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MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
Elgersburg Workshop 2013February 11–14, 2013
On the Kalman–Yakubovich–Popov Lemmaand its Application in Model Order Reduction
Peter Benner
Computational Methods in Systems and Control TheoryMax 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 Magdeburg Peter Benner, KYP and Balanced Truncation 1/35
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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
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 3/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Linear Systems
LTI Systems
Σ :
{x(t) = Ax(t) + Bu(t), x(0) = x0
y(t) = Cx(t) + Du(t),
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.
x(t) = Ax(t) + Bu(t)
y (t) = Cx(t) + Du(t)
u(t) y (t)
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 4/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Stability and Controllability
DefinitionsThe 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 signalu(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 lefthalf-plane;
controllable ⇐⇒ rank[λIn − A B
]= n for all λ ∈ C.
observable ⇐⇒ rank[λIn − AT CT
]= n for all λ ∈ C.
minimal if it is controllable and observable.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 5/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Stability and Controllability
DefinitionsThe 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 signalu(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 lefthalf-plane;
controllable ⇐⇒ rank[λIn − A B
]= n for all λ ∈ C.
observable ⇐⇒ rank[λIn − AT CT
]= n for all λ ∈ C.
minimal if it is controllable and observable.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 5/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
Then
L{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
L(Σ) :
{L{x}(s) = AL{x}(s) + BL{u}(s),
L{y}(s) = CL{x}(s) + DL{u}(s),
ThenL{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸
=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
L(Σ) :
{s(L{x}(s)− x(0)) = AL{x}(s) + BL{u}(s),
L{y}(s) = CL{x}(s) + DL{u}(s),
ThenL{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸
=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
L(Σ) :
{sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y}(s) = CL{x}(s) + DL{u}(s),
ThenL{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸
=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
L(Σ) :
{sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y}(s) = CL{x}(s) + DL{u}(s),
ThenL{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸
=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Frequency Domain Analysis
Laplace transform
L{f }(s) :=
∫ ∞0
e−st f (t)dt
Transfer function
Assume x(0) = 0. Then
L(Σ) :
{sL{x}(s) = AL{x}(s) + BL{u}(s),
L{y}(s) = CL{x}(s) + DL{u}(s),
ThenL{y}(s) = C (sIn − A)−1B︸ ︷︷ ︸
=:G(s)
L{u}(s).
The transfer function G (s) maps inputs to outputs in the frequencydomain.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 6/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 7/35
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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 supplyfunction s : Rp × Rm −→ R if there exists a storage functionV : Rn −→ R such that the dissipation inequality
V (x(t1)) ≤ V (x(0)) +
∫ t1
0
s(y(t), u(t))dt
is fulfilled for all 0 ≤ t1.
Interpretation∫ t1
0
s(y(t), u(t))dt can be seen as the energy supplied to the system
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
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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 supplyfunction s : Rp × Rm −→ R if there exists a storage functionV : Rn −→ R such that the dissipation inequality
V (x(t1)) ≤ V (x(0)) +
∫ t1
0
s(y(t), u(t))dt
is fulfilled for all 0 ≤ t1.
