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Robust finite-frequency H2 analysis of uncertain
systems with application to flight comfort
analysis
Andrea Garulli, Anders Hansson, Sina Khoshfetrat Pakazad, Alfio Masi and Ragnar Wallin
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
Andrea Garulli, Anders Hansson, Sina Khoshfetrat Pakazad, Alfio Masi and Ragnar Wallin,
Robust finite-frequency H2 analysis of uncertain systems with application to flight comfort
analysis, 2013, Control Engineering Practice, (21), 6, 887-897.
http://dx.doi.org/10.1016/j.conengprac.2013.02.003
Copyright: Elsevier
http://www.elsevier.com/
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-94316
Robust finite-frequency H2 analysis of uncertain systems with application to flight
comfort analysis
Andrea Garullia, Anders Hanssonb, Sina Khoshfetrat Pakazadb,∗, Alfio Masia, Ragnar Wallinb
aDipartimento di Ingegneria dell’Informazione Universita’ degli Studi di Siena, ItalybDivision of Automatic Control, Linkoping University of Technology, Sweden
Abstract
In many applications, design or analysis is performed over a finite-frequency range of interest. The importance of theH2
norm highlights the necessity of computing this norm accordingly. This paper provides different methods for computingupper bounds of the robust finite-frequency H2 norm for systems with structured uncertainties. An application of therobust finite-frequency H2 norm for a comfort analysis problem of an aero-elastic model of an aircraft is also presented.
Keywords: robust H2 norm, uncertain systems, robust control, flight comfort analysis.
1. Introduction
The H2 norm has been one of the pivotal design andanalysis criteria in many applications, such as structuraldynamics, acoustics, colored noise disturbance rejection,etc, Caracciolo et al. (2005); Marro and Zattoni (2005);Zattoni (2006); Alazard (2002); Banjerdpongchai and How(1998). This norm also plays an important role in thefield of robust control, where there has been a substantialamount of research on computation, analysis and designbased on this measure in the presence of uncertainty, manyof which consider the use of Linear Matrix Inequalities(LMIs) and Riccati equations for this purpose, e.g., Doyleet al. (1989); Stoorvogel (1993); Boyd et al. (1994); Iwasaki(1996); Paganini (1997, 1999a); Sznaier et al. (2002). Asurvey of methods in robust H2 analysis is provided inPaganini and Feron (2000).
Most of the methods presented in the literature considerthe whole frequency range for calculating the H2 and ro-bust H2 norm. However, in some applications it is bene-ficial to concentrate only on a finite-frequency range andcalculate the design/analysis measures accordingly. Thiscan be due to different reasons, e.g., the model is only validfor a specific frequency range or the design is targeted for aspecific frequency interval. This motivates the importanceof computing the (robust) finite-frequency H2 norm.
Frequency limitations in several analysis and designproblems relevant to control systems have been addressed
∗Corresponding Author: Sina Khoshfetrat Pakazad, Division ofAutomatic Control, Linkoping University of Technology, Sweden,Email: [email protected]
Email addresses: [email protected] (Andrea Garulli),[email protected] (Anders Hansson), [email protected](Sina Khoshfetrat Pakazad ), [email protected] (Alfio Masi),[email protected] (Ragnar Wallin)
in Iwasaki and Hara (2005), by introducing a generaliza-tion of the celebrated KYP lemma. However, it is notknown that this result can be used to compute the robustfinite-frequency H2 norm. In Gawronski (2000), a methodfor calculating the finite-frequency H2 norm for systemswithout uncertainty is presented, where the key step is tocompute the finite-frequency observability Gramian. Thisis accomplished by first computing the regular observabil-ity Gramian and then scaling it by a system dependentmatrix.
In this paper, we introduce two methods for calculat-ing an upper bound of the robust finite-frequency H2
norm for systems with structured uncertainties. We alsoassume that a LFT (Linear Fractional Transformation)representation of these systems are available, which isa common assumption in many fields concerning uncer-tain systems, e.g., in aeronautics, Cockburn and Morton(1997); Poussot-Vassal and Roos (2012); Ferreres (2011);Zhou et al. (1996). The first method combines the no-tion of finite-frequency Gramians, introduced in Gawron-ski (2000), with convex optimization tools, Boyd and Van-denberghe (2004), commonly used in robust control, and itcalculates an upper bound by solving an underlying opti-mization problem, Masi et al. (2010). The second method,provides a computationally cheaper algorithmic methodfor calculating such upper bounds. In contrast to the firstapproach, the second method performs frequency griddingand breaks the original problem into smaller problems,which are possibly easier to solve. Then it uses the ideaspresented in Roos and Biannic (2010), on computing up-per bounds of structured singular values, for solving thesmaller problems. The results of the smaller problems arethen combined to compute an upper bound over the wholedesired frequency range, Pakazad et al. (2011).This paper is structured as follows. First some of the
notations used throughout the paper are presented. Sec-
Preprint submitted to Elsevier February 1, 2013
tion 2 introduces the problem formulation. Mathematicalpreliminaries are presented in Section 3, which covers thenotion of finite-frequency Gramians and reviews the calcu-lation of upper bounds of the robust H2 norm. Sections 4and 5 provide the details of the two methods for calculat-ing upper bounds of the robust finite-frequency H2 norm.In Section 6 a numerical example is presented to illustratethe main features of the proposed techniques. The appli-cation of the two methods to a robust comfort analysisproblem for an aero-elastic model of a civil aircraft is pre-sented in Section 7. Section 8 provides more insight bydiscussing advantages and disadvantages of the proposedmethods, and finally Section 9 concludes the paper withsome final remarks.
1.1. Notation
The notation in this paper is standard. The set ofm × n real and complex matrices are denoted by Rm×n
and Cm×n, respectively. Given a matrix A, AT is its trans-pose and A∗ is its conjugate transpose. By In, we denotethe n × n identity matrix. The symbols � and ≺ denotethe inequality relation between matrices and by ln(A) wedenote the standard matrix logarithm. Given matrices Aifor i = 1, . . . , n, diag(A1, . . . , An) denotes a block-diagonalmatrix with Ais as its diagonal blocks. The min and maxrepresent the minimum and maximum of a function or aset, and similarly sup represents the supremum of a func-tion. A transfer function matrix in terms of state-spacedata is denoted
[
A B
C D
]
:= C(jωI −A)−1B +D. (1)
By ‖ · ‖2, we denote the Euclidian or 2-norm of a vector orthe norm of a matrix induced by the 2-norm. For the sakeof brevity of notation, unless needed for clarity, we dropthe dependence of functions on frequency.
