京都大学教育研究振興財団助成事業
成 果 報 告 書
平成23年 9月12日
公益財団法人京都大学教育研究振興財団
会 長

辻

井

昭

雄

様

所属部局・研究科

京都大学大学院工学研究科

職 名・学 年

博士後期課程１年

氏

細 江 陽 平名助 成 の 種 類

平成２３ 年度 ・ 若手研究者在外研究支援 ・ 国際研究集会発表助成

研 究 集 会 名

The 18th IFAC World Congress（第１８回国際自動制御連盟世界会議）

発 表 題 目

Relationship between Noncausal Linear Periodically Time-Varying Scaling andCausal Linear Time-Invariant Scaling for Discrete-Time Systems（離散時間系に対する非因果的周期時変スケーリングと因果的時不変スケーリ
ングの関係）

開 催 場 所

イタリア共和国 ロンバルディア州 ミラノ市 サクロ・クオーレ・カトリック大学

渡 航 期 間

平成 ２３年

成 果 の 概 要

８月 ２７日～平成 ２３年

９月

４日

タイトルは「成果の概要／報告者名」として、A４版２０００字程度・和文で作成し、添付して下さい。 「成果の概要」以外に添付する資料 □ 無■ 有( 発表論文 )交付を受けた助成金額

200,000 円

使 用 し た 助 成 金 額

200,000 円

返納すべき助成金額0円
航空運賃 184,240円
宿泊費の一部 15,760円

会 計 報 告助 成 金 の 使 途 内 訳

成果の概要／細江陽平
京都大学教育研究振興財団より平成２３年度国際研究集会発表助成（II 期）として渡航費に対する助成を受け，２０１１年８月２８日から９月２日の日程でイタリア共和国
ロンバルディア州ミラノ市において開催された the 18th IFAC World Congress （第１８回国際自動制御連盟世界会議）に参加した．IFAC World Congress は，３年に１度開催されるシステム制御分野における最も大規模かつ権威のある国際会議である．１９６０
年の第１回会議がモスクワ（旧ソ連）で開催されたのをはじめとして，開催都市（国）は多岐にわたっており（１９８１年には京都でも開催された），世界中から集まったシス
テム制御分野の研究者・技術者がそれぞれの最新の研究成果を互いに発表しあう貴重な場の１つとなっている．第１８回となる本国際会議は，会期中２５のオーラルセッショ
ンと１つのポスターセッションが同時に進行する形で運営され，２４００を超える数の研究発表があった．筆者は主としてロバスト制御，最適制御，線形行列不等式等をキーワードとした研究に興味を持ち，時間の許す限りさまざまな研究発表を聴講した．その中にはこれまでの研究生活において扱ったことのない概念や手法に関するものも多くあり，新たな知識を得るとともに今後の研究の新たな方向性を考える貴重な機会となった．
また，筆者は８月３０日の Robustness Analysis （ロバスト性解析）というオーラル
セッションにて，Relationship between Noncausal Linear Periodically Time-Varying
Scaling and Causal Linear Time-Invariant Scaling for Discrete-Time Systems （離散時間系に対する非因果的周期時変スケーリングと因果的時不変スケーリングの関係）と題して研究成果の発表を行った．以下にその概要を述べる．
現実の制御対象を何らかの制御理論に基づいて制御しようと試みる際，まずはその対象をモデル化するのが一般的であるが，このときモデル化誤差（すなわち不確かさ）が生じることは避けられない．不確かさが実際に生じてしまったとしても所望の性能を保証するロバスト性という考え方は，制御の実応用上の観点から極めて重要となっている．
本研究は，ロバスト性，とくにロバスト安定性の解析に関するものである．近年，解析において非因果性を導入した非因果的周期時変スケーリングという手法が提案された．
この手法は従来手法である因果的時不変スケーリングに比して容易に解析結果を改善できる可能性を秘めており，筆者らはその手法がロバスト設計応用においても有効である
ことをこれまでの研究で示した．しかしながら，その手法と従来手法との対応関係は，既存の成果のみでは十分に議論されておらず，したがって非因果的周期時変スケーリン
グの優位性に理論的な保証を与えることはこれまで容易でなかった．本研究では，それら２つの手法のそれぞれにおいてある特別なものを考えると，その間にある種の等価関係が成り立つことを初めて示した．この成果は２つの手法の間の関係を容易に考察することを可能とするものであり，本研究の今後の発展においてその中核を担うものであると確信している．
上記発表は当該セッションにおける最後の発表であり，かつその１つ前の発表が諸事
情によりキャンセルとなったため，あまり多くの方に筆者の発表を聞いていただけないのではないかと心配したが，幸いにも多くの聴講者に恵まれ，本研究を広くアピールすることができた．質疑応答では海外の研究者からリフティングを介す／介さない枠組で定義されたセパレータのクラスの包含関係に関する質問を受け，その場で可能な限り回答したのち，セッション終了後により詳細な議論を行った．また，コーヒーブレイクや
昼食の時間においても，日本から来られた研究者の方々から私の携わっている研究に関して説明を求められ，基礎的な事項について述べさせていただいた．本研究のキーアイ
ディアの１つである非因果的周期時変スケーリングは，近年提案されたばかりであり，
国内外のいずれにおいても認知度が高いとは言い難い状況にある．そのような中で，本国際会議のようなハイレベルかつ大規模な発表の場において多くの研究者に本研究をアピールできたことは，筆者の今後の研究生活において大変意義深いことであったと考える．今後も自身の研究を掘り下げると同時に制御工学に関する知識を広く身に付け，国際的に活躍できる研究者となれるよう精進していく所存である．
最後に，筆者の本国際会議への出席に対しご支援くださった京都大学教育研究振興財団に深く謝意を表す．

Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011

Relationship between Noncausal Linear
Periodically Time-Varying Scaling andCausal Linear Time-Invariant Scaling for
Discrete-Time Systems
Yohei Hosoe ∗ Tomomichi Hagiwara ∗∗Kyoto University, Nishikyo-ku, Kyoto, 615-8510 JAPAN (e-mail:hosoe@jaguar.kuee.kyoto-u.ac.jp, hagiwara@kuee.kyoto-u.ac.jp).