Interpretation∫ t1
0
s(y(t), u(t))dt can be seen as the energy supplied to the system
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
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Quadratic Supply Functions
Often, we consider
s(y(t), u(t)) =
[y(t)u(t)
]T [W SST R
] [y(t)u(t)
]with W = W T , R = RT
=
[Cx(t) + Du(t)
u(t)
]T [W SST R
] [Cx(t) + Du(t)
u(t)
]=
[x(t)u(t)
]T [CTWC CTWD + CTS
DTWC + STC DTWD + DTS + STD + R
] [x(t)u(t)
]=:
[x(t)u(t)
]T [W S
ST R
] [x(t)u(t)
]=: s(x(t), u(t)).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Quadratic Supply Functions
Often, we consider
s(y(t), u(t)) =
[y(t)u(t)
]T [W SST R
] [y(t)u(t)
]with W = W T , R = RT
=
[Cx(t) + Du(t)
u(t)
]T [W SST R
] [Cx(t) + Du(t)
u(t)
]
=
[x(t)u(t)
]T [CTWC CTWD + CTS
DTWC + STC DTWD + DTS + STD + R
] [x(t)u(t)
]=:
[x(t)u(t)
]T [W S
ST R
] [x(t)u(t)
]=: s(x(t), u(t)).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Quadratic Supply Functions
Often, we consider
s(y(t), u(t)) =
[y(t)u(t)
]T [W SST R
] [y(t)u(t)
]with W = W T , R = RT
=
[Cx(t) + Du(t)
u(t)
]T [W SST R
] [Cx(t) + Du(t)
u(t)
]=
[x(t)u(t)
]T [CTWC CTWD + CTS
DTWC + STC DTWD + DTS + STD + R
] [x(t)u(t)
]
=:
[x(t)u(t)
]T [W S
ST R
] [x(t)u(t)
]=: s(x(t), u(t)).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Quadratic Supply Functions
Often, we consider
s(y(t), u(t)) =
[y(t)u(t)
]T [W SST R
] [y(t)u(t)
]with W = W T , R = RT
=
[Cx(t) + Du(t)
u(t)
]T [W SST R
] [Cx(t) + Du(t)
u(t)
]=
[x(t)u(t)
]T [CTWC CTWD + CTS
DTWC + STC DTWD + DTS + STD + R
] [x(t)u(t)
]=:
[x(t)u(t)
]T [W S
ST R
] [x(t)u(t)
]
=: s(x(t), u(t)).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Quadratic Supply Functions
Often, we consider
s(y(t), u(t)) =
[y(t)u(t)
]T [W SST R
] [y(t)u(t)
]with W = W T , R = RT
=
[Cx(t) + Du(t)
u(t)
]T [W SST R
] [Cx(t) + Du(t)
u(t)
]=
[x(t)u(t)
]T [CTWC CTWD + CTS
DTWC + STC DTWD + DTS + STD + R
] [x(t)u(t)
]=:
[x(t)u(t)
]T [W S
ST R
] [x(t)u(t)
]=: s(x(t), u(t)).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 9/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Special Cases
Passivity
s(y(t), u(t)) =
[y(t)u(t)
]T [0 ImIm 0
] [y(t)u(t)
],
s(x(t), u(t)) =
[x(t)u(t)
]T [0 CT
C D + DT
] [x(t)u(t)
].
Contractivity
s(y(t), u(t)) =
[y(t)u(t)
]T [−Ip 00 Im
] [y(t)u(t)
],
s(x(t), u(t)) =
[x(t)u(t)
]T [−CTC −CTD−DTC Im − DTD
] [x(t)u(t)
].
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Special Cases
Passivity
s(y(t), u(t)) =
[y(t)u(t)
]T [0 ImIm 0
] [y(t)u(t)
],
s(x(t), u(t)) =
[x(t)u(t)
]T [0 CT
C D + DT
] [x(t)u(t)
].
Contractivity
s(y(t), u(t)) =
[y(t)u(t)
]T [−Ip 00 Im
] [y(t)u(t)
],
s(x(t), u(t)) =
[x(t)u(t)
]T [−CTC −CTD−DTC Im − DTD
] [x(t)u(t)
].
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 10/35
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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) =
[(sIn − A)−1B
Im
]H [W SST R
] [(sIn − A)−1B
Im
]
TheoremLet Σ be controllable. Then, Σ is dissipative with respect to
s(x(t), u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]if and only if Φ(iω) < 0 holds
for all iω ∈ iR\Λ(A).
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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) =
[(sIn − A)−1B
Im
]H [W SST R
] [(sIn − A)−1B
Im
]
TheoremLet Σ be controllable. Then, Σ is dissipative with respect to
s(x(t), u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]if and only if Φ(iω) < 0 holds
for all iω ∈ iR\Λ(A).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 11/35
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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 ispositive real, i.e.,
G (s) + GH(s) < 0 ∀s ∈ C+.
Contractivity and bounded realness
A dynamical system is contractive if and only its transfer function G isbounded real, i.e.,
Im − GH(s)G (s) < 0 ∀s ∈ C+.
RemarkIn contrast to general dissipativity, positive and bounded realness areproperties of Φ(s) in the whole open right half-plane. It can be shownthat for these cases V (x(t)) = x(t)TXx(t) for an X = XT < 0.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 12/35
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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 ispositive real, i.e.,
G (s) + GH(s) < 0 ∀s ∈ C+.
