential form):
T W
x(t)
u(t)
ST
s̃(x(t),u(t)) =
S
R
x(t)
u(t)
≥ V̇ (x(t))ẋ(t)
This obviously holds if there exists X = X T such that
T
W S
A X + XA XB
≥
.
ST R
BT X
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
16/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Consider again the dissipation inequality (in differential form):
T W
x(t)
u(t)
ST
s̃(x(t),u(t)) =
≥ V̇ (x(t))ẋ(t)
S
R
x(t)
u(t)
(set V (x(t)) = x(t)T Xx(t) with X = X T )
= 2x(t)T X (Ax(t) + Bu(t))
This obviously holds if there exists X = X T such that
T
W S
A X + XA XB
≥
.
ST R
BT X
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
16/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Consider again the dissipation inequality (in differential form):
T W
x(t)
u(t)
ST
s̃(x(t),u(t)) =
≥ V̇ (x(t))ẋ(t)
S
R
x(t)
u(t)
(set V (x(t)) = x(t)T Xx(t) with X = X T )
= 2x(t)T X (Ax(t) + Bu(t))
= x(t)T XAx(t) + x(t)T XBu(t) + x(t)T AT Xx(t) + u(t)T B T Xx(t)
This obviously holds if there exists X = X T such that
T
W S
A X + XA XB
≥
.
ST R
BT X
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
16/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Consider again the dissipation inequality (in differential form):
T W
x(t)
u(t)
ST
s̃(x(t),u(t)) =
≥ V̇ (x(t))ẋ(t)
S
R
x(t)
u(t)
(set V (x(t)) = x(t)T Xx(t) with X = X T )
= 2x(t)T X (Ax(t) + Bu(t))
= x(t)T XAx(t) + x(t)T XBu(t) + x(t)T AT Xx(t) + u(t)T B T Xx(t)
T T
x(t)
A X + XA XB x(t)
=
.
u(t)
u(t)
BT X
0
This obviously holds if there exists X = X T such that
T
W S
A X + XA XB
≥
.
ST R
BT X
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
16/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Theorem
[Willems ’72]
Let Σ be controllable. Then Σ is dissipative with respect to s(x(t), u(t))
(or equivalently Φ(iω) < 0 ∀iω ∈ iR\Λ(A)) if and only if there exists a
symmetric matrix X such that the linear matrix inequality (LMI)
T
A X + XA − W XB − S
40
−R
BT X − ST
is fulfilled.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
17/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Kalman-Yakubovich-Popov(-Anderson) Lemma
Theorem
[Willems ’72]
Let Σ be controllable. Then Σ is dissipative with respect to s(x(t), u(t))
(or equivalently Φ(iω) < 0 ∀iω ∈ iR\Λ(A)) if and only if there exists a
symmetric matrix X such that the linear matrix inequality (LMI)
T
A X + XA − W XB − S
40
−R
BT X − ST
is fulfilled.
History
’61: Popov’s criterion for stability of a feedback system with a
memoryless nonlinearity.
’62/’63: Original version of the lemma by Kalman and Yakubovich.
’67: Anderson’s positive real lemma for multivariate transfer
functions.
until today: Many generalizations and extensions.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
17/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Positive real lemma
Let Σ be controllable. Then Σ is passive (or equivalently G (s) is positive
real) if and only if there exists X = X T < 0 such that the LMI
T
A X + XA XB − C T
40
B T X − C −(D + D)T
is fulfilled.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
18/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Special Cases
Positive real lemma
Let Σ be controllable. Then Σ is passive (or equivalently G (s) is positive
real) if and only if there exists X = X T < 0 such that the LMI
T
A X + XA XB − C T
40
B T X − C −(D + D)T
is fulfilled.
Bounded real lemma
Let Σ be controllable. Then Σ is contractive (or equivalently G (s) is
bounded real) if and only if there exists X = X T < 0 such that the LMI
T
A X + XA + C T C XB + C T D
40
BT X + DT C
D T D − Im
is fulfilled.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
18/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
Max Planck Institute Magdeburg
solvable.
