Conditions for the equivalence between IQC and graph separation stability results
CConditions for the equivalence between IQC and graph separationstability results
Joaquin Carrasco a and Peter Seiler b a Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, Manchester M13 9PL, UK b Aerospace Engineering and Mechanics Department, University of Minnesota, 107 Akerman Hall, 110 Union St. SE Minneapolis, MN55455-0153
Abstract
This paper provides a link between time-domain and frequency-domain stability results in the literature. Specifically, we focus onthe comparison between stability results for a feedback interconnection of two nonlinear systems stated in terms of frequency-domainconditions. While the Integral Quadratic Constrain (IQC) theorem can cope with them via a homotopy argument for the Lurye problem,graph separation results require the transformation of the frequency-domain conditions into truncated time-domain conditions. To date,much of the literature focuses on “hard” factorizations of the multiplier, considering only one of the two frequency-domain conditions.Here it is shown that a symmetric, “doubly-hard” factorization is required to convert both frequency-domain conditions into truncatedtime-domain conditions. By using the appropriate factorization, a novel comparison between the results obtained by IQC and separationtheories is then provided. As a result, we identify under what conditions the IQC theorem may provide some advantage.
Classical multiplier theory is a well known technique toreduce the conservatism of absolute stability criteria (Zamesand Falb, 1968; Desoer and Vidyasagar, 1975). Frequency-domain and time-domain conditions are combined, and thecanonical factorization of the multiplier is the essential toolto ensure that time-domain properties can be recovered fromthe frequency-domain conditions (J¨onsson, 1996; Goh andSafonov, 1995; Goh, 1996; Carrasco et al., 2012).The IQC theorem by Megretski and Rantzer (1997) usesonly frequency-domain inequalities and provides a shortcutto avoid conditions on the existence of factorizations byusing a homotopy argument in their proof. However theoriginal IQC framework was developed using time-domainconstraints by Yakubovich (1965, 1967, 1971), so Megretskiand Rantzer (1997) have coined the terms soft and hard
IQC to establish the connection between their IQC theoremand Yakubovich’s work. Loosely speaking: Email addresses: [email protected] (JoaquinCarrasco), [email protected] (Peter Seiler). It has been shown in (Seiler et al., 2010; Seiler, 2015) that thesame IQC can be either hard or soft depending on the factorizationused to convert from frequency to time-domain; therefore the termshard and soft factorizations terminology must be introduced. • an IQC is hard when the time-domain version of the con-straint holds for any finite time interval [ , T ] ; • an IQC is soft when the time-domain version of the con-straint holds for the interval [ , ∞ ) but need not be satis-fied on finite time intervals.It may appear that a hard factorization is equivalent to thecanonical factorization in the classical multiplier theory. Inother words, one may think that a hard factorization ofan IQC is sufficient to convert frequency-domain stabil-ity conditions to equivalent time-domain conditions (Goh,1996; Seiler et al., 2010). However, it has been shown thathard factorizations are not enough to establish such equiva-lence (Veenman and Scherer, 2013; Seiler, 2015). The equiv-alence between IQC and the so-called dissipative inequalityis shown in Seiler (2015). The term hard factorization is stillused, and then an extra condition is imposed on the solutionof an LMI involving the LTI system.The graph separation framework (Safonov, 1980; Teel, 1996;Georgiou and Smith, 1997) can be seen as a generalisationof the classical multiplier theory and uses truncated time-domain conditions to obtain stability result. Recently, Car-rasco and Seiler (2015) have shown that it is possible to es-tablish a counterpart of the IQC theorem using the graph sep-aration framework. However, they rely on results in (Seiler,2015) and require one of the two systems in the intercon-nection to be LTI. Preprint submitted to Automatica 21 September 2018 a r X i v : . [ c s . S Y ] A p r (cid:54) (cid:45) (cid:27) r y ∆ G (cid:106) u (cid:106) (cid:27) r u y Fig. 1. Lurye problem
This paper builds on the results presented in (Seiler, 2015)and Carrasco and Seiler (2015). The main contribution ofthis paper is the development of the counterpart of Lemma2 in (Seiler, 2015). With this new result, we can establishthe equivalence between frequency-domain conditions andtruncated time-domain conditions from a pure input-outputpoint of view, without involving LMIs; hence the definitionof the factorization does not require one of the systems tobe LTI. This approach provides new insights, in particular,we are able to establish a formal comparison between stabil-ity results using IQC and graph separation theories for thefeedback interconnection of two nonlinear systems.The structure of the paper is as follows. Sections 2 and 3 pro-vides the IQC theorem and discusses classical hard and softfactorizations as defined by Megretski and Rantzer (1997).Section 4 states a new factorization, the so-called doubly-hard factorization, and characterises this factorization for aclass of multipliers. Section 5 demonstrates that not all hardfactorizations are doubly-hard factorizations. Section 6 de-velops two results for the stability of the feedback intercon-nection of two systems, one using the IQC theorem, andanother using the graph separation result by Teel (1996). Fi-nally, Section 7 gives the conclusions of the paper. We usethe same notation as in (Megretski and Rantzer, 1997).