Contractivity and bounded realness
A dynamical system is contractive if and only its transfer function G isbounded real, i.e.,
Im − GH(s)G (s) < 0 ∀s ∈ C+.
RemarkIn contrast to general dissipativity, positive and bounded realness areproperties of Φ(s) in the whole open right half-plane. It can be shownthat for these cases V (x(t)) = x(t)TXx(t) for an X = XT < 0.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 12/35
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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 ispositive real, i.e.,
G (s) + GH(s) < 0 ∀s ∈ C+.
Contractivity and bounded realness
A dynamical system is contractive if and only its transfer function G isbounded real, i.e.,
Im − GH(s)G (s) < 0 ∀s ∈ C+.
RemarkIn contrast to general dissipativity, positive and bounded realness areproperties of Φ(s) in the whole open right half-plane. It can be shownthat for these cases V (x(t)) = x(t)TXx(t) for an X = XT < 0.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 12/35
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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]
P
w z
K
u y
Plant P, dynamic compensator K,
noise w , estimation error z .
Goal: Find K that stabilizes the systemand minimizes the influence of w on z!( = minimizing the H∞-norm ofclosed-loop transfer function)
H∞-spaces
Hp×m∞ (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)
‖G‖H∞ = sups∈C+
σmax(G (s)) = supω∈R
σmax(G (iω))
= infγ≥0
{γ2Im − GH(iω)G (iω) < 0 ∀ω ∈ R
}.
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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]
P
w z
K
u y
Plant P, dynamic compensator K,
noise w , estimation error z .
Goal: Find K that stabilizes the systemand minimizes the influence of w on z!( = minimizing the H∞-norm ofclosed-loop transfer function)
H∞-spaces
Hp×m∞ (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)
‖G‖H∞ = sups∈C+
σmax(G (s)) = supω∈R
σmax(G (iω))
= infγ≥0
{γ2Im − GH(iω)G (iω) < 0 ∀ω ∈ R
}.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 13/35
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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]
P
w z
K
u y
Plant P, dynamic compensator K,
noise w , estimation error z .
Goal: Find K that stabilizes the systemand minimizes the influence of w on z!( = minimizing the H∞-norm ofclosed-loop transfer function)
H∞-spaces
Hp×m∞ (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)
‖G‖H∞ = sups∈C+
σmax(G (s)) = supω∈R
σmax(G (iω))
= infγ≥0
{γ2Im − GH(iω)G (iω) < 0 ∀ω ∈ R
}.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 13/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
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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 algebraicstructures such as
linear matrix inequalities,
quadratic matrix inequalities,
algebraic matrix equations (Riccati equations, Lur’e equations),
(structured matrices and matrix pencils).
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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):
s(x(t),u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]≥ V (x(t))x(t)
(set V (x(t)) = x(t)TXx(t) with X = XT )
= 2x(t)TX (Ax(t) + Bu(t))
= x(t)TXAx(t) + x(t)TXBu(t) + x(t)TATXx(t) + u(t)TBTXx(t)
=
[x(t)u(t)
]T [ATX + XA XB
BTX 0
] [x(t)u(t)
].
This obviously holds if there exists X = XT such that[W SST R
]≥[ATX + XA XB
BTX 0
].
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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):
s(x(t),u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]≥ V (x(t))x(t) (set V (x(t)) = x(t)TXx(t) with X = XT )
= 2x(t)TX (Ax(t) + Bu(t))
= x(t)TXAx(t) + x(t)TXBu(t) + x(t)TATXx(t) + u(t)TBTXx(t)
=
[x(t)u(t)
]T [ATX + XA XB
BTX 0
] [x(t)u(t)
].
This obviously holds if there exists X = XT such that[W SST R
]≥[ATX + XA XB
BTX 0
].
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 16/35
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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):
s(x(t),u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]≥ V (x(t))x(t) (set V (x(t)) = x(t)TXx(t) with X = XT )
= 2x(t)TX (Ax(t) + Bu(t))
= x(t)TXAx(t) + x(t)TXBu(t) + x(t)TATXx(t) + u(t)TBTXx(t)
=
[x(t)u(t)
]T [ATX + XA XB
BTX 0
] [x(t)u(t)
].