Abstract: In this paper, we discuss the relationship between noncausal linear periodically timevarying (LPTV) scaling and causal linear time-invariant (LTI) scaling against discrete-time LTI
closed-loop system. Noncausal LPTV scaling is naturally introduced via lifting technique, andcan induce some frequency-dependent scaling in the lifting-free (i.e., usual) framework. However,it has not been clear what classes of noncausal LPTV scaling and causal LTI scaling haveequivalent abilities in their respective frameworks. It is an important issue for sophisticating thetheoretical base of noncausal LPTV scaling, and this paper studies such a relationship.
Keywords: Discrete-time time-invariant systems, Robust stability, Lifting, Dynamic separators
1. INTRODUCTION

with causal LTI scaling, and reducing the conservativenessis a very important study.

Robustness is quite important in control because modelingerrors cannot be avoided in practice. Useful frameworks
for studying robustness such as the multiplier technique[Desoer and Vidyasagar (1975); Safonov (1980)] and integral quadratic constraints (IQC) [Megretski and Rantzer(1997)], have thus been developed. Both multiplier and
IQC approaches ensure robust stability of the closed-loopsystem through the existence of an appropriate matrix
satisfying some inequalities for a given class of uncertainties, and the relationship between these two approaches
has also been discussed in Fu et al. (2005). The IQC
approach provides unified treatment of robust stability
conditions such as the small-gain and passivity theorems,
as well as D-scaling, (D, G)-scaling and multiplier methods[Vidyasagar (1993); Zhou and Doyle (1998); Fan et al.
(1991)] by appropriately confining the matrices in the theorem called separators. The separator-type robust stability
theorem [Iwasaki and Hara (1998)] is also closely related to
the IQC approach through the topological separation notion [Safonov (1980)], and is particularly useful for dealing
with linear time-invariant (LTI) systems.

To this end, discrete-time noncausal linear periodicallytime-varying (LPTV) scaling was proposed in Hagiwara
and Ohara (2010). This approach can be naturally introduced through the application of the separator-typerobust stability theorem under the lifting treatment [Bittanti and Colaneri (2000, 2009)] of discrete-time systems.
It has been shown that noncausal LPTV scaling, evenif we confine it to being static, induces some frequencydependent (i.e., dynamic) causal LTI scaling under theinterpretation of it in the lifting-free framework. Hence,static noncausal LPTV scaling is one of less conservative
analysis approaches compared with usual static causal LTI
scaling, and its effectiveness has also been demonstrated by
Hagiwara and Ohara (2010); Hosoe and Hagiwara (2010b)(in robust stability analysis) and Hosoe and Hagiwara(2010a,c) (in robust controller synthesis).

Robust stability of discrete-time LTI systems can be analyzed by searching for separators satisfying the inequality
conditions in the separator-type robust stability theorem;
such an approach to robust stability analysis is calledcausal LTI scaling. To achieve nonconservative robust
stability analysis, however, such a search must work on
all frequency-dependent (i.e., dynamic) separators without
any constraint, but this is not feasible from a computational viewpoint. Thus, a tractable class of separators is
introduced in practice, and the search of separators satisfying the robust stability condition is carried out only onthat class. Such a restriction of separator classes generally
leads to the conservativeness in robust stability analysis
Copyright by theInternational Federation of Automatic Control (IFAC)

In spite of such a practical success, there are some unrevealed issues about a theoretical side of noncausal LPTV
scaling. As one aspect of them, we tackle in this paper the issues in the correspondence relationship between
noncausal LPTV and causal LTI scaling approaches. As
stated in the above, even static noncausal LPTV scaling
induces some dynamic causal LTI scaling in the liftingfree framework. However, preceding studies have not explicitly characterized the class of causal LTI separatorsthat we can equivalently deal with by working instead on
noncausal LPTV scaling. To reveal such a relationship is
one of the most fundamental issues for sophisticating thetheoretical base of noncausal LPTV scaling. This paper
provides a certain form of answer to this issue, and shows
the usefulness of the answer in further studies about thecomparison of a variety of noncausal LPTV and causal LTI
scaling approaches.

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The following part of this paper is organized as follows.
Section 2 states the robust stability analysis problem
we deal with, and shows the brief ideas of the analysis
with lifting-free and lifting-based approaches. Section 3
reviews the definition of noncausal LPTV scaling andsome existing result that gives a starting point of theissues we tackle in this paper. Section 4 introduces, on
the basis of the existing result, a natural class of causal
LTI scaling, and derives a class of noncausal LPTV scalingthat has an equivalent ability in robust stability analysis.
Section 5 applies such an equivalence relationship anddiscusses some further aspect of the relationship between
noncausal LPTV scaling and causal LTI scaling. The
validity of such discussions is also demonstrated with anumerical example of robust stability analysis. In addition,
it is discussed that the idea of noncausal LPTV scalingis effective even in causal LTI scaling in the lifting-freeframework, particularly in relation with the ease in thetreatment of the uncertainties ∆.
2. ROBUST STABILITY THEOREM ANDDISCRETE-TIME LIFTINGThis section first states the robust stability analysis problem that we deal with in this paper, and then reviews two
approaches to robust stability analysis. One is to analyze
robust stability of systems by directly applying a robust
stability theorem, and the other is to analyze it by applying such a theorem via lifting treatment. Based on theidea of the latter approach, noncausal LPTV scaling [Hagiwara and Ohara (2010)] can naturally be introduced. In
this paper, we discuss the relationship between noncausal
LPTV scaling introduced in the lifted framework andusual frequency-dependent scaling treated in the liftingfree framework (i.e., causal LTI scaling).
2.1 Robust stability analysis problem
This paper studies the robust stability problem of thediscrete-time closed-loop system Σ shown in Fig. 1 consisting of the nominal system G and the uncertainty ∆.
The nominal system G is assumed to be internally stable,
finite-dimensional, LTI, and represented by
xk+1 = Axk + Buk , yk = Cxk + Duk
n