Peter Benner, KYP and Balanced Truncation
19/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
solvable.
m
Quadratic Matrix Inequality
AT X + XA − W + (XB − S) R −1 B T X − S T 4 0, X = X T
Max Planck Institute Magdeburg
solvable.
Peter Benner, KYP and Balanced Truncation
19/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
solvable.
m
Quadratic Matrix Inequality
AT X + XA − W + (XB − S) R −1 B T X − S T 4 0, X = X T
solvable.
m
Algebraic Riccati Equation
AT X + XA − W + (XB − S) R −1 B T X − S T = 0,
Max Planck Institute Magdeburg
X = XT
solvable.
Peter Benner, KYP and Balanced Truncation
19/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
Max Planck Institute Magdeburg
solvable.
Peter Benner, KYP and Balanced Truncation
20/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
solvable.
m
Quadratic Matrix Inequality
AT X + XA − W + (XB − S) R −1 B T X − S T 4 0,
X = XT
cannot be formulated!
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
20/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality
T
A X + XA − W XB − S
4 0, X = X T
−R
BT X − ST
solvable.
m
Lur’e Equation
AT X + XA − W = −K T K ,
XB − S = −K T L,
−R = −LT L,
X = XT
solvable for (X , K , L) ∈ Rn×n × Rp×n × Rp×m and p as small as possible.
first formulated in [Lur’e ’57]
More on KYP and Lur’e equations in M. Voigt’s talk on Wednesday!
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
20/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Some Remarks on Numerical Aspects
In the control literature, one often finds statements:
We have reduced the problem to an LMI =⇒ problem solved!
Good reference for LMI formulaion of control problems:
V. Balakrishnan, L. Vandenberghe, ”Semidefinite programming duality and linear
time-invariant systems”, IEEE TAC, 2003.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
21/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Some Remarks on Numerical Aspects
In the control literature, one often finds statements:
We have reduced the problem to an LMI =⇒ problem solved!
Good reference for LMI formulaion of control problems:
V. Balakrishnan, L. Vandenberghe, ”Semidefinite programming duality and linear
time-invariant systems”, IEEE TAC, 2003.
True for small dimensions, say n < 10.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
21/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Some Remarks on Numerical Aspects
In the control literature, one often finds statements:
We have reduced the problem to an LMI =⇒ problem solved!
Good reference for LMI formulaion of control problems:
V. Balakrishnan, L. Vandenberghe, ”Semidefinite programming duality and linear
time-invariant systems”, IEEE TAC, 2003.
True for small dimensions, say n < 10.
But: numerical solution of LMIs requires Semidefinite Programming
(SDP) methods, this requires generically O(n6 ) floating point
operations (flops), with some tricks and exploiting structures
O(n4.5 ).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
21/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Some Remarks on Numerical Aspects
In the control literature, one often finds statements:
We have reduced the problem to an LMI =⇒ problem solved!
Good reference for LMI formulaion of control problems:
V. Balakrishnan, L. Vandenberghe, ”Semidefinite programming duality and linear
time-invariant systems”, IEEE TAC, 2003.
True for small dimensions, say n < 10.
But: numerical solution of LMIs requires Semidefinite Programming
(SDP) methods, this requires generically O(n6 ) floating point
operations (flops), with some tricks and exploiting structures
O(n4.5 ).
Methods based on Lyapunov or Riccati equations, invariant
subspaces of Hamiltonian matrices or even pencils generically require
only O(n3 ) flops, and can be implemented in O(nmp) flops for some
large-scale problems with sparse state matrix A.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
21/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Some Remarks on Numerical Aspects
Complexity of Numerical Linear Algebra (NLA) and SDP Solutions to Control Problems
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
21/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Conclusions
References
Peter Benner, KYP and Balanced Truncation
22/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Idea
(
ẋ(t) = Ax(t) + Bu(t),
with A stable, i.e., Λ (A) ⊂ C− ,
y (t) = Cx(t) + Du(t),
is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov
equations
Σ:
AP + PAT + BB T = 0,
AT Q + QA + C T C = 0,
satisfy: P = Q = diag(σ1 , . . . , σn ) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Idea
(
ẋ(t) = Ax(t) + Bu(t),
with A stable, i.e., Λ (A) ⊂ C− ,
y (t) = Cx(t) + Du(t),
is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov
equations
Σ:
AP + PAT + BB T = 0,
AT Q + QA + C T C = 0,
satisfy: P = Q = diag(σ1 , . . . , σn ) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1 , . . . , σn } are the Hankel singular values (HSVs) of Σ.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Idea
(
ẋ(t) = Ax(t) + Bu(t),
with A stable, i.e., Λ (A) ⊂ C− ,
y (t) = Cx(t) + Du(t),
is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov
equations
Σ:
AP + PAT + BB T = 0,
AT Q + QA + C T C = 0,
satisfy: P = Q = diag(σ1 , . . . , σn ) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1 , . . . , σn } are the Hankel singular values (HSVs) of Σ.
