Technical report from Automatic Control at Linköpings universitet
Robust finite-frequency
H
2
analysis of
uncertain systems
Andrea Garulli, Anders Hansson, Sina Khoshfetrat Pakazad,
Alfio Masi, Ragnar Wallin
Division of Automatic Control
E-mail: garulli@ing.unisi.it, hansson@isy.liu.se,
sina.kh.pa@isy.liu.se, masi@dii.unisi.it,
ragnarw@isy.liu.se
13th May 2011
Report no.: LiTH-ISY-R-3011
Submitted to Automatica
Address:
Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.
Inmanyapplications,designoranalysisisperformedovera nite-frequency
rangeofinterest. TheimportanceoftheH2/robustH2normhighlightsthe
necessityofcomputingthisnormaccordingly. Thispaperprovidesdierent
methods for computing upper bounds on the robust nite-frequency H
2
normforsystemswithstructureduncertainties. Anapplicationoftherobust
nite-frequencyH
2normforacomfortanalysisproblem ofanaero-elastic
modelofanaircraftis alsopresented.
Robust finite-frequency H
2
analysis of uncertain systems
Andrea Garulli
a, Anders Hansson
b, Sina Khoshfetrat Pakazad
b, Alfio Masi
a,
Ragnar Wallin
baDipartimento di Ingegneria dell’Informazione Universita’ degli Studi di Siena, Italy bDivision of Automatic Control, Link¨oping University of Technology, Sweden
Abstract
In many applications, design or analysis is performed over a finite-frequency range of interest. The importance of the H2/robust
H2 norm highlights the necessity of computing this norm accordingly. This paper provides different methods for computing
upper bounds on the robust finite-frequency H2norm for systems with structured uncertainties. An application of the robust
finite-frequency H2 norm for a comfort analysis problem of an aero-elastic model of an aircraft is also presented.
Key words: Robust H2 norm, Uncertain systems, Robust control.
1 Introduction
The H2/robust H2 norm has been one of the pivotal
design and analysis criteria in many applications, such as structural dynamics, acoustics, colored noise distur-bance rejection, etc, [22], [9], [3]. Due to the importance of the H2/robust H2norm, there has been a substantial amount of research on computation, analysis and design based on these measures, many of which consider the use of Linear Matrix Inequalities (LMIs) and Ricatti equa-tions for this purpose, e.g. [5], [19], [1], [11], [12], [4], [20]. A survey of recent methods in robust H2analysis is
pro-vided in [14].
Most of the methods presented in the literature consider the whole frequency range for calculating the H2/robust
H2norm. However, in some applications it is beneficial to concentrate only on a finite-frequency range of inter-est and calculate the design/analysis measures accord-ingly. This can be due to different reasons, e.g. the model is only valid for a specific frequency range or the design is targeted for a specific frequency interval. This
moti-∗ Corresponding Author: Sina Khoshfetrat Pakazad, Link¨oping University of Technology, Email: sina.kh.pa@isy.liu.se
Email addresses: garulli@ing.unisi.it(Andrea Garulli), hansson@isy.liu.se (Anders Hansson), sina.kh.pa@isy.liu.se(Sina Khoshfetrat Pakazad), masi@dii.unisi.it(Alfio Masi), ragnarw@isy.liu.se (Ragnar Wallin).
vates the importance of computing the (robust) finite-frequency H2norm.
In [6], a method for calculating the finite-frequency H2norm for systems without uncertainty is presented, where the key step is to compute the finite-frequency observability Gramian. This is accomplished by first computing the regular observability Gramian and then scaling it by a system dependent matrix.
This paper introduces two methods for calculating an upper bound on the robust finite-frequency H2norm for
systems with structured uncertainties. The first method combines the notion of finite-frequency Gramians, intro-duced in [6], with convex optimization tools, [2], com-monly used in robust control and calculates the upper bound by solving an underlying optimization problem [10]. The second method, provides a computationally cheaper algorithmic method for calculating the desired upper bound. In contrast to the first approach, the sec-ond method performs frequency gridding and breaks the original problem into smaller problems, which are pos-sibly easier to solve. Then it uses the ideas presented in [18] on computing upper bounds on structured singular values, for solving the smaller problems. The results of the smaller problems are then combined to compute the upper bound on the whole desired frequency range, [15].
This paper is structured as follows. First some of the notations used throughout the paper are presented. Sec-tion 2 introduces the problem formulaSec-tion. Mathemati-cal preliminaries are presented in Section 3, which covers
the notion of finite-frequency Gramians and reviews the calculation of upper bounds on the robust H2norm. Sec-tions 4 and 5 provide the details of the two methods for calculating upper bounds on the robust finite-frequency H2norm. In Section 6 numerical examples are presented. Section 7 provides more insight to the proposed meth-ods by investigating the advantages and disadvantages of them, and finally Section 8 concludes the paper with some final remarks.
1.1 Notation
The notation in this paper is standard. The min and max represent the minimum and maximum of a function or a set, and similarly sup represents the supremum of a function. The symbols and ≺ denote the inequality relation between matrices. A transfer matrix in terms of state-space data is denoted
A B C D := C(jωI − A)−1B+ D. (1)
With k·k2, we denote the Euclidian or 2-norm of a vector or the norm of a matrix induced by the 2-norm. Further-more RH∞ represents real rational functions bounded
on Re(s) = 0 including ∞. For the sake of brevity of notation, unless necessary, we drop the dependence of functions on frequency.
2 Problem formulation
2.1 H2norm of a system
Consider the following system in state space form
˙x = Ax + Bu
y= Cx (2)
and define G(s) as the corresponding transfer function. Then the H2 norm of the system in (2) is defined as follows kGk2 2= Z ∞ −∞ Tr {G(jω)∗ G(jω)} dω 2π. (3)
This can also be written as
kGk2 2= Z ∞ 0 TrnBTeATtCTCeAtBodt = Tr BT Z ∞ 0 eATtCTCeAtdt B = TrBTW oB , (4)
Fig. 1. Uncertain system with structured uncertainty
where Wois the observability Gramian of the system.
