Stability analysis with integral quadratic constraints: A
dissipativity based proof
Joost Veenman∗ and Carsten W. Scherer∗
∗ Department
of Mathematics, University of Stuttgart, Germany
{joost.veenman,carsten.scherer}@mathematik.uni-stuttgart.de
Abstract— In this paper we formulate a dissipativity based
proof of the well-known integral quadratic constraint (IQC)
theorem under mild assumptions. For general dynamic (frequency dependent) IQC-multipliers it is shown that, once the
conditions of the IQC-theorem are satisfied, it is possible to
construct a nonnegative Lyapunov function that satisfies a
dissipation inequality. This not only shows that IQCs can be
interpreted as dynamic supply functions, but also opens the
way to merge frequency-domain techniques with time-domain
conditions known from Lyapunov-theory. The proof relies on (i)
a reformulation of the IQC-theorem, which is of independent
interest; (ii) a symmetric Wiener-Hopf factorization; (iii) the
gluing lemma, which describes how certain operations on
frequency domain conditions can be performed in the statespace; and (iv) on dissipativity theory.
I. I NTRODUCTION
An important framework for the systematic analysis of
uncertain systems is the integral quadratic constraint (IQC)
approach [1]. IQCs are very useful in capturing the properties
a diverse class of uncertainties and/or nonlinearities. One
could think of time-invariant or time-varying parametric uncertainties, possibly with bounds on the rate-of-variation, linear time-invariant (LTI) dynamic uncertainties, time-varying
delay uncertainties or static nonlinearities such as sectorbounded and slope-restricted nonlinearities. We refer the
reader to [1], [2], [3], [4], [5], [6], [7], [8] and references
therein. Apart from analysis, the IQC-framework can be
employed for synthesis purposes too. Recent years have
witnessed a variety of synthesis results allowing not only for
static (frequency independent) (see e.g. [9], [10], [11], [12],
[13]) but also for general dynamic (frequency dependent)
IQC-multipliers (see e.g. [14], [15], [16], [17], [18], [8]).
Another central notion in systems theory is dissipativity
[19], [20]. Roughly speaking, a dissipative dynamical system
is characterized by the property that, at any time, the amount
of energy that can be stored in the system never exceeds the
amount of energy that has been supplied to the system. This
can be nicely formalized for a general class of systems by
means of the so-called dissipation inequality which involves
a storage and a supply function. An appealing aspect for the
analysis of linear dissipative systems is that linear matrix
inequalities (LMIs) very naturally emerge. This makes the
framework attractive from a computational point-of-view. In
fact, one can often relatively easy generalize the notions from
linear to time-varying, uncertain and/or nonlinear dissipative
system.
It is relevant to note that there is a common interest in
finding a connection between the IQC-framework and the
dissipativity approach [21], [22], [23], [24], [25], [26], [27],
[28] (see also [29] for another link between the IQC and
the multiplier approach [2], [30]). A connection between the
two approaches is of particular importance, because it would
open the way to merge frequency-domain techniques with
time-domain conditions known from Lyapunov-theory. This
could lead to generalizations that would be hard to obtain
with frequency domain conditions. So far, a link has only
been established for the special case of static and a rather
restrictive class of dynamic IQC-multipliers that are satisfied
on all finite time horizons [21], [22], [23], [24], [25], [26],