Compute balanced realization of the system via state-space
transformation
T : (A, B, C , D)
7→
=
Max Planck Institute Magdeburg
(TAT −1 , TB, CT −1 , D)
„»
– »
–
ˆ
A11 A12
B1
,
, C1
A21 A22
B2
C2
˜
«
,D
.
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Idea
(
ẋ(t) = Ax(t) + Bu(t),
with A stable, i.e., Λ (A) ⊂ C− ,
y (t) = Cx(t) + Du(t),
is balanced, if system Gramians, i.e., solutions P, Q of the Lyapunov
equations
Σ:
AP + PAT + BB T = 0,
AT Q + QA + C T C = 0,
satisfy: P = Q = diag(σ1 , . . . , σn ) with σ1 ≥ σ2 ≥ . . . ≥ σn > 0.
{σ1 , . . . , σn } are the Hankel singular values (HSVs) of Σ.
Compute balanced realization of the system via state-space
transformation
T : (A, B, C , D)
7→
=
Truncation
Max Planck Institute Magdeburg
(TAT −1 , TB, CT −1 , D)
„»
– »
–
ˆ
A11 A12
B1
,
, C1
A21 A22
B2
C2
˜
«
,D
.
(Â, B̂, Ĉ , D̂) = (A11 , B1 , C1 , D).
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Motivation:
HSV are system invariants: they are preserved under T and determine
the energy transfer given by the Hankel map
H : L2 (−∞, 0) 7→ L2 (0, ∞) : u− 7→ y+ .
”functional analyst’s point of view”
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Motivation:
HSV are system invariants: they are preserved under T and determine
the energy transfer given by the Hankel map
H : L2 (−∞, 0) 7→ L2 (0, ∞) : u− 7→ y+ .
”functional analyst’s point of view”
In balanced coordinates, energy transfer from u− to y+ is
R∞
E :=
sup
u∈L2 (−∞,0]
x(0)=x0
y (t)T y (t) dt
0
R0
=
u(t)T u(t) dt
n
1 X 2 2
σj x0,j .
||x0 ||2 j=1
−∞
”engineer’s point of view”
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Motivation:
HSV are system invariants: they are preserved under T and determine
the energy transfer given by the Hankel map
H : L2 (−∞, 0) 7→ L2 (0, ∞) : u− 7→ y+ .
”functional analyst’s point of view”
In balanced coordinates, energy transfer from u− to y+ is
R∞
E :=
sup
u∈L2 (−∞,0]
x(0)=x0
y (t)T y (t) dt
0
R0
=
u(t)T u(t) dt
n
1 X 2 2
σj x0,j .
||x0 ||2 j=1
−∞
”engineer’s point of view” =⇒ Truncate states corresponding to “small” HSVs
=⇒ analogy to best approximation via SVD, therefore
balancing-related methods are sometimes called SVD methods.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Implementation: SR Method
1
Compute (Cholesky) factors of the solutions of the Lyapunov
equations,
P = S T S, Q = R T R.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Implementation: SR Method
1
Compute (Cholesky) factors of the solutions of the Lyapunov
equations,
P = S T S, Q = R T R.
2
Compute SVD
"
SR
Max Planck Institute Magdeburg
T
= [ U1 , U2 ]
#
Σ1
Σ2
V1T
V2T
.
