Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

Parameteriza tion of Seed Functions f or Equiv alent Representat ions of Time-V arying Delay Systems Sengiyumva Kisole 1 , Jungba e Chun 2 , Peter Seiler 2 , and Matthew M. Peet 1 Abstract — Abel’ s classic transforma tion shows th at any well- posed system with time-va rying delay is equivalent t o a parameter -varying system with fixed delay . Th e existence of such a parameter -varying constant delay re presentation then simplifies the problems of stability analysis and optimal control. Unfortunately , the method for construction of such transfor - mations has been ad -hoc – requirin g an iterative t i me-stepping approach to constructi ng the transfo rmation beginning wit h a seed function su bject to boun dary-value constraints. Moreov er , a poor choice of seed fu nction often results in a constant delay repre sentati on with large time-va riations in system parameters – obviating the benefits of su ch a representation. In this paper , we show h ow the set of all f easib l e seed functions can b e parameterized usin g a basis for L 2 . This parameteriza tion is then used to search fo r seed fun ctions f or which the correspond- ing time-transforma tion results i n smaller parameter va riation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed fun ctions affect t he boun dedness of a time transfo rmation. I . I N T RO D U C T I O N T ime-varying delays appear in networked co ntrol, teleop- eration, and biolog ical regulatio n, where even small time- variations can alter stability and per forman ce. Howe ver , analysis and contr ol of systems with time-varying delay is complicated by the typ ically associated time-varying notion of state-space [1 ]–[4]. T o address this issue, a classical remedy , dating back to Abel- type transform ations, shows that any well-posed system with a time-varying delay can be recast as a par a meter-v arying system with a fixed d elay; with a change of time v ariables, o ne regains a time-inv ar iant state space [5]–[ 11]. In th e fixed-delay rep resentation, stand ard tools— Lin ear Matrix In equalities (LMI s), Integral Quadr a tic Constraints (IQCs), and the Partial In tegral Equation s (PIEs) framework, can then be u sed for analysis and contro l de- sign [1 2]–[1 5]. The con version of a system with time varying delay to parameter-varying system with fixed delay is based on a giv en time-tr a nsformatio n t 7→ h ( t ) which must satisfy certain recursive con stra in ts (L em. 1). Howe ver , this time- transform ation is not uniq u e, but ra th er is p arameterized by a choic e o f seed fun ction , φ wh ic h specifies the transfor- mation on an initial tim e interval. This seed f unction m ust This materia l is based upon work supported by the N ational S cience Founda tion under Grants NSF EPCN-2337751 and NSF EPCN-2337752 . 