[27], [28]. In case of general dynamic IQCs, which only
need to hold on infinite time horizons, it remains unclear
how to proceed. Here we would like to emphasize that [28]
comprises a technical glitch.
As a first contribution, we provide (to the best of our
knowledge) a novel reformulation of the IQC-theorem which
is of independent interest. We will exploit the result for
the main contribution of this paper, which is a dissipativity
based proof of the IQC-theorem. Indeed, for a rather general
class of dynamic IQC-multipliers it will be shown that,
once the conditions of the IQC-theorem are satisfied, it is
possible to construct a nonnegative Lyapunov function that
satisfies a dissipation inequality. The proof relies on (i) the
reformulation of the IQC-theorem; (ii) a symmetric WienerHopf factorization [31], which is guaranteed to exist and can
be easily constructed through the solution of an algebraic
Riccati equation (ARE) [32] (see also [33]); (iii) the gluing
lemma, which describes how certain operations on frequency
domain conditions can be performed in the state-space [34];
and (iv) on dissipativity theory.
The paper is organized as follows: After a short recap
of stability analysis with IQCs in Section II-A−II-C, we
formally state the problem in Section II-D. Subsequently, in
Section III we present a reformulation of the IQC-theorem
which is of independent interest and which serves as a
preparation of the main results in Section IV: A dissipativity
based proof of the IQC-theorem. We conclude the paper
by giving a brief sketch of (possible) generalizations and
applications in Section V.
Notation: We denote by L2e the extended space of
vector-valued locally square integrable functions on [0, ∞),
while L2 ⊂ L2e is the subspace of functions with finite
energy, equipped with the standard inner product h., .i and
the corresponding norm k·k. Further we denote by RL m×n
∞
(RH m×n
) the space of real-rational and proper (and stable)
∞
matrix functions without poles on the extended imaginary
axis C0 := iR∪{∞} (in the closed right-half complex-plane).
The induced-L2-norm of a bounded and causal operator is
denoted by k · kLh2 and
i realizations of LTI operators are
B
denoted by G = CA D
or G = (A, B, C, D). Finally, if
necessary, we abbreviate G∗ + G and G∗ P G as He(G) and
(⋆)∗ P G respectively.
II. A
SHORT RECAP ON ROBUST INPUT- OUTPUT
STABILITY
A. The basic setup
Consider the standard feedback interconnection
(
q = Gp + µ
p = ∆(q) + η
(1)
where G ∈ RH ∞ , ∆ is a bounded and causal operator
on L2e and µ, η ∈ L2e are exogenous disturbance inputs.
Typically G represents a nominal and fixed LTI model, while
∆ is viewed as the trouble-making component that captures
the uncertainties and/or nonlinearities which are allowed to
vary in a certain class ∆. Given G and the class ∆, the
central task in robust stability analysis is to characterize
whether the feedback interconnection (1) remains stable for
all ∆ ∈ ∆. For that purpose, let us rewrite the feedback
interconnection (1) as
!
!
!
! !
q
−Gp
µ
I −G q
:=
+
=
(2)
p
−∆(q)
p
η
−∆ I
{z
}
|
I(G,∆)
and introduce the following definitions.
Definition 1 (Well-posedness): The feedback interconnection (1) is well-posed if for each col(µ, η) ∈ L2e there exists
a unique response col(q, p) ∈ L2e satisfying (2) such that
the map col(µ, η) → col(q, p) is causal.
Definition 2 (Stability): The feedback interconnection (1)
is stable if it is well-posed and if the L2 -gain of the map
col(µ, η) → col(q, p) is bounded.
Definition 3 (Robust stability): The feedback interconnection is robustly stable if it is stable for all ∆ ∈ ∆.
B. The IQC-framework
As mentioned in the introduction, the IQC-framework