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Implementation: SR Method
1
Compute (Cholesky) factors of the solutions of the Lyapunov
equations,
P = S T S, Q = R T R.
2
Compute SVD
"
SR
3
Set
T
= [ U1 , U2 ]
−1/2
W = R T V1 Σ1
4
#
Σ1
Σ2
,
V1T
V2T
.
−1/2
V = S T U1 Σ1
.
Reduced model is (W T AV , W T B, CV ).
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Implementation: SR Method
1
Compute (Cholesky) factors of the solutions of the Lyapunov
equations,
P = S T S, Q = R T R.
2
Compute SVD
"
SR
3
Set
T
= [ U1 , U2 ]
−1/2
W = R T V1 Σ1
4
#
Σ1
Σ2
,
V1T
V2T
.
−1/2
V = S T U1 Σ1
.
Reduced model is (W T AV , W T B, CV ).
1
Note: T := Σ− 2 V T R yields balancing
–state-space transformation with
»
ˆ
˜
WT
−1
T
− 12
T = S UΣ , so that T =
and T −1 = V ∗ .
∗
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Properties:
Reduced-order model is stable with HSVs σ1 , . . . , σr .
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Properties:
Reduced-order model is stable with HSVs σ1 , . . . , σr .
Adaptive choice of r via computable error bound:
Xn
||y − ŷ ||2 ≤ 2
σk ||u||2 .
k=r +1
{z
}
|
=:δ
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Relation to KYP
Structural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
ˆ
E (s) = C
−Ĉ
˜
„
»
sIn+r −
A
–«−1 » –
“
”−1
B
=: C̃ sIn+r − Ã
B̃.
Â
B̂
is a stable transfer function, i.e., E ∈ H∞ .
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Relation to KYP
Structural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
ˆ
E (s) = C
−Ĉ
˜
„
»
sIn+r −
A
–«−1 » –
“
”−1
B
=: C̃ sIn+r − Ã
B̃.
Â
B̂
is a stable transfer function, i.e., E ∈ H∞ . Hence,
kE kH∞ < δ
⇐⇒
Φδ (iω) < 0 ∀ω
for Popov function
»
Φδ (s) =
Max Planck Institute Magdeburg
(sIn+r − Ã)−1 B̃
Im
–H »
−C̃ T C̃
0
0
δ Im
2
–»
–
(sIn+r − Ã)−1 B̃
.
Im
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Model Reduction for LTI Systems
Balanced truncation for linear systems
Relation to KYP
Structural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
ˆ
E (s) = C
−Ĉ
˜
„
»
sIn+r −
A
–«−1 » –
“
”−1
B
=: C̃ sIn+r − Ã
B̃.
Â
B̂
is a stable transfer function, i.e., E ∈ H∞ . Hence,
kE kH∞ < δ
⇐⇒
Φδ (iω) < 0 ∀ω
for Popov function
»
Φδ (s) =
(sIn+r − Ã)−1 B̃
Im
–H »
−C̃ T C̃
0
0
δ Im
2
–»
–
(sIn+r − Ã)−1 B̃
.
Im
Using KYP and properties of balanced realizations, one can prove
existence of symmetric solution of corresponding LMI.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
23/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
Max Planck Institute Magdeburg
Conclusions
References
Peter Benner, KYP and Balanced Truncation
24/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
25/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
2
Use Frequency-limited Gramians: recall that the Gramians of stable
systems satisfy
AP + PAT + BB T = 0
AT Q + QA + C T C = 0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
25/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
2
Use Frequency-limited Gramians: recall that the Gramians of stable
systems satisfy
Z ∞
T
AP + PAT + BB T = 0 ⇔ P =
e At BB T e A t dt
Z0 ∞
T
T
T
A Q + QA + C C = 0 ⇔ P =
e A t C T Ce At dt
0
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
25/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
2
Use Frequency-limited Gramians: recall that the Gramians of stable
systems satisfy
Z ∞
1
AP + PAT + BB T = 0 ⇔ P =
(ωI − A)−1 BB T (ωI − A)−H dt
2π −∞
Z ∞
1
(ωI − A)−H C T C (ωI − A)−1 dt
AT Q + QA + C T C = 0 ⇔ Q =
2π −∞
Max Planck Institute Magdeburg
Peter Benner, KYP and Balanced Truncation
25/35
Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
2
Use Frequency-limited Gramians:
Z $
1
P($) :=
(ωI − A)−1 BB T (ωI − A)−H dt
2π −$
Z $
1
Q($) :=
(ωI − A)−H C T C (ωI − A)−1 dt
2π −$
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Motivation
Disadvantages of Balanced Truncation
Global error bound can be pessimistic in relevant frequency bands, e.g., in
mechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) are
relevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be of
interest.
