Stability Analysis for Linear Time-Delay Systems. Described by Fractional Parameterized. Models Possessing Multiple Internal. Constant Discrete Delays

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1 Appled Mathematcal Sceces, Vol. 3, 29, o. 23, 5-25 Stablty Aalyss fo Lea me-delay Systems Descbed by Factoal Paametezed Models Possessg Multple Iteal Costat Dscete Delays Mauel De la Se Isttuto de Ivestgacó y Desaollo de Pocesos Campus of Leoa, Bzkaa, Spa mauel.delase@ehu.es Abstact. hs pape dscusses lea factoal epesetatos (LFR of paamete-depedet olea systems wth eal-atoal oleates ad pot-delayed dyamcs. Suffcet codtos fo obust global asymptotc stablty both depedet of ad depedet o the delays ae vestgated va lea matx equaltes. Such equaltes ae obtaed fom the values of the tme-devatves of appopate Lyapuov fuctos at all the vetces of the polytope whch cotas the paametzed ucetates Itoducto me-delay systems ae vey commo atue lke, fo stace, elated to taspotato poblems, populato gowg ad sgal tasmsso methods (see, fo stace, [-2] ad efeeces thee. he stablty ad stablzato of those systems have bee studed the lteatue coecto, fo stace, wth Lyapuov theoy. Some of the elated esults ae efeed to ethe as beg depedet of the szes of the delays o as depedet of those szes. Wth ths last class of esults, they met specal atteto those elated to the chaactezato of the fst teval of admssble delay szes allowg stablzato. O the othe had, the most volved goup of esults to obta s that elated to teal delays (.e. the state sce ts assocate dyamcs possess ftely may modes geeal. I ths

2 6 M. De la Se pape, we cosde a paamete-depedet ( geeal, olea ad tme-vayg system subject to a fte set of pot delays whch may be, geeal, defed by eal-atoal oleates, whose paamete vecto H s estcted to le a polytope Θ R cotag the og, hs s called a so-called polytopc delayed system followg the omeclatue used fo delay-fee systems [3]]. he esults developed the followg mght be stll appled f the set Θ s ot a polytope afte eplacg t by some polytope Θ poly Θ. he ma agumets used to develop the fomalsm ae based o the fact that the polytope whee the paametes belog to defes affe fucto matces of vetces whch may be calculated fom those of the ogal polytope Θ R of paametzed ucetates. I the followg, the obust global asymptotc stablty of such a polytopc delayed system subject to pot delays s vestgated va Lyapuov theoy. m.. Notato. m ( C R s the set of eal (complex m matces ad P = P > stads fo a eal symmetc postve-defte matx. - Fo a gve set S, oe defes σ S = { σ s s S } f σ s a postve umbe. m - he covex hull of complex m matces ( ϑ, ϑ, ϑ, ϑ C s Co 2... l l, 2 l = ϑ ϑ = λ ϑ, λ, λ = = { ϑ ϑ,... ϑ } - I m s the m-detty matx wth the subscpt beg omtted f ts sze follows dectly fom cotext. - If Θ s a polytope cotag the og ad Δ ( θ ( =,, beg a tege, ae eal-valued atoal matx fuctos of ay ode of θ the ( k Δ = { Δ ( θ θ Δ } ad Δ = Δ Δ...Δ ae polytopes of v vetces Δ, k =, v ;, ( k, k,..., k ( k ( k Δ = Δ... Δ = ; ad, espectvely, whee deotes the Catesa poduct of matces (cosdeed as sets. I ou cotext, Θ s the polytope whee the system paametes belog to whle Δ s the polytope whee the atoal matx fucto A ( θ, defg the dyamcs of the -th delay h ( A ( θ descbes the delay-fee dyamcs;.e. h = as the paamete vecto θ us ove Θ ; =,. l 2. Lea factoal descptos Cosde the paamete depedet system subject to pot delays h (, = x& (t = A ( θ(t x ( t h + B( θ(t u(t (.a y(t = = C( θ(t x ( t + D( θ (t u(t (.b