p

(1)

p

where xk ∈ R , uk ∈ R , yk ∈ R . The uncertainty ∆is assumed to belong to some given set ∆ satisfying thefollowing assumption.
Assumption 1. Every ∆ ∈ ∆ is internally stable, finitedimensional and LTI, and ∆ is a connected set such that0 ∈ ∆.
The transfer matrices of G and ∆ are denoted by G(ζ) and∆(ζ), respectively.
∆ uG
Fig. 1. Closed-loop system Σy2.2 Separator-type robust stability theorem
To tackle the robust stability problem of Σ , we review therobust stability theorem given in Iwasaki and Hara (1998).
Theorem 1. Suppose that G is internally stable. If Σ iswell-posed, ∀∆ ∈ ∆, then Σ is robustly stable with respectto ∆ if and only if there exists Θ(ζ) = Θ(ζ)∗ (ζ ∈ ∂D)
such that


∗

I
I
≤ 0 (∀ζ ∈ ∂D)
(2)
Θ(ζ)
G(ζ)
G(ζ)





∗
∆(ζ)
∀∆ ∈ ∆,
∆(ζ)
Θ(ζ)
>0
(3)
I
∀ζ ∈ ∂D
I
where ∂D := {ζ ∈ C : |ζ| = 1}.
See, e.g., Zhou and Doyle (1998) for the definition of wellposedness in the above theorem. The Hermitian matrix
Θ(ζ) in (2) and (3) is called a separator. We can study therobust stability of the closed-loop system Σ by searchingfor separators Θ(ζ) satisfying (2) and (3) against the given∆. We call this approach causal LTI scaling. In the next
subsection, we show an alternative approach to robust
stability analysis through the lifting technique.
2.3 Robust stability theorem with lifting treatment
In this subsection, we consider applying the separator-typerobust stability theorem to the closed-loop system Σ via
lifting treatment. We first give a brief review of the lifting
technique [Bittanti and Colaneri (2000, 2009)].
The operation of constructing new signal representaTtions ubκ := uTκN , uTκN +1 , · · · , uTκN +N −1
and ybκ :=
 T
T
T
T
yκN , yκN +1 , · · · , yκN +N −1 from the discrete-time signals uk and yk is called lifting of signals, where N ≥ 2 isa positive integer. This converts the treatment of systems
with input uk and output yk into that of systems withlifted input ubκ and lifted output ybκ , and such treatment
is called lifting of systems. The resulting lifted representations of systems are called N -lifted systems. The lifting
technique is often applied to periodic systems throughchoosing N to be equal to the period of the systems.
However, it is obvious that we can also apply the lifting
technique to the LTI systems G and ∆ with any N ≥ 2.
By defining xbκ := xκN , we describe the N -lifted nominalb bysystem G
bxκ + Bbb uκ , ybκ = Cbx
b uκ .
xbκ+1 = Abbκ + Db(4)b can be constructed withAll the coefficient matrices of G
the coefficient matrices in (1). The lifted nominal systemb is also LTI, and we denote its transfer matrix by G(z);bG
it is called the N -lifted transfer matrix of G. We can alsob and N -lifted transfer
obtain the N -lifted representation ∆bmatrix ∆(z)from ∆. Through these ideas, we can obtainb (Fig. 2) from the closed-loop
the lifted representation Σsystem Σ . We remark that ζ has been used for the symbol
for the z-transform of the original discrete-time signals,
while z is used for that of lifted signals.

It is known from the property of lifting that Σ is robustly
b is. In this subsection, we consider
stable if and only if Σ
analyzing the robust stability of the original LTI system
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The approach to robust stability analysis based on causal
LPTV separators defined above is called causal LPTV
scaling. Even though it has been described in the framework of lifting, such scaling corresponds to the conventional lifting-free scaling with causal LPTV systems.

b 
∆

u
bb
G

yb

b
Fig. 2. Lifted closed-loop system Σb through the following theorem,Σ via analyzing that of Σwhich follows immediately from Theorem 1.
Theorem 2. Suppose that G is internally stable. If Σ iswell-posed, ∀∆ ∈ ∆, then Σ is robustly stable with respectbb ∗ (z ∈ ∂D)
to ∆ if and only if there exists Θ(z)= Θ(z)
such that


∗

I
I
b
≤ 0 (∀z ∈ ∂D)
(5)
Θ(z)
b
b
G(z)
G(z)


∗



b
b
∀∆ ∈ ∆,
∆(z)
∆(z)
b
Θ(z)
>0
(6)
∀z ∈ ∂DI
I
As in the previous subsection, we can analyze the robustbstability of Σ by searching for separators Θ(z)satisfying(5) and (6) against the given ∆.

On the other hand, a noncausal LPTV separator [Hagiwara and Ohara (2010)] has been defined by generalizing
Definition 1.
b
Definition 2. A separator given by Θ(z)= Vb (z)∗ Γ Vb (z)is called a noncausal LPTV separator, where Vb (z) is thetransfer matrix of a causal LTI system Vb with 2N p inputs
defined on the lifted time axis and Γ = Γ ∗ is a constantmatrix of compatible size. In particular, if Vb is static, thenthe corresponding separator is called a static noncausal
LPTV separator.
The approach to robust stability analysis based on noncausal LPTV separators defined above is called noncausal
LPTV scaling.

The above theorem leads to an alternative approach to
the robust stability analysis of Σ . On the basis of thisapproach, we can naturally introduce noncausal LPTV
scaling. In the next section, we review the definition of
noncausal LPTV scaling and an existing result about
the relationship between noncausal LPTV scaling andconventional causal LTI scaling.