Similarly the finite-frequency H2 norm of (2) is defined
as kGk2 2,¯ω= Z ω¯ − ¯ω Tr {G(jω)∗ G(jω)} dω 2π. (5)
2.2 Robust H2norm of a system
Consider the uncertain state space system
˙x = Ax + Bqq+ Bww p= Cpx+ Dpqq z= Czx+ Dzqq q= ∆p (6)
where x ∈ Rn, w ∈ Rm, z ∈ Rl and p, q ∈ Rd. Also
∆ ∈ Cd×d, which represents the uncertainty present in
(6), has the following structure
∆ = diaghδ1Ir1 · · · δLIrL ∆L+1 · · · ∆L+F
i , (7)
where δi ∈ R for i = 1, · · · , L, ∆L+j ∈ Cmj×mj for
j= 1, · · · , F and L X i=1 ri+ F X j=1 mj = d. Also in addition to
that and without loss of generality it is assumed that ∆ ∈ B∆ where B∆ is the unit ball for the induced 2-norm.
This structure of ∆ can represent both real parametric uncertainties (δiIri) and un-modeled system dynamics
(∆L+j).
The transfer matrix for the uncertain system in (6) is defined as below, see Figure 1,
M(jω) = " M11 M12 M21 M22 # = A Bq Bw Cp Dpq 0 Cz Dzq 0 . (8) where M ∈ C(d+l)×(d+m), M 11∈ Cd×d, M12 ∈ Cd×m, M21∈ Cl×dand M22∈ Cl×m.
The following definition of this transfer matrix will also be used later in the upcoming sections
M(jω) =hM1 M2 i = A Bq Bw C D 0 , (9) where M1∈ C(d+l)×(d), M 2∈ C(d+l)×(m)and C= " Cp Cz # , D= " Dpq Dzq # . (10)
In analysis of uncertain systems, the transfer function between the signals w(t) and z(t) is of interest. This transfer function is given by the upper LFT representa-tion
(∆ ∗ M ) = M22+ M21∆(I − M11∆)−1M12. (11)
Having (11), the robust H2norm of the system in (6) is
defined as below sup ∆∈B∆ k∆ ∗ M k2 2 = sup ∆∈B∆ Z ∞ −∞ Tr {(∆ ∗ M )∗ (∆ ∗ M )} dω 2π. (12)
Similarly the robust finite-frequency H2norm of the sys-tem in (6) is defined as sup ∆∈B∆ k∆ ∗ M k2 2,¯ω = sup ∆∈B∆ Z ω¯ − ¯ω Tr {(∆ ∗ M )∗ (∆ ∗ M )} dω 2π. (13)
This paper proposes methods for calculating upper bounds on (13).
3 Mathematical preliminaries
3.1 Finite-frequency observability Gramian
As was mentioned in Section 1, one of the ways of cal-culating the H2 norm of the system in (2) is by using
the observability Gramian of the system, see (4). Com-putation of the observability Gramian can be done by solving the following Lyapunov equation
ATWo+ WoA+ CTC= 0, (14)
where Wo ∈ Rn×n is the observability Gramian.
Us-ing the Parseval’s identity and (4), the observability Gramian can also be expressed as
Wo= Z ∞ −∞ H(jω)∗ CTCH(jω)dω 2π, (15) where H(jω) = (jωI − A)−1.
To continue with the study of the robust finite-frequency H2problem, next the notion of finite-frequency observ-ability Gramian is introduced, as proposed in [6]. The finite-frequency observability Gramian is defined as
Wo(¯ω) = Z ω¯ − ¯ω H(jω)∗ CTCH(jω)dω 2π. (16)
The next lemma provides a way to compute Wo(¯ω) in
terms of the observability Gramian, Wo.
Lemma 1 The finite-frequency observability Gramian can be computed as
Wo(¯ω) = L(A, ¯ω)∗Wo+ WoL(A, ¯ω), (17)
where Wo is defined by (14) or equivalently (15) and
L(A, ¯ω) = Z ω¯ − ¯ω H(jω)dω 2π = j
2πln[(A + j ¯ωI)(A − j ¯ωI)
−1]. (18)
PROOF. See [6, page 100]. 2
3.2 An upper bound on the robust H2norm
Let X represent Hermitian, block diagonal positive def-inite matrices that commute with ∆, i.e. every X ∈ X has the following structure
X = diaghX
1 · · · XL xL+1Im1 · · · xL+FImF
i . (19)
where the blocks in X have compatible dimensions with their corresponding blocks in ∆. The following condition plays a central role throughout this section.
Condition 1 Consider the system in (6). There exists X(ω) ∈ X where X ⊆ Rd×d, Hermitian Y (ω) ∈ Rm×m and > 0 such that
M(jω)∗ " X(ω) 0 0 I # M(jω) − " X(ω) 0 0 Y(ω) # " −I 0 0 0 # . (20)
The set of operators X are often called scaling matrices. In many cases it is customary to use constant scaling ma-trices to make the problem easier to handle. However the results achieved based on constant scaling matrices can be conservative. One of the ways to reduce the conserva-tiveness and keep the computational complexity reason-able is to use special classes of dynamic scaling matrices. This will be investigated in more detail in Section 3.2.2.
Next, two methods for computing upper bounds on ro-bust H2 norm of systems with structured uncertainties are presented. The first method explicitly defines Y (ω) in Condition 1 and uses Y (ω) to construct the upper bound on the robust H2 norm of the system. This method will
be referred to as explicit upper bound calculation. The second method calculates the upper bound through com-puting the observability Gramian via solving a set of LMIs. This method is referred to as Gramian based up-per bound calculation.
