Hinrichsen, D.

Ilchmann, Achim

Pritchard, A. J.

Digitale Bibliothek Thüringen

               Hinrichsen, D. ; Ilchmann, Achim ; Pritchard, A. J.:  A stability radius for time-varying linear systems                     Aus: The mathematics of control theory : based on the proceedings of a Conference on Control Theory / organized by the Institute of Mathematics and Its Applications and held at the University of Strathclyde in September 1988. Ed. by N. K. Nichols .. - Oxford : Clarendon Press, 1992. (The Institute of Mathematics and its Applications conference series ; N.S., 37). - ISBN 0-19-853640-2, S. 281-290.  A stability radius for time-varying linear systemsD. I l inr ichsen A. I lchmann A. J.  Pr i tchardNotat ionI R 1  = { z e m , l z 2 0 }C - = { z € O l l ? e z < 0 }a(,4) spectmm of ,4 € C'x'GL,"(C,) t,he set of all invcrtiblc matrices T € C"x"llrll Euclidean orm of u € O'lf Dll induced operator norm for D eC^,Pl lD( . ) l l r_  =  s r rp  { l lD( t ) l l }  fo r  D(  )  €  pC ( ( to , r r ) ;  Omxr )l o ( l ( l rL2(ts , ty ;  C^)  space of  funct ions u :  ( ts , l r )  -  C-  s .  t .,  *  l l r , ( t ) l l2  is  in tegrable over  ( t6, t1)PC(( to, t r ) ;  C" t - )  set  of  p iecewise cont inuous matr ix  f i rnct ionsD ( . ) : ( t s , l r ) * ( D " -PCb(( lo , l t ) ;  O""*)  set  o[a l l  bounded matr ix  funct ions inPc ( ( t o , t r ) ;  C " ' - )PCl( (o,11) ;  G1,"(C))  set  of  a l l  p iecewise cont inuously d i f ferent iablefunct ions D(. )  :  ( ts , t1)  -  GL^(C)Cl(tn,t; C"t-) set of all continuously differentiableD ( . )  : ( 1 6 , l r ) - O ' " -Y282 D. I l inr ichsen el aI1 fntroductionIn recent years problems of robust stability have received a good deal of attention.Most of the work on time-invariant linear systems - including the successful .F/--approach (see [4], [12]) - is based on transform techniques. However, in [7], [8] astate space approach viathe concept of stabilily radiusis proposed. In the presentpaper this approach is extended to a time-varying setting.Consider a nominal svstem of the formi ( t ) = A ( t ) x ( t ) ,  t > 0 ( 1 . 1 )where l( .)  € PC(IR+,O"^). Assume that (1.1) is exqtonential ly slable, i .e. theree x i s t M , u ) 0 s o t h a tl l d ( t , " ) l l  <  l l [ e - u ( t - ' )  f o r a l l J  >  s > 0 ( l  2 )where d(r,") denotes the transition matrix of (1.1). l\Iany arrthors (see [l], [2],[3], [5], [10]) have determined bounds 6 > 0 so that exponential stabil ity of thedistrrrbed systemt ( , )  = [ ,4(/)  + D(t) ] . r( t )  , ,  > 0 (  1 .3 )is preserved wheneverl lP( ' ) l lz -  < o for  ,D( . )  €  PC(IR+,  O'"" ) . ( 1 . 4 )Tlrese bounds are conservative. Our problem is to dctcrmine a sharp rrppcr bound.We call this bound the (complex\l stability radius and define it byrc(,4) - inf { l l r( ) l l r_ lD € pCö(IR+, C'* ') (  1 .5 )and (1.3) is not exponentially stable )We afso consider the case where,4 is subjected Lo slraclured pertubations, so thatthe perturbed system ist ( r)  = [ ,4(,)  + ß(t)D(t)C(t) l  x( t) ,  r  > 0 (  1 .6 )where D(.) € PCü(IR+,C-to) is a.n znlcnou;n bounded time-varying disturbancemat, r ix  and B(. )  € PC(IR+,C" ' - ) ,  C(  )  e  PC(m,+,6rxn)  are g i len "scal ing ma-trices" defining the "structure" of the perturbation. Then the slruclurcd slabil i lylThe real stabil ity radius is dcfined analogously. In spite of its prime importance it is notstudied here, since even in the time-invariant setrrp only rudimentary results are available.Time-targing l inear sgslemsradius isrq (A;B,C)  =  in f  { l lp ( . ) l l r_ lD €  PG( IR+,C- 'p )and (1.6) is not exponential ly stable )(  1 . 