Definitions and results related with the IQC framework aregiven in this section.
Definition 1
A stable and causal system ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) is said to satisfy the IQC defined by a bounded,measurable Hermitian-valued function Π : j R → C ( m + l ) × ( m + l ) if (cid:90) ∞ − ∞ (cid:34) (cid:98) u ( j ω ) (cid:99) ∆ u ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34) (cid:98) u ( j ω ) (cid:99) ∆ u ( j ω ) (cid:35) d ω ≥ , (1) for any u ∈ L m [ , ∞ ) . Theorem 1 (IQC theorem (Megretski and Rantzer, 1997))
Let G ∈ RH m × l ∞ , let ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) be a boundedcausal operator, and let Π : j R → C ( m + l ) × ( m + l ) be abounded, measurable Hermitian-valued function. Assumethat:(1) for every τ ∈ [ , ] , the interconnection of G and τ ∆ iswell-posed;(2) for every τ ∈ [ , ] , the IQC defined by Π is satisfiedby τ ∆ ; (3) there exists ε > such that (cid:34) G ( j ω ) I (cid:35) ∗ Π ( j ω ) (cid:34) G ( j ω ) I (cid:35) < − ε I ∀ ω ∈ R . (2) Then, the feedback interconnection of G and ∆ in Fig. 1 isstable. The multiplier Π is normally defined as a block 2-by-2 ma-trix, i.e. Π = (cid:34) Π Π Π Π (cid:35) . (3)where Π is m × m and Π is l × l . Then Π ( j ω ) is calleda positive-negative multiplier if there exists ε > Π ( j ω ) ≥ ε I m and Π ( j ω ) ≤ − ε I l ∀ ω ∈ R .In this note we restrict our attention to positive-negativerational multipliers Π ∈ RL ( m + l ) × ( m + l ) ∞ . The IQC in Equation 1 can be expressed in the time-domainand this leads to a characterization of the IQC as soft or hard.Specifically, let Π ( j ω ) = Ψ (cid:62) ( − j ω ) M Ψ ( j ω ) where Ψ is acausal and stable transfer function. Such factorizations arenot unique but can be computed with state-space methods(Scherer and Weiland, 2000). With some abuse of notationwe will use the same notation for the transfer function and itscorresponding stable operator. The IQC-factorization ( Ψ , M ) is said to be soft if (cid:90) ∞ (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) dt ≥ , (4)for any u ∈ L m [ , ∞ ) . The frequency-domain constraintof Inequality 1 implies the time-domain soft constraint ofInequality 4 by Parseval’s theorem. The factorization is saidto be hard if (cid:90) T (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) dt ≥ , (5)for any u ∈ L m e [ , ∞ ) and any T >
0. This condition for ahard factorization is more restrictive. Specifically, all factor-izations of Π are soft but only certain factorizations are hard.It is now clear that the factorization step, i.e. Π = Ψ ∼ M Ψ , is Note that the dependence of time has been suppressed inEquation 4 for simplicity. More precisely, this soft IQC is (cid:82) ∞ y (cid:62) ( t ) My ( t ) dt ≥ y : = Ψ [ u ∆ u ] . The time dependence willsimilarly be dropped in other time-domain IQCs. Π can have hard and soft factoriza-tions. These are called ( Ψ , M ) -hard and ( Ψ , M ) -soft factor-izations (Seiler et al., 2010; Seiler, 2015). The terms com-plete and conditional IQCs by Megretski (2010) are gener-alizations of hard and soft IQCs. The hard/soft terminologywill be used here.There are simple sufficient conditions for the existence of ahard factorization (Goh, 1996). For positive-negative mul-tipliers, it is always possible to find a hard factorization ( Ψ , J m , l ) : Ψ = (cid:34) Ψ Ψ Ψ (cid:35) and J m , l = (cid:34) I m − I l (cid:35) (6)where Ψ , Ψ − , Ψ are stable rational transfer functions.This ensures that the truncation of the IQC will preserve itssign (Goh, 1996). This fact is shown via a simple argumentas (cid:90) ∞ (cid:34) Ψ u ( Ψ + Ψ ∆ ) u (cid:35) (cid:62) J m , l (cid:34) Ψ u ( Ψ + Ψ ∆ ) u (cid:35) dt = (cid:107) Ψ u (cid:107) − (cid:107) ( Ψ + Ψ ∆ ) u (cid:107) ≥ . (7)Any input u can be truncated on [ , T ] and extended overthe positive real line by selecting an artificial input z ([ T , ∞ )) such that Ψ ˜ u ( t ) = t > T where the piecewise input˜ u is defined by ˜ u ( t ) = (cid:26) u ( t ) if t ≤ T , z ( t ) if t > T . (8)The pair ( ˜ u , ∆ ˜ u ) satisfies the infinite horizon constraint ofInequality 7. Hence, by construction, the pair ( u , ∆ u ) satisfiesthe constraint over the finite horizon [ , T ] . The key point inthis construction is the stability of Ψ − since it ensures thatthe artificial input ˜ u belongs to L and (cid:107) Ψ u (cid:107) T = (cid:107) Ψ ˜ u (cid:107) T .It may initially appear that this truncation is sufficient tocomplete a dissipativity (or graph separation) proof for sta-bility. However, the role of the second IQC condition seemsunderappreciated in the literature. Specifically Equation 2 inthe IQC theorem is equivalent to the following second time-domain IQC condition (because both G and Ψ are LTI): (cid:90) ∞ (cid:32) Ψ (cid:34) Guu (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) Guu (cid:35)(cid:33) dt < − ε (cid:107) u (cid:107) . (9)All operators in this IQC condition are stable LTI systemsand hence the condition can be checked via an equivalentfrequency-domain condition. However, this does not implythat the sign of this inequality will be preserved under finite-horizon truncations in the time-domain. In particular, the key difficulty is observed if we use the triangular factorizationalong with the truncation arguments introduced above: − (cid:90) ∞ (cid:34) Ψ Gu ( Ψ G + Ψ ) u (cid:35) (cid:62) J m , l (cid:34) Ψ Gu ( Ψ G + Ψ ) u (cid:35) dt = (cid:107) ( Ψ G + Ψ ) u (cid:107) − (cid:107) Ψ Gu (cid:107) > ε (cid:107) u (cid:107) . (10)We now see the difficulties in creating an extension of the in-put once a truncation u ([ , T ]) has been selected. The exten-sion of the piecewise input on [ T , ∞ ) must cancel ( Ψ G + Ψ ) ˜ u for any time after the truncation. It may be possi-ble in some cases, but in general this leads to piecewise˜ u (cid:54)∈ L [ , ∞ ) since Ψ − is not stable. This problem is linkedto the well known difficulties of applying feedback lineari-sation to non-minimum phase systems (Isidori, 2013). It is possible to show that positive-negative multipliers havea more useful factorization for the purposes of stability anal-ysis. It is shown by Seiler (2015) that J-spectral factoriza-tions can be constructed for positive-negative multipliers, i.e. Π ( j ω ) = Ψ (cid:62) ( − j ω ) J m , l Ψ ( j ω ) where Ψ and Ψ − are bothstable transfer functions. Moreover, this factorization allowsus to ensure that the signs of both IQCs are preserved undertruncation.To the best of our knowledge this duality property of thefactorization has been overlooked. The argument by Seiler(2015), where the J -spectral factorization is given, was basedon storage function and dissipativity