1 School for the Engineering of Matter , Tra nsport and Energy , Arizona State Univ ersity , T empe, AZ 85298, USA. sengi.kisole@ asu.edu, m peet@asu.edu 2 Departmen t of Electrica l E ngineering and Computer Sc ience , Univ ersity of Michigan, Ann Arbor , MI 48109, USA. jungbaec@umich.e du, pseiler@umich .edu itself satisfy ce r tain boun dary constraints as deter mined by the tim e -varying delay . Then , for any valid seed functio n, the cor respond ing time-transfo rmation may b e constructed iterativ ely . Unfortu n ately , howe ver , as is demonstrated in Sec- tion V, not all seed func tio ns result in well-behaved time- transform ations. Spec ifica lly , we show that many choices of seed f unction result in a tim e-transfor mation wh ose deriv ati ve, ˙ h gr ows over time. This growth in ˙ h translates directly into large parameter variations in the re su lting fixed- delay represen tation – an attemp t to qua n tify this growth fo r systems with periodic time-varying delays via pertur b ativ e expansion can be fo und in [1 6]. This par ameter variation then prevents the use of techniques such as normalization [11], [17] to th e fixed-delay representatio n. The goal of this paper, th en, is to prop o se a param- eterization o f seed functio n s which th en allows one to find time-transfo rmations with desirab le p roperties such as periodicity an d boun dedness. Sp ecifically , we p rovide an L 2 parameteriz a tion of the set o f ad missible seed fu nctions ( S τ ), defining a affine operato r T : L 2 → S such that for any ν ∈ L 2 , φ = T ν ∈ S . Fu rthermo re, we show th at for any φ ∈ S , T φ ′′′ = φ – establishing a one- to-one m ap fr om L 2 to S . By working directly with the seed parameters, ν , rather than the seed itself, o n e can enforce c o nstraints on th e seed function s (and re su lting time-tran sformation s). For examp le, using a polyno m ial basis for L 2 , one ca n use Sum of Squares (SoS) [ 18] to enf orce no n-negativity of φ ′ . T o establish the map from L 2 to seed fu n ctions, φ , to time-transfo rmation, h , to con stan t delay r e presentation , we begin in Section III by defining th e class of admissible time-transfo rmations an d showing that for any such time- transform ation, solutio ns of the resulting fixed- d elay and time-varying delay systems are equ i valent – a slight exten- sion of classical results in [ 5]–[7 ]. W e then d efine the set of ad missible seed function s an d show how any such seed function defines a resu lting time - transform ation, h . In Section IV, we then establish th e inv ertible mapping between seed para m eters an d seed fun ctions – also showing that the well-known quadr a tic seed function is a spec ial case and that m onoton icity of th e seed f unction can b e enfo rced using SoS c o nstraints o n th e seed param eter . Finally , in Section V, we motiv ate the p roposed p arameter- ization by demonstrating th e impact o f a seed function cho ice on the d eriv ati ve of the time transformatio n, ˙ h — co mparing the effect of quadr atic, exponential, and af fine plus sinusoidal seed function s o n ˙ h for a time-varying sinusoid al delay . I I . N OTA T I O N N n , R n , and R n + denote the spac e of n − d imensional vec- tors of natural, r e a l, and po siti ve rea l numb ers, r espectiv