offers a systematic way to study the stability properties of
complex system interconnections. The basic procedure can
be summarized as follows [1]: Two signals v, w ∈ L2
are said to satisfy the IQC defined by the IQC-multiplier
Π = Π∗ ∈ RL ∞ if
!
!+
*
!
v
v
Π11 Π12
≥ 0. (3)
Σ(Π, v, w) :=
,
Π∗12 Π22
w
w
IQCs can be employed for capturing the properties of uncertainties and/or nonlinearities by imposing suitable constraints
on the IQC-multiplier Π such that Σ(Π, q, ∆(q)) ≥ 0 holds
for all q ∈ L2 . If we assume Π to belong to the set
Π = {Π = Π∗ ∈ RL ∞ : Π11 < 0, Π22 4 0 on C0 },
then we can state the following version of the well-known
results of [1].
Theorem 1: Assume that:
1) the feedback interconnection of G and ∆ is well-posed;
2) the IQC defined by Π ∈ Π is satisfied for ∆;
3)
!∗
!
G
G
≺ 0 on C0 .
(4)
Π
I
I
Then the feedback interconnection (1) is stable.
Recall that the results in [1] have been proven by means of
a homotopy method. This necessitates the first two conditions
in Theorem 1 to be satisfied with τ ∆ for all τ ∈ [0, 1]. This
implies that Π11 < 0 on C0 . If, in addition, Π22 4 0 on C0
then Σ(Π, q, ∆(q)) ≥ 0 for all q ∈ L2 automatically implies
that Σ(Π, q, τ ∆(q)) ≥ 0 holds for all τ ∈ [0, 1) and for
all q ∈ L2 . If compared to the general case, without any
inertia constraints on Π22 , the hypothesis on Π22 might be
restrictive. Nevertheless, we still cover most practical IQCmultipliers that are found in the literature. Indeed, apart
from certain full-block multiplier relaxation schemes, all
IQC-multipliers presented in [1], [2], [3], [4], [5], [6], [7],
[8] satisfy the above-mentioned inertia constraints. Without
too much loss of generality we can hence assume that
Π ∈ Π. Further note that we only require the feedback
interconnection of G and τ ∆ to be well-posed for τ = 1.
Consequently, it hence is possible to consider more general
uncertainty sets, and even allow for singletons.
C. Robust stability
A key feature of the IQC-framework involves the computational search for a whole family of IQC-multipliers that
are parameterized as ΠP = Ψ∗ PΨ ⊆ Π with a suitable
(LMIable) set of symmetric matrices P and with a typically
tall Ψ ∈ RH ∞ such that the IQC defined by Π = Ψ∗ P Ψ,
P ∈ P, holds for all ∆ ∈ ∆. Robust stability can then be
characterized as follows.
Corollary 1: Suppose that the first two conditions in Theorem 1 hold for all ∆ ∈ ∆, with Π replaced by ΠP . Then
the feedback interconnection of G and ∆ is robustly stable if
there exists some P ∈ P for which the following frequency
domain inequality (FDI) holds:
!∗
!
G
G
Ψ∗ P Ψ
≺ 0 on C0 .
(5)
I
I
If P has a nice description, condition (5) can be easily
checked numerically by exploiting the KYP-lemma [35].
Indeed, if we assume Ψcol(G, I) to admit the realization
(A, B, C, D) with eig(A) ⊂ C− , then (5) is equivalent to the
existence of a symmetric matrix V for which the following
matrix inequality holds:
!
⊤ V A + A⊤ V V B
+
P
≺ 0. (6)
C
D
C
D
B⊤V
0
D. Key difficulties to resolve
Let us briefly discuss the main difficulties which occur
when one tries to prove Theorem 1 with dissipation arguments. For that purpose, consider the open sets X ⊂ Rn ,
U ⊂ Ru , Y ⊂ Ry and define the sufficiently smooth
mappings f : X × U → Rn , g : X × U → Y and
s : U × Y → R. Then we can formally introduce the
following definition of dissipativity [19].
Definition 4 (Dissipativity): The system ẋ = f (x, u), y =
g(x, u) is dissipative with respect to the supply rate s(·, ·) if
there exists a storage function V : X → R such that
Z t2
s(u(t), y(t))dt ≤ V (x(t1 ))
(7)
V (x(t2 ))+
t1
holds for all trajectories with x(t) ∈ X, u(t) ∈ U , y(t) ∈ Y
for t ∈ [t1 , t2 ] and for all t1 < t2 .