2. Problem formulation
Consider the following stable system in state space form{
x = Ax+Bu
y = Cx(2)
and define G(s) as its corresponding transfer function.Then the H2 norm of the system in (2) is defined as
‖G‖22 =
∫ ∞
−∞
Tr {G(jω)∗G(jω)}dω
2π, (3)
which can also be expressed as
‖G‖22 =
∫ ∞
0
Tr{
BT eAT tCTCeAtB
}
dt
= Tr
{
BT(∫ ∞
0
eAT tCTCeAt dt
)
B
}
= Tr{
BTWoB}
= Tr{
CWcCT}
,
(4)
where Wo and Wc are the observability and controllabilityGramians of the system, respectively.As can be seen from the equation in (3), computing the
H2 norm requires the integration over the whole frequencyrange. We define the finite-frequency H2 norm of the sys-tem by limiting the integration bounds to finite values as
‖G‖22,ω =
∫ ω
−ω
Tr {G(jω)∗G(jω)}dω
2π, (5)
where the integration interval Id = [−ω , ω] representsthe frequency range of interest. Similarly, we can extendthis notion to uncertain systems. Consider the followinguncertain system in state space form
x = Ax+Bqq +Bww
p = Cpx+Dpqq
z = Czx+Dzqq
q = ∆p
(6)
where x ∈ Rn, w ∈ R
m, z ∈ Rl and p, q ∈ R
d, see Fig-ure 1. The perturbation block ∆, which represents the un-certainty in (6), is a causal linear time invariant operator,bounded in the L2 induced norm, and has the followingstructure
∆ = diag[
δ1Ir1 · · · δLIrL ∆L+1 · · · ∆L+F
]
, (7)
where δi ∈ R, for i = 1, . . . , L, and ∆L+i ∈ Cmi×mi ,
for i = 1, . . . , F , such that∑L
i=1 ri +∑F
i=1mi = d. Itis assumed that ∆ ∈ B∆ where B∆ is the unit ball forthe induced L2 norm, i.e., B∆ = {∆ : ‖∆‖2 ≤ 1}. Thisstructure of ∆ is standard in robust control and allows oneto represent both real parametric uncertainties and un-modeled system dynamics using real and complex blocks,respectively.
Remark 1. The assumption on ∆ can be relaxed tocope with more general uncertainty models, such as time-varying or nonlinear operators. In such cases, the notionof the H2 norm of a system must be carefully reconsid-ered (see Paganini (1999b) and Sznaier et al. (2002) for athorough discussion on this issue).
The transfer function matrix for the uncertain systemin (6) is defined as below, see Figure 1,
M(jω) =
[
M11 M12
M21 M22
]
=
A Bq BwCp Dpq 0Cz Dzq 0
, (8)
where M ∈ C(d+l)×(d+m), M11 ∈ Cd×d, M12 ∈ Cd×m,M21 ∈ Cl×d and M22 ∈ Cl×m. In the upcoming sections,we also utilize the following partitioning of this transferfunction matrix
M(jω) =[
M1 M2
]
=
[
A Bq BwC D 0
]
, (9)
2
M11 M12M21 M22
∆
p(t) q(t)
z(t) w(t)
Figure 1: Uncertain system with structured uncertainty
where M1 ∈ C(d+l)×(d),M2 ∈ C(d+l)×(m) and
C =
[
CpCz
]
, D =
[
Dpq
Dzq
]
. (10)
In analysis of uncertain systems, the transfer function be-tween the signals w(t) and z(t) is of interest. This transferfunction is given by the upper LFT representation
(∆ ∗M) =M22 +M21∆(I −M11∆)−1M12. (11)
which is a special case of the so-called Redheffer product,Zhou et al. (1996). Having (11), the robust H2 norm ofthe system in (6) is defined as
sup∆∈B∆
‖∆ ∗M‖22
= sup∆∈B∆
∫ ∞
−∞
Tr {(∆ ∗M)∗(∆ ∗M)}dω
2π, (12)
and similarly the robust finite-frequency H2 norm of thesystem in (6) is defined as
sup∆∈B∆
‖∆ ∗M‖22,ω
= sup∆∈B∆
∫ ω
−ω
Tr {(∆ ∗M)∗(∆ ∗M)}dω
2π. (13)
This paper proposes methods for calculating upper boundsof the robust finite-frequency H2 norm of such systems.Next section provides the mathematical background forthese methods.
3. Mathematical preliminaries
3.1. Finite-frequency observability Gramian
As was mentioned in Section 1, one of the ways of com-puting the H2 norm of the system in (2) is by using its ob-servability Gramian, see (4). We can compute the observ-ability Gramian by solving the following Lyapunov equa-tion
ATWo +WoA+ CTC = 0. (14)
Using Parseval’s identity and (4), the observabilityGramian can also be expressed as
Wo =
∫ ∞
−∞
H(jω)∗CTCH(jω)dω
2π, (15)
where H(jω) = (jωI − A)−1. This allows us to definethe finite-frequency observability Gramian, as proposed inGawronski (2000), as
Wo(ω) =
∫ ω
−ω
H(jω)∗CTCH(jω)dω
2π. (16)
The next lemma provides a way to expressWo(ω) in termsof the observability Gramian, Wo.
Lemma 1. The finite-frequency observability Gramiancan be expressed as
Wo(ω) = L(A, ω)∗Wo +WoL(A, ω), (17)
where Wo is defined by (14) or equivalently by (15) and
L(A, ω) =
∫ ω
−ω
H(jω)dω
2π
=j
2πln[(A+ jωI)(A− jωI)−1].
(18)
Proof. See (Gawronski, 2000, page 100). �
Remark 2. Note that by following the ideas in Gawron-ski (2000), it is also possible to compute the finite-frequency observability Gramian for general frequency in-tervals, e.g., [ω , ω] .
Remark 3. From (5), (16) and Lemma 1, it is straight-forward to observe that the finite-frequency H2 norm ofthe system in (2) can be expressed as
‖G‖22,ω = Tr{
BTWo(ω)B}
. (19)
3.2. An upper bound of the robust H2 norm
Let X represent Hermitian, block diagonal positive defi-nite matrices that commute with ∆, i.e., every X ∈ X hasthe following structure
X = diag[
X1 · · · XL xL+1Im1· · · xL+F ImF
]
.