Comparing Definition 2 with Definition 1, we readily
see that the structure of noncausal LPTV separators is
generalized from that of causal LPTV separators. Since
noncausal LPTV scaling is more flexible than causal LPTV
scaling in this way, the former is known to be generally
more effective for robust stability analysis than the latter.
In this section, we review an existing result about therelationship between noncausal LPTV scaling with lifting
treatment and usual scaling without lifting treatment.

3. DEFINITION OF NONCAUSAL LPTV SCALING
AND REVIEW OF EXISTING RESULT

3.2 Causal LTI scaling induced by noncausal LPTV
scaling

In this section, we review the definition of noncausal
LPTV scaling, and the fact shown in Hagiwara and Ohara(2010) that noncausal LPTV scaling induces some class ofdynamic causal LTI scaling in the lifting-free framework.

In the previous subsection, we reviewed the definitionsof causal LPTV scaling and its generalized form called
noncausal LPTV scaling. In this subsection, we further
review an existing result showing that noncausal LPTV
scaling with lifting treatment induces some dynamic causal
LTI scaling in the lifting-free framework even if the noncausal LPTV scaling is static. More precisely, we have thefollowing theorem [Hagiwara and Ohara (2010)].
bb 0 (z) satisfies (5) andTheorem 3. If a separator Θ(z)=Θ(6) in the N -lifted framework, then the separatorb 0 (ζ N )T (ζ)Θ(ζ) = T (ζ)∗ Θ(8)

3.1 Definition of noncausal LPTV scalingb
The separators Θ(z)in Theorem 2 have been classified into
two types (causal or noncausal type) from the causality
viewpoint. The noncausal type is more general and thus
effective than the causal type, and the present paper is
primarily concerned with the noncausal type. To makeclearer the feature of the robust stability analysis with
the noncausal type, however, we begin by reviewing thedefinitions of both types of separators, as well as theassociated two types of LPTV scaling. First, causal LPTV
separators are defined as follows [Hagiwara and Ohara(2010)].
Definition 1. A separator given by

∗ 

b
b Vb1 (z) Vb2 (z)Θ(z)= Vb1 (z) Vb2 (z) Λ(7)is called a causal LPTV separator, where Vb1 (z) and Vb2 (z)are the N -lifted transfer matrices of causal N -periodicb=Λb∗systems V1 and V2 with p inputs, respectively, and Λb = diag[Λ1 , · · · , ΛN ]is a constant matrix of the form Λwith the size of Λi being the same for all i = 1, · · · , N andcompatible with V1 and V2 . In particular, if V1 and V2 arestatic, then the corresponding separator is called a static
causal LPTV separator.

satisfies (2) and (3) in the lifting-free framework, where −(N−1) 
ζ
Ip


.
..


T (ζ) := diag[Tp (ζ), Tp (ζ)], Tp (ζ) := . (9)
 ζ −1 Ip 
Ipbb 0 (z)This theorem implies that if we obtain Θ(z)= Θsatisfying (5) and (6) in the lifted framework, then we immediately obtain Θ(ζ) satisfying (2) and (3). In particular,b 0 (z) is constrained to asince Theorem 3 is true even if Θb 0 ), even static noncausal
static one (i.e., constant matrix Θ
LPTV scaling induces some frequency-dependent scaling(i.e., dynamic causal LTI scaling) if it is interpreted in
the lifting-free framework. This is one of the advantages of
noncausal LPTV scaling, and demonstrated in Hagiwara
and Ohara (2010) and Hosoe and Hagiwara (2010b) to lead

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Milano (Italy) August 28 - September 2, 2011

to effectiveness for reducing the conservativeness of robust
stability analysis, and also for robust controller synthesis
in Hosoe and Hagiwara (2010a,c).

Lifting-free framework
How large gap, if any?

We have reviewed in the above an existing result clarifying
to some extent the relationship between noncausal LPTV
scaling and causal LTI scaling. However, having only sucha result is not yet enough as a study of the correspondence
relationship between these two types of scaling approaches.
Next section states detailed reasons why the above result
alone is not enough as a comparison of these scaling
approaches, and provides a further result about theirmutual relationship.

Θ(ζ)

4. RELATIONSHIP BETWEEN NONCAUSAL LPTVSCALING AND CAUSAL LTI SCALING
In this section, we first describe the issue to be clarified
further about the relationship between noncausal LPTV
scaling and causal LTI scaling. We then provide a certain
result for such an issue. More specifically, we derive as oneof the main results of this paper some class of dynamic
noncausal LPTV separators that has an ability equivalent
to a natural class (with a simple parametrization) of
causal LTI separators, where the latter contains the classof separators induced on the lifting-free framework byb static of static noncausal LPTV separators. A
the class Θtheoretical implication of such a result is also discussed.
4.1 Overview of the relationship given by existing result
In this paper, we study the relationship between noncausal
LPTV scaling and causal LTI scaling, and from such a
point of view, we reviewed Theorem 3 in the previous
section giving a partial result on the relationship. It showsthat noncausal LPTV scaling with lifting treatment induces some dynamic causal LTI scaling when it is interpreted in the lifting-free framework, even if the noncausal
LPTV scaling is static. However, it is not generally clear
if the inverse assertion of Theorem 3 is also true. That is,bb 0 satisfies (5) and (6)it is not clear if a static Θ(z)=Θ
whenever it together with (8) leads to Θ(ζ) satisfying (2)and (3). More precisely speaking, when we defineb static }b0 ∈ Θ
(10)
Θ(ζ) := {Θ(ζ) given by (8) Θ(which depends on N and corresponds to a subset of all
causal LTI separators), it is not clear whether we canreduce the problem of searching for Θ(ζ) ∈ Θ(ζ) equivalently into the problem of searching for static noncausalbb 0 . We give a schematic picture
LPTV separators Θ(z)=Θof the above relationship in Fig. 3, where the circles denotethe classes of separators we consider here for lifting-freeand lifted frameworks. If we could succeed in revealing
in more details the correspondence relationships between
both frameworks (e.g., what class of causal LTI scalingcorresponds to some class of noncausal LPTV scaling andvice versa), we would be able to understand more deeply
about the advantages and drawbacks of different types ofscaling approaches. Furthermore, making the relationships
clear is quite an important issue also for further development of the theory of noncausal LPTV scaling. Aiming at
taking a step forward to this issue, this paper introduces aclass of dynamic noncausal LPTV separators denoted byb
Θ(z),and shows that it has an equivalent ability to what