arguments. The factor-ization there was still referred to as a hard factorization witha focus on the IQC condition for ∆ , but the second condi-tion was established in terms of the resulting LMI. Here wepropose a more symmetric and convenient definition wherewe do not require the construction of the LMI, so we areable to establish the properties of the factorization withoutinvoking the linearity of one of the systems.In the graph framework, it is standard to use the graph andthe inverse graph (Safonov, 1980; Teel, 1996). The standardIQC notation uses the graph of the system ∆ . To develop asymmetric formulation, we define the IQC over the inversegraph, henceforward “inverse-graph IQC” as follows: Definition 2 (Inverse-graph IQC)
A stable and causalsystem ∆ : L l e [ , ∞ ) → L m e [ , ∞ ) is said to strictly satisfythe inverse-graph IQC defined by a bounded, measurableHermitian-valued function Π : j R → C ( m + l ) × ( m + l ) if thereexists ε > such that (cid:90) ∞ − ∞ (cid:34)(cid:99) ∆ u ( j ω ) (cid:98) u ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34)(cid:99) ∆ u ( j ω ) (cid:98) u ( j ω ) (cid:35) d ω ≤ − ε (cid:107) u (cid:107) , (11) for any u ∈ L l [ , ∞ ) . ∆ is linear, then (11) is equivalent to (2) by using ∆ insteadof G .Then we can state the definition of the factorization whichwill lead to an equivalence between frequency-domain con-ditions and truncated time-domain conditions: Definition 3 (Doubly-hard factorization)
For a given Π : j R → C ( m + l ) × ( m + l ) , a factorization ( Ψ , M ) is said to be a doubly-hard IQC factorization if the following two condi-tions hold:(1) for any bounded and causal ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) , the IQC condition (cid:90) ∞ − ∞ (cid:34) (cid:98) u ( j ω ) (cid:100) ∆ u ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34) (cid:98) u ( j ω ) (cid:100) ∆ u ( j ω ) (cid:35) d ω ≥ , (12) for all u ∈ L m [ , ∞ ) implies that (cid:90) T (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) dt ≥ . (13) for any u ∈ L m e [ , ∞ ) and any T > , and(2) for any bounded and causal ∆ : L l e [ , ∞ ) → L m e [ , ∞ ) , the inverse-graph IQC condition (cid:90) ∞ − ∞ (cid:34)(cid:100) ∆ u ( j ω ) (cid:98) u ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34)(cid:100) ∆ u ( j ω ) (cid:98) u ( j ω ) (cid:35) d ω ≤ − ε (cid:107) u (cid:107) , (14) for all u ∈ L l [ , ∞ ) implies that (cid:90) T (cid:32) Ψ (cid:34) ∆ uu (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) ∆ uu (cid:35)(cid:33) dt ≤ − ε (cid:107) u (cid:107) T , (15) for any u ∈ L l e [ , ∞ ) and any T > . Finally, we show that the key property to obtain a doubly-hard factorization is the stability of both Ψ and Ψ − . Thisresult requires Lemma 2 in Seiler (2015), and the develop-ment of a result for the inverse-graph condition (14).Let Ψ ∼ A B v B w C D v D w . (16)Define the functional J on v ∈ L [ , ∞ ) , w ∈ L [ , ∞ ) and x ∈ R n as J ( v , w , x ) = (cid:90) ∞ z ( t ) (cid:62) Mz ( t ) dt (17) subject to˙ x ( t ) = Ax ( t ) + B v v ( t ) + B w w ( t ) , x ( ) = x ; z ( t ) = Cx ( t ) + D v v ( t ) + D w w ( t ) . Define the upper value ¯ J ( x ) as¯ J ( x ) : = inf v ∈ L [ , ∞ ) sup w ∈ L [ , ∞ ) J ( v , w , x ) , and the lower value J ( x ) asJ ( x ) : = sup w ∈ L [ , ∞ ) inf v ∈ L [ , ∞ ) J ( v , w , x ) . Lemma 1 (Seiler (2015))