(20)where the blocks in X have compatible dimensions withtheir corresponding blocks in ∆. The following condition,taken from Paganini (1999a), plays a central role through-out this section.
Condition 1. Consider the system in (6). There existsX (ω) ∈ X, Hermitian Y (ω) ∈ Cm×m and ǫ > 0 such that
M(jω)∗[
X (ω) 00 I
]
M(jω)−
[
X (ω) 00 Y (ω)
]
�
[
−ǫI 00 0
]
.
(21)
The set of matrices X are the so-called D-scaling matri-ces. In many cases it is customary to use constant scalingmatrices to make the problem easier to handle, Fan et al.(1991), Packard and Doyle (1993). However, it is well
3
known that the results achieved based on constant scalingmatrices can be conservative, Iwasaki and Hara (1998).One of the ways to reduce the conservativeness and keepthe computational complexity reasonable is to use specialclasses of dynamic D-scaling matrices, Scherer and Kose(2007, 2008). This will be investigated in more detail inSection 3.2.2. Also, even less conservative scaling matricescan be considered, like D − G scalings, Fan et al. (1991)or LFT scalings, Iwasaki and Hara (1998).
Next, two methods for computing upper bounds of therobust H2 norm of systems with structured uncertaintiesare reviewed. The first method explicitly defines Y (ω) inCondition 1 and uses Y (ω) to construct an upper boundof the robust H2 norm of the system. This method willbe referred to as explicit upper bound calculation. Thesecond method calculates an upper bound through com-puting the observability Gramian via solving a set of LMIs.This method is referred to as Gramian based upper boundcalculation.
3.2.1. Explicit upper bound calculation
Consider Condition 1. This condition can be restatedas follows.
Lemma 2. If there exists X (ω) ∈ X such that
M∗11X (ω)M11 +M∗
21M21 −X (ω) ≺ 0, (22)
then Condition 1 is satisfied if and only if there existsY (ω) = Y (ω)∗ such that,
M∗12X (ω)M12 +M∗
22M22 − (M∗12X (ω)M11 +M∗
22M21)×
(M∗11X (ω)M11 +M∗
21M21 −X (ω))−1×
(M∗12X (ω)M11 +M∗
22M21)∗ � Y (ω).
(23)
Proof. See Appendix A.
Using Condition 1 and Lemma 2, the following theoremprovides an upper bound of the gain of the system for allfrequencies and will be used to accommodate an upperbound of the robust H2 norm for systems with structureduncertainty.
Theorem 1. If there exists X (ω) ∈ X such that (22) issatisfied for all ω and if we define Y (ω) as below
Y (ω) =M∗12X (ω)M12 +M∗
22M22−
(M∗12X (ω)M11 +M∗
22M21)×
(M∗11X (ω)M11 +M∗
21M21 −X (ω))−1×
(M∗12X (ω)M11 +M∗
22M21)∗,
(24)
then (∆ ∗M)(jω)∗(∆ ∗M)(jω) � Y (ω) ∀ω.
Proof. See Appendix B. �
Corollary 1. If there exists X (ω) ∈ X and a frequencyinterval centered at ωi, I(ωi) = [ωi + ωmin ωi + ωmax],such that
M∗11XM11 +M∗
21M21 −X ≺ 0 ∀ω ∈ I(ωi), (25)
and if we consider Y (ω) as defined in (24) for the men-tioned frequency interval, then∫
ω∈I(ωi)
Tr {(∆ ∗M)∗(∆ ∗M)}dω
2π≤
∫
ω∈I(ωi)
Tr {Y (ω)}dω
2π,
(26)
for all ∆ ∈ B∆, and specifically if I(ωi) covers all frequen-cies
sup∆∈B∆
‖∆ ∗M‖22 ≤
∫ ∞
−∞
Tr {Y (ω)}dω
2π. (27)
As a result, using the inequality in (27), it is possible togenerate an upper bound of the robust H2 norm of thesystem via numerical integration.
3.2.2. Gramian-based upper bound calculation
In this section, we consider a class of dynamic scalingmatrices with the following structure
X (ω) = ψ(jω)Xψ(jω)∗
=[
Cψ(jωI −Aψ)−1 I
]
X[
Cψ(jωI −Aψ)−1 I
]∗,
(28)
where Aψ ∈ Rnψ×nψ and Cψ ∈ Rd×nψ are fixed ma-trices with appropriate dimensions such that Aψ is sta-ble and (Cψ , Aψ) is observable. Also note that X ∈R
(d+nψ)×(d+nψ) is a free basis for the parameters such thatX (s) ∈ X. In order to derive an upper bound of the robustH2 norm relying on scaling matrices of the form (28), thefollowing technical result taken from Giusto (1996) will beuseful.
Lemma 3. Consider the partitioning M =[
M1 M2
]
,defined in (9), for the transfer matrix of system in (6). Byreplacing X (ω) with X (ω)−1 in (21), the condition in (21)can be restated as
M1(jω)X (ω)M1(jω)∗ −
[
X (ω) 00 I
]
M2(jω)
M2(jω)∗ −Y (ω)
� 0.
(29)
Proof. See (Giusto, 1996, Lemma 1). �
The upper left block of (29) can be expressed, up to itssign, as
C11 :=
[
X (ω) 00 I
]
−M1(jω)X (ω)M1(jω)∗
=
[
ψ 00 I
] [
X 00 I
] [
ψ 00 I
]∗
−
[
M11ψ
M21ψ
]
X
[
M11ψ
M21ψ
]∗
=
[
M11ψ ψ 0M21ψ 0 I
]
−X 0 00 X 00 0 I
[
M11ψ ψ 0M21ψ 0 I
]∗
.
(30)
4
By introducing the following transfer matrix
C(jωI − A)−1Bq + D =
[
M11ψ ψ
M21ψ 0
]
, (31)
and setting Γ =[
0 I]T
, (30) can be reformulated as
C11 =
(
[
C(jωI − A)−1 I]
[
Bq 0
D Γ
])
−X 0 00 X 00 0 I
×
(
[
C(jωI − A)−1 I]
[
Bq 0
D Γ
])∗
,
(32)
with
A =
A BqCψ 00 Aψ 00 0 Aψ
, Bq =
0 Bq 0 0I 0 0 00 0 I 0
,
C =
[
C DCψ
[
Cψ0
]]
, D =
[
0 D
[
0 I
0 0
]]
,
(33)
where A ∈ Rn×n, Bq ∈ Rn×d, C ∈ R(l+d)×n and D ∈
R(l+d)×d, with n = 2nψ + n and d = 2nψ + 2d.