This obviously holds if there exists X = XT such that[W SST R
]≥[ATX + XA XB
BTX 0
].
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 16/35
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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):
s(x(t),u(t)) =
[x(t)u(t)
]T [W SST R
] [x(t)u(t)
]≥ V (x(t))x(t) (set V (x(t)) = x(t)TXx(t) with X = XT )
= 2x(t)TX (Ax(t) + Bu(t))
= x(t)TXAx(t) + x(t)TXBu(t) + x(t)TATXx(t) + u(t)TBTXx(t)
=
[x(t)u(t)
]T [ATX + XA XB
BTX 0
] [x(t)u(t)
].
This obviously holds if there exists X = XT such that[W SST R
]≥[ATX + XA XB
BTX 0
].
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 16/35
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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 asymmetric matrix X such that the linear matrix inequality (LMI)[
ATX + XA−W XB − SBTX − ST −R
]4 0
is fulfilled.
History
’61: Popov’s criterion for stability of a feedback system with amemoryless nonlinearity.
’62/’63: Original version of the lemma by Kalman and Yakubovich.
’67: Anderson’s positive real lemma for multivariate transferfunctions.
until today: Many generalizations and extensions.
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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 asymmetric matrix X such that the linear matrix inequality (LMI)[
ATX + XA−W XB − SBTX − ST −R
]4 0
is fulfilled.
History
’61: Popov’s criterion for stability of a feedback system with amemoryless nonlinearity.
’62/’63: Original version of the lemma by Kalman and Yakubovich.
’67: Anderson’s positive real lemma for multivariate transferfunctions.
until today: Many generalizations and extensions.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 17/35
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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 positivereal) if and only if there exists X = XT < 0 such that the LMI[
ATX + XA XB − CT
BTX − C −(D + D)T
]4 0
is fulfilled.
Bounded real lemma
Let Σ be controllable. Then Σ is contractive (or equivalently G (s) isbounded real) if and only if there exists X = XT < 0 such that the LMI[
ATX + XA + CTC XB + CTDBTX + DTC DTD − Im
]4 0
is fulfilled.
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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 positivereal) if and only if there exists X = XT < 0 such that the LMI[
ATX + XA XB − CT
BTX − C −(D + D)T
]4 0
is fulfilled.
Bounded real lemma
Let Σ be controllable. Then Σ is contractive (or equivalently G (s) isbounded real) if and only if there exists X = XT < 0 such that the LMI[
ATX + XA + CTC XB + CTDBTX + DTC DTD − Im
]4 0
is fulfilled.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
m
Quadratic Matrix Inequality
ATX +XA−W +(XB − S) R−1(BTX − ST
)4 0, X = XT solvable.
m
Algebraic Riccati Equation
ATX +XA−W +(XB − S) R−1(BTX − ST
)= 0, X = XT solvable.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
m
Quadratic Matrix Inequality
ATX +XA−W +(XB − S) R−1(BTX − ST
)4 0, X = XT solvable.
m
Algebraic Riccati Equation
ATX +XA−W +(XB − S) R−1(BTX − ST
)= 0, X = XT solvable.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R nonsingular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
m
Quadratic Matrix Inequality
ATX +XA−W +(XB − S) R−1(BTX − ST
)4 0, X = XT solvable.
m
Algebraic Riccati Equation
ATX +XA−W +(XB − S) R−1(BTX − ST
)= 0, X = XT solvable.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 20/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
m
Quadratic Matrix Inequality
ATX + XA−W + (XB − S) R−1(BTX − ST
)4 0, X = XT
cannot be formulated!
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Other Algebraic Characterizations — R singular
Linear Matrix Inequality[ATX + XA−W XB − S
BTX − ST −R
]4 0, X = XT solvable.
m
Lur’e Equation
ATX + XA−W = −KTK ,
XB − S = −KTL,
−R = −LTL,
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!
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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 pointoperations (flops), with some tricks and exploiting structuresO(n4.5).
Methods based on Lyapunov or Riccati equations, invariantsubspaces of Hamiltonian matrices or even pencils generically requireonly O(n3) flops, and can be implemented in O(nmp) flops for somelarge-scale problems with sparse state matrix A.
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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 pointoperations (flops), with some tricks and exploiting structuresO(n4.5).