Let Π be a multiplier and ( Ψ , M ) any factorization with Ψ stable. Assume ∆ is a causalbounded operator such that (cid:90) ∞ − ∞ (cid:34) (cid:98) v ( j ω ) (cid:98) w ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34) (cid:98) v ( j ω ) (cid:98) w ( j ω ) (cid:35) d ω ≥ , (18) for any v ∈ L [ , ∞ ) and w = ∆ v. Then for all T ≥ ,for all v ∈ L [ , ∞ ) , and w = ∆ v, the signal defined byz = Ψ ( j ω ) (cid:34) (cid:98) v ( j ω ) (cid:98) w ( j ω ) (cid:35) satisfies (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≥ − ¯ J ( x ( T )) , (19) where x ( T ) denotes the state of the system Ψ at the instant Twhen driven by the inputs ( v , w ) with null initial conditions. Lemma 2
Let Π be a multiplier and ( Ψ , M ) any factoriza-tion with Ψ stable. Assume ∆ is a causal bounded operatorsuch that (cid:90) ∞ − ∞ (cid:34) (cid:98) v ( j ω ) (cid:98) w ( j ω ) (cid:35) ∗ Π ( j ω ) (cid:34) (cid:98) v ( j ω ) (cid:98) w ( j ω ) (cid:35) d ω ≤ − ε ( (cid:107) v (cid:107) + (cid:107) w (cid:107) ) , (20) for any w ∈ L [ , ∞ ) and v = ∆ w. Then for all T ≥ , for allw ∈ L [ , ∞ ) , and v = ∆ w, the signal defined by z = Ψ (cid:34) vw (cid:35) satisfies (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:107) z (cid:107) T − J ( x ( T )) , (21) where x ( T ) denotes the state of the system Ψ at the instant Twhen driven by the inputs ( v , w ) with null initial conditions. Proof:
See Appendix. (cid:50) heorem 2 Given a positive-negative multiplier Π ∈ RL ∞ ,the factorization ( Ψ , M ) is doubly-hard if Ψ , Ψ − ∈ RH ∞ . Proof: If Ψ ∈ RH ∞ , Lemma 1 and Lemma 2 hold. More-over, if the multiplier Π is positive-negative and Ψ − ∈ RH ∞ then ¯ J ( x ) = J ( x ) = x ∈ R n (see Lemma 5in (Seiler, 2015)). As a result the factorization ( Ψ , M ) isdoubly-hard. (cid:50) Therefore all J -spectral factorizations are doubly-hard fac-torizations. vs J -spectral factorization This section provides a simple example highlighting thedistinction between triangular and J -spectral factorizations.Consider a simple feedback interconnection of the static sys-tem G = and an operator ∆ . Define an positive-negativemultiplier Π ∈ RL × ∞ by Π ( s ) = (cid:34) − s + s + − s − s − − s + s − (cid:35) (22)Assume the interconnection of G and τ ∆ is well-posed forall τ ∈ [ , ] . Also assume that τ ∆ satisfies the IQC definedby Π for all τ ∈ [ , ] . It can be verified that (cid:2) G (cid:3) ∼ Π (cid:2) G (cid:3) = − . <
0, i.e. G satisfies the IQC constraint with Π . Thusthe frequency domain IQC conditions in Theorem 1 are sat-isfied and the feedback interconnection is stable.As noted above, the factorization of Π is not unique. Herewe construct two different factorizations. First, a stable tri-angular factorization ( Ψ , M ) of Π is given by: M = (cid:2) − (cid:3) and Ψ = (cid:104) s − s + (cid:105) . (23)Note that Ψ is stable but the (2,2) entry of Ψ is non-minimumphase. The multiplier satisfies the positive-negative condi-tions and hence it also has a J -spectral factorization ( ˜ Ψ , ˜ J ) :˜ J = (cid:2) − (cid:3) and ˜ Ψ = (cid:20) − . . s − . s + − . − . s − . s + (cid:21) . (24)Note that for this factorization ˜ Ψ and ˜ Ψ − are both stable.Figure 2 shows the IQC evaluated on [ , T ] versus the finitehorizon time T for the input signal u ( t ) = .
458 sin ( t ) for t ∈ [ , ] and u ( t ) = .