3.2.1 Explicit upper bound calculation
Consider Condition 1. This condition can be restated as follows
Lemma 2 If there exists X (ω) ∈ X such that
M∗
11X(ω)M11+ M21∗M21− X(ω) ≺ 0, (21)
then Condition 1 is satisfied if and only if there exists Y(ω) = Y (ω)∗ such that, M12∗X(ω)M12+ M∗ 22M22−(M ∗ 12X(ω)M11+ M ∗ 22M21)× (M11∗X(ω)M11+ M∗ 21M21− X(ω))−1× (M∗ 12X(ω)M11+ M22∗M21)∗Y(ω). (22)
PROOF. See Appendix A.
Using Condition 1 and Lemma 2, the following theorem provides upper bounds on the gain of the system for all
frequencies and will be used to provide an upper bound on the robust H2 norm for systems with structured un-certainty.
Theorem 1 If there exists X (ω) ∈ X such that (21) is satisfied ∀ω and we define Y (ω) as below
Y(ω) = M∗ 12X(ω)M12+ M ∗ 22M22− (M∗ 12X(ω)M11+ M22∗M21)× (M∗ 11X(ω)M11+ M ∗ 21M21− X(ω))−1× (M12∗X(ω)M11+ M∗ 22M21) ∗ , (23) then (∆ ∗ M )(jω)∗(∆ ∗ M )(jω) Y (ω) ∀ω.
PROOF. See Appendix B.
Corollary 1 If there exists X (ω) ∈ X and a frequency interval centered at ωi, I(ωi) = [ωi+ ωmin ωi+ ωmax],
such that
M∗
11XM11+ M21∗M21− X ≺0 ∀ω ∈ I(ωi), (24)
and we consider Y (ω) as defined in (23) for the men-tioned frequency interval then
Z ω∈I(ωi) Tr {(∆ ∗ M )∗(∆ ∗ M )} dω 2π ≤ Z ω∈I(ωi) Tr {Y (ω)} dω 2π, (25)
for all ∆ ∈ B∆, and specifically if I(ωi) covers all
fre-quencies sup ∆∈B∆ k∆ ∗ M k2 2≤ Z ∞ −∞ Tr {Y (ω)} dω 2π. (26)
3.2.2 Gramian-based upper bound calculation
In this section a class of dynamic scaling matrices with the following structure will be considered
X(ω) = ψ(jω)Xψ(jω)∗ =hCψ(jωI − Aψ)−1 I i XhCψ(jωI − Aψ)−1 I i∗ , (27)
where Aψ ∈ Rnψ×nψ and Cψ ∈ Rd×nψ are fixed
ma-trices with appropriate dimensions such that Aψ is
R(d+nψ)×(d+nψ) is a free basis for the parameters such
that X (s) ∈ X. As shown in [7] using this class of scaling matrices, Condition 1 can be rewritten as follows
Lemma 3 Consider the partitioning M = hM1 M2 i
, defined in (9), for the transfer matrix of system in (6). By replacing X (ω) with X (ω)−1in (20), it can be restated as
M1(jω)X (ω)M1(jω)∗− " X(ω) 0 0 I # M2(jω) M2(jω)∗ −Y(ω) 0. (28)
PROOF. See [7, Lemma 1]. 2
The upper left block of (28) can be expressed, up to its sign, as C11:= " X(ω) 0 0 I # −M1(jω)X (ω)M1(jω)∗ = " ψ 0 0 I # " X 0 0 I # " ψ 0 0 I #∗ − " M11ψ M21ψ # X " M11ψ M21ψ #∗ = " M11ψ ψ 0 M21ψ 0 I # −X 0 0 0 X 0 0 0 I " M11ψ ψ 0 M21ψ 0 I #∗ . (29)
By introducing the following transfer matrix
˜ C(jωI − ˜A)−1B˜q+ ˜D= " M11ψ ψ M21ψ 0 # , (30) and setting Γ =h0 Ii T , (29) can be reformulated as C11= h ˜C(jωI − ˜A)−1 Ii" ˜Bq 0 ˜ D Γ #! −X 0 0 0 X 0 0 0 I × h ˜ C(jωI − ˜A)−1 Ii" ˜Bq 0 ˜ D Γ #!∗ , (31) where ˜ A= A BqCψ 0 0 Aψ 0 0 0 Aψ , B˜q= 0 Bq 0 0 I 0 0 0 0 0 I 0 , ˜ C= " C DCψ " Cψ 0 ## , D˜ = " 0 D " 0 I 0 0 ## , (32) where ˜A ∈ Rnט˜ n, ˜B q ∈ Rn× ˜˜ d, ˜C ∈ R(l+d)טn and ˜D ∈ R(l+d)× ˜d, with ˜n= 2n ψ+ n and ˜d= 2nψ+ 2d.
Let Π(X, ˜Bq, ˜D) be an affine function of X defined as
below Π(X, ˜Bq, ˜D) = " ˜Bq 0 ˜ D Γ # −X 0 0 0 X 0 0 0 I " ˜Bq 0 ˜ D Γ #T = " Π11 Π12 ΠT 12 Π22 # , (33) where Π11 ∈ Rnט˜ n,Π 12 ∈ R˜n×(l+d) and Π22 ∈ R(l+d)×(l+d).
The following theorem taken from [11] can be used to calculate upper bounds on the robust H2norm.
Theorem 2 If there exist matrix X such that X (ω) in (27) satisfies X (ω) ∈ X, and Hermitian matrices P−, P+∈ Rnט˜ n, Q ∈ Rnψ×nψ, ˜Wo∈ Rnט˜ n, such that P−, Q 0, AψQ+ QATψ QCψT CψQ 0 −X ≺0, ˜ AP−+ P−A˜T P−C˜T ˜ CP− 0 −Π(X, ˜Bq, ˜D) ≺ 0, ˜ AP++ P+A˜T P +C˜T ˜ CP+ 0 −Π(X, ˜Bq, ˜D) ≺ 0, ˜ Wo I I P+−P− 0, Tr BT w 0 ˜ Wo Bw 0 < γ2, (34)
then X (ω) satisfies (28) and the system (∆ ∗ M ) defined in (11) has robust H2norm less than γ2.