3 Stablty aalyss fo lea tme-delay systems 7 whee h =, x (t R, u(t R u y, y(t R, output sgals, espectvely, ad ( fuctos of tme-vayg paamete vecto (t ( t = ( θ (t, θ (t,..., θ (t 2 m m R, t R ae the state, put ad measuable A =, B, C ad D ae eal- valued atoal θ wth θ Θ fo all t wth θ such that the eal vecto assocated fucto Θ θ ( [ m fo all x ( t s defed such a way that ( has a mld soluto o [,t t fo ay absolutely cotuous fucto [ h, ] R ϕ( t, t [ h, ] h= Max h. Oe defes wth ( ϕ of tal codtos - he ufoced system ( s obustly globally asymptotcally stable f x(t ufomly bouded ad lm x(t = t f u fo ay bouded x(. he system ( s obustly stablzable f thee exsts a output-feedback ealzable cotol law u( t = K( y, θ( t, t such that the closed-loop system s obustly globally asymptotcally stable. Fo temology smplcty, sce o cofuso s expected, we efe the sequel to obust global asymptotc stablty smply as obust stablty. - he obust stablty (stablzablty mag of ( fo a ucetaty set s σ m ( ρ m = Sup { σ (ρ System ( s obustly stable (stablzable ove γ Θ fo all γ, σ ( γ, ρ. Now, fst cosde the ufoced veso of ( gve by [ ] [ ] x& (t = A ( θ(t x( t h ; y(t = C( θ(t x( t (2 = 2. Fst LFR Sce ( (t =, s a eal-valued atoal matx fucto of θ(t, the LFR descpto of each matx fucto A ( θ(t exsts fo some appopate matces =, A A θ ( A o, q B, pq D ( ( d pq p ( θ (t = A + B Δ ( θ(t I D Δ ( θ(t C q ( d pq t whee I d s the d detty matx (, fo ay Δ ((θ (t such that the well-posedess codto Det I D Δ ( θ(t θ Θ, all s, =. I the followg, the explct depedece of θ (t o tme s omtted the otato fo the shake of smplcty whe o cofuso s expected. If (2 s quadatcally stable the A ( =, ae stctly Huwtza (.e. wth all the egevalues Re s <. A state-space ealzato of the state evoluto of the dyamcal system (2 usg s ( A x( t h + B q ( t x& (t = q h ; = p (t = C px q (t ( pq C p x (t ( I D Δ θ C x (t ( t + D q (t = I D Δ ( θ pq = Δ ( θ p (t = Δ ( θ ( ; (4 pq p

4 8 M. De la Se d whee d q R ; p R ( =, whee s = Max ( s k (, Δ = { Δ ( θ θ Θ } (, (k delay h ad the polytope of v vetces Δ ( k,v ; =, A ( θ ( =, = ad the polytope = ae, espectvely, the -th LFR degee wth espect to the = fo paametzatos of. I patcula, s s the LFR degee of the delay-fee system (4 (.e., fo the case whe A =;=, of paametzato wth a polytope Δ of v vetces ( k Δ ( k=. All those vetces wll become cucal the subsequet,v stablty aalyss the case of covex poblems sce t would be suffcet to check stablty codtos fom matx equaltes at all the set of dstct vetces. Note that the ole of the sgals q (. s that of omalzed equvalet puts fom the equvalet outputs p (. tough omalzato matces Δ (. whch deped o the values of the ucetaty paamete vecto. he LFR epesetato (4 wll be the ma tool of the obust stablty aalyss ad stablzato pocedue poposed ths pape. Note also that the LFR (4 of (2 s vald, patcula, fo the case of commesuate delays = ~ h = h. (, =. he ucetaty Δ t h Dag( Δ ( t h,..., Δ ( t h s absobed the fowad loop whle the detty opeato plays the ole of the ucetaty. A alteatve sgle-put LFR to (4 fo the state of the system (2 s poposed the followg 2.2 Secod LFR Decompose the atoal eal-valued matx fucto A( θ(t = A ( θ(t U t h, U (t beg the uty Heavsde fucto, as A = ( ( ( ( ( ( θ(t = A + B Δ ( θ(t I D Δ( θ(t C U t h = q pq wth A (., B q, D p q ad C p(. ae eal matces of appopate szes. Ude wellposedess;.e. Det( I D pq Δ( θ, ths leads to the sgle-put mult-output LFR of the state equato of (2 ( t h B q (t x (t = A x + = & ; q(t ( p(t q = Δ p (t = C p x( t h + D pq q(t θ = Dag θ I s, θ 2 I s,..., θ 2 m I = d d d d whee q R, p R, B q R, C p R (, θ p ; ( ( s m Δ (6 d d =, ad Δ( θ R s a matx fucto of the tme-vayg paamete vecto θ (t Θ ad s = Max( s s the m LFR degee of (6. he polytope Δ = { Δ ( θ θ Θ } has v vetces Δ ( ; =, v, v v= [ ] v =, whch depeds o that of the polytope Θ whch paametzes the system, ad the paamete vecto θ Θ paametzes the whole dyamcs of A( θ ( t. A equvalet descpto to -(6 s