458 isselected to normalize the signal (cid:107) u (cid:107) =
1. As t → ∞ , bothIQCs converge to − .
25. This value is consistent with theconstraint (cid:2) G (cid:3) ∼ Π (cid:2) G (cid:3) = − . <
0. Thus both factoriza-tions satisfy the time-domain constraint as t → ∞ . However,the lower triangular factorization goes positive on the ap-proximate interval [ , . ] . Thus the lower triangular factor-ization can violate the constraint over finite horizons. On I Q C on [ , T ] Lower TriangJ−spectral
Fig. 2. (cid:82) T (cid:0) Ψ (cid:2) G (cid:3) u (cid:1) T M (cid:0) Ψ (cid:2) G (cid:3) u (cid:1) dt versus time T . the other hand, the J -spectral factorization remains negativeand hence satisfies the constraint over all finite horizons.It can be shown that lower triangular factorizations have a ( , ) entry that is non-minimum phase in general. Specifi-cally, if Ψ is lower triangular and Π = Ψ ∼ J Ψ then the en-tries of Ψ satisfy: Π = Ψ ∼ Ψ − Ψ ∼ Ψ Π = − Ψ ∼ Ψ Π = − Ψ ∼ Ψ These conditions imply that if Π has poles in the left halfplane then Ψ must be non-minimum phase. Specifically,if Π is a positive-negative multiplier then there exists ε > − Π ( j ω ) ≥ ε I ∀ ω ∈ R . Hence it can be factorizedas − Π = H ∼ H where H ∈ RH and H − is anti-stable. Inother words H is stable and anti-minimum phase. This fac-torization can be constructed from the normal stable, min-imum phase spectral factorization (Youla, 1961). Next, let { p i } Ni = denote the poles of Π in the left half plane. Define Ψ and Ψ as Ψ ( s ) : = H ( s ) (cid:32) N ∏ i = s + ¯ p i s − p i · I m (cid:33) (25) Ψ ( s ) : = (cid:0) − Π ( s ) Ψ − ( s ) (cid:1) ∼ (26)By construction, Ψ is stable and anti-minimum phase. Theinclusion of the Blaschke products in the definition of Ψ does not impact the value of Ψ ∼ Ψ on the imaginaryaxis. Thus Π = Ψ ∼ Ψ on the imaginary axis by con-struction of H . This choice of Ψ is required to ensure that Π Ψ − is anti-stable and hence Ψ is stable. Moreover, Ψ ∼ Ψ = − Π . A stable, stably invertible Ψ can then beconstructed from a spectral factorization of Π + Ψ ∼ Ψ . See (Partington, 2004) for a definition.
5n this construction, any LHP poles of Π appear as RHPzeros in Ψ . In this section we develop two stability results for the feed-back interconnection of two nonlinear systems. One of theseresults will be obtained using graph separation methods.For completeness, we state an IQC version of Corollary 5.1in (Teel, 1996) as follows:
Theorem 3 (Teel (1996))
Let ∆ and ∆ be two causal andbounded systems. Let Ψ be a stable linear system. Assumethat:(1) the feedback interconnection of G and ∆ is well-posed;(2) the time-domain IQC (cid:90) T (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) u ∆ u (cid:35)(cid:33) dt ≥ , (27) is satisfied for any T > and u ∈ L e [ , ∞ ) ;(3) the time-domain inverse-graph IQC (cid:90) T (cid:32) Ψ (cid:34) ∆ uu (cid:35)(cid:33) (cid:62) M (cid:32) Ψ (cid:34) ∆ uu (cid:35)(cid:33) dt < − ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:34) ∆ uu (cid:35)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T , (28) is satisfied for any T > and u ∈ L e [ , ∞ ) .Then the feedback interconnection between ∆ and ∆ is L -stable. In the spirit of J¨onsson (2011), we can establish the followingcorollary for the interconnection of two nonlinear systems:
Corollary 1 (Corollary of Theorem 1)
Let ∆ : L l e [ , ∞ ) → L m e [ , ∞ ) and ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) be boundedcausal operators, and let Π ∈ RL ( m + l ) × ( m + l ) ∞ . Assume that:(I) for every τ ∈ [ , ] , the feedback interconnection of τ ∆ and τ ∆ is well-posed;(II) for every τ ∈ [ , ] , τ ∆ satisfies the IQC defined by Π ;(III) for every τ ∈ [ , ] , τ ∆ strictly satisfies the inverse-graph IQC defined by Π .Then, the feedback interconnection of ∆ and ∆ is stable. Proof:
The result follows from the application of the IQCtheorem using ∆ = (cid:34) ∆ ∆ (cid:35) and G = (cid:34) II (cid:35) , (29) and the following augmented multiplier: Π a = Π Π − Π − ε I − Π ∗ Π ∗ Π − Π − Π ∗ − ε I (30)Some straightforward algebra is required to show that theconditions in Theorem 1 are satisfied. (cid:50) Using Theorem 2, then it is possible to remove the homotopycondition in the above result if the matrix Π is positive-negative. Formally we can state the following result: Corollary 2 (Corollary of Theorem 3)