Let us first consider the special case of static IQCmultipliers (i.e. Ψ = I). For any trajectory of (1), with
initial condition x(0), we can right and left-multiply (6) with
col(x(t), p(t)) and its transpose. Then, after integration on
[0,T], we obtain
Z T
⊤
x(T ) V x(T ) +
y(t)⊤ P y(t)dt ≤ x(0)⊤ V x(0) ∀T ≥ 0.
0
(8)
Here y = col(Gp, p) = col(q, ∆(q)). Now observe that the
left-upper block of P is nonnegative just because Ψ = I.
Hence, (6) implies that RV ≻ 0 if and only if eig(A) ⊂
T
C− . Further note that 0 y(t)⊤ P y(t)dt ≥ 0 holds with
col(q, ∆(q)) for all T > 0, again, just because Ψ = I.
We conclude that all the classical arguments apply in case
one tries to proof Theorem 1 for the special case of static
IQC-multipliers. On the other hand, for the general case of
dynamic IQC-multipliers (i.e. Ψ is an arbitrary tall and stable
transfer matrix), one cannot directly relate the trajectories of
Ψcol(G, I) to the trajectories of the feedback interconnection
(1), since Ψ is not stably invertible. Moreover, P is in general
indefinite matrix. Hence, eig(A) ⊂ C− neither implies, nor
by positive definiteness of V . Further note that
RisTimplied
⊤
y(t)
P
y(t)dt
is in general only nonnegative for T = ∞.
0
All this prevents us to prove stability. In the sequel we will
show how to resolve these difficulties.
III. A
REFORMULATION OF
T HEOREM 1
As a preparation for the proof of Theorem 1, we will
first reformulate the feedback interconnection (1) as well
as the conditions in Theorem 1. For that purpose, let us
introduce the signals q = q1 = p1 and p = p2 = q2 with
p1 , q1 , p2 , q2 ∈ L2e . Then (1) can be expressed as
!
!
!
!
p1
−I 2G
2µ1
q1
=
+
,
p2
0
0 I
q2
| {z } | {z } | {z } | {z }
qe
p
µs
Ge
! e!
!
!
p1
0
I 0
q1
+
=
.
p2
2η2
2∆ −I
q2
{z
} | {z }
| {z } |
pe
∆e (qe )
ηs
Moreover, by replacing the structured external disturbances
µs and ηs by the unstructured external disturbances µe :=
col(µ1 , µ2 ) and ηe := col(η1 , η2 ) respectively, we obtain the
following extended feedback interconnection:
(
qe = Ge pe + µe
(9)
pe = ∆e (qe ) + ηe
If we also introduce the extended IQC-multiplier
!
!
!
!
0 Πε
Π11ε Π12
Π11 Π12
εI 0
Πe :=
, Πε =
=
+
Πε 0
Π∗12 Π22
Π∗12 Π22
0 0
| {z }
Tε
for ε ≥ 0, we can state the following result.
Theorem 2:
1) The feedback interconnection (1) is stable iff the
feedback interconnection (9) is stable.
2) For all ε ≥ 0 the IQC defined by Πε is satisfied for ∆
iff the IQC defined by Πe is satisfied for ∆e .
3) The FDI (4) is satisfied iff there exists a small ε > 0
such that
!
!∗
Ge
Ge
≺ 0 on C0 .
Πe
(10)
I
I
Proof: Let us start proving the first statement. For
this purpose, observe that I(Ge , ∆e ) can be written as
I(Ge , ∆e ) = T2 T3 T4 = T2 T3 T5 T5 T4 , with
!
!
!
−2I 0
Ge T1 −Ge
I(G, ∆) 0
T1 =
, T2 =
, T3 =
,
0 2I
0
I
0
−I
!
!
I 0
I 0
T4 =
, T5 =
.
∆e −I
0 −I
If I(G, ∆) has a causal inverse on L2e , then one can readily
verify that
I(Ge , ∆e )−1 = T4−1 T3−1 T2−1 = (T5 T4 )−1 (T3 T5 )−1 T2−1 .
Note that there exist constants κ2 , κ2i , κ4 , κ4i > 0 such that
kT2 kL2 ≤ κ2 , kT2−1kL2 ≤ κ2i , kT4 kL2 ≤ κ4 , kT4−1kL2 ≤ κ4i .
Hence, if we assume that (1) is well-posed and that there
exists some κ1 > 0 such that kI(G, ∆)−1 kL2 ≤ κ1 , then
k(T5 T3 )−1 kL2 ≤ κ1 +1
⇒ kT2−1kL2 k(T5 T3 )−1 kL2 k(T5 T4 )−1 kL2 ≤ (κ1 +1)κ2i κ4i
⇒ k(T5 T4 )−1 (T3 T5 )−1 T2−1 kL2 ≤ (κ1 +1)κ2i κ4i
⇒ kI(Ge , ∆e )−1 kL2 ≤ (κ1 +1)κ2i κ4i .
Therefore, stability of (9) is implied by stability of (1).
Conversely, suppose that (9) is well-posed and that there
exists some κ3 > 0 such that kI(Ge , ∆e )−1 kL2 ≤ κ3 , then
kT4−1 T3−1 T2−1 kL2 ≤ κ3
⇒ kT4 kL2 kT4−1 T3−1 T2−1 kL2 kT2 kL2 ≤ κ2 κ3 κ4
⇒ kT3−1 kL2 ≤ κ2 κ3 κ4
⇒ kI(G, ∆)−1 kL2 ≤ κ2 κ3 κ4 + 1.