Remedies
1
Frequency-weighted BT (FWBT): aim at minimizing kGo (G − Ĝ )Gi kH∞ ,
where Gi , Go are rational transfer functions, e.g., lowpass/highpass filters.
2
Use Frequency-limited Gramians:
Z $
1
P($) :=
(ωI − A)−1 BB T (ωI − A)−H dt
2π −$
Z $
1
Q($) :=
(ωI − A)−H C T C (ωI − A)−1 dt
2π −$
Both approaches yield good local approximation properties, but error bounds
are still global and stability preservation often requires some modifications!
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Theorem
[Iwasaki/Hara ’05]
−1
Consider G (ω) = C (ωI − A) B + D, $ ∈ R such that $ is not a
pole of G , and let Π = ΠT ∈ Rn×n . Then TFAE:
∗ G ($)
G ($)
a)
Π
4 0.
I
I
b) There exist symmetric matrices P and Q 0 of appropriate
dimensions, satisfying
A
C
I
0
−Q
P + $Q
P − $Q −$2 Q
B 0
B
+
Π
D I
D
"
Note: in standard KYP, we used −Π =
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W
S
ST
R
A
C
0
I
I
0
T
T
4 0.
#
.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
A family of frequency-dependent systems
Given , $ ∈ R, we define
ẋ(t)
=
A$ x(t) + B$ u(t),
y (t)
=
C$ x(t) + D$ u(t),
by
A$
:= $I − (( + $)I − A)−1 ($I − A),
B$
:= (( + $)I − A)−1 B,
C$
:= C (( + $)I − A)−1 ,
D$
:= D + C (( + $I ) − A)−1 B.
The associated transfer function is
G$ (ω) = C$ (ωI − A$ )−1 B$ + D$ .
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
c) (A, B) controllable =⇒ (A$ , B$ ) controllable.
d) (A, C ) observable =⇒ (A$ , C$ ) observable.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
c) (A, B) controllable =⇒ (A$ , B$ ) controllable.
d) (A, C ) observable =⇒ (A$ , C$ ) observable.
e) (A, B, C , D) is a minimal realization of G =⇒
(A$ , B$ , C$ , D$ ) is a minimal realization of G$ .
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
c) (A, B) controllable =⇒ (A$ , B$ ) controllable.
d) (A, C ) observable =⇒ (A$ , C$ ) observable.
e) (A, B, C , D) is a minimal realization of G =⇒
(A$ , B$ , C$ , D$ ) is a minimal realization of G$ .
f) G$ ($) = G ($).
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
c) (A, B) controllable =⇒ (A$ , B$ ) controllable.
d) (A, C ) observable =⇒ (A$ , C$ ) observable.
e) (A, B, C , D) is a minimal realization of G =⇒
(A$ , B$ , C$ , D$ ) is a minimal realization of G$ .
f) G$ ($) = G ($).
g) kG kH∞ ≤ γ =⇒ kG$ kH∞ ≤ γ.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all > 0.
b) If G is unstable, then G$ is stable for 0 < < ˆ$ , where
($ − =(λu ))2
ˆ$ =
min
+ <(λu ) .
<(λu )
λu ∈Λ (A)∩C+
0
c) (A, B) controllable =⇒ (A$ , B$ ) controllable.
d) (A, C ) observable =⇒ (A$ , C$ ) observable.
e) (A, B, C , D) is a minimal realization of G =⇒
(A$ , B$ , C$ , D$ ) is a minimal realization of G$ .
f) G$ ($) = G ($).
g) kG kH∞ ≤ γ =⇒ kG$ kH∞ ≤ γ.
h) kG$ kH∞ ≤ γ$ =⇒ σmax (G ($)) ≤ γ$ .