Let ∆ : L l e [ , ∞ ) → L m e [ , ∞ ) and ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) be boundedcausal operators, and let Π ∈ RL ( m + l ) × ( m + l ) ∞ . Assume that:(i) the feedback interconnection of ∆ and ∆ is well-posed;(ii) ∆ satisfies the IQC defined by Π ;(iii) ∆ strictly satisfies the inverse-graph IQC defined by Π ;(iv) Π is a positive-negative multiplier.Then, the feedback interconnection of ∆ and ∆ is stable. Proof: If Π is a positive-negative multiplier, then there ex-ists a factorization ( Ψ , M ) such that Ψ and Ψ − are bothstable (Seiler, 2015). Therefore the factorization ( Ψ , M ) isdoubly-hard as it satisfies the conditions in Theorem 2.The frequency-domain conditions (ii) and (iii) can be trans-formed into truncated time-domain conditions by using thefactorization ( Ψ , M ) . As a result, Theorem 3 can be used toestablish the stability of feedback interconnection between ∆ and ∆ . (cid:50) Remark 4
It would not be possible to prove Corollary 2 byusing triangular factorizations as it fails to guarantee thatcondition (iii) is equivalent to the truncated time-domaincondition (28).6.2 Discussion
A na¨ıve comparison of the results would suggest that condi-tion (iv) in Corollary 2 an extra condition over the conditionsof Corollary 1. It is well known that the homotopy conditionin (II) is satisfied if Π is positive. Similarly, the homotopycondition in (III) is satisfied if Π is negative. Hence onecan think of a superiority of Corollary 1 over Corollary 2.However, if ∆ and ∆ are both nonlinear, the IQC theoremrequires homotopy conditions for both systems. If condition(II) holds, the requirement of the condition to be true when τ = Π ( j ω ) ≥ ω ∈ R . Similarly, if con-dition (III) holds, the same argument when τ = Π ( j ω ) ≤ − ε I for some ε > Π ( j ω ) ≥ Π with ¯ Π ( j ω ) ≥ δ I for some δ >
0, hence ¯ Π can be factorised: Lemma 3
Let G ∈ RH m × l ∞ , let ∆ : L m e [ , ∞ ) → L l e [ , ∞ ) be a bounded causal operator. If conditions (2) and (3) inTheorem 1 are satisfied for some Π , then there exists some δ > such that conditions (2) and (3) are satisfied for ¯ Π = (cid:34) Π + δ I m Π Π Π (cid:35) . Proof:
See Appendix. (cid:50)
Remark 5
The counterpart result for Corollary 1 is triviallyobtained as the only required condition that is the bound-edness of ∆ . As a result, we can consider without loss of generality thatCorollary 1 can only be satisfied if Π is positive-negative. Inconclusion, the IQC theorem may only provide better resultsover the graph separation theory when (a) ∆ is linear and(b) Π is non-negative. Otherwise, graph separation andIQC theories lead to the same stability result for rationalmultipliers. The aim of this paper is to complete the classification ofIQC-factorizations. It concludes previous work presentedin (Seiler, 2015; Carrasco and Seiler, 2015), establishing anovel connection between IQC and graph separation theo-ries. Here we propose the term doubly-hard factorizations,where both frequency conditions can be transformed intotruncated time-domain conditions. We show that the stan-dard triangular factorization is hard factorization but failsto be a doubly-hard. Then it cannot be used to establish anequivalence between the IQC theorem and separation resultsin the truncated time-domain. We have shown that ( Ψ , M ) is a doubly-hard factorization if Ψ and Ψ − are both stable.The new results allow us to compare both theories for thefeedback interconnection two nonlinear systems. As a resultwe conclude that the IQC theorem for two nonlinear systemdoes not provide any significant advantage over its counter-part result derived using graph separation tools. However,the IQC theorem may provide some advantages when oneof the system is linear and the term Π is non-negative. Acknowledgement
The first author acknowledges William Heath for fruitfuldiscussions and comments.