5 Stablty aalyss fo lea tme-delay systems 9 A ( ( ( = q ( θ(t = A + B Δ ( θ(t I D Δ( θ(t C U t h whch leads to the multput sgle-output LFR (t = = ( A x( t h + B q (t h x& ; q(t = Δ( θ p(t p (t q pq = C x( t D q(t ; Δ ( θ = Dag( θ I, θ I,..., θ (7 p + pq s I 2 s 2 m s m d d d d whee q R, p R, B q R, C p = ewtte the fom (4 by defg eal matces q [ B q,b q,..., B q ] C p = C BlockDag [ D,D,..., D ] p p R (, p B = ; [,C,..., C ] pq pq pq pq p. he LFR s (6-(7 may be [ ] D = ; Δ( θ = Dag Δ ( θ, Δ ( θ,... Δ ( θ (8.a ad p (t = [ p ] (t, p (t,...,p (t ; (t [ q ] (t, q (t,...,q (t to yeld A ( θ ( t = + Δ ( θ ( t I Δ ( θ ( t =, q = (8.b ( ( C p U( t A B q d D pq 2.3 Remaks. ( he LFR s (6 ad (7 ae equvalet, the fst oe descbg the delayed dyamcs though equvalet output delays whle the secod oe descbes t though equvalet put delays. (2 Eqs. 8 pove that the secod LFR may be equvaletly ewtte the fom (4, assocated wth the dyamcs epesetatos. he equvalece ases fom the fact that ethe each A may be paametzed by a patcula paamete vecto θ Θ wth the matx fucto A (θ(t = A ( θ(t U( t h θ = ( θ, θ,..., θ Θ = Θ Θ... Θ = h beg paametzed by Sce the secod LFR may be ewtte as the fst oe, the fomalsm peseted ths mauscpt wll be developed fo the fst LFR wth o loss geealty. Note that the oveall umbe of dstct vetces of polytopes v to be checked s v v= [ ] v =. hat equalty may be stct depedg o the poblem at had sce, depedg o the paametzato, some of the matces defg ( may evetually geeate commo vetces. A smple case mplyg v v s whe two of those (. matces ae detcal so that the assocated set of vetces Δ (. become detcal. o smplfy the otato, we cosde the followg v = v wth o loss geealty otg that the stablty fo commo vetces oly eque to be tested oce.