PROOF. See [11]. 2
Theorem 2 includes the problem with constant scaling matrices as a special case. Let
ˆ A= A, ˆBq = h Bq 0 i , ˆC= C, ˆD= " D " Id 0 ## . (35)
Then the following Corollary is a restatement of Theo-rem 2 for constant scaling matrices, i.e. X (ω) = X.
Corollary 2 If there exist matrix X ∈ X and symmetric matrices P−, P+, Z ∈ Rn×nsuch that
P−, X 0, " ˆAP−+ P−AˆT P −CˆT ˆ CP− 0 # −Π(X, ˆBq, ˆD) ≺ 0, " ˆAP ++ P+AˆT P+CˆT ˆ CP+ 0 # −Π(X, ˆBq, ˆD) ≺ 0, Z I I P+−P− 0, TrBT wZBw < γ2. (36)
then X (ω) = X satisfies (28) and the system (∆ ∗ M ) defined in (11) has robust H2norm less than γ2.
PROOF. See [11]. 2
Aside from upper bounds on the robust H2 norm of
the system, Theorem 2 also provides additional informa-tion that will be used in the upcoming secinforma-tions. These additional information are highlighted in the following Lemma.
Lemma 4 Let P−, P+, X, Q and ˜Wo satisfy (34), and
C11 be defined as in (29). Then C11 0 with spectral factor ˜N such that ˜N , ˜N−1 ∈ RH
∞, i.e. C11 = ˜N ˜N∗.
Also let the scaled M be defined as
ˆ M = " X(ω)12 0 0 I # M " X(ω)−12 0 0 I # , (37) and be partitioned as ˆM =hMˆ1 Mˆ2 i . Then k ˜N−1Mˆ2k2 2<
γ2. A state space realization for ˜N−1Mˆ
2 is given by ˜ N−1Mˆ2= ˜ A −(Π12−P−C˜T)Π−122C˜ B˜w Π−12 22 C˜ 0 . (38)
Moreover Wois the observability Gramian of ˜N−1Mˆ2.
PROOF. See [11]. 2
4 Gramian-based upper bound on the robust finite-frequency H2norm
In this section the first method for calculating an up-per bound on the robust finite-frequency H2 norm of system in (6) is presented. The following theorem com-bines the ideas presented in Section 3.1, regarding the finite-frequency observability Gramians, with the results of Section 3.2.2, and computes the upper bound on the robust finite-frequency H2 norm for (6). Hereafter this method is referred to as Method 1.
Theorem 3 Let P−, P+, X, Qand ˜Wo be a solution to
(34), then sup ∆∈B∆ k∆ ∗ M k22,¯ω ≤Tr " Bw 0 #T L( ˜A,ω)¯ ∗W˜o+ ˜WoL( ˜A,ω)¯ " Bw 0 # (39)
where L( ˜A,ω) is defined in (18) and ˜¯ A = ˜A −(Π12−
P−C˜T)Π−122C.˜
PROOF. See Appendix C.
As was mentioned in Section 3.2, by using dynamic scal-ing matrices and increasscal-ing the order of these scalscal-ing matrices, it is possible to reduce the conservativeness of the results. In order to further reduce the conservative-ness of the bounds and improve the numerical properties of the optimization problems, it is useful to perform un-certainty partitioning. In this approach, for each of the uncertainty partitions, the upper bound on the robust finite-frequency H2norm of the system is computed and the maximum of these bounds is considered as the final result.
5 Frequency gridding based upper bound on the robust finite-frequency H2norm
In this section the second method to compute upper bounds on the robust finite-frequency H2 norm is pre-sented.
The following corollary to Theorem 1 plays a central role in the proposed algorithm.
Corollary 3 Let I(ωi) for i = 1, . . . , m be disjoint
frequency intervals such that Sm
i=1I(ωi) =
h −ω¯ ω¯
i . Also let the constant matrices Xi for i = 1, . . . , m be
the scaling matrices for which M∗
11XiM11+ M21∗M21−
Xi≺ 0 ∀ω ∈ I(ωi). Then, it holds that
sup ∆∈B∆ k∆ ∗ M k2 2,¯ω ≤ sup ∆∈B∆ m X i=1 Z ω∈I(ωi) Tr {(∆ ∗ M )∗ (∆ ∗ M )} dω 2π ≤ m X i=1 Z ω∈I(ωi) Tr {Yi(ω)} dω 2π, (40)
where Yi(ω) is defined as in (23), with X (ω) = Xi.
Corollary 3 provides a sketch for calculating upper bounds on the robust finite-frequency H2norm via fre-quency gridding. However calculating a suitable scaling matrix Xi requires checking M11∗XiM11 + M21∗M21 −
Xi ≺ 0 for an infinite number of frequencies in I(ωi).
Next a method is proposed to solve this issue. Consider the following two LMIs
M11(jω)∗X (ω)M11(jω)+M21(jω)∗ M21(jω)−X (ω) ≺ 0, (41) " M11(jω) 0 M21(jω) 0 #∗ ¯ X(ω) " M11(jω) 0 M21(jω) 0 # − ¯X(ω) ≺ 0. (42) Then ¯Xi= " Xi 0 0 I # ∈ R(d+l)×(d+l)satisfies (42) for ω =
ωi, if and only if Xisatisfies (41) for the same frequency.