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A Proof of Lemma 3
If condition (2) is satisfied for Π , then it is trivial that it isalso satisfied for ¯ Π since (cid:34) (cid:98) u ( j ω ) (cid:99) ∆ u ( j ω ) (cid:35) ∗ (cid:34) δ I m
00 0 (cid:35) (cid:34) (cid:98) u ( j ω ) (cid:99) ∆ u ( j ω ) (cid:35) = δ | (cid:98) u ( j ω ) | ≥ ω ∈ R . If condition (3) is satisfied for Π , then thereexists ε > (cid:34) G ( j ω ) I (cid:35) ∗ Π ( j ω ) (cid:34) G ( j ω ) I (cid:35) ≤ − ε I ∀ ω ∈ R . (A.2)Moreover, for any δ >
0, it follows (cid:34) G ( j ω ) I (cid:35) ∗ (cid:34) δ I m
00 0 (cid:35) (cid:34) G ( j ω ) I (cid:35) = δ ( G ( j ω ) ∗ G ( j ω )) ≤ δ (cid:107) G (cid:107) ∞ I , (A.3)for all ω . As a result, taking δ = ε (cid:107) G (cid:107) ∞ , (cid:34) G ( j ω ) I (cid:35) ∗ ¯ Π ( j ω ) (cid:34) G ( j ω ) I (cid:35) ≤ − ( ε − ε ) I = − ε I ∀ ω ∈ R , (A.4) B Proof of Lemma 2
For any T ≥
0, the frequency-domain inequality (20) canbe converted into time-domain (by Parserval’s theorem) and re-arranged as (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) ∞ z ( t ) (cid:62) z ( t ) dt − (cid:90) ∞ T z ( t ) (cid:62) Mz ( t ) dt (B.1)Note that − ε (cid:82) ∞ T z ( t ) (cid:62) z ( t ) dt ≤ (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt − (cid:90) ∞ T z ( t ) (cid:62) Mz ( t ) dt (B.2)Next let ˜ w ∈ L [ , ∞ ) be any signal satisfying ˜ w T = w T .Define ˜ v = ∆ ˜ w and let ˜ z = Ψ [ ˜ v ˜ w ] be the response of Ψ withnull initial condition. By causality of ∆ and Ψ , w T = ˜ w T implies v T = ˜ v T and z T = ˜ z T . Hence for all ˜ w , it holds (cid:90) T z ( t ) (cid:62) Mz ( t ) dt = (cid:90) T ˜ z ( t ) (cid:62) M ˜ z ( t ) dt . Moreover, the IQC holds for any input/output pairs of ∆ .In particular, Equation B.2 holds with z replaced by ˜ z . As aresult, any ˜ w ∈ L satisfying ˜ w T = w T can be used to upperbound the integral (cid:82) ∞ z ( t ) (cid:62) Mz ( t ) dt obtained with w : (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt − (cid:90) ∞ T ˜ z ( t ) (cid:62) M ˜ z ( t ) dt (B.3)Minimizing over all feasible ˜ w yields the upper bound (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt + inf ˜ w ∈ L , ˜ w T = w T (cid:18) − (cid:90) ∞ T ˜ z ( t ) (cid:62) M ˜ z ( t ) dt (cid:19) , (B.4)The suitable set of signals ˜ w can be rewritten as˜ w ( t ) = (cid:26) w ( t ) if t ≤ Tw f ( t ) if t > T for any w f ∈ L [ T , ∞ ) . We can rewrite the minimisation as (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt + inf w f ∈ L [ T , ∞ ) (cid:18) − (cid:90) ∞ T ˜ z ( t ) (cid:62) M ˜ z ( t ) dt (cid:19) , (B.5)such that ˜ v = ∆ ˜ w and ˜ z = Ψ (cid:34) ˜ v ˜ w (cid:35) . The dependence on ∆ can be removed following similar arguments to those givenin (Seiler, 2015). Partition ˜ v = ∆ ˜ w as:˜ v ( t ) = (cid:26) ∆ w ( t ) if t ≤ Tv f ( t ) if t > T (B.6)8he bound in Equation B.5 only involves ˜ z defined on [ T , ∞ ) .This signal can be computed from the state of Ψ at time T ,i.e. x T , as well as the signals w f and v f . Note that x ( T ) = x T is the same for any feasible choice of ˜ w because ˜ w T = w T and ˜ v T = v T . The dependence on ∆ is removed, with someconservatism, by simply maximizing over all possible futuresignals v f defined on [ T , ∞ ) instead of using ˜ v = ∆ w . Inother words, (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt + inf w f ∈ L [ T , ∞ ) sup v f ∈ L [ T , ∞ ) (cid:18) − (cid:90) ∞ T ˜ z ( t ) (cid:62) M ˜ z ( t ) dt (cid:19) , (B.7)This is subject to constraint x ( T ) = x T . This can be rewrittenusing the cost function J as: (cid:90) T z ( t ) (cid:62) Mz ( t ) dt ≤ − ε (cid:90) T z ( t ) (cid:62) z ( t ) dt − J ( x T ) ,,