Methods based on Lyapunov or Riccati equations, invariantsubspaces of Hamiltonian matrices or even pencils generically requireonly O(n3) flops, and can be implemented in O(nmp) flops for somelarge-scale problems with sparse state matrix A.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 21/35
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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 pointoperations (flops), with some tricks and exploiting structuresO(n4.5).
Methods based on Lyapunov or Riccati equations, invariantsubspaces of Hamiltonian matrices or even pencils generically requireonly O(n3) flops, and can be implemented in O(nmp) flops for somelarge-scale problems with sparse state matrix A.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 21/35
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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 pointoperations (flops), with some tricks and exploiting structuresO(n4.5).
Methods based on Lyapunov or Riccati equations, invariantsubspaces of Hamiltonian matrices or even pencils generically requireonly O(n3) flops, and can be implemented in O(nmp) flops for somelarge-scale problems with sparse state matrix A.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 21/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Some Remarks on Numerical AspectsComplexity of Numerical Linear Algebra (NLA) and SDP Solutions to Control Problems
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 21/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Idea
Σ :
(x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),with A stable, i.e., Λ (A) ⊂ C−,
is balanced, if system Gramians, i.e., solutions P,Q of the Lyapunovequations
AP + PAT + BBT = 0, AT Q + QA + CT 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-spacetransformation
T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)
=
„»A11 A12
A21 A22
–,
»B1
B2
–,ˆ
C1 C2
˜,D
«.
Truncation (A, B, C , D) = (A11,B1,C1,D).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 23/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Idea
Σ :
(x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),with A stable, i.e., Λ (A) ⊂ C−,
is balanced, if system Gramians, i.e., solutions P,Q of the Lyapunovequations
AP + PAT + BBT = 0, AT Q + QA + CT 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-spacetransformation
T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)
=
„»A11 A12
A21 A22
–,
»B1
B2
–,ˆ
C1 C2
˜,D
«.
Truncation (A, B, C , D) = (A11,B1,C1,D).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 23/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Idea
Σ :
(x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),with A stable, i.e., Λ (A) ⊂ C−,
is balanced, if system Gramians, i.e., solutions P,Q of the Lyapunovequations
AP + PAT + BBT = 0, AT Q + QA + CT 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-spacetransformation
T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)
=
„»A11 A12
A21 A22
–,
»B1
B2
–,ˆ
C1 C2
˜,D
«.
Truncation (A, B, C , D) = (A11,B1,C1,D).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 23/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Idea
Σ :
(x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),with A stable, i.e., Λ (A) ⊂ C−,
is balanced, if system Gramians, i.e., solutions P,Q of the Lyapunovequations
AP + PAT + BBT = 0, AT Q + QA + CT 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-spacetransformation
T : (A,B,C ,D) 7→ (TAT−1,TB,CT−1,D)
=
„»A11 A12
A21 A22
–,
»B1
B2
–,ˆ
C1 C2
˜,D
«.
Truncation (A, B, C , D) = (A11,B1,C1,D).
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 23/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Motivation:HSV are system invariants: they are preserved under T and determinethe energy transfer given by the Hankel map
H : L2(−∞, 0) 7→ L2(0,∞) : u− 7→ y+.
”functional analyst’s point of view”
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Motivation:HSV are system invariants: they are preserved under T and determinethe 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
E := supu∈L2(−∞,0]
x(0)=x0
∞R0
y(t)T y(t) dt
0R−∞
u(t)T u(t) dt
=1
||x0||2
nXj=1
σ2j x2
0,j .
”engineer’s point of view”
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Motivation:HSV are system invariants: they are preserved under T and determinethe 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
E := supu∈L2(−∞,0]
x(0)=x0
∞R0
y(t)T y(t) dt
0R−∞
u(t)T u(t) dt
=1
||x0||2
nXj=1
σ2j x2
0,j .
”engineer’s point of view” =⇒ Truncate states corresponding to “small” HSVs
=⇒ analogy to best approximation via SVD, thereforebalancing-related methods are sometimes called SVD methods.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Implementation: SR Method1 Compute (Cholesky) factors of the solutions of the Lyapunov
equations,P = STS , Q = RTR.
2 Compute SVD
SRT = [ U1, U2 ]
[Σ1
Σ2
] [V T
1
V T2
].