Let Π(X, Bq, D) be an affine function of X , defined asbelow
Π(X, Bq, D) =
[
Bq 0
D Γ
]
−X 0 00 X 00 0 I
[
Bq 0
D Γ
]T
=
[
Π11 Π12
ΠT12 Π22
]
,
(34)
where Π11 ∈ Rn×n,Π12 ∈ Rn×(l+d) and Π22 ∈R(l+d)×(l+d). Then the following theorem taken from Pa-ganini (1997) can be used to calculate an upper bound ofthe robust H2 norm.
Theorem 2. If there exist matrix X such that X (ω)in (28) satisfies X (ω) ∈ X, and Hermitian matricesP−, P+ ∈ Rn×n, Q ∈ Rnψ×nψ , Wo ∈ Rn×n, such that
P−, Q ≻ 0,[
AψQ+QATψ QCTψCψQ 0
]
−X ≺ 0,
[
AP− + P−AT P−C
T
CP− 0
]
−Π(X, Bq, D) ≺ 0,
[
AP+ + P+AT P+C
T
CP+ 0
]
−Π(X, Bq, D) ≺ 0,
[
Wo I
I P+ − P−
]
≻ 0,
Tr
{
[
BTw 0]
Wo
[
Bw
0
]}
< γ2,
(35)
then X (ω) satisfies (29) and the system (∆ ∗M) definedin (11) has robust H2 norm less than γ2.
Proof. See Paganini (1997). �
Theorem 2 includes the problem with constant scalingmatrices as a special case. Let
A = A, Bq =[
Bq 0]
, C = C, D =
[
D
[
Id0
]]
. (36)
Then the following Corollary is a restatement of Theorem 2for constant scaling matrices, i.e., for X (ω) = X .
Corollary 2. If there exist matrix X ∈ X and symmetricmatrices P−, P+, Z ∈ Rn×nsuch that
P−, X ≻ 0,[
AP− + P−AT P−C
T
CP− 0
]
−Π(X, Bq, D) ≺ 0,
[
AP+ + P+AT P+C
T
CP+ 0
]
−Π(X, Bq, D) ≺ 0,
[
Z I
I P+ − P−
]
≻ 0,
Tr{
BTwZBw}
< γ2.
(37)
then X (ω) = X satisfies (29) and the system (∆ ∗ M)defined in (11) has robust H2 norm less than γ2.
Proof. See Paganini (1997). �
4. Gramian-based upper bound of the robust
finite-frequency H2 norm
In this section the first method for calculating an upperbound of the robust finite-frequency H2 norm of the sys-tem in (6) is presented. The following theorem combinesthe ideas presented in Section 3.1, regarding the finite-frequency observability Gramians, with the results of Sec-tion 3.2.2, and computes an upper bound of the robustfinite-frequency H2 norm for (6). Hereafter this method isreferred to as Method 1 .
Theorem 3. Let P−, P+, X,Q and Wo be a solutionto (35), then
sup∆∈B∆
‖∆ ∗M‖22,ω
≤ Tr
{
[
Bw0
]T(
L(A, ω)∗Wo + WoL(A, ω))
[
Bw0
]
}
(38)
where L(A, ω) is defined in (18) and A = A − (Π12 −P−C
T )Π−122 C.
Proof. See Appendix C. �
Remark 4. By Remark 2, the integral in (C.4) can berestated for a generic frequency range, e.g., [ω , ω]. Thisallows us to compute an upper bound of robust finite-frequency H2 norm for general frequency ranges.
5
As was mentioned in Section 3.2, by using dynamic scalingmatrices and increasing the order of these scaling matri-ces, it is possible to reduce the conservativeness of theresults. In order to further reduce the conservativeness ofthe bounds and improve the numerical properties of theoptimization problems, it is useful to perform uncertaintypartitioning. In this approach, for each of the uncertaintypartitions, an upper bound of the robust finite-frequencyH2 norm of the system is computed and the maximum ofthese bounds is considered as the final result.
5. Frequency gridding based upper bound of the
robust finite-frequency H2 norm
In this section the second method to compute upperbounds of the robust finite-frequency H2 norm is pre-sented. The following corollary to Theorem 1, which isa straightforward extension of Corollary 1, plays a centralrole in the proposed algorithm.
Corollary 3. Let I(ωi) for i = 1, . . . ,m be disjoint fre-quency intervals such that Id = [−ω , ω] =
⋃m
i=1 I(ωi).Also let the constant matrices Xi for i = 1, . . . ,m bethe scaling matrices for which M∗
11XiM11 + M∗21M21 −
Xi ≺ 0 ∀ω ∈ I(ωi). Then, it holds that
sup∆∈B∆
‖∆ ∗M‖22,ω
≤ sup∆∈B∆
m∑
i=1
∫
ω∈I(ωi)
Tr {(∆ ∗M)∗(∆ ∗M)}dω
2π
≤m∑
i=1
∫
ω∈I(ωi)
Tr {Yi(ω)}dω
2π,
(39)
where Yi(ω) is defined as in (24), with X (ω) = Xi.
Corollary 3 provides a sketch for computing upperbounds of the robust finite-frequency H2 norm via fre-quency gridding. However, calculating a suitable scalingmatrixXi requires checkingM
∗11XiM11+M
∗21M21−Xi ≺ 0
for an infinite number of frequencies in I(ωi). Next amethod is proposed to solve this issue. Consider the fol-lowing two LMIs
M11(jω)∗X (ω)M11(jω) +M21(jω)
∗M21(jω)−X (ω) ≺ 0,(40)
[
M11(jω) 0M21(jω) 0
]∗
X (ω)
[
M11(jω) 0M21(jω) 0
]
− X (ω) ≺ 0. (41)
Then Xi =
[
Xi 00 Il
]
satisfies (41) for ω = ωi, if and only if
Xi satisfies (40) for the same frequency. The following the-orem taken from Roos and Biannic (2010), solves the issueof infinite dimensionality of the problem in Corollary 3 byproviding a way to extend the validity of a scaling matrixthat satisfies M∗
11XiM11 +M∗21M21 −Xi ≺ 0 for a single
frequency, e.g., ω = ωi, to a frequency interval, I(ωi).