Therefore, stability of (1) is implied by stability of (9).
In order to prove the second statement, we need to show
that, for all ε ≥ 0,
Σ(Πε , q, ∆(q)) ≥ 0 ∀q ∈ L2
(11)
is equivalent to
Σ(Πe , qe , ∆e (qe )) ≥ 0 ∀qe ∈ L2 .
(12)
For that purpose, let us introduce the Fourier-transforms q̂1
\
and ∆(q
1 ) of the signals q1 = q and p2 = ∆(q1 ), respectively,
and observe that (11) is equivalent to (Parseval)
Z ∞
\
q̂1∗ Π11ε q̂1 + 2Re q̂1∗ Π12 ∆(q
1) +
persists to hold. With Π11ε = Π11 + εI we can then apply
the Schur complement in order to obtain
!
−Π11ε
Π11ε G
≺ 0 on C0 .
(15)
G∗ Π11ε G∗ Π12 + Π∗12 G + Π22
Now (15) can be equivalently written as
!
−Π11ε 2Π11ε G + Π12
=
He
−Π∗12 2Π∗12 G + Π22
!
!!
−I 2G
Π11ε Π12
≺ 0 on C0 ,
= He
0 I
Π∗12 Π22
−∞
∗
\
\
+∆(q
1 ) Π22 ∆(q1 )dω ≥ 0.
(13)
Since Π22 4 0 on C0 we infer that
∗
0
\
\
Π22 ∆(q
∆(q
1 ) − q̂2 ≤ 0 a.e. on C
1 ) − q̂2
for all q1 , q2 ∈ L2 . Hence for all q1 , q2 ∈ L2 it holds that
∗
∗
∗
\
\
\
∆(q
1 ) Π22 ∆(q1 ) ≤ 2Re q̂2 Π22 ∆(q1 ) − q̂2 Π22 q̂2
a.e. on C0 . Therefore (13) implies
Z ∞
\
q̂1∗ Π11ε q̂1 + 2Re q̂1∗ Π12 ∆(q
1) +
−∞
and thus
Z ∞
He
−∞
∗
\
+2Re q̂2∗ Π22 ∆(q
1 ) − qˆ2 Π22 qˆ2 dω ≥ 0
q̂1
q̂2
!∗
Π11ε Π12
Π∗12 Π22
!
q̂1
\
2∆(q1 ) − q̂2
!
dω ≥ 0.
By defining the Fourier transforms q̂e = col(q̂1 , q̂2 ) and
p̂e = col(p̂1 , p̂2 ) of the signals qe = col(q1 , q2 ) and pe =
col(p1 , p2 ), respectively, and observing that
!
q̂
1
\
∆
e (qe ) =
\
2∆(q
1 ) − q̂2
is nothing but the Fourier transform of pe = ∆e (qe ), we can
finally infer that (12) is indeed satisfied for all ε ≥ 0.
In order to show that (12) implies (11), set q̃e =
col(q, ∆(q)) for q ∈ L2 . By the definition of ∆e we have
p̃e = ∆e (q̃e ) = col(q, ∆(q)) = q̃e . Hence, for qe = q̃e the
lifted IQC (12) simplifies into
Σ(Πe , q̃e , q̃e ) ≥ 0, q̃e = ∆e (q̃e ), q̃e ∈ L2 .
Direct inspection reveals that this is nothing but (11).
It remains show the third statement. For this purpose,
observe that the FDI (4) trivially be written as
G∗ Π11 G + G∗ Π12 + Π∗12 G + Π22 ≺ 0 on C0 .
Clearly there exists some small ε > 0 such that
G∗ (Π11 + εI)G + G∗ Π12 + Π∗12 G + Π22 ≺ 0 on C0 (14)
which is (10). The reverse direction directly follows from
!∗
!∗
!!
!
G
Ge
Ge
G
≺ 0 on C0 .
Πe
I
I
I
I
Note that Theorem 2 is an auxiliary result of independent interest.
For example, IQC-multipliers of the form
Π = Φ0∗ Φ0 have emerged in a number of interesting
gain-scheduling and distributed controller synthesis problems
[32], [36], [37]. Since the augmented IQC-multiplier Πe
possesses the very same structure, Theorem 2 might be
useful for generalizations to larger classes of dynamic IQCmultipliers. Further, we will exploit the result in the sequel,
in order to proof of Theorem 1.
IV. M AIN
RESULT: A DISSIPATIVITY BASED PROOF OF
THE IQC- THEOREM
As discussed in Section II-C, a key feature of the IQCframework involves the computational search of a symmetric
matrix P ∈ P that satisfies the FDI in Corollary 1 in order
to verify whether or not the feedback interconnection (1) is
robustly stable for a certain class ∆. It hence makes sense to
prove Theorem 1 by taking Corollary 1 as a starting point.
For this reason, let us assume that the first two conditions
in Theorem 1 hold for all ∆ ∈ ∆, with Π replaced by ΠP ,
and that there exists some P ∈ P for which the FDI (5) is
satisfied. Let us also assume, for notational simplicity, that
Π11 ≻ 0 on C0 such that ε in Theorem 2 can be set to zero.
By Theorem 2 the FDI (5) is then equivalent to
!
!∗
!
ΨGe
ΨGe
0 P
≺ 0 on C0 .
(16)
Ψ
P 0
Ψ
If we assume Ψ and Ge to admit the realizations
(AΨ , BΨ , CΨ , DΨ ) and (Ae , Be , Ce , De ), with eig(AΨ ) ⊂
C− and eig(Ae ) ⊂ C− , we can define the realization