ely . L 2 ([ a, b ] , R ) is the space of squar e–integrable fun ctions; H 3 ([ a, b ] , R ) is th e Sobo lev space of f unctions with th ir d (weak) deriv ati ve in L 2 . Composition is written ( g ◦ f )( x ) = g ( f ( x )) . Moreover, f ◦ k denotes k -fold compo sition, and ( f − 1 ) ◦ k the k -fold composition o f th e inverse (when it exists). For co mpact Ω ⊂ R , C k (Ω) d enotes the spac e of k -times co ntinuou sly differentiable function s f : Ω → R with norm k f k = sup t ∈ Ω k f ( t ) k 2 . I I I . E Q U I V A L E N C E B E T W E E N V A R I A B L E - A N D F I X E D - D E L AY D D E S Consider a d elay differential equation (DDE) o f the form: ˙ x ( t ) = A 0 x ( t ) + A 1 x  t − τ ( t )  , t ≥ 0 , (1) x ( t ) = ζ ( t ) , t ∈ [ − τ (0) , 0] , where A 0 , A 1 ∈ R n x × n x . W e assume the initial fu nction ζ ∈ C ([ − τ (0) , 0] , R n x ) is con tinuous, and the time- varying delay τ ∈ C 1 ( R , R + ) is bou nded with ˙ τ ( t ) < 1 fo r all t ≥ 0 . A. T ime T ransforma tio ns A time transformation , h , m ay be used to convert a DDE with time varying delay to a parameter-varying DDE with fixed time delay . This fu nction is invertible and chang es th e time variable t to a new time variable defined as λ := h − 1 ( t ) . The time transforma tio n h ( λ ) , m ust be a strictly in creasing function constru cted to satisfy a n Abel equ ation. This Ab el equation ensur es that the d elay τ ( t ) a t time t = h ( λ ) aligns with a con stant shift τ ∗ in the new time λ . This is form alized in the fo llowing lem m a which is a slight mod ification of [17 , Thm. 1]. Lemma 1. Supp ose τ ∗ ∈ R + and τ ∈ C 1 ( R , R + ) with ˙ τ ( t ) < 1 . Let h ∈ C 1 ([ − τ ∗ , ∞ ) , [ − τ (0) , ∞ )) be a strictly incr easing u n boun ded fun ction, with h (0) = 0 and where h ( λ ) − τ ( h ( λ ) ) = h ( λ − τ ∗ ) , ∀ λ ≥ 0 . (2) F or any ζ ∈ C [ − τ (0) , 0] , if x ( t ) satisfies ˙ x ( t ) = A 0 x ( t ) + A 1 x ( t − τ ( t )) , ∀ t ≥ 0 x ( t ) = ζ ( t ) , ∀ t ∈ [ − τ (0) , 0 ] (3) then ¯ x ( λ ) = x ( h ( λ )) and ¯ ζ = ζ  h ( λ )  satisfy ˙ ¯ x ( λ ) = ˙ h ( λ ) A 0 ¯ x ( λ ) + ˙ h ( λ ) A 1 ¯ x ( λ − τ ∗ ) , ∀ λ ≥ 0 ¯ x ( λ ) = ¯ ζ  λ  , ∀ λ ∈ [ − τ ∗ , 0] . (4) Con versely , if ¯ ζ ∈ C [ − τ ∗ , 0] , an d ¯ x satisfies Eqn . (4) , then x ( t ) = ¯ x ( h − 1 ( t )) and ζ  t  = ¯ ζ ( h − 1 ( t )) satisfy E q n. ( 3) . Pr oo f: By the inv erse functio n theor em, h − 1 ∈ C 1 and ( h − 1 ) ′ ( t ) = 1 ˙ h  h − 1 ( t )  , t ∈ [ − τ (0) , ∞ ) . ( ⇒ ) L et ζ ∈ C ([ − τ (0) , 0]) and su p pose x so lves (3 ). De fin e ¯ x ( λ ) := x  h ( λ )  . For λ ≥ 0 , ˙ ¯ x ( λ ) = ˙ x  h ( λ )  ˙ h ( λ ) = ˙ h ( λ ) h A 0 x  h ( λ )  + A 1 x  h ( λ ) − τ ( h ( λ ) )  i . Using (2 ), we hav e: ˙ ¯ x ( λ ) = ˙ h ( λ ) h A 0 ¯ x ( λ ) + A 1 ¯ x ( λ − τ ∗ ) i . For the initial segment, if λ ∈ [ − τ ∗ , 0] then h ( λ ) ∈ [ h ( − τ ∗ ) , h (0)] = [ − τ (0) , 0] , hence ¯ x ( λ ) = x  h ( λ )  = ζ  h ( λ )  . T hus ( 4 ) h olds. ( ⇐ ) Let ¯ ζ ∈ C ([ − τ ∗ , 0]) and supp ose ¯ x satisfies ( 4). Define x ( t ) := ¯ x  h − 1 ( t )  for t ≥ 0 . Th en ˙ x ( t ) = ˙ ¯ x ( h − 1 ( t )) ( h − 1 ) ′ ( t ) = ˙ ¯ x ( h − 1 ( t )) 1 ˙ h ( h − 1 ( t )) = A 0 ¯ x ( h − 1 ( t )) + A 1 ¯ x ( h − 1 ( t ) − τ ∗ ) From (2) , we o b tain h − 1  t − τ ( t )  = h − 1 ( t ) − τ ∗ , so ¯ x ( h − 1 ( t ) − τ ∗ ) = ¯ x  h − 1 ( t − τ ( t ))  = x  t − τ ( t )  . Hen c e ˙ x ( t ) = A 0 x ( t ) + A 1 x  t − τ ( t )  , which is the d ifferential equation in (3). For the initial segment, if t ∈ [ − τ (0) , 0 ] then for h − 1 ( t ) ∈ [ − τ ∗ , 0] , an d thus for ζ ( t ) = ¯ ζ ( h − 1 ( t )) , x ( t ) = ¯ x  h − 1 ( t )  = ¯ ζ  h − 1 ( t )  =: ζ ( t ) , so (3) h olds. Lemma 1 establishes equivalence of solutions between a system w ith time- varying delay and a fixed-d elay system with par ameter-v arying un certainty . W e note, howe ver , th at the choice o f time-transfo r mation, h , may not be un iquely defined. Nor ha ve we suggested any appr o ach to constru cting such a transform ation. In the following subsection, we sh ow that such tim e -transform ations may b e parameter ized through the choice of seed fun ction, φ . B. Recu rsive Construction of the T ime T r ansformation Lemma 1 defines a map from a set of admissible time- transform ations to a set o f equivalent fixed- delay repre- sentations of a system with time-varying d elay . I n this subsection, we show how such tim e-transfor mations may be parameteriz e d b y cho ice of admissible seed function , φ . The set of admissible seed fun ctions is d efined next b a sed on [17, Thm. 1 ], [ 11, Sec. 2.3 ]. Definition 1 . W e say that φ is a admissible seed fun ction associated to τ ∗ , τ (0) > 0 , ˙ τ (0) < 1 if φ is strictly incr easing and φ ∈ S ˙ τ (0) ,τ (0) ,τ ∗ wher e S ˙ τ 0 ,τ 0 ,τ ∗ := (5) ( φ ∈ H 3 [ − τ ∗ , 0] : φ (0) = 0 , φ ( − τ ∗ ) = − τ 0 , ˙ φ (0) = ˙ φ ( − τ ∗ ) 1 − ˙ τ 0 ) For a ny given admissible seed fun c tio n, we may co nstruct an associated time transfo rmation h u sing the recursion defined in [1 1], [17]. Spe cifically , given τ ( t ) and φ ∈ S ˙ τ (0) ,τ (0) ,τ ∗ , let h ∈ C 1 ([ − τ ∗ , ∞ ) , R + ) b e defined as: h ( λ ) := ( φ ( λ ) , λ ∈ [ − τ ∗ , 0] θ − 1  h ( λ − τ ∗ )  , λ ≥ 0 (6) where θ ( t ) := t − τ ( t ) is strictly in creasing and unbou nded and hence is invertible. Eq uiv alently , for k ∈ N , h ( λ ) := ( θ − 1 ) ◦ k  φ ( λ − k τ ∗ )  λ ∈ [( k − 1) τ ∗ , k τ ∗ ] . (7) Moreover , for k ∈ N and λ ∈ [( k − 1) τ ∗ , k τ ∗ ] , ˙ h ( λ ) = ˙ φ ( λ − k τ ∗ ) Q k n =1 ˙ θ  ( θ − 1 ) ◦ ( k − n +1) ( φ ( λ − nτ ∗ ))  . (8) As shown in [11] , [1 7], for any ad missible seed f u nction, this recursion results in a tim e-transfor mation, h which satisfies the con d itions of Lemma 1 . Howev er , th e seed function for a giv en time-varying delay is n o t u n iquely defined. It is restricted only by the values of τ (0) an d ˙ τ (0) (and the choice of τ ∗ ). Fu rthermo re, we note that the c o rrespon ding fixed delay r epresentation in Eqn . (4) includes the multiplicative time-varying p arameter, ˙ h ( λ ) . From Eqn. ( 8) we see that ˙ h ( λ ) will d e pend on th e ch oice of seed fun c tion. Thus, while all seed functions will result in equiv alent system