6 2 M. De la Se Note that f the ufoced system s globally asymptotcally stable depedet of delays the A ad A ae both stablty matces sce Θ cotas the og θ = = (descbg the ucetaty-fee system ad such a system has to be stable fo h ad h = (, =. 3. Ma esults he followg stablty esult s coceed wth the fst LFR eqs. 4. Its poof s omtted. 3. heoem. he (ufoced system (2 s globally asymptotcally stable depedet of the delays f thee exst eal matces P = P >, S S = > ( =, such that, fo evey Δ ( θ Δ, θ Θ ( =,, thee exst complex matces d G ( Δ( θ C d ad ( d H Δ( θ C such that the squae ( + + d Δ = d = Δ d eal symmetc matx Q (Δ (θ defed as Q ( Δ ( θ = Block Matx [ Q j ( ( θ ;, j,3] Q Δ <, whee = * A P+ PA + S + G Δ C p + C p G Δ = A = ; Q = Q = [ PA, PA,..., P ] Q = [ ( ( (,...,P( Δ ] Q 3 Q 3 = P B H * q Δ + G Δ D pq Δ G Δ+ C p Δ M P B q Δ B q [ G C + C G S M... M G C + C G ] * * 22 = BlockDag Δ p p Δ Δ p p Δ S Q = Q = Block Dag [ G Δ( D...G ( D G C H * H * pq Δ G Δ + C p Δ M M Δ pq Δ Δ + p Δ ] * * Q 33 = BlockDag [ H Δ ( D p qδ + ( D p qδ H Δ H Δ H Δ MH ( D Δ + ( D Δ H H H ] * *... Δ p q p q Δ Δ Δ M (9 If G Δ (. ad H Δ (. ae estcted to have specal foms such that Q (Δ (θ s covex fo Θ beg covex, the t suffces that Q (Δ (θ < fo ts evaluato at each (k patcula j-th vetex Δ ( k =, v ;=, geeated fom the vetces of Θ. Also, f the ucetaty paamete vecto s θ = the a well-kow smplfed veso of heoem fo Δ ( = ; =,, fom the last equato (4, guaatees global asymptotc stablty depedet of the delays of the (ucetaty-fee omal system (2. Fom smple matx ak cotuty agumets, that popety s stll guaateed a obustess stablty cotext wth a ceta ope eghbohood of θ =. he subsequet Coollaes to heoem follow.

7 Stablty aalyss fo lea tme-delay systems Coollay. he (ufoced system (2 s globally asymptotcally stable depedet of the delays f thee exst eal matces P P = >, S = S > ( =, such that, fo evey Δ ( θ Δ, θ Θ ( =,, thee exst complex matces d G ( Δ( θ C ad H Δ ( Δ(θ C d d such that the squae ( + eal Δ symmetc matx Q ' (θ defed as Q ' ( Δ ( θ = Block Matx, whee ' ' Q = A P+ PA + S Q ~ + = ' ' Q 22 BlockDag[ S,..., S ] + Q ~ 2 2 whee Q ~ ' 2 2 Q ~ ' ' ; Q = Q = [ PA,PA 2,...,PA ] + 2 Q ' ( Δ( θ ;,j=, 2 j = (.a ' = G Δ [ B * * Δ C p + C p G + P ( q Δ + G Δ ( D pq Δ G Δ + C p H Δ ] ( I D Δ C + C ( I D Δ ( B Δ P+ D Δ G Q ' 2 = Q 2 pq p p pq * * [ q ( pq Δ G Δ + H C ] [ C ] ' = P( B Δ ( I D Δ C M... M P ( B Δ ( I D Δ q p q ' ( Q ~ G C + C G + G ( D Δ G + C ( I D p q Δ C p * * 22 = Δ p p Δ Δ pq Δ p H Δ p * * * + C p ( I D pqδ [( D p q Δ G Δ G Δ + H Δ C p ] * * + C p ( I D Δ [ H Δ( D p q Δ + ( D p q Δ H Δ H H Δ ] ( I D p qδ C p pq q Δ p q p Δ p < ( =, (.b 3.3 Coollay 2. Assume that the system (4 s tme-vaat, omally paametzed at θ = Θ wth o paametcal ucetates;.e. θ (τ = θ (τ = fo all tme. he, t s globally asymptotcally stable depedet of the delays f the eal symmetc (+ squae matx (2 (2 Q BlockMatx Q j ;, j =,2 < =, whee ( 2 ( 2 Q = + + A P PA S ; Q ( 2 2 = Q 2 = Q 2= [ PA,PA 2,...,PA ] = ( 2 Q BlockDag [ S,..., ] 22= S ( fo some eal P = P > ; S = S > ( =,. A ecessay codto fo Q (2 < s that A be a stablty matx (.e. of all ts egevalues of egatve eal pats. Also, fo some eal R >, thee exsts a ope egbohood N (, R of Θ (the polytope of paametcal ucetates of adus R such that the system (4 (ad the the