3 Set, V = STU1Σ
−1/21 .
4 Reduced model is (W TAV ,W TB,CV ).
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Implementation: SR Method1 Compute (Cholesky) factors of the solutions of the Lyapunov
equations,P = STS , Q = RTR.
2 Compute SVD
SRT = [ U1, U2 ]
[Σ1
Σ2
] [V T
1
V T2
].
3 Set, V = STU1Σ
−1/21 .
4 Reduced model is (W TAV ,W TB,CV ).
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Implementation: SR Method1 Compute (Cholesky) factors of the solutions of the Lyapunov
equations,P = STS , Q = RTR.
2 Compute SVD
SRT = [ U1, U2 ]
[Σ1
Σ2
] [V T
1
V T2
].
3 SetW = RTV1Σ
−1/21 , V = STU1Σ
−1/21 .
4 Reduced model is (W TAV ,W TB,CV ).
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Implementation: SR Method1 Compute (Cholesky) factors of the solutions of the Lyapunov
equations,P = STS , Q = RTR.
2 Compute SVD
SRT = [ U1, U2 ]
[Σ1
Σ2
] [V T
1
V T2
].
3 SetW = RTV1Σ
−1/21 , V = STU1Σ
−1/21 .
4 Reduced model is (W TAV ,W TB,CV ).
Note: T := Σ−12 V T R yields balancing state-space transformation with
T−1 = ST UΣ−12 , so that T =
»W T
∗
–and T−1 =
ˆV ∗
˜.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Properties:
Reduced-order model is stable with HSVs σ1, . . . , σr .
Adaptive choice of r via computable error bound:
||y − y ||2 ≤(
2∑n
k=r+1σk
)︸ ︷︷ ︸
=:δ
||u||2 .
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Properties:
Reduced-order model is stable with HSVs σ1, . . . , σr .
Adaptive choice of r via computable error bound:
||y − y ||2 ≤(
2∑n
k=r+1σk
)︸ ︷︷ ︸
=:δ
||u||2 .
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Relation to KYPStructural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
E(s) =ˆC −C
˜„sIn+r −
»A
A
–«−1 »B
B
–=: C
“sIn+r − A
”−1
B.
is a stable transfer function, i.e., E ∈ H∞.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Relation to KYPStructural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
E(s) =ˆC −C
˜„sIn+r −
»A
A
–«−1 »B
B
–=: C
“sIn+r − A
”−1
B.
is a stable transfer function, i.e., E ∈ H∞. Hence,
‖E‖H∞ < δ ⇐⇒ Φδ(iω) < 0 ∀ω
for Popov function
Φδ(s) =
»(sIn+r − A)−1B
Im
–H »−CT C 00 δ2Im
– »(sIn+r − A)−1B
Im
–.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Model Reduction for LTI SystemsBalanced truncation for linear systems
Relation to KYPStructural properties of reduced-order models can be proved using KYP.
Error bound can be proved using KYP as follows:
E(s) =ˆC −C
˜„sIn+r −
»A
A
–«−1 »B
B
–=: C
“sIn+r − A
”−1
B.
is a stable transfer function, i.e., E ∈ H∞. Hence,
‖E‖H∞ < δ ⇐⇒ Φδ(iω) < 0 ∀ω
for Popov function
Φδ(s) =
»(sIn+r − A)−1B
Im
–H »−CT C 00 δ2Im
– »(sIn+r − A)−1B
Im
–.
Using KYP and properties of balanced realizations, one can proveexistence of symmetric solution of corresponding LMI.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians:
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians: recall that the Gramians of stablesystems satisfy
AP + PAT + BBT = 0
AT Q + QA + CT C = 0
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 25/35
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians: recall that the Gramians of stablesystems satisfy
AP + PAT + BBT = 0 ⇔ P =
Z ∞0
eAtBBT eAT t dt
AT Q + QA + CT C = 0 ⇔ P =
Z ∞0
eAT tCT CeAt dt
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians: recall that the Gramians of stablesystems satisfy
AP + PAT + BBT = 0 ⇔ P =1
2π
Z ∞−∞
(ωI − A)−1BBT (ωI − A)−Hdt
AT Q + QA + CT C = 0 ⇔ Q =1
2π
Z ∞−∞
(ωI − A)−HCT C(ωI − A)−1dt
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians:
P($) :=1
2π
Z $
−$(ωI − A)−1BBT (ωI − A)−Hdt
Q($) :=1
2π
Z $
−$(ωI − A)−HCT C(ωI − A)−1dt
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Motivation
Disadvantages of Balanced TruncationGlobal error bound can be pessimistic in relevant frequency bands, e.g., inmechanical systems, often only frequencies 0 ≤ 2πω ≤ 1000 (in Hz) arerelevant, in VLSI design only an operating frequency, e.g., 2.6 GHz, may be ofinterest.