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
The Frequency-dependent KYP Lemma
Properties of the frequency-dependent systems
Theorem 2
Suppose the LTI system (A, B, C , D) is Hurwitz and minimal, and denote
its controllability, observability, and balanced Gramians as P, Q, Σ, then
for any $-dependent extended system (A$ , B$ , C$ , D$ ) with Gramians
P$ , Q$ , Σ$ :
a) P P$ ,
Q Q$ ,
b) limε→0 P$ = 0,
c) limε→∞ P$ = P,
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Σ Σ$ .
limε→0 Q$ = 0,
limε→0 Σ$ = 0.
limε→∞ Q$ = Q,
limε→∞ Σ$ = Σ.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency-dependent Balanced Truncation (FDBT)
Apply the generic balancing procedure to (A$ , B$ , C$ , D$ ), i.e., solve
H
A$ P$ + P$ AH$ + B$ B$
= 0,
H
AH
$ Q$ + Q$ A$ + C$ C$ = 0,
and compute the balancing transformation T$ so that
H
−H
−1
T$ P$ T$
= T$
Q$ T$
= Σ$ = diag (σ$,1 , . . . , σ$,n ),
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with σ$,k ≥ σ$,k+1 .
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency-dependent Balanced Truncation (FDBT)
Apply the generic balancing procedure to (A$ , B$ , C$ , D$ ), i.e., solve
H
A$ P$ + P$ AH$ + B$ B$
= 0,
H
AH
$ Q$ + Q$ A$ + C$ C$ = 0,
and compute the balancing transformation T$ so that
H
−H
−1
T$ P$ T$
= T$
Q$ T$
= Σ$ = diag (σ$,1 , . . . , σ$,n ),
with σ$,k ≥ σ$,k+1 .
Balance the system:
−1
−1
(T$ A$ T$
, T$ B$ , C$ T$
, D$ )
„»
– »
–
ˆ
A$,11 A$,12
B$,1
=
,
, C$,1
A$,21 A$,22
B$,2
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C$,2
˜
«
, D$
.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency-dependent Balanced Truncation (FDBT)
Balance the system:
−1
−1
(T$ A$ T$
, T$ B$ , C$ T$
, D$ )
„»
– »
–
ˆ
A$,11 A$,12
B$,1
=
,
, C$,1
A$,21 A$,22
B$,2
C$,2
˜
«
, D$
.
Reduced-order model is then obtained by truncation and back transformation:
select r such that σ$,r > σ$,r +1 and set
Â
=
B̂
=
Ĉ
=
D̂
=
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$Ir − ($Ir − A$,11 ) (( − $)Ir + A$,11 )−1 ,
1
(( + $)Ir − Â)B$,1 ,
1
C$,1 (( + $)Ir − Â),
1
D$ − 2 C$,1 (( + $)Ir − Â)B$,1 .
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Frequency-dependent Balanced Truncation (FDBT)
Reduced-order model is then obtained by truncation and back transformation:
select r such that σ$,r > σ$,r +1 and set
Â
=
B̂
=
Ĉ
=
D̂
=
$Ir − ($Ir − A$,11 ) (( − $)Ir + A$,11 )−1 ,
1
(( + $)Ir − Â)B$,1 ,
1
C$,1 (( + $)Ir − Â),
1
D$ − 2 C$,1 (( + $)Ir − Â)B$,1 .