represen tations, some seed f unctions may result in fixed-delay rep r esentations with large or unboun ded parameter variation. I n the following section, we consider a parameteriz a tion of seed fu nctions whic h will allow u s to search for seed fu nctions which result in time tr ansforma tio ns with certain desirable pro perties. I V . P A R A M E T E R I Z A T I O N O F S E E D F U N C T I O N S In this section, we provide an inv ertible af fine m a p ( T ) between L 2 and the set of seed fun ctions S τ ′ 0 ,τ 0 ,τ ∗ . The key observation here is that any seed fun ction, φ , is uniq uely determined by its third derivati ve, φ ′′′ . Following the main result in Thm. 2 , w e provid e SoS co nditions fo r φ to be increasing and sh ow that the p reviously used quadratic seed function is a special case of the propo sed parameteriz a tion. Define o perator T as fo llows. ( T ν )( λ ) = − τ 0 + τ 0 β ( λ ) + Z 0 − τ ∗ K ( λ, η ) ν ( η ) dη + Z λ − τ ∗ ( λ − η ) 2 2 ν ( η ) dη , (9) where β ( λ ) = 2(1 − τ ′ 0 ) (2 − τ ′ 0 ) τ ∗ ( λ + τ ∗ ) + τ ′ 0 (2 − τ ′ 0 )( τ ∗ ) 2 ( λ + τ ∗ ) 2 , (1 0) and K ( λ, η ) = 1 − τ ′ 0 2 − τ ′ 0  − η − η 2 τ ∗  ( λ + τ ∗ ) − 1 2 − τ ′ 0  τ ′ 0 η 2 2( τ ∗ ) 2 − 1 − τ ′ 0 τ ∗ η  ( λ + τ ∗ ) 2 . (11) The following theore m sho ws that T de fin es an invertible map from L 2 to S τ ′ 0 ,τ 0 ,τ ∗ . Theorem 2. Given τ ∗ , τ 0 > 0 a nd τ ′ 0 < 1 . Let S τ ′ 0 ,τ 0 ,τ ∗ be defined as in Eqn. (5 ) and T be de fi ned as in Eq n. (9) . Th en: 1) F or an y φ ∈ S τ ′ 0 ,τ 0 ,τ ∗ , T φ ′′′ = φ . 2) F or an y ν ∈ L 2 [ − τ ∗ , 0] , T ν ∈ S τ ′ 0 ,τ 0 ,τ ∗ . 3) F or an y ν ∈ L 2 [ − τ ∗ , 0] , ( T ν ) ′′′ = ν . Pr oo f: For 1), since φ ∈ S τ ′ 0 ,τ 0 ,τ ∗ ⊂ H 3 [ − τ ∗ , 0] ⊂ C 2 [ − τ ∗ , 0] , we ap ply the T ay lor formula with integral re- minder [ 19]: for any f ∈ H k +1 ([ a, b ]) and f or any x ∈ [ a, b ] , f ( x ) = f ( a ) + f ′ ( a )( x − a ) + f ′′ ( a ) 2! ( x − a ) 2 + · · · + f ( k ) ( a ) k ! ( x − a ) k + Z x a f ( k +1) ( ξ ) ( x − ξ ) k k ! dξ . (12) Applying (12) with f = φ , a = − τ ∗ and k = 2 yeilds φ ′ (0) = φ ′ ( − τ ∗ ) + τ ∗ φ ′′ ( − τ ∗ ) − Z 0 − τ ∗ η φ ′′′ ( η ) dη , (13) φ (0) = φ ( − τ ∗ ) + τ ∗ φ ′ ( − τ ∗ ) + 1 2 ( τ ∗ ) 2 φ ′′ ( − τ ∗ ) + Z 0 − τ ∗ η 2 2 φ ′′′ ( η ) dη , and (14) φ ( λ ) = φ ( − τ ∗ ) + ( λ + τ ∗ ) φ ′ ( − τ ∗ ) + 1 2 ( λ + τ ∗ ) 2 φ ′′ ( − τ ∗ ) + Z λ − τ ∗ ( λ − η ) 2 2 φ ′′′ ( η ) dη . (15) φ ∈ S τ ′ 0 ,τ 0 ,τ ∗ implies φ ′ (0) = φ ′ ( − τ ∗ ) (1 − ˙ τ 0 ) , w h ich co mbined with (13) y ields φ ′′ ( − τ ∗ ) = 1 τ ∗  τ ′ 0 1 − τ ′ 0 φ ′ ( − τ ∗ ) + Z 0 − τ ∗ η φ ′′′ ( η ) dη  . (16) Like wise comb ining φ (0) = 0 and φ ( − τ ∗ ) = − τ 0 with (1 4) imply φ ′′ ( − τ ∗ ) = 2 ( τ ∗ ) 2  τ 0 − τ ∗ φ ′ ( − τ ∗ ) − Z 0 − τ ∗ η 2 2 φ ′′′ ( η ) dη  . (17) Substituting (1 7) into (1 3) and rearr a n ging yields φ ′ ( − τ ∗ ) = 1 − τ ′ 0 2 − τ ′ 0  2 τ 0 τ ∗ − 1 τ ∗ Z 0 − τ ∗ η 2 φ ′′′ ( η ) dη − Z 0 − τ ∗ η φ ′′′ ( η ) dη  . (18) Substituting (1 8) into (1 6), we get: φ ′′ ( − τ ∗ ) = 1 τ ∗  τ ′ 0 2 − τ ′ 0  2 τ 0 τ ∗ − 1 τ ∗ Z 0 − τ ∗ η 2 φ ′′′ ( η ) dη − Z 0 − τ ∗ η φ ′′′ ( η ) dη  + Z 0 − τ ∗ η φ ′′′ ( η ) dη  = 2 τ ′ 0 2 − τ ′ 0 τ 0 ( τ ∗ ) 2 − τ ′ 0 2 − τ ′ 0 1 ( τ ∗ ) 2 Z 0 − τ ∗ η 2 φ ′′′ ( η ) dη + 2(1 − τ ′ 0 ) 2 − τ ′ 0 1 τ ∗ Z 0 − τ ∗ η φ ′′′ ( η ) dη . (19) Now , substitutin g (18 ) an d (19) into (15) and using the bound ary condition φ ( − τ ∗ ) = − τ 0 , we ob tain : φ ( λ ) = − τ 0 + ( λ + τ ∗ ) 1 − τ ′ 0 2 − τ ′ 0  2 τ 0 τ ∗ − 1 τ ∗ Z 0 − τ ∗ η 2 φ ′′′ ( η ) dη − Z 0 − τ ∗ η φ ′′′ ( η ) dη  + ( λ + τ ∗ ) 2 2  2 τ ′ 0 2 − τ ′ 0 τ 0 ( τ ∗ ) 2 − τ ′ 0 2 − τ ′ 0 1 ( τ ∗ ) 2 Z 0 − τ ∗ η 2 φ ′′′ ( η ) dη + 2(1 − τ ′ 0 ) 2 − τ ′ 0 1 τ ∗ Z 0 − τ ∗ η φ ′′′ ( η ) dη  + Z λ − τ ∗ ( λ − η ) 2 2 φ ′′′ ( η ) dη = − τ 0 + τ 0 β ( λ ) + Z 0 − τ ∗ K ( λ, η ) φ ′′′ ( η ) dη + Z λ − τ ∗ ( λ − η ) 2 2 φ ′′′ ( η ) dη = ( T φ ′′′ )( λ ) , for all λ ∈ [ − τ ∗ , 0] . Next, we establish 3) by showing that for any ν ∈ L 2 [ − τ ∗ , 0] , if φ = T ν , then φ ′′′ = ν . Recall from (9) that ( T ν )( λ ) = − τ 0 + τ 0 β ( λ ) + Z 0 − τ ∗ K ( λ, η ) ν ( η ) dη + Z λ − τ ∗ ( λ − η ) 2 2 ν ( η ) dη . From the defin itio ns of β in (1 0 ) and K in ( 11) , we observe that β ( λ ) and K ( λ, η ) are at most quad ratic in λ . Hence, β ′′′ ( λ ) = 0 an d ∂ 3 λ K ( λ, η ) = 0 . Next, we note th at d dλ Z λ − τ ∗ ( λ − η ) 2 2 ν ( η ) dη = Z λ − τ ∗ ( λ − η ) ν ( η ) dη , ⇒ d 2 dλ 2 Z λ − τ ∗ ( λ − η ) 2 2 ν ( η ) dη = Z λ − τ ∗ ν ( η ) dη , ⇒ d 3 dλ 3 Z λ − τ ∗ ( λ − η ) 2 2 ν ( η ) dη = ν ( λ ) . Thus, we c onclude that ( T ν ) ′′′ = ν . For 2 ), suppose ν ∈ L 2 [ − τ ∗ , 0] . As p er Eqn. (5), we need to show tha t T φ ∈ H 3 , ( T φ )(0) = 0 , ( T φ )( − τ ∗ ) = − τ 0 , and ( T φ ) ′ (0)(1 − ˙ τ ) = ( T φ ) ′ ( − τ ∗ ) . First, 3) implies φ = T ν ∈ H 3 . Next, since β ( − τ ∗ ) = 0 an d K ( − τ ∗ , η ) = 0 , we h av e ( T ν )( − τ ∗ ) = − τ 0 . T hird, since β (0) = 1 and K (0 , η ) = − η 2 2 , we have ( T ν )(0) = − τ 0 + τ 0 + Z 0 − τ ∗ − η 2 2 ν ( η ) dη + Z 0 − τ ∗ η 2 2 ν ( η ) dη = 0 . For the final condition , differentiating (9), we o btain: ( T ν ) ′ ( λ ) = τ 0 β ′ ( λ ) + Z 0 − τ ∗ ∂ λ K ( λ, η ) ν ( η ) dη + Z λ − τ ∗ ( λ − η ) ν ( η ) dη , ∀ λ ∈ [ − τ ∗ , 0] , ( 20) where β ′ ( λ ) = 2  1 − τ ′ 0   2 − τ ′ 0  τ ∗ + 2 τ ′ 0  2 − τ ′ 0  ( τ ∗ ) 2 ( λ + τ ∗ ) and ∂ λ K ( λ, η ) = 1 − τ ′ 0 2 − τ ′ 0  − η − η 2 τ ∗  − 2 2 − τ ′ 0  τ ′ 0 η 2 2( τ ∗ ) 2 − 1 − τ ′ 0 τ ∗ η  ( λ + τ ∗ ) . W e also note that β ′ ( − τ ∗ ) = 2(1 − τ ′ 0 ) (2 − τ ′ 0 ) τ ∗ and ∂ λ K ( − τ ∗ , η ) = 1 − τ ′ 0 2 − τ ′ 0  − η − η 2 τ ∗  which implies ( T ν ) ′ ( − τ ∗ ) = τ 0 β ′ ( − τ ∗ ) + Z 0 − τ ∗ ∂ λ K ( − τ ∗ , η ) ν ( η ) dη = 1 − τ ′ 0 2 − τ ′ 0  2 τ 0 τ ∗ − 1 τ ∗ Z 0 − τ ∗ η 2 ν ( η ) dη − Z 0 − τ ∗ η ν ( η ) dη  . (21) Similarly , since β ′ (0) = 2 (2 − τ ′ 0 ) τ ∗ and ∂ λ K (0 , η ) = 1 2 − τ ′ 0  (1 − τ ′ 0 ) η − η 2 τ ∗  , we have ( T ν ) ′ (0) = τ 0 β ′ (0) + Z 0 − τ ∗ ∂ λ K (0 , η ) ν ( η ) dη − Z 0 − τ ∗ η ν ( η ) dη = 1 2 − τ ′ 0  2 τ 0 τ ∗ − 1 τ ∗ Z 0 − τ ∗ η 2 ν ( η ) dη − Z 0 − τ ∗ η ν ( η ) dη  . (22) Combining Eqn. (2 1) a n d ( 2 2), we ob tain: ( T ν ) ′ ( − τ ∗ ) ( T ν ) ′ (0) = 1 − τ ′ 0 . W e co nclude that T ν ∈ S τ ′ 0 ,τ 0 ,τ ∗ . Having