Theorem 4. Let M =
[
M11 0M21 0
]
=
[
A B
C D
]
, and let
D = X1
2
i , where Xi satisfies the LMI in (41) for ω = ωi.Define
G = AX −BXD−1X CX , (42)
where
AX =
[
AG 0−C∗
GCG −A∗G
]
, BX =
[
−BGC∗GDG
]
,
CX =[
D∗GCG B∗
G
]
, DX = I −D∗GDG,
(43)
in which
G =
[
AG BGCG DG
]
=
[
A− jωiI BD−1
DC DDD−1
]
, (44)
and define ωlow
and ωhigh
as
ωlow
=
{
−ωi, if jG has no positive real eigenvalue
max{λ ∈ R− : det(λI + jG) = 0}, otherwise
ωhigh
=
{
∞, if jG has no negative real eigenvalue
min{λ ∈ R+ : det(λI + jG) = 0}, otherwise
Then Xi satisfies (41) for all ω such that ω ∈ I(ωi) =[
ωi + ωlow
, ωi + ωhigh
]
.
Proof. See Appendix D. �
Using Corollary 3 and Theorem 4, the following algo-rithm can be used for calculating an upper bound of therobust finite-frequencyH2 norm. This algorithmic methodis referred to as Method 2.
Algorithm 1. Let Id = [−ω , ω] denote the frequencyrange of interest. Then
(I) Divide Id into a desired number of disjoint partitions,I(ωi), where ωi is the center of the respective parti-tion.
(II) For each of the partitions, compute Xi such that itsatisfies (40) for ω = ωi. If there is a partition forwhich there exists no feasible solution, exit the algo-rithm.
(III) Construct Xi from the achieved Xi in (II).
(IV) Using Theorem 4 calculate the valid frequency range,I(ωi), for the LMIs in (41). If I(ωi) 6⊆ I(ωi), go backto (I) and choose a finer partitioning for Id.
(V) Define Yi(ω) using (24) with X (ω) = Xi.
(VI) Use numerical integration to calculate∫
ω∈I(ωi)
Tr {Yi(ω)}dω
2π.
(VII) By Corollary 3, compute the upper bound by summingup the integrals computed in (VI).
6
Remark 5. As can be seen from Step (II) of Algorithm 1,in case there exists a partition for which it is impossible tofind Xi that satisfies (40) for ω = ωi this algorithm fails toproduce an upper bound. Note that this can stem eitherfrom the fact that the system is not robustly stable orfrom conservatism of the robust stability condition basedon (40).
The second step of Algorithm 1, requires computationof constant scaling matrices that satisfy (40) for ω = ωi foreach of the partitions. This can be accomplished throughdifferent approaches. Considering the expression in (39)and the importance of Tr {Yi(ω)} in the tightness of theproposed upper bound, it seems intuitive to calculate thescaling matrices while aiming at minimizing Tr {Yi(ωi)}.The following two approaches utilize this in the process ofcomputing suitable scaling matrices.
Approach 1. Compute Xi in Step (II) of Algorithm 1 asthe solution of the following optimization problem
minimizeXi,Yi
Tr {Yi}
subj. to (21) with ω = ωi.(45)
Remark 6. The idea of frequency gridding was also pre-sented in Paganini (1999b), where the authors considerthe H2 performance problem for discrete time systems. Inthat paper, an optimization problem similar to (45) forfrequencies 0 = ω0 . . . ωN = 2π is formulated and then the
integral∫ 2π
0 trace(Y (ω)) dω2π is approximated by the follow-ing Riemann sum expression
1
2π
N∑
i=1
Tr{Yi}(ωi − ωi−1), (46)
where 0 = ω0 . . . ωN = 2π. However, this approach doesnot necessarily provide a guaranteed upper bound of therobust H2 norm of the system.
For any Xi satisfying the LMI in (40) for ω = ωi let
f(α) = Tr{M∗12αXiM12 +M∗
22M22−
(M∗12αXiM11 +M∗
22M21)×
(M∗11αXiM11 +M∗
21M21 − αXi)−1×
(M∗12αXiM11 +M∗
22M21)∗}.
(47)
This function is convex with respect to α. Next, follow-ing the same objectives as in Approach 1, an alternativemethod for calculating suitable scaling matrices is intro-duced.
Approach 2. Compute Xi in Step (II) of Algorithm 1using the following sequential method
(I) Find Xi such that it satisfies the LMI in (40) for ω =ωi.
(II) Minimize f(α), in (47), with the achieved Xi withrespect to all α such that αXi still satisfies the LMIin (40) for ω = ωi.
Denote α∗ as the minimizing α. Then α∗Xi will be usedwithin the remaining steps of Algorithm 1. In order toassure that α∗Xi satisfies (40) the search for α should besubject to the constraint α > αmin, where
αmin = (48)
1
min
{
eig
([
Λ−1
2 00 I
]
U(−M∗11XiM11 + Xi)U∗
[
Λ−1
2 00 I
])},
in which U , a unitary matrix, and Λ, are defined by a
singular value decomposition M∗21M21 = U∗
[
Λ− 1
2 00 0
]
U .
It is important to note that for some problems it mightbe required to perform many iterations between the firstand the fourth steps of Algorithm 1. One of the ways toalleviate this issue and even compute better upper bounds,is to modify the proposed approaches by augmenting newconstraints for other frequencies from the partition underinvestigation. In this case the cost function can also bemodified accordingly. As an example, Approach 1 can bemodified as follows
minimizeXi,Yi
Tr {Yi}
subj. to M(jω)∗[
Xi 00 I
]
M(jω)−
[
Xi 00 Yi
]
� 0
for ω = ωj ∈ I(ωi), j = 1, . . . , Ni,
(49)
or alternatively as
minimizeXi,Y
j
ij=1,...,Ni
Ni∑
j=1
Tr{
Yji
}
subj. to M(jω)∗[
Xi 00 I
]
M(jω)−
[
Xi 0
0 Yji
]
� 0
for ω = ωj ∈ I(ωi), j = 1, . . . , Ni.
(50)
Remark 7. In case we use either of the formulationsin (49) and (50) for the second step of Algorithm 1 andfail to find a feasible solution, it does not necessarily meanthat we cannot use this algorithm. In this case it is pos-sible to return to the first step of the algorithm and try afiner partitioning for the frequency range of interest.
Similar to Method 1, uncertainty partitioning improvesthe quality of the calculated upper bound using thismethod too.