8 22 M. De la Se ufoced system (2 s globally asymptotcally stable depedet of the delays f θ { N(, R I Θ } fo all tme t. 3.4 Coollay 3. he (ufoced system (2 s globally asymptotcally stable depedet of the delays f thee exst f thee exst [ ] Q fo all [ ;,j=,3 ] ( k,k,..., k BlockMatx Q ( k,k,..., k k,v ; =, j v matces = v = = < = fo some eal -matces P P = A P+ PA + S C p M C p = Q + 3 ( k,...,k Q ( k,...,k Q = 3 = >, M M = > ;, = whee ; Q = Q = Q = [ PA,PA,...,P ] A ( k ( k = P B Δ q + Cp M D pqδ M ( k ( k P B Δ q,...,p B Δ q Q 22 = BlockDag [ C p M C S,...,C p M C p S ] p ( k,..., k Q ( k,..., k Q = ( k = Δ ( k Block Dag C Δ p M D pq,...,c p M D pq Q (k (k k,..., k 33 =Block Dag M + Δ Δ D M.. M pq M D pq (k (k M + Δ Δ D pq M D pq (2 3.5 Remak 4. Note that Coollay 3 may be tested fo ay set of eal symmetc M =, ad holds, patcula, f the stablty test postve defte matces ( becomes postve fo the v-matces ( k,...,k matces M M> ( =, Q < fo some detcal symmetc =. Coollay 3 adopts the followg paallel fom, ude weake codtos, f the LFR degees ae uty wth espect to all the delays. 3.6 Coollay 4. Assume that s = ; =,. he, the ufoced system (4 s globally asymptotcally stable depedet of the delays f thee exst P = P > ad v eal postve defte symmetc matces M( k ; k =, v ;=, such that the v eal symmetc matces [ ;,j =, 3 ] ( 4 ( k,k,..., k Block Matx Q ( k,k,..., k ( 4 Q = j < whee the block matces have the same stuctues as Coollay 3 wth the eplacemets M M ( k ; k =,v ;=,. It s obvous that Coollay 4 s stoge tha

9 Stablty aalyss fo lea tme-delay systems 23 Coollay 3 sce the v matx egatve defteess tests mght be tested wth M ( k dstct matces (oe pe vetex. Paallel esults wee fst obtaed fo the delayfee case [3]. 4. Asymptotc stablty depedet o ad depedet of the delays Paallel esults to those the above Secto II may be obtaed depedg o the fst tevals fo delays esug global asymptotc stablty;.e [, h ] h ( =, ad h = h =. he esults ae obtaed fom Lyapuov s fucto caddates that clude tegal tems fo the effects of coupled combed delays. he ad-hoc veso of heoem fo ths stuato s 4. heoem 2. he (ufoced system (2 s globally asymptotcally stable fo all the delays [, ] ( f thee exst eal matces P = P >, S j = S j > h h =, ( =, ;j =, such that, fo evey Δ ( θ Δ, θ Θ ( =,, thee exst complex d matces G ( Δ( θ C d ad ( d H Δ( θ C such that the squae Δ j Δ j ( + + d d = d eal symmetc matx Q ˆ (Δ (θ defed as = ( Δ ( θ = Block Matx [ ( Δ ( θ ;,j,3 ] < j = whee j j Q ~ˆ = + j ;, j =, ; + = A P+ P A = = = j= 2 2 = [ h PA Â,...,h PA Â ] [ h R,..., h ] h S j = ; = = [ P( B Δ,P( B Δ,..., P( Δ ] 22 BlockDag R 3 3 q q B q = (4 Q ˆ 23 = Q ˆ 32 = ; Q ˆ 33 = Â [ ] ( + A,A,..., = R ; [ ] ( + ( + R S,S,..., S A = R, fo =, = ; 2 = 2 3 G * = 3 = [ P( B q Δ + G Δ( D pq Δ G Δ+ C p H Δ P( B q,, {..., 4243 Δ ( {... P B q Δ,,..., 4243 q q q q [ ( *... * 23= 32 = BlockDag G Δ D pq Δ G Δ + C p H Δ G ( D p q Δ G Δ + C p H Δ... ( * * Δ D p q Δ G Δ + C p H Δ... G Δ ( D p q Δ G Δ + C p H Δ ] Δ