defined a parameter ization o f seed function s T ν for ν ∈ L 2 , we now ob serve that the simp lest such seed function (fo r ν = 0 ) is precisely the quadr atic seed f unction derived in [17, Thm. 1 ]. Remark 1. I f ν ( η ) = 0 for all η ∈ [ − τ ∗ , 0] , then the inte gral terms in T ν (See Eq n. ( 9) ) van ish, and for λ ∈ [ − τ ∗ , 0] , φ ( λ ) = − τ (0) + τ (0) β ( λ ) , (23) wher e β ( λ ) in (10) is a qua d ratic p olynomial in λ . This quadratic seed fu n ction coincides with the choice used in [17, Th m. 1]. Although Thm. 2 ensures that fo r any ν ∈ L 2 , φ = T ν ∈ S τ ′ 0 ,τ 0 ,τ ∗ , admissibility of a seed f u nction further require s φ ′ to be non-negative. Such a conditio n can be imposed upo n the seed p arameter, ν as fo llows. (a) (b) (c) Fig. 1: The derivati ve h ′ ( λ ) o f the time transfor mation f or (a) qu a dratic seed, (b ) affine plus sinu soidal seed, and (c) exponential seed, with different maximum values of h ′ for horizon of 10 0 – illustrating the im pact of seed fu nction choice on stability margins for a delay τ ( t ) = ( 1 2 π − 0 . 001) sin(2 πt ) + ( 1 2 π + 0 . 001) . Corollary 2.1. Give n τ ′ 0 , τ 0 , τ ∗ , let T be as in E qn. (9) . F or any ν ∈ L 2 [ − τ ∗ , 0] , ( T ν ) ′ ( λ ) > 0 for all λ ∈ [ − τ ∗ , 0] if and only if τ 0 β ′ ( λ ) + Z 0 − τ ∗ ∂ λ K ( λ, η ) ν ( η ) dη + Z λ − τ ∗ ( λ − η ) ν ( η ) dη > 0 , for all λ ∈ [ − τ ∗ , 0] . Pr oo f: Follows from the definition of T in Eqn. (20 ). If the seed parame ters are po lynomial, then SoS o ptimization can be u sed to enfo rce the non-negativity conditio n in Cor . 2.1 . No te that sin c e the non-n egati vity co ndition here is univ ariate, the correspond ing univ ariate SoS n o n-negativity test is b oth necessary and sufficient . V . N U M E R I C A L C O M PA R I S O N O F S E E D F U N C T I O N S For a gi ven tim e-varying delay τ ( t ) , the ch o ice of seed function significantly influences the time transfo rmation h and its deriv ati ve h ′ , potentially complica ting analy sis o f the fixed-delay , par ameter-v arying representatio n. In this section, we compa r e the effect of three choices of seed paramete r on the deriv ati ve of the time-transfo rmation, h ′ . Specifically , w e con sid e r two examples o f per iodic time- varying d elay , and for each example, examine three candi- date seed parameter s. For each resulting seed functio n, we construct h ′ ( λ ) over an extended interval. Specifically , rec a ll th at for θ ( t ) = t − τ ( t ) , h ( λ ) := ( θ − 1 ) ◦ k  φ ( λ − k τ ∗ )  λ ∈ [( k − 1) τ ∗ , k τ ∗ ] . (a) (b) (c) Fig. 2: The derivati ve h ′ ( λ ) o f the time transfor mation f or (a) qu a dratic seed, (b ) affine plus sinu soidal seed, and (c) exponential seed, with different maximum values of h ′ for horizon of 1 00 for the delay τ ( t ) = 1 + 0 . 