The following theorem taken from [18], solves the issue of infinite dimensionality of the problem in Corollary 3
by providing a way to extend the validity of a scaling matrix that satisfies M∗
11XiM11+ M21∗M21−Xi≺0 for
a single frequency, e.g. ω = ωi, to a frequency interval,
I(ωi). Theorem 4 Let ˜M = " M11 0 M21 0 # = ˜ A B˜ ˜ C D˜ , and let D= ¯X12
i , where ¯Xisatisfies the LMI in (42). Define
G= AX−BXD−1 X CX, (43) where AX = " AG 0 −C∗ GCG −A∗G # , BX= " −BG C∗ GDG # , CX= h D∗ GCG B∗G i , DX= I − DG∗DG (44) in which G= AG BG CG DG = ˜ A − jωiI BD˜ −1 D ˜C D ˜DD−1 , (45)
and define ωlow and ωhigh as
ωlow = (
−ωi, if jG has no positive real eigenvalue
max{λ ∈ R−: det(λI + jG) = 0}, otherwise
(46)
ωhigh= (
∞, if jG has no negative real eigenvalue min{λ ∈ R+: det(λI + jG) = 0}, otherwise
(47)
Then ¯Xisatisfies (42) ∀ω ∈ ¯I(ωi) = ωi+ ωlow , ωi+ ωhigh .
PROOF. See Appendix D.
Using Corollary 3 and Theorem 2, the following algo-rithm can be used for calculating an upper bound on the robust finite-frequency H2 norm. This algorithmic
Algorithm 1 (Computation of an upper bound on the robust finite-frequency H2 norm)
(I) Divide the frequency interval of interest into a de-sired number of disjoint partitions, I(ωi), where ωi
is the center of the respective partition.
(II) For each of the partitions, compute Xi such that
it satisfies (41) for ω = ωi. In case there exist a
partition for which there exists no feasible solution, the system is not robustly stable and this method cannot be applied to this system.
(III) Construct ¯Xifrom the achieved Xiin (II).
(IV) Using Theorem 4 calculate the valid frequency range for the mentioned LMIs in (II). If the achieved fre-quency range does not cover the respective frefre-quency partition, i.e. I(ωi) 6⊆ ¯I(ωi), go back to (I) and
choose a finer partitioning for the frequency interval of interest.
(V) Define Yi(ω) using (23) with X (ω) = Xi.
(VI) Use numerical integration to calculate R
ω∈I(ωi)Tr {Yi(ω)}
dω 2π.
(VII) By Corollary 3, calculate the upper bound by sum-ming up the integrals computed in (VI).
The second step of Algorithm 1, requires computation of constant scaling matrices that satisfy (41) for ω = ωi for each of the partitions. This can be accomplished
through different approaches. However, considering the expression in (40) and the importance of Tr {Yi(ω)} in
the quality (closeness to the actual value) of the proposed upper bound on the robust finite-frequency H2norm in (40), it seems intuitive to calculate the scaling matrices while aiming at minimizing Tr {Yi(ωi)}. The following
two approaches utilize this in the process of computing suitable scaling matrices.
Approach 1 Compute Xi in Step (II) of Algorithm 1
as the solution of the following optimization problem
minimize
Xi,Yi
Tr {Yi}
subj. to
(20) with ω = ωi. (48)
Remark 1 The idea of frequency gridding was also pre-sented in [13], where the authors consider the H2
perfor-mance problem for discrete time systems. In that paper, an optimization problem similar to (48) for frequencies 0 = ω0. . . ωN = 2π is formulated and then the integral
R2π
0 trace(Y (ω)) dω
2π is approximated by the following
Rie-mann sum expression
1 2π N X i=1 trace(Yi)(ωi−ωi−1), (49) where 0 = ω0. . . ωN = 2π.
However, this approach does not necessarily provide a guaranteed upper bound on the robust H2 norm of the
system.
For any Xi satisfying the LMI in (41) for ω = ωilet
f(α) = Tr{M∗ 12αXiM12+ M ∗ 22M22− (M∗ 12αXiM11+ M22∗M21)× (M∗ 11αXiM11+ M21∗M21−αXi)−1× (M∗ 12αXiM11+ M ∗ 22M21) ∗} . (50)
This function is convex with respect to α. Next, follow-ing the same objectives as in Approach 1, an alternative method for calculating suitable scaling matrices is intro-duced.
Approach 2 Compute Xi in Step (II) of Algorithm 1
using the following sequential method
(I) Find Xi such that it satisfies the LMI in (41) for
ω= ωi.
(II) Minimize f (α), in (50), with the achieved Xiwith
respect to all α such that αXistill satisfies the LMI
in (41) for ω = ωi.
Denote α∗ as the minimizing α. Then α∗
Xi will be used
within the remaining steps of Algorithm 1. In order to assure that α∗
Xisatisfies (41) the search for α should be
subject to the constraint α > αmin, where
αmin= (51) 1 min eig Λ−1 2 0 0 I U(−M ∗ 11XiM11+ Xi)U∗ Λ−1 2 0 0 I ,
in which U , a unitary matrix, and Λ, are defined by the singular value decomposition M∗
21M21= U∗ " Λ−1 2 0 0 0 # U.
It is important to note that for some problems it might be required to perform many iterations between the first and the fourth steps of Algorithm 1. One of the ways
to alleviate this issue and even calculate better upper bounds, is to modify the proposed approaches by aug-menting new constraints for other frequencies from the partition under investigation. In this case the cost func-tion can also be modified accordingly. As an example, Approach 1 can be modified as follows
minimize Xi,Yi Tr {Yi} subj. to M(jω)∗ " Xi 0 0 I # M(jω) − " Xi 0 0 Yi # 0 for ω = ωj∈ I(ωi), j = 1, . . . , Ni, (52) or alternatively as minimize Xi,Yijj=1,...,Ni Ni X j=1 TrnYijo subj. to M(jω)∗ " Xi 0 0 I # M(jω) − " Xi 0 0 Yij # 0 for ω = ωj ∈ I(ωi), j = 1, . . . , Ni. (53)
Similar to Method 1, uncertainty partitioning improves the quality of the calculated upper bound on this method too.