AΨ 0 BΨ Ce BΨ De

! 
 0 AΨ 0
BΨ 


ΨGe
=
0 0 Ae
Be 
.

Ψ


 CΨ 0 DΨ Ce DΨ De 
0 CΨ 0
DΨ
By the KYP-lemma (16) is equivalent to the existence of a
matrix Ve = Ve⊤ for which the following LMI is satisfied [35]:


I 0
0
0


0
0 
0 I



I
0 
0 Ve 0 0 
0 0




⊤Ve 0 0 0 AΨ 0 BΨ Ce BΨ De
(⋆) 
(17)

 ≺ 0.
 0 0 0 P  0 AΨ 0
BΨ 


Be 
0 0 P 0 
 0 0 Ae



CΨ 0 DΨ Ce DΨ De
0 CΨ 0
DΨ
Here Ve is assumed to be partitioned as


V11 V12 V13

 ⊤
Ve = V12
V22 V23  ,
⊤
⊤
V13
V23
V33
were V11 , V22 and V33 have compatible dimensions with AΨ ,
AΨ and Ae respectively.
Now observe that Ve is in general an indefinite matrix. As
discussed in Section II-D, this is one of the main problems
that prevented us from concluding stability of the feedback
interconnection (1). In order to overcome this difficulty we
need to factorize the IQC-multiplier Ψ∗ P Ψ as Ψ̂∗ P̂ Ψ̂, with
P̂ = P̂ ⊤ , Ψ̂, Ψ̂−1 ∈ RH ∞ . For that reason, let us collect
some existing insights from the literature [32].
n×n
Lemma 1: Let Q ∈ RL ∞
. Then Q satisfies Q = Q∗
iff it admits a realization with the structure


⊤
−A⊤
Q EQ CQ


(18)
Q =  0 AQ BQ  ,
⊤
−BQ
CQ DQ
⊤
⊤
with DQ = DQ
, EQ = EQ
and eig(AQ ) ⊂ C− .
Lemma 2: Let S ∈ RH ∞ and Q = Q∗ ∈ RL ∞ satisfy
He(QS) ≺ 0 on C0 . Then there exists a TQ ∈ RH ∞ with
TQ (∞) = I, TQ−1 ∈ RH ∞ such that Q = TQ∗ Q(∞)TQ . If Q
is realized as in (18), then DQ is invertible and the ARE
⊤ −1
⊤
A⊤
Q ZQ + ZQ AQ + EQ − (⋆) DQ (BQ ZQ + CQ ) = 0 (19)
⊤
has a unique stabilizing solution ZQ = ZQ
, i.e. AQ−BQ C̃Q is
−1
⊤
Hurwitz for C̃Q := DQ (BQ ZQ +CQ ). A symmetric canonical Wiener-Hopf factorization is given by Q = TQ∗ Q(∞)TQ
with TQ = (AQ , BQ , C̃Q , I).
Now observe that


⊤
⊤
−A⊤
Ψ CΨ P CΨ CΨ P DΨ


Ψ∗ P Ψ =  0
(20)
AΨ
BΨ 
⊤
⊤
⊤
−BΨ DΨ P CΨ DΨ P DΨ
has precisely the structure of (18). Hence, by Lemma 2,
feasibility of (17) implies the existence of the stabilizing
solution Z of the ARE (19) corresponding to realization (20)
which can be expressed as



!⊤
I 0
0Z 0
C̃Ψ


⊤
P̃ C̃Ψ I , (21)
(⋆) Z 0 0 AΨ BΨ  =
I
CΨ DΨ
0 0P
We can thus define the realization Ψ̃ = (AΨ , BΨ , C̃Ψ , I)
such that Ψ∗ P Ψ = Ψ̃∗ P̃ Ψ̃, with Ψ̃, Ψ̃−1 ∈ RH ∞ .
It is now possible to merge (21) and (17). Indeed, by
applying the gluing-lemma [34] it is not hard to verify that
(17) is equivalent to


I 0
0
0


0
0 
 0 I



0 Ṽe 0 0 
I
0 
 0 0




⊤  Ṽe 0 0 0   AΨ 0 BΨ Ce BΨ De 
(⋆) 

 ≺ 0, (22)
 0 0 0 P̃   0 AΨ 0
BΨ 


Be 
0 0 P̃ 0 
 0 0 Ae



De 
 C̃Ψ 0 Ce
0 C̃Ψ 0
I
with

V11 V12 −Z V13

 ⊤
Ṽe =  V12
−Z V22 V23  .
⊤
⊤
V13
V23
V33

Moreover, it is straightforward to show that Ṽe is in fact
positive definite: A simple congruence transformation reveals
that (22) is equivalent to
!
Ṽe Ãe Ṽe B̃e
≺ 0,
(23)
He
P̃ C̃e P̃ D̃e
where