7 )In the unstructured case 16(,4) is simply the distance of (l.l) from the set ofnot exponentially stable systems with respect to the l--norm.Rcmark 1.1 The fol lowing propert ies are easi ly obtained:(a) 16(r l)  = 0 €=+ (1.1) is not exponential ly stable(b) 16(4,,1) = arc,(A) for al l  o ) 0(.) ,a(.) -  16(,4) is continuous on PCo(lR+, C"')2 Bohl exponent and Bohl transformationFor the stabil ity behaviour of (1.1) the numbcrk6(A):= in f  l -w € lRl f i / -  > 0 :( 2 . 1 )t  2  s  > 0 + l l / ( t ,  s ) l l  <  M.e-- ( ' - " )  )introduced by Bohl [ l ]  is useful.  \ \ 'e cal l  ks(,4) the Bohl erponent of ( l . l ) .  I t  ispossible that ß6(,4) = *oo. The following properties are easily seen.Proposit ion 2.L Let.4(.) € PC(I&, O'x').  Thcn(a) ß6(,a) < 0 +=+ (1.1) is exponential ly stable(b)  I f  / ( . )  =  Ä  €  (D"x '  thenkB(A) = max l?e);(,4), where );(,4) are the eigenvalues of .4.(c) In the scalar case, i .e. n = 1, we have16(4)  =  -ka  . (A)(d) For the matrix case only an inequality is valid:re (A)  <  -ka(A)284 D. I l inr ichsen el alRemark 2.2We want to emphasize that kn(A) may be a bad indicator for therobustness margin of (1.1). Consider,  | . p  r - 3 1  | -  1  0 l. 4 * = - | . ö  1 . J ' D * = r ' - ' l _ i  ; ] ,  r o r f r € t Nfnenohl1kn(Ar)  = -oo.  I lowever ,  o( i l *D*)  = { i , i -2 f t }a l thorrghol 'J i l lOr l l  =0. Thus Ji lLro(e*) = O.The following properties of the Bohl exponent can be found in [3].Proposition 2.3 Let ,4(.) e PC(IR.1 , C""). Then(a) fr6(,a) is f inite if ,a(.) is bounded.(b)  kB(,4)  is  f in i te  i t r  supo<;r -s ;<r  l l / ( t ,s) l l  <  oo.(c) It ßp(,a) ( oo thcnk B ( A\ :,',1T"'lg l* f d-'t" "l tt\ \ 'e now analyse the effect of t ime-varying l inear coordinate transformationsz(t)  = T(t)- t  x(t) ,  "( . )  € pcl(rR+, Gt"(c))  (2.2)on the  sys tem (1 .1 )  wh ich  y ie lds2(t) = 4111t111, where ,i - T-1AT - T-li (2 3)These transformations wil l  not,,  in general,  preserve exponential stabi l i ty. Thcre-fore we introduce the set of ßohl transformalions ßn , i .e. l ,he set of al l7 ( . )  €  PCt ( IR+,  Gr" (O) )  such tha ti n f {e€  t r t l lM,  >  0  :V t ,s  )  0+  l l r ( t ) - t l l  l l ? . ( " ) l l  ( ,4 I .c ' l t - ' I1  -  0  (2  4 )Remark 2.4 It, is obvious that(a) the set Bn lorms a group wil,h respect to (point,u'ise) multiplication(b) B" contains the group of Lyapunou lransforma.lions, i.e. all7 ( . )  €  PCr ( IR+ ,GL" (C) )  so  t ha t  7 ( . ) , 7 ( ' ) - t , I ( . )  a re  bo r rnd r :d ,(c) As(,a) = kB(T-r AT - T-ti) for all T € BnTim e- tt ary in g lin e ar sy sl emsThe following proposition shows that even in the time-invariant case a similaritytransformation may drastically change the stabil ity radius.Proposition 2.5 l7l It A e O'x' with o(A) C C- thcn {rc(T-rAT); T eGI"(O)) is equal to the interval (0,-maxl?e);(,4)] with possible exception ofthe right extremum.In