Remedies
1 Frequency-weighted BT (FWBT): aim at minimizing ‖Go(G − G)Gi‖H∞ ,where Gi ,Go are rational transfer functions, e.g., lowpass/highpass filters.
2 Use Frequency-limited Gramians:
P($) :=1
2π
Z $
−$(ωI − A)−1BBT (ωI − A)−Hdt
Q($) :=1
2π
Z $
−$(ωI − A)−HCT C(ωI − A)−1dt
Both approaches yield good local approximation properties, but error boundsare 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]
Consider G (ω) = C (ωI − A)−1B + D, $ ∈ R such that $ is not apole of G , and let Π = ΠT ∈ Rn×n. Then TFAE:
a)
[G ($)
I
]∗Π
[G ($)
I
]4 0.
b) There exist symmetric matrices P and Q � 0 of appropriatedimensions, satisfying[
A IC 0
] [−Q P + $Q
P − $Q −$2Q
] [A IC 0
]T
+
[B 0D I
]Π
[B 0D I
]T
4 0.
Note: in standard KYP, we used −Π =
"W S
ST R
#.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
The Frequency-dependent KYP LemmaA family of frequency-dependent systems
Given ε,$ ∈ R, we define
x(t) = A$x(t) + B$u(t),
y(t) = C$x(t) + D$u(t),
by
A$ := $I − ε((ε+ $)I − A)−1($I − A),
B$ := ε((ε+ $)I − A)−1B,
C$ := εC ((ε+ $)I − A)−1,
D$ := D + C ((ε+ $I )− A)−1B.
The associated transfer function is
G$(ω) = C$(ωI − A$)−1B$ + D$.
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
The Frequency-dependent KYP LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 1
a) G stable =⇒ G$ is stable for all ε > 0.
b) If G is unstable, then G$ is stable for 0 < ε < ε$, where
ε$ = minλu∈Λ (A)∩C+
0
{($ −=(λu))2
<(λu)+ <(λu)
}.
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) ‖G‖H∞ ≤ γ =⇒ ‖G$‖H∞ ≤ γ.
h) ‖G$‖H∞ ≤ 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 LemmaProperties of the frequency-dependent systems
Theorem 2
Suppose the LTI system (A,B,C ,D) is Hurwitz and minimal, and denoteits controllability, observability, and balanced Gramians as P,Q,Σ, thenfor any $-dependent extended system (A$,B$,C$,D$) with GramiansP$,Q$,Σ$:
a) P � P$, Q � Q$, Σ � Σ$.
b) limε→0 P$ = 0, limε→0 Q$ = 0, limε→0 Σ$ = 0.
c) limε→∞ P$ = P, 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
A$P$ + P$AH$ + B$BH
$ = 0, AH$Q$ + Q$A$ + CH
$C$ = 0,
and compute the balancing transformation T$ so that
T$P$T H$ = T−H
$ Q$T−1$ = Σ$ = diag (σ$,1, . . . , σ$,n), 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
A$P$ + P$AH$ + B$BH
$ = 0, AH$Q$ + Q$A$ + CH
$C$ = 0,
and compute the balancing transformation T$ so that
T$P$T H$ = T−H
$ Q$T−1$ = Σ$ = diag (σ$,1, . . . , σ$,n), with σ$,k ≥ σ$,k+1.