Remark 2 Note that although the calculated value for the upper bound using Algorithm 1 has a decreasing trend with respect to the number of partitions, this trend is not necessarily monotonically decreasing. This is due to the fact that the calculated upper bound not only is dependent on the number of partitions but also on the quality of the calculated scaling matrices and how they affect the numerical integration procedure.
6 Numerical examples
In this section the proposed methods are tested on the-oretical and practical examples. The chosen thethe-oretical example can be solved analytically, i.e. the robust H2
norm for this example can be computed via routine cal-culations. The achieved results for this example are re-ported in Section 6.1.
As a practical example, an application to the comfort analysis problem for a civil aircraft model is discussed. Due to its more complex uncertainty structure, this ex-ample is computationally more challenging. Section 6.2
presents the analysis results for this example. It should be pointed out that all the computations, for both ex-amples, are conducted using the Yalmip toolbox [8] with the SDPT3 solver [21]. The platform used for the sim-ulations uses a Dual Core AMD OpteronT M Processor 270 as the CPU and 4 GB of RAM.
6.1 Theoretical Example
Consider the uncertain system in (6) with the following system matrices A= −2.5 0.5 0 −50 0 0 −1 0.5 0 0 0 −0.5 0 0 0 0 0 0 −5 100 0 0 0 −100 0 , Bq= 0.25 −0.5 0 0 0 0 0 0 0 0 , Bw= 0 5 0 0 5 , C= " Cp Cz # = 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 D= " Dpq Dzq # = 0 0 1 0 0 0 . (54)
In this example ∆(δ) = δI2 with −1 ≤ δ ≤ 1. This
system is known to have robust H2 norm, as defined in (12), equal to 1.5311 which is attained for δ = 0.25.
Figure 2 illustrates the gain plots of the system for dif-ferent values of the uncertain parameter. The aim is to calculate the robust finite-frequency H2norm of the sys-tem and avoid the peak occurring at 100 rad/s. This is motivated by Figure 3 which presents the calculated finite-frequency H2norm of the system in (54), with re-spect to different values for the uncertain parameter and frequency bounds. As can be seen from this figure and the jump at ¯ω= 100 rad/s, the contribution of this peak to the robust finite-frequency H2 norm cannot be ne-glected. In order to avoid this peak, the frequency bound that has been considered for this example is ¯ω = 50 rad/s. The actual value for the robust finite-frequency H2norm for (54) with this frequency bound is 0.8919.
Method 1, presented in Section 4, utilizes the following class of dynamic scaling matrices
10−2 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 ω[rad s] k ∆ ∗ M k 22 ¯ ω= 50
Fig. 2. Gain plot versus different values for the uncertain parameter. 10−2 100 102 −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ¯ ω[rad s] δ k∆ ∗ M k 22, ¯ω
Fig. 3. Finite-frequency H2 norm versus different values of
the uncertain parameter and frequency bounds.
ψ(s) =h(s−p)(s−p)nψ −1nψ I2 (s−p) nψ −2 (s−p)nψ −1I2 . . . 1 (s−p)I2 I2 i (55)
with p = 150, and via Theorem 3 it calculates the up-per bound on the robust finite-frequency H2 of the sys-tem. For this particular example dynamic scaling matri-ces with order higher than 3 do not produce any better upper bounds, so only scaling matrices up to order 3 are considered.
Method 2, presented in Section 5, has been applied to this example with Approaches 1 and 2. The number of frequency partitions is increased until either the per-formance matches the perper-formance of Method 1 or the
10−3 10−2 10−1 100 101 102 103 0 0.5 1 1.5 2 2.5 ¯ ω[rad s] k ∆ ∗ M k 2 2,¯ω
Fig. 4. Robust finite-frequency H2norm and the computed
upper bounds on it for different frequency bounds. The solid line illustrates the actual value for the robust finite-frequency H2norm. The dashed and dashed-dotted lines represent the
achieved upper bounds using Methods 1 and 2 for different orders of scaling matrix and numbers of frequency partition numbers, respectively.
improvement in the computed upper bound is not dis-cernible anymore.
Figure 4 illustrates the achieved upper bounds on dif-ferent frequency bounds, ¯ω. The curve marked with the solid line reports the actual values for the robust finite-frequency H2 norm of the system. The dashed lines
present the achieved upper bounds using Method 1. As can be seen from the figure as the order of the dynamic scaling matrices increases the computed upper bound becomes tighter. Note that the upper bounds computed using scaling matrices with nψ ≥ 1 are practically
in-distinguishable. The bounds presented with the dashed-dotted lines are results achieved by applying Method 2 to this example. As can be seen from Figure 4, Method 2 with Approach 1 can produce better upper bounds than the second approach and can match the performance of Method 1 with 40 partitions. Figure 6 illustrates the cal-culated upper bounds on the systems gains, Y (ω), for different frequencies. Table 1 presents a summary of the achieved results.
So far the presented results are achieved without any un-certainty partitioning. In order to illustrate the effect of uncertainty partitioning on the performance of the pro-posed methods, Method 1 and Method 2 with Approach 1 are applied to this example with uncertainty partition-ing. Figures 6 and 7 present the achieved upper bounds on robust finite-frequency H2 norm of the system with ¯
ω= 50 rad/s using Methods 1 and 2, respectively. These figures illustrate the upper bound with respect to num-ber of uncertainty partitions and order of dynamic scal-ing matrices, for Method 1, and number of frequency
10−2 10−1 100 101 102 103 0 0.5 1 1.5 2 ω[rad s ] k∆ ∗ M k 2 2 ¯ ω= 50 Fig. 5. Magnitudes of k∆ ∗ M k2
2for different uncertainty
val-ues (solid lines), and the calculated upper bound on each frequency point. The dashed and dashed-dotted lines repre-sent the achieved upper bounds using Methods 1 and 2 for different orders of scaling matrix and numbers of frequency partition numbers, respectively.
grid points, for Method 2. As can be seen from the fig-ures and considering the actual robust finite-frequency H2norm of the system, the computed upper bounds us-ing both methods are extremely tight. A summary of the results from this analysis is presented in Tables 2 and 3.