the scalar case we can proveProposition 2.5 Ifr ( t )  = a( t ) r ( / ) ,  o( . )  e  PC(I fu,  C)  (2.5)has a slricl Rohl exponent, i.e.r^g l ld( r ,  s) l lß 6 ( a )  = , , , ] l j l _  , _ "then there ex is ts  O € 6r  so that  : ( t )  = O( l ) -1r( t )  conver ts  (2.5)  in to2( t )  = ks(a)z( t )3 The perturbation operatorIn  the  t ime- invar ian tse tup ,  where  (A ,R,C)  €  O 'x 'xCnxm xCP" ' ,  the  s t ruc tu rcdstabi l i ty radius can be characterized via the convolut ion operatorL 6  :  L 2 ( 0 , o o ; C ^ )  +  1 , 2 ( 0 ,  o o ; O P )ü ( . )  H  ( t  -  i  c e A ( t - " )  ß u ( s ) d s )  ( 3 ' t )0as followsProposi t ion 3.1 [8]  I f  a( ,a)  C C- and G(s) : -  C(sIn- .4)-1.8 then(  | -  t - r16(A,B,c)  =  {  l l ro l l - '=  l f f f l l c ( tu ' ) l l l  i r  G +0[  -  i f  G = 0In order to explore the possibil i t,y of obtaining similar results for t imc-varying sys-tems, we consider the pa.ramctrizcd family of perrurbalion opcrarors (tf l)r"em1defined byD. Il inrichsen et alL!"  :  L2( ts ,oo;C-)  + .L2(16,  oo;  CP),  lo  2 0(3 .2 ),r(.) H (t * i C(t)ö(t,s)B(s)rr(.s)ds)l sassociated withE  =  ( . 4 ,  B , C )  e  P C ( I R + , C n x n )  x  P C 6 ( I R 1 , C o ' - ) t(3.3)xPC6(IRa, (DP"), ßB(,,1) < 0Basic properties of LE^ are summarized in the followingProposition 3.2 [6](") fä is a bounded operator(b) to * l l t l" l l  is monotonically decreasing on lR1(.) l l fP"ll =llL|l l  for all to,tr e IRa if Ä, B,C are periodic with a commonperiod(d )  l l re l l - '  <  rs (A ;B,C)(e) For the unstructured case, i.e. B(.) = C(.) = 1,, if Af,ro ) 0 satisfy(1.2)  then# < l l rBl l - '  s ,JyLl l r i l l l - '  < rc(A)As opposed t,o the time-invariant case llIfl l l-1 or,jF* llri l l l-t do not necessarilycoincide with 16(,4; B,C). Even in the simple case when o(.) e PC(IRa,IR) isperiodic and ö = c = 1, we have worked out an example in [6] for whichl l re l l - t  =  l l r i l l - t  <  ' " (o)  for  a l l  to , l r  €  rR+Ilowever notc that scalar Bohl tra.nsformations O € 81 do not change the stabilityradius but wil l change the norm of the perturbation operator. LetDs := ( .4 -  $4;  n,c1By using Proposition 2.5 and 3.1 one can showProposition 3.3 Suppose a(.) e PC(IR1, O) has a strict Bohl exponent [s(o) < 0and Ö,c€ C.  Then there ex is ts  a O eBt  such thatre(a;b,c)  = l l l , i " " l l -1  for  a l l  to  2 0.Tim e-unry in g l inear  sgstemsFor the matrix case we have the followingConjccture 3.4 Suppose E = (A,ß,C) satisfies (3.3), thent : (A :ß ,C)=  s r rp  {  r i -  t t r ,A " t t - t }'  " ' b '  l t s * -  "  ' o  "  )4 The associated parametrized differential Ric-cati equationIn the t ime-invariantsetupanothcr useful characterization of 16(,r1; R,C)is possiblevia the parametrized algehraic R,iccati  equation, AREpA*P +  PA-  pC-C -  PBB*  P =0,  p  €  IRPropos i t i on  4 .1  [ 8 ]  Suppose  (A ,B ,C)  €  0 ' x ' x  Onxm xCp ' ^  and  a ( ,4 )  C  O- :(a)  I f  -  oo < p < r$(A;B,C) then there ex is ts  a unique stabi l iz ingTlermitian solution Pp of AR.E,, that is a solut,ion Pp = Piwhich satisfieso(A -  BB.  P")  C A_.If p = rzo then there exist,s a unique Hcrmitian solution Pr6 ol ARE',having the property o(A - BB'P,6 ) C e -(b) If there exists a l{ermitian solution Po of ARE, then necessarilyp 3 r 2 o ( A ; B , C ) .Guided by t,his result we study, in the t, ime-varying setting, l,he paramctrized dif-ferential Riccati equation, DREpP(,)  + A.