Remark 8. Although the calculated value for the upperbound using Algorithm 1 has usually a decreasing trend
7
with respect to the number of partitions, this trend is notnecessarily monotonic. This is due to the fact that the cal-culated upper bound not only is dependent on the numberof partitions but also on the quality of the calculated scal-ing matrices and how they affect the numerical integrationprocedure.
Remark 9. Note that in Corollary 3, we can choose ID =[ω , ω]. This allows us to use this method for computingupper bounds of the robust finite-frequency H2 norm overgeneral frequency ranges.
6. Numerical example
In this section the proposed methods are tested on atheoretical example. The example has been chosen de-liberately simple, so that the exact robust H2 norm canbe computed via routine calculations. All the computa-tions are conducted by using the Yalmip toolbox, Lofberg(2004), with the SDPT3 solver, Toh et al. (1999). Theplatform used for the simulations uses a Dual Core AMDOpteronTM Processor 270 as CPU and 4 GB of RAM.Consider the uncertain system in (6) with the following
system matrices
A =
−2.5 0.5 0 −50 00 −1 0.5 0 00 −0.5 0 0 00 0 0 −5 1000 0 0 −100 0
, Bq =
0.25 −0.50 00 00 00 0
,
Bw =
05005
, C =
[
Cp
Cz
]
=
1 0 0 0 00 0 0 0 01 0 0 0 0
D =
[
Dpq
Dzq
]
=
0 01 00 0
.
(51)
In this example ∆(δ) = δI2 with −1 ≤ δ ≤ 1. This sys-tem is known to have robust H2 norm, as defined in (12),equal to 1.5311 which is attained for δ = 0.25. Figure 2illustrates the gain plots of the system for different valuesof the uncertain parameter. The aim is to calculate the ro-bust finite-frequency H2 norm of the system and avoid thepeak occurring at 100 rad/s. This is motivated by Figure 3which presents the calculated finite-frequency H2 norm ofthe system in (51), with respect to different values for theuncertain parameter and frequency bounds. As can beseen from this figure and the jump at ω = 100 rad/s, thecontribution of this peak to the robust finite-frequency H2
norm cannot be neglected. In order to avoid this peak, thefrequency bound that has been considered for this exam-ple is ω = 50 rad/s. The actual value for the robust finite-frequency H2 norm for (51) with this frequency bound is0.8919.
10−2
10−1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
ω[rads
]
‖∆∗
M‖2 2
ω = 50
Figure 2: Gain plot over frequency for different values for the uncer-tain parameter.
In Method 1, presented in Section 4, we consider thefollowing structure for scaling matrices
ψ(s) =[
(s−p)nψ−1
(s+p)nψ
I2(s−p)
nψ−2
(s+p)nψ−1 I2 . . . 1
(s+p)I2 I2
]
,
(52)
where nψ is the order of the scaling matrix and we havechosen p = 150. This choice of scaling matrices has beeninspired by Laguerre basis functions, Wahlberg (1991)(similar approaches has also been considered in Schererand Kose (2006, 2008)). For this particular example dy-namic scaling matrices with order higher than 3 do notproduce any better upper bounds, so only scaling matri-ces up to order 3 are considered.Method 2, presented in Section 5, has been applied to
the example with Approaches 1 and 2. The number of fre-quency partitions is increased until either the performancematches the performance of Method 1 or the improvementin the computed upper bound is not discernible anymore.Figures 4 and 5 illustrate the achieved upper bounds fordifferent frequency bounds, ω, using methods 1 and 2.In both figures, the curve marked with the solid line re-ports the actual values for the robust finite-frequency H2
norm of the system. In Figure 4, the dashed lines presentthe achieved upper bounds using Method 1. As the or-der of the dynamic scaling matrices increases, the com-puted upper bound becomes tighter. These upper boundshave been computed for nψ = 0, 1, 2, 3. Note that the up-per bounds computed using scaling matrices with nψ ≥ 1are practically indistinguishable. Hence, in Figure 4, thedashed line furthest from the solid line represents the up-per bound computed with nψ = 0 and the ones closest tothe solid line are those computed with nψ = 1, 2, 3. InFigure 5, the bounds presented with the dashed lines areresults achieved by applying Method 2 to this example.The dashed curve furthest from the solid line correspondto the bound computed using Approach 2. Hence, as can
8
10−2
100
102
−1
−0.5
0
0.5
10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ω[rads
]δ
‖∆
∗M
‖2 2,ω
Figure 3: Finite-frequency H2 norm versus different values of theuncertain parameter and frequency bounds.
be seen from Figure 5, Method 2 with Approach 1 canproduce better upper bounds than the second approachand can match the performance of Method 1 (the plottedcurve refers to the case npar = 40). Table 1 presents asummary of the achieved results.
Table 1: Numerical example: comparison of Methods 1 and 2.
Method Estimated ElapsedBound Time[sec]
M.1, nψ = 0 1.2609 11M.1, nψ = 1 1.1972 10M.1, nψ = 2 1.1944 12M.1, nψ = 3 1.1911 13
M.2, App.1, npar = 40 1.189 44M.2, App.1, npar = 200 1.186 144M.2, App.2, npar = 200 1.3184 552
So far the presented results are achieved without anyuncertainty partitioning. In order to illustrate the effectof uncertainty partitioning on the performance of the pro-posed methods, Method 1 and Method 2 with Approach 1are applied to this example with uncertainty partitioning.Figures 6 and 7 present the achieved upper bounds ofthe robust finite-frequency H2 norm of the system withω = 50 rad/s using Methods 1 and 2, respectively. Thesefigures illustrate the upper bound with respect to the num-ber of uncertainty partitions and the order of dynamic scal-ing matrices, for Method 1, and the number of frequencygrid points, for Method 2. As can be seen from the fig-ures and considering the actual robust finite-frequency H2
norm of the system, the computed upper bounds usingboth methods are extremely tight. A summary of the re-sults from this analysis is presented in tables 2 and 3. As itcan be observed, although both methods produce equally
10−2
10−1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ω[ r a ds]
‖∆
∗M
‖2 2,ω
Figure 4: Robust finite-frequency H2 norm and the computed upperbounds for different frequency bounds, using Method 1. The solidline illustrates the actual value of the robust finite-frequency H2
norm. The dashed lines represent the achieved upper bounds usingMethod 1 for different orders of the scaling matrix.
Table 2: Numerical example: effect of partitioning on Method 1.
nψ No. Uncer. Estimated ElapsedPar. Bound Time[sec]
2 1 1.1944 122 20 0.8928 434
tight upper bounds, Method 1 achieves this goal with lowercomputational time.