Lifted frameworkb static of
The class Θ
static noncausal LPTV
b0
separators Θ

Simple characterization
unknown
Fig. 3. Schematic picture of the correspondence relationships among different classes of separators
the class of causal LTI separators Θ(ζ) possesses in thelifting-free framework. We also discuss the properties ofb
the scaling with such Θ(z).
4.2 Equivalence relationship between some classes of
noncausal LPTV scaling and causal LTI scaling

As we have seen in the preceding discussions, even static
noncausal LPTV scaling generally induces dynamic (andthus frequency-dependent) causal LTI scaling if it is interpreted in the lifting-free framework. This might suggest
the use of static noncausal LPTV scaling as an alternative
tool for searching for dynamic scaling in the lifting-freeframework, and such an approach is indeed favorable dueto the ease in the search of static separators. However, evenb 0 induces
though the static noncausal LPTV separator ΘΘ(ζ) ∈ Θ(ζ) as seen from Theorem 3 and (10), it is notclear if such an approach is equivalent (i.e., not conservative) to the conventional lifting-free treatment with theseparator Θ(ζ) considered in the whole class Θ(ζ); see
Fig. 3. Stimulated by this question, we introduce in thissubsection a class of noncausal LPTV separators denotedbby Θ(z),and show that noncausal LPTV scaling withsuch a separator class is the one that has an equivalentability to the lifting-free LTI scaling with the separator
class Θ(ζ). The following theorem plays a crucial role forb
the construction of such Θ(z);the proof is omitted becauseof limited space.
b 0 , the separator
Theorem 4. Given a Hermitian matrix ΘΘ(ζ) given by (8) satisfies (2) and (3) if and only if theseparator given by
N −1
1 X l ∗b lb
Θ(z)=(S ) Θ0 S
(11)
N
l=0

satisfies (5) and (6) in the N -lifted framework, where S is
the z-dependent matrix given by0
z −1 Ip. (12)
S := diag[Sp , Sp ], Sp := Sp (z) =I(N −1)p 0

b
In view of (11), we define Θ(z)as follows.
b static }.
bb0 ∈ Θb
Θ(z):= {Θ(z)given by (11) Θ(13)Then, the aforementioned “equivalence” between Θ(ζ)
band Θ(z)follows immediately. A schematic picture of thesituation is given in Fig. 4. We remark that this figure,unlike Fig. 3, does not represent the classes of separators
but represents (the level of) the abilities of different types

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Lifting-free framework

5. FURTHER DISCUSSION ABOUT RELATIONSHIPBETWEEN THE TWO SCALING APPROACHES

Lifted framework
easier to characterize

Static noncausal
LPTV scaling
Causal LTI scaling
Noncausal LPTV scalingbwith Θ(ζ)
with Θ(z)Fig. 4. Schematic picture of the relationships among theabilities of different types of scaling approaches
of scaling approaches with the sizes of the circles; noteb static of static noncausal LPTV separatorsthat the class Θbis not a subset of Θ(z)but the ability of the former does
not exceed that of the latter, as indicated in the figure. Letb
us further observe a relationship between the classes Θ(z)band Θ static . As a direct consequence of Theorems 3 and4, we can see that if a static noncausal LPTV separatorbbb 0 satisfies (5) and (6), then Θ(z)
b
Θ(z)
= Θ
∈ Θ(z)bgiven by (11) with the same Θ0 also satisfies (5) and(6) again. However, it should be noted that it is notclear, in general, if the converse is true. That is, even ifbba Θ(z)∈ Θ(z)
satisfies (5) and (6), it is not clear if thesame inequalities hold also with any (or even a suitably
b 0 leading tochosen) static noncausal LPTV separator Θbthat Θ(z) through (8). This corresponds to the possibleb
gap between the scaling with the separator class Θ(z)andbthat with the class Θ static , as shown in Fig. 4 with respect
to the lifted framework. This gap corresponds to that (i.e.,
our initial concern in the lifting-free framework) in Fig. 3,
where the former gap is expected to be more tractable
than the latter because each of the corresponding classesof separators has been characterized explicitly in the liftedframework.
This consequence gives a partial answer to the unresolved
issues left behind Theorem 3 about the properties andability of static noncausal LPTV scaling. The significanceof Theorem 4 lies in leading to such a consequence, withwhich we can advance the study about the theoreticalaspects of noncausal LPTV scaling in the following section,and hopefully in more details in the future.
Remark 1. The sufficiency assertion of Theorem 4 could
be proved through direct but tedious computations bybapplying Theorem 3, i.e., by substituting Θ(z)given byb 0 (z) in (8). The necessity assertion, however,(11) into Θ
does not follow from Theorem 3, and thus this assertion
corresponds to the theoretical advance that Theorem 4 has
made over Theorem 3, as initially intended. Our omitted
proof is based on the properties of the transfer matrices of
N -lifted systems and the N th root of 1, and reversing thearguments leads to the sufficiency proof.
Remark 2. The assertion of Theorem 4 remains valid evenb 0 is replaced by a dynamic separator Θb 0 (z). However,if Θ
simply because the interest of this paper lies mostly in theb 0 has been assumed to beclass Θ(ζ) introduced in (10), Θ
static accordingly.