# AΨ −BΨ De C̃Ψ BΨCe BΨDe
"
BΨ 
Ãe B̃e 
 0 AΨ −BΨ C̃Ψ 0

=
= Ψ̃Ge Ψ̃−1=: G̃e .
Ae
Be 
−Be C̃Ψ
C̃e D̃e  0
C̃Ψ −DC̃Ψ
Ce
De
(24)
Clearly, since AΨ , AΨ − BΨ C̃Ψ and Ae are all Hurwitz we
can conclude the same for Ãe . Moreover, the left-upper block
of (23) reads as He(Ṽe Ãe ) ≺ 0 which indeed implies Ṽe ≻ 0.
Now recall from Theorem 2 that Σ(Π, q, ∆(q)) ≥ 0 is
equivalent to Σ(Πe , qe , ∆e (qe )) ≥ 0. With Π = Ψ̃∗ P̃ Ψ̃ and
!
!
!
q̃e
Ψ̃ 0
qe
=
,
p̃e
0 Ψ̃
pe
˜ e (q̃e )) ≥ 0, where ∆
˜ e = Ψ̃∆e Ψ̃−1
we obtain Σ(P̃ , q̃e , ∆
defines a bounded and causal operator on L2e . We conclude
that
Z ∞
˜ e (q̃e )(t)dt ≥ 0 ∀q̃e ∈ L2 .
q̃e (t)⊤ P̃ ∆
0
Moreover, for any q̃e ∈ L2e we infer that the truncated signal
(i.e. q̃eT = q̃e on [0, T ] and q̃eT = 0 on (T, ∞)) satisfies
˜ e we infer
q̃eT ∈ L2 . By causality of ∆
Z ∞
˜ e (q̃eT )(t)dt ≥ 0
q̃eT (t)⊤ P̃ ∆
0
⇒
Z
0
∞
˜ e (q̃eT )T (t)dt ≥ 0
q̃eT (t)⊤ P̃ ∆
⇒
Z
∞
˜ e (q̃e )T (t)dt ≥ 0
q̃eT (t)⊤ P̃ ∆
0
⇒
Z
T
˜ e (q̃e )T (t)dt ≥ 0 ∀ T ≥ 0. (25)
q̃eT (t)⊤ P̃ ∆
0
RT
Now recall from Section II-D that 0 y(t)⊤ P y(t)dt is in
general only nonnegative for T = ∞. This was another
important reason that prevented us from concluding stability
of the feedback interconnection (1). On the other hand, it is
nice to see that (25) does satisfy the nonnegativity property
for all T ≥ 0. This resolves the last obstacle for proving
stability of the feedback interconnection (1). Indeed, all
previous step were a preparation in order to transform the
augmented feedback interconnection (9) into the feedback
interconnection
(
q̃e = G̃e p̃e + µ̃e
(26)
˜ e (q̃e ) + η̃e ,
p̃e = ∆
with the external disturbances related by
!
!
!
µ̃e
Ψ̃ 0
µe
=
.
η̃e
0 Ψ̃
ηe
The overall procedure is illustrated in Figure 1. Note that
G̃e
Ge
Ψ̃−1 - ? - Ψ̃−1 -
∆e
-
Ψ̃
µe
-
Ψ̃
µ̃e
q̃e
Fig. 1.
p̃e
η̃e
6
Ψ̃
ηe
ỹe
Ψ̃
˜e
∆
Transformed feedback interconnection
these are very classical arguments [2], [30]. The key twist is
to perform them in the state-space, starting with the original
parametrization of the IQC-multiplier Π = Ψ∗ P Ψ. For one
reason, this might be very useful in case one would like to
impose hard time-domain constraints on the original data.
Furthermore, observe that stability of (9) is equivalent to the
stability of (26). Indeed, let Ψ̃e = diag(Ψ̃, Ψ̃) and suppose
˜ e )−1 kL2 ≤ γ1 .
that there exists a γ1 > 0 such that kI(G̃e , ∆
−1
−1
With γ1 = γ2 (kΨ̃e kL2 kΨ̃e kL2 ) we can then infer that
˜ e )−1 kL2 kΨ̃e kL2 kΨ̃−1 kL2 ≤ γ2 ⇒
kI(G̃e , ∆
e
˜ e )−1 Ψ̃−1
kΨ̃e I(G̃e , ∆
e kL2 ≤ γ2 ⇒
kI(Ge , ∆e )−1 kL2 ≤ γ2 .
The converse statement follows from identical arguments.
We are now ready to prove Theorem 1. Due to all our
preparations, it remains to show that there exists a γ > 0
˜ e )−1 kL2 ≤ γ for all ∆ ∈ ∆. For this
such that kI(G̃e , ∆
reason, let us define a linear fractional representation of (26)
as follows:
 
  
ỹe
G̃e I G̃e
q̃e
 
  
˜ e (q̃e ).
(27)
 q̃e  = G̃e I G̃e   µ̃e  , ỹe = ∆
η̃e
I 0 I
p̃e
By recalling (24) it is straightforward to see that