Balance the system:
(T$A$T−1$ ,T$B$,C$T−1
$ ,D$)
=
„»A$,11 A$,12
A$,21 A$,22
–,
»B$,1B$,2
–,ˆ
C$,1 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:
(T$A$T−1$ ,T$B$,C$T−1
$ ,D$)
=
„»A$,11 A$,12
A$,21 A$,22
–,
»B$,1B$,2
–,ˆ
C$,1 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
A = $Ir − ε($Ir − A$,11) ((ε− $)Ir + A$,11)−1 ,
B =1
ε((ε+ $)Ir − A)B$,1,
C =1
εC$,1((ε+ $)Ir − A),
D = D$ −1
ε2C$,1((ε+ $)Ir − A)B$,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
A = $Ir − ε($Ir − A$,11) ((ε− $)Ir + A$,11)−1 ,
B =1
ε((ε+ $)Ir − A)B$,1,
C =1
εC$,1((ε+ $)Ir − A),
D = D$ −1
ε2C$,1((ε+ $)Ir − A)B$,1.
Theorem 3 (Local Error Bound)
The reduced-order transfer function G (s) = C (sIr − A)−1B + D satisfies:
σmax
(G ($)− G ($)
)≤ 2
n∑k=r+1
σ$,k .
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 Conclusions References
1 Linear Systems Basics
2 Dissipativity and Structural PropertiesDissipative SystemsDissipativity in the Frequency Domain
3 The Kalman-Yakubovich-Popov Lemma
4 Model Reduction for LTI SystemsBalanced truncation for linear systems
5 Frequency-dependent KYP Lemma and Model ReductionThe Frequency-dependent KYP LemmaFrequency-dependent Balanced Truncation
6 Numerical ExamplesRLC ladder networkButterworth filter
7 Conclusions and Future Work
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Numerical ExamplesRLC 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 ExamplesRLC ladder network
Comparison of FDBT and BT (ω = 0, ε = 1)
r FDBT BTbound true error bound true error
4 1.2201× 10−7 1.2201× 10−7 0.0006 0.00063 8.7426× 10−5 8.7182× 10−5 0.1752 0.17402 5.5028× 10−4 3.7568× 10−4 0.3914 0.04211 0.0584 0.0582 0.6311 0.1975
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Numerical ExamplesRLC ladder network
Comparison of FDBT and BT (ω = 0, varying ε)
0 100 200 300 4000
0.2
0.4
0.6
0.8
ε
r=1
0 100 200 300 4000
0.1
0.2
0.3
0.4
ε
r=2
FDBTBT
0 100 200 300 4000
0.05
0.1
0.15
0.2
ε
r=3
FDBTBT
0 100 200 300 4000
2
4
6
x 10−4
ε
r=4
FDBTBT
FDBTBT
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Basics Dissipativity KYP Lemma Model Reduction for LTI Systems FD-KYP Numerical Examples Conclusions References
Numerical ExamplesButterworth 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 ExamplesButterworth filter
Bandstop filter [A,B,C,D]= butter(50,[90 110],stop,s)
Transfer functions
60 70 80 90 100 110 120 130 140
0
0.5
1
1.5
2
2.5
ω
The sigma plot of the continuous−time bandstop filter and the reduced systems
OriginalBT
FDBT ε=100
FDBT ε=1000
MMMultipoint MM (90,100,110)Multipoint MM (80,100,120)
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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 newfrequency-dependent balanced truncation (FDBT) method.
FDBT offers alternative to interpolation-based method if good localapproximation quality is desired.
Continuous- and discrete-time FDBT derived.
FDBT is stability preserving and has local error bound, which isoften 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.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 34/35
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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 newfrequency-dependent balanced truncation (FDBT) method.
FDBT offers alternative to interpolation-based method if good localapproximation quality is desired.
Continuous- and discrete-time FDBT derived.
FDBT is stability preserving and has local error bound, which isoften 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.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 34/35
![Page 98: On the Kalman–Yakubovich–Popov Lemma and its Application ... · MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Elgersburg Workshop 2013 February 11{14,](https://reader030.vdocument.in/reader030/viewer/2022041216/5e059af3360b7849293b0a59/html5/thumbnails/98.jpg)
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, January2013.
T. Iwasaki and S. Hara.Generalized KYP lemma: unified frequency domain inequalities with designapplications.IEEE Transactions on Automatic Control 50(1):41–59, 2005.
J. Willems.Dissipative dynamical systems.Archive for Rationale Mechanics Analysis 45:321–393, 1972.
Max Planck Institute Magdeburg Peter Benner, KYP and Balanced Truncation 35/35