On the basis of Theorem 4, we have introduced in thebprevious section the class Θ(z)of noncausal LPTV separators that has a completely equivalent ability to theclass Θ(ζ) of causal LTI separators given by (8) withb static . This section first shows a structural property
b0 ∈ Θ
Θbbof the noncausal LPTV separators Θ(z)∈ Θ(z),
andthen further discusses the relationship between noncausal
LPTV scaling and causal LTI scaling. In particular, we
introduce some new classes of separators other than Θ(ζ)
band Θ(z)both in the lifting-free and lifted frameworks,
and demonstrate the importance of Theorem 4 in revealing
the mutual relationships among the scaling approacheswith such separator classes.
b
5.1 Interpretation of the separator class Θ(z)via timing
shift
In Subsection 3.2, we reviewed Theorem 3 about the
implication of (static) noncausal LPTV scaling interpreted
in the framework of conventional lifting-free treatment.
This led us to the introduction of the class Θ(ζ) of causal
LTI separators given by (10). In addition, by consideringba noncausal LPTV separator Θ(z)in the form of (11)binstead of the static noncausal LPTV separator Θ(z)=b 0 , we gave, as Theorem 4, a sort of converse assertion ofΘb
Theorem 3. This led to the introduction of the class Θ(z)of noncausal LPTV separators given in (13), as a class thatis equivalent to the class Θ(ζ) of causal LTI separators forthe lifting-free treatment. With this in mind, we study inthis subsection a structural property of noncausal LPTVbb
separators Θ(z)∈ Θ(z).
We begin with the properties of Sp in (12). It is relatedwith the timing shift matrix Zp introduced in Hosoe andHagiwara (2010b) by Sp = Zp−1 , and satisfies

Sp Sp∗ = Sp∗ Sp = I, SpN = z −1 IpN (z ∈ ∂D).
(14)
We call this Sp a timing back-shift matrix. Applying thesimilarity transformation by Sp on the transfer matrix of
the N -lifted LTI representation of an N -periodic systemcorresponds to lifting that N -periodic system with its
input and output signals advanced by a unit discrete-time,
and then introducing the transfer matrix of the resulting
lifted LTI system. Hence, such a similarity transformation
leads to a different lifted transfer matrix, in general.
However, if it is applied on the lifted transfer matricesof LTI systems such as G and ∆ in the present paper,then it induces no variations of the transfer matrices,
i.e., the transfer matrices remain invariant. More precisely,
−1 b
b
b
b
Sp−1 G(z)Sp = G(z) and Sp ∆(z)Sp = ∆(z).

From these relations, post-multiplying Sp and pre-multiplying Sp−1 = Sp∗ on (5) and (6) lead to the fact thatbb 0 (z) satisfies these inequalities if and only(a) Θ(z)= Θbb 0 (z)S satisfies the same inequalities. In
if Θ(z) = S ∗ Θ
b static satisfies (5) and (6)bb0 ∈ Θ
particular, (b) Θ(z)=Θbb0S
if and only if the timing-shifted separator Θ(z)= S ∗Θ
satisfies the same inequalities. Repeating this argument,

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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011b static satisfies (5) and (6) if andbb0 ∈ Θ
we see that Θ(z)=Θl∗bb 0 S l satisfies the same inequalities
only if Θ(z)= (S ) Θfor all l = 0, 1, · · · . However, considering all l = 0, 1, · · · isredundant if we note from (14) thatb 0 S l+N = (S l )∗ Θ
b 0 S l , z ∈ ∂D
(S l+N )∗ Θ
(15)

and it is enough to consider only l = 0, · · · , N − 1 in theabove observation. We can further see that the separatorbΘ(z)given in Theorem 4 is nothing but the average ofthese N timing-shifted separators. In particular, it follows
again from (15) that the average is such a special separator
that is invariant under the congruence transformation bybbS, i.e., S ∗ Θ(z)S
remains the same as Θ(z).
For simplicity,
we say that such separators are shift-invariant. It can beinterpreted that the success of Theorem 4 in providing asort of converse assertion of Theorem 3 stems from theintroduction of such a shift-invariant separator. Indeed, inb static has been considered in theb0 ∈ Θ
Theorem 3, only Θlifted framework, but it is not shift-invariant, in general,and thus the above observation (b) leads to a new but
equally effective timing-shifted separator. The price wehave paid to arrive at the above “converse” (to be able
to ensure the equivalence of the robust stability analysis
in the lifted framework and that with the separator classΘ(ζ) in the lifting-free framework) is that we are ledbto the “S-invariant” class Θ(z)of dynamic noncausal
b static . In this regard, it
LPTV separators instead of Θ
would be worth noting that, in spite of the existence ofb
S l , l = 0, · · · , N − 1 in (11), the class Θ(z)consists of
only such separators represented as linear combinationsof z i , i = −1, 0, 1. This is because Sp (z)l has the same
property as can be verified from the definition of Sp (seealso (14)).
5.2 Comparison of causal LTI and noncausal LPTV
scaling approaches based on equivalence relationshipIn the previous subsection, we briefly discussed the propb
erties of noncausal LPTV scaling based on Θ(z)fromthe viewpoint of timing shift introduced in Hosoe andHagiwara (2010b). In this subsection, using the structuralbproperty of Θ(z)given by (11), we discuss further relationships between noncausal LPTV scaling and causal LTI
scaling and clarify some relative advantages and drawbacksof the use of these approaches.
We first consider introducing some classes of separators inbaddition to Θ(ζ) and Θ(z).
We begin by introducing theclass of the separators given by
∗

 −1

z IN p b 2N z −1 IN p
b
,
(16)
Θ0,ij
Θ(z)
=
IN p
IN p
i,j=1,2b 2N
b 2N )i,j=1,2 ∈ Θ
b 2N = (ΘΘstatic0,ij02Nbwhere Θstatic is the class of static noncausal LPTV separators defined in the 2N -lifted framework (we consider
it even though we do remain at the moment in the N lifted framework; thus (16) is indeed a separator for the
N -lifted framework as seen from its size). Since everybb
Θ(z)∈ Θ(z)
is represented as a linear combination ofionly z , i = −1, 0, 1 from the definition of Sp (z), we canbsee that the class Θ(z)is included in the class of theLifting-free framework