As can be observed from Tables 2 and 3, although both methods produce equally tight upper bounds, Method 1 achieves this goal with lower computation time.
Table 1
Numerical results for the theoretical example.
Method Estimated Elapsed Upper bound Time[sec] M.1, nψ= 0 1.2609 11 M.1, nψ= 1 1.1972 10 M.1, nψ= 2 1.1944 12 M.1, nψ= 3 1.1911 13 M.2, App.1, npar= 40 1.189 44 M.2, App.1, npar= 200 1.186 144 M.2, App.2, npar= 200 1.3184 552 Table 2
Numerical results for the theoretical example Using Method 1.
nψ No. Uncer. Estimated Elapsed
Par. Upper bound Time[sec]
2 1 1.1944 12 2 20 0.8928 434 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
No. Uncertainty Partitions
Ψ k (∆ ∗ M )k 22,5 0
Fig. 6. The achieved upper bound on robust finite-frequency H2 norm with ¯ω = 50 rad/s with respect to number of
uncertainty partitions and order of dynamic scaling matrices.
0 5 10 15 20 25 0 20 40 60 0.8 0.850.9 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 Partitions No. Uncertainty Partitions
k (∆ ∗ M )k 22,5 0
Fig. 7. The achieved upper bound on robust finite-frequency H2 norm with ¯ω = 50 rad/s with respect to number of
uncertainty partitions and number of frequency grid points.
6.2 Comfort Analysis Application
The problem considered in this section involves a model of a civil aircraft, including both rigid and flexible modes. This model is to be used to evaluate the effect of wind turbulence on different points of the aircraft, and is
re-Table 3
Numerical results for the theoretical example Using Method 2.
No. freq. No. Uncer. Estimated Elapsed Grids Par. Upper bound Time[sec]
20 1 1.1945 30
10−2 10−1 100 101 102 103 0 0.5 1 1.5 2 2.5 3 ω[rad s] k∆ ∗ M k 2 2 ¯ ω= 15
Fig. 8. Gain plot versus different values for the uncertain parameter.
ferred to as comfort analysis. This problem can be refor-mulated as an H2 performance analysis problem for an extended system, including the model of the aircraft, a Von Karman filter, modeling the wind spectrum, and an output filter, accounting for the turbulence field, [16]. In the provided aircraft model the uncertain parameter δ corresponds to the level of fullness of the fuel tanks and it is normalized to vary within the range [−1, 1]. The overall extended system is presented in LFT form, as in (6), with n = 21 states and an uncertainty block size of d= 14.
The provided aircraft model is valid for frequencies up to 15 rad/s and beyond that does not have any physical meaning [17]. This motivates performing finite-frequency H2 performance analysis, limited to this frequency range.
Figure 8, illustrates the gain plots of the system under consideration as a function of frequency. Different curves in this figure correspond to different uncertainty values. As can be seen from the figure, the frequency bound at 15 rad/s is necessary to avoid the peak at approximately 20 rad/s which is outside the validity range of the model.
The methods considered for performing comfort analy-sis are Methods 1 and 2 with the use of constant scaling matrices and Approach 1, respectively. Tables 4 and 5 summarize the achieved results using Methods 1 and 2, respectively. As can be seen from the tables, both meth-ods perform equally accurate in estimating the robust finite-frequency H2norm of the system. However, in
con-trast to the example in Section 6.1, Method 2 is faster in calculating the upper bound with equal accuracy.
Similar to Section 6.1, it is possible to improve the com-puted upper bounds via uncertainty partitioning. This can be observed from Tables 4 and 5.
7 Discussion and General remarks
This section highlights the advantages and disadvan-tages of the proposed methods and provides insight on how to improve the performance of the methods consid-ering the characteristics of the problem at hand.
7.1 The observability Gramian based method
This method considers the frequency interval of interest as a whole and calculates an upper bound on the robust finite-frequency H2 norm of the system in one shot or
one iteration by solving an SDP. However the dimen-sion of this optimization problem grows rapidly with the number of states and/or size of the uncertainty block. This limits the capabilities of this method in handling medium or large sized problems, i.e. analysis of systems with high number of states or large uncertainty blocks.
The most apparent possibility to improve the accuracy of the computed upper bound using this method is to increase the order of the dynamic scaling matrices. This comes at the cost of rapidly increasing the number of optimization variables in the underlying SDP and affects the computational tractability of the method.
Another way of improving the computed upper bound is to perform uncertainty partitioning, which proved to be effective through the examples presented in Section 6. However, this improvement comes at the cost of a much higher computational burden, see Table 4.
7.2 The frequency gridding based method
This method starts with an initial partitioning of the de-sired frequency interval and calculates the upper bound on the robust finite-frequency H2 norm by solving the
corresponding SDP for each of the partitions.
Table 4
Numerical results for the theoretical example Using Method 1.
nψ No. Uncer. Estimated Elapsed
Par. Upper bound Time[h] 0 50 1.2434 8.62 0 450 0.7970 59.24 Table 5
Numerical results for the theoretical example Using Method 2.
No. freq. No. Uncer. Estimated Elapsed Grids Par. Upper bound Time[h] 80 1 1.2382 0.5611 80 10 0.7911 4.25
The size of the underlying SDPs in this method is smaller than the previous method and is mainly depen-dent on the size of the uncertainty block. Consequently, this method can handle larger problems. However for large problems, the algorithm might require some iter-ations between steps IV and I of the algorithm, to be able to produce consistent results. Another issue with this method is the requirement to perform numerical integration on a rational function in step VI of the al-gorithm. This can become slightly problematic for high order systems.
There are two main ways to improve the computed upper 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 in the partition and/or adding more variables to the SDPs corresponding to the partitions. This proved to scale bet-ter considering the computation time, as compared to Method 1 see Table 5.