3 sin ( t ) . Because there is n o an alytic exp r ession for θ − 1 , the h ( λ ) a n d ˙ h ( λ ) are computed p o intwise in λ . That is, for every given λ ∈ [( k − 1) τ ∗ , k τ ∗ ] w ith k ∈ N , we initialize h 0 := φ ( λ − k τ ∗ ) an d for j = 0 , · · · , k , c ompute h j +1 := ( θ − 1 )( h j ) where θ − 1 is e valuated numerica lly using Newton iteration – yieldin g h j = ( θ − 1 ) ◦ j  φ ( λ − k τ ∗ )  and h k = h ( λ ) . The h j are th e n used to comp ute h ′ ( λ ) as h ′ ( λ ) = φ ′ ( λ − k τ ∗ ) Q k j =1 θ ′ ( h j ) For all examples, a time hor izon of λ ∈ [0 , 1 00] is used. a) Exa mple 1: First consider time-varying d elay τ ( t ) = γ 0 sin(2 π t ) + γ 1 , w h ere γ 0 = 1 2 π − 0 . 00 1 and γ 1 = 1 2 π + 0 . 001 . Here τ (0) = γ 1 , τ ′ (0) = 2 π γ 0 and we choo se τ ∗ = 1 , correspo n ding to th e per iod of delay . W e n ow select thr ee seed parame ter s ν ∈ L 2 : ν 1 ( λ ) = 0 , ν 2 ( λ ) = 8 γ 1 1 − e − 2 e 2 λ ν 3 ( λ ) = − Λ 1  π 2  3 sin ( π ( λ + 1)) where Λ 1 = 4 π γ 0 γ 1 π − 2 π 2 γ 0 + 2 γ 1 . The correspo nding qu adratic, expon ential and affine plus sinusoidal seed fu nctions are then ( T ν 1 )( λ ) = − τ (0)+ τ (0) β ( λ ) , ( T ν 2 )( λ ) = τ (0)( e 2 λ − 1) 1 − e − 2 and ( T ν )( λ ) = Λ 2 − τ (0) + Λ 2 λ + Λ 1  1 − cos  π 2 ( λ + 1)  , where Λ 2 = 0 . 002 π 2 γ 1 0 . 002 π 2 + 2 γ 1 and wher e β ( λ ) is defined in Eqn. (10) f o r τ 0 = γ 1 and τ ′ 0 = 2 π γ 0 . I t is easily verified that the seed s T ( ν i ) ∈ S τ ′ 0 ,τ 0 ,τ ∗ are m o noton ic. b) Exa mple 2 : In the secon d example, we use a time- varying delay τ ( t ) = 1 + 0 . 3 s in( t ) . Here τ (0) = 1 , τ ′ (0) = 0 . 3 and we c h oose τ ∗ = 1 , whic h is a fractio n 1 2 π the period of d elay . In this case, we again u se sligh tly mod ifed version of ν 1 , ν 2 and ν 3 from the first example: ν 1 ( λ ) = 0 , ν 2 ( λ ) = 8 1 − e − 2 e 2 λ ν 3 ( λ ) = − Λ 1  π 2  3 sin ( π ( λ + 1)) where Λ 1 = 2 . 6 0 . 7 π + 2 , Λ 2 = 0 . 6 0 . 7 π + 2 . c) Ana lysis: Th e de r iv ative of the time-transfo rmation, h ′ ( λ ) cor respond in g to the tim e-delays and seed param e ters from Examp le 1 c an be fo und in Figure 1. For Ex ample 2, h ′ ( λ ) is shown in Figur e 2. Among the tested seed function s, th e affine-plus-sinusoidal parameter p roduc e s the smallest b ounds on h ′ , followed by the quadr atic function , while the exponen tial fu nction results in the largest b ounds. The suitability of the affi ne-plus-sinu soidal may be due to structural similarity betwe e n seed fun ction and time-varying delay , suggestin g the po ssibility of seed func tio ns wh ose correspo n ding time- transform ation ad mit global boun ds on the deriv ati ve. These results also high lig ht th e imp ortance of careful selection of seed fun ction to minimize par a meter variation, thereby enh ancing stability analysis in small-gain or I QC fr amew orks. V I . C O N C L U S I O N Systems with time-varying d elay can be eq u iv ale n tly rep- resented by systems with fixed delay and mu ltiplicative pa- rameter variation. Ho wev er , this r epresentation is no t unique, being param eterized by a time-transfo r mation which is, in turn, d efined by a seed fu nction. The choice o f seed functio n has a significan t effect on the p arameter variation of the resulting co nstant-delay r epresentation . 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