Causal LPTV
scaling with Θ(ζ)

Lifted framework
Noncausal LPTV
b ext (z)scaling with ΘNoncausal LPTVbscaling with Θ(z)(16)

Causal LPTV
scaling with Θ νN (ζ)

Noncausal LPTVb νN (z)scaling with Θ
Fig. 5. Schematic picture of further relationships among
the abilities of other different types of scaling approaches
b ext (z).
separators given by (16), which we denote by ΘHence, it is obvious that the robust stability analysis based
b ext (z) is less (more precisely, aton the separator class Θbleast no more) conservative than that with Θ(z).
Then,
b
b
b
by regarding Θ(z) ∈ Θ ext (z) as Θ0 (z) in (8) and applying
Theorem 3, we can see that the noncausal LPTV scalingb ext (z) also induces some classwith the separator class Θof causal LTI scaling in the lifting-free framework (see thedashed circled in Fig. 5, which is again meant to represent
the abilities of different types of scaling approaches, as
in Fig. 4). By direct computation of (8) resulting fromthe above argument, we see that such an induced class is
included in the class of lifting-free causal LTI scaling with
Θ 2N (ζ), which is defined by (10) with the underlying N
replaced by 2N (this corresponds to the outermost circle
in the lifting-free framework with ν = 2). Moreover, byapplying Theorem 4, we can obtain the class of separatorsb 2N (z) (see the outermost circle in the lifted framework)Θsuch that the noncausal LPTV scaling with it has an equivalent ability to the causal LTI scaling with Θ 2N (ζ). Here,b 2N (z) is a class defined (andit should be noted that Θthus to be used) in the 2N -lifted framework. We remarkthat the above arguments are limited to the case of ν = 2,
for simplicity, but readily generalize to the arguments inthe νN -lifted framework. This situation is also shown in
Fig. 5.
To summarize, by applying Theorem 3 and 4, we can reveal
the equivalences and differences in the abilities of various
types of scaling approaches introduced here, in terms of theconservativeness of robust stability analysis. Aside fromsuch a theoretical aspect, however, the computation load isalso an important factor for the comparison of these scaling
approaches. We next discuss their relationships from sucha viewpoint, but for simplicity, we confine ourselves to theb static and Θb ext (z), andnoncausal LPTV scaling with Θ
2N
the causal LTI scaling with Θ(ζ) and Θ (ζ). Each of
the separator class above is parametrized by a constantmatrix, and as stated in Appendix A, we can have an exact
search method for such a constant matrix through the
KYP lemma [Rantzer (1996)], provided that we conform

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Table 2. Comparison of the analysis results(N = 2)

Table 1. Comparison of computation load
Scaling type

(i)

(ii)

(iii)

b static
noncausal, Θcausal, Θ(ζ)
b ext (z)noncausal, Θcausal, Θ 2N (ζ)

2pN
2pN
4pN
4pN

n
pN +n − ppN +n
2pN +n−ppN + n3pN +n−2p
4pN +n6pN +n−2p

to the standard technique in robust stability analysis withrespect to the treatment of the inequalities (3) and (6)about the uncertainties ∆. Such a technique leads to LMI
conditions with which the parameter matrix in each of theb static , Θ
b ext (z), Θ(ζ) and Θ 2N (ζ) can be searchedclasses Θfor.

Now, there are three main factors affecting the computab static (or Θb 2N )b0 ∈ Θ
b 2N ∈ Θtion load; (i) the size of Θ
static0
we essentially search for, (ii) the order of the “nominal
system” GT (ζ) that arises in the process of reduction to
LMI (see Appendix A), which determines the size of the
Lyapunov matrix in the resulting LMI, and (iii) the sizeof the resulting LMI itself. These three factors can bestudied easily, and can be summarized as shown in Table 1.
According to this table, we can see that a lower line hasa larger value in each of (i), (ii) and (iii) (since we may
reasonably assume N > 1) and thus a less conservative
approach (recall the relation in Fig. 5) is considered totend to take a more computation time. This suggests anatural consequence that, to obtain a sharper result inrobust stability analysis with whatever scaling approachin the lifting-free or lifted framework, it basically has tobe allowed for a more computation time.

In spite of the above observation, however, there clearly
exists an advantage in the idea of noncausal LPTV scaling.
Indeed, there is an important qualitative fact concealed behind the quantitative comparison in Table 1 but suggesting
the effectiveness of the idea of noncausal LPTV scalingeven in the search for the separators in the classes Θ(ζ)
and Θ 2N (ζ) for causal LTI scaling. Such an important
fact can be described as follows. In causal LTI scaling,Θ(ζ) should satisfy (3), and thus the parameter matrix
b 2N in Θ(ζ) must be confinedb static or Θ
b0 ∈ Θ
b 2N ∈ ΘΘstatic0
to a convex set such that (3) is satisfied regardless of∆ ∈ ∆; otherwise, the LMI condition derived from (2)b0through the KYP lemma cannot be solved easily for Θ2Nbor Θ0 . Construction of such a convex set for the liftingfree framework is actually deeply related to the seemingly
irrelevant noncausal LPTV scaling approach in the liftedframework, and can be carried out easily with the ideaof noncausal LPTV scaling (see Appendix A for moredetails).
As we have seen in the above, we can explicitly compare avariety of scaling approaches in both lifting-free and liftedframeworks by making use of Theorem 4. This clearly
demonstrates the significance of Theorem 4 as our main
result in this paper.
5.3 Numerical example of robust stability analysis
This subsection numerically confirms the facts stated in
the previous subsection, i.e., the relationship among thedifferent types of scaling approaches in terms of their