Ãe B̃e 0 B̃e
G̃e I G̃e


  C̃e D̃e I D̃e 

.
G̃e I G̃e  = 
 C̃e D̃e I D̃e 
I 0 I
0 I 0 I
(28)
Moreover, it is not hard to show that there exists a sufficiently
large γ > 0 such that the inequality (23), corresponding to
the system (24), can be augmented with the performance
channels col(µ̃e , η̃e ) → col(q̃e , p̃e ) as


Ṽe Ãe Ṽe B̃e 0 Ṽe B̃e
!
 P̃ C̃ P̃ D̃ P̃ P̃ D̃ 
e 1
e
e

⊤ C̃e D̃e I D̃e
≺ 0,
He
+ (⋆)
 0
0 I 0 I
0 − γ2 I 0  γ
0
0
0 − γ2 I
(29)
corresponding to the open-loop generalized plant (28). Further note that for some small ǫ > 0 we can replace the
right-hand side of (29) by −ǫI, without violating (29).
For any trajectory of (26) with initial condition zero, we
can right- and left-multiply the resulting inequality with
col(x(t), ỹe (t), µ̃e (t), η̃e (t)) and its transpose. Then we obtain
d
x(t)⊤ Ṽe x(t) + 2q̃e (t)⊤ P̃ ỹe (t) + γ1 (kq̃e (t)k2 + kp̃e (t)k2 )
dt
+ǫ(kx(t)k2 + kỹe (t)k2 ) ≤ (γ − ǫ)(kµ̃e (t)k2 + kη̃e (t)k2 )
and after integration on [0, T ]
Z T
x(T )⊤ Ṽe x(T ) + 2
q̃e (t)⊤ P̃ ỹe (t)dt
0
+ γ1
Z
T
kq̃e (t)k2 +kp̃e (t)k2 dt + ǫ
0
Z
T
kx(t)k2 +kỹe (t)k2 dt
0
≤ (γ − ǫ)
Z
T
kµ̃e (t)k2 + kη̃e (t)k2 dt ∀ T ≥ 0.
0
˜ e (q̃e ), we can exploiting (25) to get
In view of ỹe = ∆
Z T
Z T
2
2
1
˜ e )(t)k2 dt+
kq̃
(t)k
+kp̃
(t)k
dt
+
ǫ
kx(t)k2 +k∆(q̃
e
e
γ
0
0
Z T
x(T )⊤ Ṽe x(T ) ≤ (γ − ǫ) kµ̃e (t)k2 + kη̃e (t)k2 dt ∀ T ≥ 0.
0
Since the right-hand size is bounded for T → ∞ we can infer
the same (due to Ṽe ≻ 0) for the left-hand side. This shows
˜ e ) ∈ L2 . Robust stability of the feedback
that q̃e , p̃e , x, ∆(q̃
interconnection (1) is finally proven by observing that for all
µ̃e , η̃e ∈ L2 we have
Z T
1
kq̃e (t)k2 + kp̃e (t)k2 dt + x(T )⊤ Ṽe x(T ) ≤
γ
0
γ
Z
T
kµ̃e (t)k2 + kη̃e (t)k2 dt ∀ T ≥ 0.
0
˜ e )−1 kL2 ≤ γ for all ∆ ∈ ∆
Clearly this implies that kI(G̃e , ∆
and the proof is finished.
V. A
BRIEF SKETCH OF POSSIBLE GENERALIZATIONS
AND CONSEQUENCES OF THE MAIN RESULT
Let us conclude the paper by briefly discussing the main
results. We have established a connection between the IQCframework and the dissipativity approach. This not only
shows that IQCs can be interpreted as dynamic supply
functions, but also opens the way to merge frequencydomain techniques with time-domain conditions known from
Lyapunov-theory. For example, one could think of incorporation hard time-domain constraints, such as L1 or generalized
H2 constraints, which could only be applied if the IQCs
were valid on all finite horizons. This might lead to nice
applications in the field of anti-windup compensator design
(see e.g. [13] and references therein). Further, it might also
be possible establish a connection with parameter dependent
Lyapunov functions. We refer the reader to [28] for some
more concrete applications. However, note that the inertia
assumption on the storage function in [28, Theorem 3] is by
no means true in general. As shown in Section IV it is required to perform a shifting operation on the storage function
through the solution of an algebraic Riccati equation (ARE).
Therefore, the storage function does not solely consists of
original data, which might cause difficulties in finding any
of the listed generalizations. This could be resolved by means
of an LMI characterization corresponding to the ARE [24].
VI. C ONCLUSIONS
In this paper we have given a dissipativity based proof of
the well-known integral quadratic constraint (IQC) theorem
under mild assumptions. This opens the way to merge
frequency-domain techniques with time-domain conditions
known from Lyapunov-theory.
ACKNOWLEDGEMENTS
The authors would like to thank the German Research Foundation (DFG)
for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.
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