Theorem 3 (Local Error Bound)
The reduced-order transfer function Ĝ (s) = Ĉ (sIr − Â)−1 B̂ + D̂ satisfies:
n
X
σmax G ($) − Ĝ ($) ≤ 2
σ$,k .
k=r +1
Proof: use proof for BT error bound based on FD-KYP instead of KYP.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
1
Linear Systems Basics
2
Dissipativity and Structural Properties
Dissipative Systems
Dissipativity in the Frequency Domain
3
The Kalman-Yakubovich-Popov Lemma
4
Model Reduction for LTI Systems
Balanced truncation for linear systems
5
Frequency-dependent KYP Lemma and Model Reduction
The Frequency-dependent KYP Lemma
Frequency-dependent Balanced Truncation
6
Numerical Examples
RLC ladder network
Butterworth filter
7
Conclusions and Future Work
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Conclusions
References
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Numerical Examples
RLC ladder network
Simple example of electronic circuit from
[Sorensen ’05]
input ≡ voltage u, output ≡ current y ,
scaled inductances, capacities, and resistance:
Lj = 1, Cj = 1 for all j; R1 = 0.5 , R2 = 0.2.
n = 5, m = p = 1.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Numerical Examples
RLC ladder network
Comparison of FDBT and BT (ω̄ = 0, ε = 1)
r
4
3
2
1
FDBT
bound
true error
−7
1.2201 × 10
1.2201 × 10−7
−5
8.7182 × 10−5
8.7426 × 10
−4
5.5028 × 10
3.7568 × 10−4
0.0584
0.0582
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bound
0.0006
0.1752
0.3914
0.6311
BT
true error
0.0006
0.1740
0.0421
0.1975
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Numerical Examples
RLC ladder network
Comparison of FDBT and BT (ω̄ = 0, varying ε)
r=1
r=2
0.8
0.4
0.6
0.3
FDBT
BT
0.4
0.2
0
FDBT
BT
0.2
0.1
0
100
200
ε
300
400
0
0
100
−4
r=3
300
400
r=4
x 10
0.2
200
ε
6
0.15
FDBT
BT
0.1
2
0.05
0
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FDBT
BT
4
0
100
200
ε
300
400
0
0
100
200
ε
300
400
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Numerical Examples
Butterworth filter
Bandstop filter [A,B,C,D]= butter(50,[90 110],stop,s)
Hankel singular values
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Numerical Examples
Butterworth filter
Bandstop filter [A,B,C,D]= butter(50,[90 110],stop,s)
Transfer functions
The sigma plot of the continuous−time bandstop filter and the reduced systems
2.5
2
Original
BT
FDBT ε=100
FDBT ε=1000
MM
Multipoint MM (90,100,110)
Multipoint MM (80,100,120)
1.5
1
0.5
0
60
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70
80
90
100
ω
110
120
130
140
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Conclusions and Future Work
Summary:
Relations of KYP lemma to balanced truncation.
Frequency-dependent KYP lemma suggests new
frequency-dependent balanced truncation (FDBT) method.
FDBT offers alternative to interpolation-based method if good local
approximation quality is desired.
Continuous- and discrete-time FDBT derived.
FDBT is stability preserving and has local error bound, which is
often much better than global BT bound.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
Conclusions and Future Work
Summary:
Relations of KYP lemma to balanced truncation.
Frequency-dependent KYP lemma suggests new
frequency-dependent balanced truncation (FDBT) method.
FDBT offers alternative to interpolation-based method if good local
approximation quality is desired.
Continuous- and discrete-time FDBT derived.
FDBT is stability preserving and has local error bound, which is
often much better than global BT bound.
Future work:
Details for non-minimal systems.
Large-scale implementation and testing.
Computational feasible method for frequency bands.
Extension to descriptor systems.
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Basics
Dissipativity
KYP Lemma
Model Reduction for LTI Systems
FD-KYP
Numerical Examples
Conclusions
References
References
V. Balakrishnan and L. Vandenberghe.
Semidefinite programming duality and linear time-invariant systems.
IEEE Transactions on Automatic Control 48(1):30–41, 2003.
doi: 10.1109/TAC.2002.806652
X. Du, P. Benner, G. Yang, and D. Ye.
Balanced Truncation of Linear Time-Invariant Systems at a Single Frequency.
Max Planck Institute Magdeburg Preprints, MPIMD/13–02, January
2013.
T. Iwasaki and S. Hara.
Generalized KYP lemma: unified frequency domain inequalities with design
applications.
IEEE Transactions on Automatic Control 50(1):41–59, 2005.
J. Willems.
Dissipative dynamical systems.
Archive for Rationale Mechanics Analysis 45:321–393, 1972.
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