Scaling typeδComputation timeb static
noncausal, Θcausal, Θ(ζ)
b ext (z)noncausal, Θcausal, Θ 2N (ζ)

1.23601.49381.4942
1.49424.81 sec
6.41 sec
8.21 sec
13.94 sec

conservativeness as shown in Fig. 5, and that in termsof their computation load as discussed in Table 1.
We consider the internally stable LTI system G given by


0
1
0 0
0
0 0
 0
0
1 0
0
0 0 

 

0
0 1
0
0 0 
 0
A B
 0
0
0 0
1
0 0.1 =
. (17)C D
 −0.2 −0.62 0.01 0.6 −0.7 0.1 0 


 1
1
1 0
0 0.1 0.2 
0
0
0 1
1
0 0
In addition, we assume the corresponding uncertainties ∆are static LTI and structured as given by ∆ = diag[δ1 δ2 ].
The purpose here is to compute (a lower bound of) the
maximum δ such that the closed-loop system Σ is robustly
stable against the uncertainty set ∆ = {∆ : k∆k <δ}. More precisely, we apply the four scaling approaches
in Table 1 and compare the analysis results and thecomputation times.
The results for N = 2 are shown in Table 2, where
we confined ourselves to the class of (D, G)-scaling typeseparators [Fan et al. (1991)] in each of those approaches.
Since larger resulting δ implies that less conservative
analysis has been achieved, Table 2 shows that a scaling
approach in a lower row leads to less conservative robust
stability analysis. We see that this is indeed consistent withthe discussions in the preceding subsection (see Fig. 5).
At the price for the improvement of the accuracy in theanalysis, we also see that the computation times grow.
We see that this is also consistent with the observation in
Table 1.
6. CONCLUSION
In this paper, we aimed at revealing some aspect of
the relationship between noncausal LPTV scaling in thelifted framework and causal LTI scaling in the lifting-freeframework. On the basis of an existing result showingthat even static noncausal LPTV scaling generally induces
dynamic (i.e., frequency-dependent) scaling in the liftingfree framework, we first introduced the class Θ(ζ) of causal
LTI separators in the lifting-free framework such thatthe induced equivalent dynamic scaling can be related
to a separator Θ(ζ) ∈ Θ(ζ). We then introduced thebclass Θ(z)of noncausal LPTV separators in the liftedframework and proved that noncausal LPTV scaling withb
Θ(z)has an equivalent ability to causal LTI scaling with
Θ(ζ) in robust stability analysis. Moreover, based on sucha newly derived equivalence relationship, we explicitly
compared some classes of causal LTI and noncausal LPTV
scaling approaches in terms of the conservativeness ofrobust stability analysis and the computation load. The
validity of such discussions has been confirmed by a3390

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numerical example. In addition, we also discussed an
important issue about the effectiveness of the idea of
noncausal LPTV scaling even in the lifting-free frameworkwith causal LTI separators, particularly in relation withthe ease in the treatment of the uncertainties ∆ and theconstruction of an appropriate separator class in such aframework.
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ζ −1 Ip
..
.









−(N−1)


I
Ip
ζ
I
p
0
p 
b

Θ0
≤ 0, GT (ζ) =  (A.1)GT (ζ)GT (ζ)
G(ζ)




..


.
−(N−1)
ζG(ζ)
by introducing the permutation matrix P such thatb0 =
[Tp (ζ)T , (Tp (ζ)G(ζ))T ]T = P [Ip , GT (ζ)T ]T , where Θ0
T b
P Θ0 P . Hence, we can regard the problem stated aboveb 0 for GT (ζ) viewed as theas that of searching for static Θ0nominal system. Such a problem can be solved exactly
through the well-known KYP lemma [Rantzer (1996)] asb 0 ) is handledb 0 (or Θlong as the other constraint (3) on Θ
0properly. Conforming to the standard technique in robust
b 0 to astability analysis, it is reasonable for us to restrict Θbbsubset Θ static,∆ of Θ static , where such a subset is requiredbb0 ∈ Θ
to have the property that every Θstatic,∆ together
with (8) yields Θ(ζ) that satisfies (3) for any ∆ ∈ ∆.



∗





Construction of such a subset might look an intricate
problem particularly because we must deal with the dynamic separator Θ(ζ) and thus (3) does not reduce to astatic inequality even if we were to consider only static
uncertainties ∆. Nevertheless, such a subset can actually
be constructed rather easily, and Theorem 3 plays a significant role in the construction; it follows immediately fromb
this theorem that we may take Θ
static,∆ to be a subsetbbbbof Θ static such that Θ(z) = Θ0 ∈ Θstatic,∆ satisfies (6).
Such a subset has been introduced in Hosoe and Hagiwara(2010a) and Hosoe and Hagiwara (2010c) in an explicit
form, in accordance with the block diagonal structure of∆ consisting not only of static but also dynamic subblocks. The construction of such a class viewed in the liftedframework is in fact fairly easy as an extension of similar
techniques in µ-analysis [Zhou and Doyle (1998)]. Thisimplies an important fact that the idea of noncausal LPTV
scaling is quite useful even if we were to carry out robust
stability analysis within the lifting-free framework throughthe separator class Θ(ζ), particularly with respect to thetreatment of the uncertainties ∆ and thus the inequality(3).
It is obvious that we can also search for dynamic separatorsb
Θ(z)given by (11) or (16) (more precisely, the constantb 0 or Θ
b 2N ) basically with the same idea, if we notematrix Θ
0that (11) is a special case of (16) as stated in Subsection 5.2
and that (16) can be regarded as a special case of (8) with
N set to 2 and ζ replaced by z.

Appendix A. A KEY IDEA FOR THE SEARCH OF
DYNAMIC SEPARATORS SATISFYING ROBUSTSTABILITY CONDITIONS VIA KYP LEMMA
b static such thatb0 ∈ Θ
Suppose that we are to search for ΘΘ(ζ) given by (8) satisfies (2) and (3). Substituting (8)into (2) leads to3391