8 Conclusion
This paper has provided two methods for calculating upper bounds on the robust finite-frequency H2 norm.
Through the paper different guidelines for improving the performance of the proposed methods have been pre-sented and their effectiveness has been illustrated using both a theoretical and a practical example.
The proposed methods consider different formulations for calculating a consistent upper bound on the robust finite-frequency H2 norm. Due to this, although both
methods can produce equally tight upper bounds, they have different computational properties. Method 1 is more suitable for small-sized problems and produce re-sults faster than the second method for this type of prob-lems. On the other hand, Method 2 can handle larger problems and produce results more rapidly for this class of problems.
Acknowledgements
The authors wish to thank involved personnel form AIR-BUS, Cl´ement Roos and Carsten D¨oll from ONERA and Simon Hecker and Andras Varga from DLR for provid-ing the model of the civil aircraft used in Section 6.2.
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A Proof of Lemma 2 Let C11= M∗ 11X(ω)M11+ M21∗M21− X(ω), C12= M∗ 11X(ω)M12+ M ∗ 21M22, C21= M∗ 12X(ω)M11+ M22∗M21, C22= M12∗X(ω)M12+ M22∗M22. (A.1)
Then the left hand side of Condition 1 can be written as
M∗(jω) " X(ω) 0 0 I # M(jω) − " X(ω) 0 0 Y(ω) # = " C11 C12 C21 C22−Y(ω) # . (A.2)
Now if we assume that there exists X (ω) ∈ X such that C11≺0, then Lemma 2 is the direct outcome of Schur’s lemma. 2
B Proof of Theorem 1
If the assumptions of the theorem are satisfied, then by Lemma 2, Condition 1 is valid, i.e. (20) holds. Define
ˆ M = " X(ω)12 0 0 I # M " X(ω)−12 0 0 I # . (B.1)
Then (20) can be rewritten as
ˆ M∗M −ˆ " I 0 0 Y (ω) # " −I 0 0 0 # . (B.2) As a result ˆ M∗M ˆ " I 0 0 Y (ω) # . (B.3) Define ¯q(jω) = X (ω)12q(jω) and ¯p(jω) = X (ω)12p(jω).
By pre and post multiplying both sides of (B.3) by h ¯ q(jω)∗ w(jω)∗iand " ¯ 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)−12, and hence
¯
q= X12q= X12∆X−12p¯= ∆¯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. 2
C Proof of Theorem 3
Let P−, P+, X, Qand ˜Wosatisfy (34). Define
˜
Y = ( ˜N−1Mˆ2)∗
( ˜N−1Mˆ2) = ˆM2∗C−1
11Mˆ20. (C.1)
where ˜N and ˆM2 are defined in Lemma 4. From (C.1) C11 0. If we set Y = ˆM∗ 2C11−1Mˆ2, by Schur’s lemma it follows that " −C11 Mˆ2 ˆ M∗ 2 −Y # 0 (C.2)
By replacing X (ω) with X (ω)−1 in (B.1) and using
Lemma 3, (C.2) is equivalent to (B.2). In other words
M(jω)∗ " X(ω)−1 0 0 I # M(jω)− " X(ω)−1 0 0 Y(ω) # " −I 0 0 0 # . (C.3)
By (C.3) and the same arguments as in the proof of Theorem 1, (∆∗M )(jω)∗(∆∗M )(jω) Y (ω) ∀ω, ∀∆ ∈
B∆. As a result by using lemmas 1 and 4, (39) follows.
2
D Proof of Theorem 4
Consider the LMI in (42) with ¯X(ω) = ¯Xi. This LMI
¯ X−12 i M˜ ∗¯ X12 i X¯ 1 2 i M ¯˜X −1 2 i −I ≺0. (D.1) Let G(jω) = ¯X12 i M˜(j(ω + ωi)) ¯X −1 2
i . It now follows that
G= AG BG CG DG
. In this theorem we are looking for the
largest frequency interval, for which the LMI in (D.1) is valid. On the boundary of this interval I − G(jω)∗
G(jω) becomes singular, i.e. det(I − G(jω)∗G(jω)) = 0.
By (44) and (45), I − G(jω)∗G(jω) = AX BX CX DX .
Us-ing Sylvester’s determinant lemma and some simple ma-trix manipulation we have
det(I − G(jω)∗ G(jω)) = 0 ⇔ det(I + D−12 X CX(jωI − AX)−1BXD −1 2 X ) = 0 ⇔ det(I + (jωI − AX)−1BXD−1X CX). (D.2)
By using the matrix determinant lemma and the defini-tion of G it is also straight forward to establish equiva-lence between the following expressions
det(I + (jωI − AX)−1BXDX−1CX) ⇔
det(jωI − (AX−BXD−1X CX)) = 0 ⇔ det(ωI + jG) = 0,
(D.3)
Division,Department
DivisionofAutomaticControl
DepartmentofElectricalEngineering
Date 2011-05-13 Språk Language Svenska/Swedish Engelska/English Rapporttyp Reportcategory Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrigrapport
URLförelektroniskversion
http://www.control.isy. liu .se
ISBN
ISRN
Serietitelochserienummer
Titleofseries,numbering
ISSN
1400-3902
LiTH-ISY-R-3011
Titel
Title
Robust nite-frequencyH2 analysisofuncertainsystems
Författare
Author
AndreaGarulli,AndersHansson,SinaKhoshfetratPakazad,Al oMasi,RagnarWallin
Sammanfattning
Abstract
Inmanyapplications,designoranalysisisperformedovera nite-frequencyrangeofinterest.
TheimportanceoftheH2/robustH2normhighlightsthenecessityofcomputingthisnorm
accordingly.Thispaperprovidesdierentmethodsforcomputingupperboundsontherobust
nite-frequency H2 normforsystemswithstructureduncertainties. Anapplication ofthe
robust nite-frequency H2 normforacomfortanalysisproblemofanaero-elasticmodelof