7. Application to flight comfort analysis
As a practical application, a comfort analysis problemfor a civil aircraft model is considered. The derivation ofuncertainty models from aircraft physical models is gen-erally a hard task; the resulting models are usually highdimensional and therefore difficult to handle through stan-dard robust analysis tools (see e.g., Poussot-Vassal andRoos (2012)). In this paper we refer to the model devel-oped in Roos (2009). Due to the model size, the problemis computationally much more challenging than the oneaddressed in Section 6. The objective is to provide an es-timate of the energy of the oscillations induced by distur-bances like wind gusts or turbulences at different positionsalong the fuselage of the aircraft. The considered problem
Table 3: Numerical example: effect of partitioning on Method 2.
No. freq. No. Uncer. Estimated ElapsedGrids Par. Bound Time[sec]
20 1 1.1945 3020 20 0.8924 532
9
10−3
10−2
10−1
100
101
102
103
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ω[ r a ds]
‖∆
∗M
‖2 2,ω
Figure 5: Robust finite-frequency H2 norm and the computed upperbounds for different frequency bounds, using Method 2. The solidline illustrates the actual value of the robust finite-frequency H2
norm. The lines represent the bounds obtained using Method 2 withthe two different approaches considered.
involves a model of a civil aircraft, including both rigidand flexible modes, along with parametric uncertainty.
Such a problem can be reformulated as an H2 perfor-mance analysis problem for an extended system, includ-ing the model of the aircraft, a so-called Von Karman fil-ter (modeling the wind spectrum), and an output filter,accounting for the turbulence field, Papageorgiou et al.(2011). In this aircraft model the uncertain parameter δcorresponds to the level of fullness of the fuel tanks and itis normalized to vary within the range [−1, 1]. The overallextended system is presented in LFT form, as in (6), withn = 21 states and an uncertainty block size of d = 14.
The aircraft model is valid for frequencies up to 15 rad/s,and beyond that it does not have any physical meaning,Roos (2009). This motivates performing finite-frequencyH2 performance analysis, limited to this frequency range.Figure 8, illustrates the gain plots of the system as a func-tion of frequency. Different curves in this figure correspondto different uncertainty values. As can be seen from thefigure, the frequency bound at 15 rad/s is necessary toavoid the peak at approximately 20 rad/s which is outsidethe validity range of the model.
The methods considered for performing comfort analy-sis are methods 1 and 2 with the use of constant scalingmatrices and Approach 1, respectively. Tables 4 and 5summarize the achieved results using methods 1 and 2, re-spectively. As can be seen from the tables, both methodsperform equally accurate in estimating the robust finite-frequency H2 norm of the system. However, in contrastto the example in Section 6, Method 2 is faster in cal-culating the upper bound with equal accuracy. Similarto Section 6, it is possible to improve the computed upperbounds via uncertainty partitioning. This can be observedfrom tables 4 and 5. A possible way to further reduce thecomputational times could be to apply adaptive partition-
0
5
10
15
20
25
00.5
11.5
22.5
3
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
No. Uncertainty PartitionsΨ
‖(∆
∗M
)‖2 2,5
0
Figure 6: The achieved upper bounds of robust finite-frequency H2
norm for ω = 50 rad/s with respect to the number of uncertaintypartitions and the order of dynamic scaling matrices.
ing techniques, like those proposed for example in Oishiand Fujioka (2009), Garulli et al. (2011).
8. Discussion and General remarks
This section highlights the advantages and disadvan-tages of the proposed methods and provides insight onhow to improve the performance of these methods.
8.1. The observability Gramian based method
This method considers the whole frequency interval andcalculates an upper bound of the robust finite-frequencyH2 norm of the system in one shot or one iteration by solv-ing an SDP (SemiDefinite Program). However, the dimen-sion of this optimization problem grows rapidly with thenumber of states and/or size of the uncertainty block. Thislimits the capabilities of this method in handling mediumor large sized problems, i.e., analysis of systems with highnumber of states and/or large uncertainty blocks.The most apparent possibility to improve the accuracy
of the computed upper bound using this method is to in-crease the order of the dynamic scaling matrices. Thiscomes at the cost of higher number of optimization vari-ables in the underlying SDP and affects the computationaltractability of the method.
Table 4: Numerical results for the flight comfort application
using Method 1.
nψ No. Uncer. Estimated ElapsedPar. Bound Time[h]
0 50 1.2434 8.620 450 0.7970 59.24
10
05
1015
2025
0
20
40
60
0.80.850.9
0.951
1.051.1
1.151.2
1.251.3
1.351.4
1.451.5
1.551.6
1.651.7
1.751.8
1.851.9
1.952
2.052.1
2.152.2
No. Frequency PartitionsNo. Uncertainty Partitions
‖(∆
∗M
)‖2 2,5
0
Figure 7: The achieved upper bound of the robust finite-frequencyH2 norm for ω = 50 rad/s with respect to the number of uncertaintypartitions and the number of frequency grid points.
Table 5: Numerical results for the flight comfort application
using Method 2.
No. freq. No. Uncer. Estimated ElapsedGrids Par. Bound Time[h]
80 1 1.2382 0.561180 10 0.7911 4.25
Another way of improving the computed upper boundis to perform uncertainty partitioning, which proved to beeffective for the examples presented in sections 6 and 7.However, this improvement comes at the cost of a muchhigher computational burden, see Table 4.
8.2. The frequency gridding based method
This method starts with an initial partitioning of thedesired frequency interval and calculates an upper boundof the robust finite-frequency H2 norm by solving the cor-responding SDP for each of the partitions. The sizes of theunderlying SDPs in this method are smaller than the onesof the previous method and are mainly dependent on thesize of the uncertainty block. Consequently, this methodcan handle larger problems. However for large problems,the algorithm might require some iterations between stepsIV and I of the algorithm to be able to produce consistentresults. Another issue with this method is the requirementto perform numerical integration on a rational function instep VI of the algorithm. This can become slightly prob-lematic for high order systems.There are two main ways to improve the computed up-
per bounds using this method, namely increasing the num-ber of partitions, and augmenting the SDP for each par-tition with more constraints for other frequency points inthe partition and/or adding more variables to the SDPs
10−2
10−1
100
101
102
103
0
0.5
1
1.5
2
2.5
3
ω[rads
]
‖∆
∗M
‖2 2
ω = 15
Figure 8: Gain plots for different values of the uncertain parameter.
corresponding to the partitions. This proved to scale bet-ter considering the computational time, as compared toMethod 1, see tables 4 and 5.