10 24 M. De la Se 33= BlockDag * H Δ H Δ H [ ( ( * H Δ D D pq Δ + pq Δ H Δ * * H Δ H Δ H Δ M... M... M H Δ ( D p q Δ + ( D p q Δ M ( ( * Δ D p q Δ + D p q Δ H Δ HΔ H Δ M... M H ( Δ + ( Δ H H H ] * Δ D p q (6 Note that, sce < the the delay-fee system, z (t A & = z(t s globally = expoetally stable f heoem 2 holds. Sce (4 s a LFR of (2, the global asymptotc stablty of, depedet o the delays, may be aalyzed fo all (costat o tme-vayg θ Θ fom that of the LFR (4 by cosdeg the vetces of the polytope Δ. heefoe, the followg Coollaes 5-6 to heoem 2 may be stated ad poved a smla way as Coollaes 3-4 to heoem. 4.2 Coollay 5. he (ufoced system (2 s globally asymptotcally stable D p q depedet of the delays f thee exst f thee exst (at most [ ] [,3 ] ( k, k,..., k BlockMatx j ( k,k,..., k ;, j k =,v ;, fo all the delays h [,h ](, = fo all = matces P P > = K < =, M M = > ;, =, whee = * Δ = v = Δ v matces * Δ wth h = h ad some eal - = = A P+ P A + h S j+ C p M C p = = = j= = ; (5 ( k,..., k = KQ ( k,..., (5 2 2 = Q 2 = (defed eqs k 3 ( k,...,k ( k ( k P B C M D = = Δ + Δ, q p pq 3 3 ( k ( k P B Δ q,...,p B qδ = BlockDag [ C p M C p h S, S,..., S M... M C p M C p h S, S,..., S ] 22 ( k,...,k 23 = 32 = BlockDag ( k,...,k Q ( k,..., k { { ( k Δ ( k C Δ p M D pq M M... MC p M D pq M = (7 Ackowledgemets. he autho s gateful to MCY by ts patal suppot of ths wok tough Gat DPI ad to the Basque Govemet by ts facal suppot though Gat GIC743-I

11 Stablty aalyss fo lea tme-delay systems 25 Refeeces [] NICULESCU, S.I. Delay Effects o Stablty. A Robust Cotol Appoach. Lectue Notes Cotol ad Ifomato Sceces, M. homa, M. Moa Eds. No. 269, Bel 2. [2] DE LA SEN M., O some stuctues of stablzg cotol laws fo lea ad tme-vaat systems wth bouded pot delays, It. J. of Cotol, Vol. 59, pp , 994. [3] WANG F. ad BALAKRISHNAN V., Impoved stablty aalyss ad ga scheduled cotolle sythess fo paamete-depedet systems, IEEE as. Automat. Cot., Vol. 47, No. 5, pp , May 22. [4] SCHERER C., Mxed H 2 / H Cotol. I eds Cotol. A Euopea Pespectve, A. Isdo, Ed., Spge-Velag, Bel- Hedelbeg, 995. Receved Novembe, 28