9. Conclusion
This paper has provided two methods for calculatingupper bounds of the robust finite-frequency H2 norm.Through the paper different guidelines for improving theperformance of the proposed methods have been presentedand their effectiveness has been illustrated using both atheoretical and a practical example.The proposed methods consider different formulations
for calculating upper bounds of the robust finite-frequencyH2 norm. Both methods can produce equally tight upperbounds, but they have different computational properties.While Method 1 is more suitable for small-sized problemsand produce results faster than the second method for thistype of problems, Method 2 can handle larger problemsand produce results more rapidly for this type of problems.
Acknowledgements
The authors wish to thank involved personnel from AIR-BUS, Clement Roos and Carsten Doll from ONERA andSimon Hecker and Andras Varga from DLR for providingthe model of the civil aircraft used in Section 7.
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Appendix A. Proof of Lemma 2
Let
C11 =M∗11X (ω)M11 +M∗
21M21 −X (ω),
C12 =M∗11X (ω)M12 +M∗
21M22,
C21 =M∗12X (ω)M11 +M∗
22M21,
C22 =M∗12X (ω)M12 +M∗
22M22.
(A.1)
Then the left hand side of Condition 1 can be written as
M∗(jω)
[
X (ω) 00 I
]
M(jω)−
[
X (ω) 00 Y (ω)
]
=
[
C11 C12
C21 C22 − Y (ω)
]
. (A.2)
Now if we assume that there exists X (ω) ∈ X such thatC11 ≺ 0, then Lemma 2 is the direct outcome of Schur’slemma. �
Appendix B. Proof of Theorem 1
If the assumptions of the theorem are satisfied, then byLemma 2, Condition 1 is valid, i.e., (21) holds. Define
M =
[
X (ω)1
2 00 I
]
M
[
X (ω)−1
2 00 I
]
. (B.1)
Then (21) can be rewritten as
M∗M −
[
I 00 Y (ω)
]
�
[
−ǫI 00 0
]
. (B.2)
12
As a result
M∗M �
[
I 00 Y (ω)
]
. (B.3)
Define q(jω) = X (ω)1
2 q(jω) and p(jω) = X (ω)1
2 p(jω). By
pre and post multiplying both sides of (B.3) by
[
q(jω)w(jω)
]∗
and
[
q(jω)w(jω)
]
, respectively, we have
| z(jω) |2 + | p(jω) |2≤| q(jω) |2 +w(jω)∗Y (ω)w(jω).(B.4)
For all frequencies ∆ commutes with X (ω)− 1
2 , and hence
q = X1
2 q = X1
2∆X− 1
2 p = ∆p. Considering the fact that∆ ∈ B∆, it now follows from (11) and (B.4) that
| z(jω) |2 = w(jω)∗(∆ ∗M)(jω)∗(∆ ∗M)(jω)w(jω)
≤ w(jω)∗Y (ω)w(jω), (B.5)
which completes the proof. �
Appendix C. Proof of Theorem 3
Let us first introduce a technical lemma, taken fromPaganini (1997), which is instrumental for proving Theo-rem 3.
Lemma 4. Let P−, P+, X,Q and Wo satisfy (35), and C11be defined as in (30). Then, C11 ≻ 0 and there exists aspectral factor N such that N and N−1 are stable, C11 =NN∗, and ‖N−1M2‖
22 < γ2, where M2 is defined in (9).
A state space realization for N−1M2 is given by
N−1M2 =
A− (Π12 − P−CT )Π−1
22 C
[
Bw0
]
Π− 1
2
22 C 0
.
(C.1)Moreover, Wo is the observability Gramian of N−1M2.
Proof. See Paganini (1997). �
Now, we are ready to prove Theorem 3. Let P−, P+, X,Q
and Wo satisfy (35). Define
Y = (N−1M2)∗(N−1M2) =M∗
2 C−111 M2 � 0. (C.2)
where N andM2 are defined in Lemma 4. From (C.2) andC11 ≻ 0, by Schur’s lemma it follows that
[
−C11 M2
M∗2 −Y
]
� 0 (C.3)
which corresponds to (29). By using Lemma 3, it turns outthat (C.3) is equivalent to (21). By appplying the samereasoning as in the proof of Theorem 1, (B.1)-(B.3) holdand one gets
(∆ ∗M)(jω)∗(∆ ∗M)(jω) � Y (ω) ∀ω, ∀∆ ∈ B∆.
As a result, by using (C.2), Lemma 4 and Lemma 1, onegets
sup∆∈B∆
‖∆ ∗M‖22,ω ≤
∫ ω
−ω
Tr {Y (ω)}dω
2π
=
∫ ω
−ω
Tr{
(N−1(jω)M2(jω))∗(N−1(jω)M2(jω))
} dω
2π
= Tr
{
[
Bw0
]T(
L(A, ω)∗Wo + WoL(A, ω))
[
Bw0
]
}
(C.4)
which corresponds to (38). �
Appendix D. Proof of Theorem 4
Consider the LMI in (41) with X (ω) = Xi. This LMIcan be rewritten as
X−
1
2
i M∗X1
2
i X1
2
i MX−
1
2
i − I ≺ 0. (D.1)
Let G(jω) = X1
2
i M(j(ω + ωi))X−
1
2
i . It now follows that
G =
[
AG BGCG DG
]
. In this theorem we are looking for the
largest frequency interval, for which the LMI in (D.1) isvalid. On the boundary of this interval I −G(jω)∗G(jω)becomes singular, i.e., det(I −G(jω)∗G(jω)) = 0.
By (43) and (44), I − G(jω)∗G(jω) =
[
AX BXCX DX
]
.
Using Sylvester’s determinant theorem and some simplematrix manipulations we have
det(I −G(jω)∗G(jω)) = 0 ⇔
det(I +D− 1
2
X CX(jωI −AX)−1BXD− 1
2
X ) = 0 ⇔
det(I + (jωI −AX)−1BXD−1X CX) = 0. (D.2)
By using the matrix determinant lemma and the definitionof G it is also straight forward to establish equivalencebetween the following expressions
det(I + (jωI −AX)−1BXD−1X CX) = 0 ⇔
det(jωI − (AX −BXD−1X CX)) = 0 ⇔ det(ωI + jG) = 0,
(D.3)
which completes the proof. �
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