( t )P( t )  + P( t )A( t )  -  pC.( t )C( t )  -  P( t )B( t )8. ( l )P( t )  = 0,  I  )  toassociated with the systemi ( t )  =  A ( t ) x ( r )  +  13 ( t )u ( r ) ,  r ( 10 )  -  " o  €  C"  ly \ t ) = c ( t ) x ( t ) ,  r ) r o ) r  i  ( 4 ' l )Throughorrt, this sect,ion we assume E = (,4, B, C) satisfies (3.3).Kalman [9] and R,eghis anil l l fegan [l l ] among others, have studietl diffcrent,ialRiccat,i eqrrations, however their results cannot be applied to DRD, if p > 0.Just as in the time-invariant case we consider the parametrized opt,imal controlproblem OCPr:288 D. I l inr ichsen eI alN { i n i r n i z e  o v e r  ? ,  €  L 2 Q 6 , t 1 ; C , ^ )J o(x s ,  ( ts , r '  ) , , ( . ) )  =  |11; ,1r ;  ;1 '  -  p l ly (s) l l? l r /st swhcre  y ( . )  i s  dc f incd  v ia  (4 .1 )  and p  €  IR.  I f  p  <  0  th is  i s  the  usua l  l inear  quadra t icregulator problem LQR., whereas in our situation p ) 0, so that the stat,e penalt 'y isnegative. To consider a cost functional with negative state penalty is qrr i te naturalin this context since we are concerned with a minimum norm destabi l izat ion problemwhile the classical LQR problem is concerned with st,abi l izat ion.The analysis of the DR.E, and i ts relat ion to t .he OCP, is qrr i te involvcd, de-tai ls may be found in [6].  I lcre we only state the main results.Propos i t ion  4 .2  ( f in i t c  t ime)  I f  p  <  l lL ] j1 ' ,  0  (  to  (  11  (  oo ,  then(a) there exists a unique l lermit ian solut ion P' '( .)  of DREeo n  [ t 6 , 1 1 ]  w i t h  P " ( t 1 )  =  g( b )  P t ' ( , ) ( 0  ( r c s p .  2 0 )  f o r a l l  l € [ r r , t 0 ]  i f  2 ) 0  ( r e s p .  p S 0 ) .(c) the minimal cost o[ OCPo is" . t , , i | , f ,  ; t : ^ , J c ( , . o ' ( 1 0 ' / r ) ' r r ( ' ) )  =  <  r o ' P t ' ( / o ) " 0  >(d) the optimal control is given byu ( l )  =  - 4 . ( t ) P " ( t ) " r ( l )where  r ( . )  so lves  i ( t )  =  [ .4  -  I ]B*  P t t l ( t ) r ( t ) ,  r ( te )  =  16 .The next proposit ion is obtaincd by studying what happens i f  l l  * qo.Proposit ion 4.3 ( inf ini te t ime) I f  p < l lL?.l l - t ,  ls ) 0, thcn(a) P+(t) = ,JF_ P"(l)  exists for al l  t  )  ls and yields a bounded l lcrmit iansolut ion of DR.Eo;( b )  P + (  )  i s  t h e  o n l y  s o l u t i o n  s o  t h a l ,  k n ( A -  R t - P + )  <  0 ;(c )  fo r  any  o ther  bounded I l c rn r i t ian  so lu t ion  8( . )  €  Ct ( t !o ,x ;C" ' " ) ,t ä > l o  o f  D R E '  w e h a v e  Q ( ) < P + ( t )  f o r a l l  l ) 1 / o ;(d) the minimal cost is, . " , , 1 1 , L , o -  , J ' ( " ' ( 1 0 ' o o ) ' " ( ' ) )  =  <  , ' 6 ' P + ( 1 6 ) 1 6  > ;(e) the optimal control isu ( / )  =  - / 1 . ( / ) P + ( / ) r ( / ) ,  t ) t owhcre r(') ;ff: IA - ßR.p+](r)r(r), c(rs) = 16, r ) 16Time-aarging l inear syslemsAs a partial converse of Proposition 4.3 we haveProposit ion 4.4 \t  QO € c1(ro,oo; c"" ')  is a bounded l lernri t ian sorution ofDREe on (ls, oo) then necessarily p < llL],ll-,.While the previous two proposit ions yield a complcte characterization of thenorm l l I f l l l  in terms of the associated parametrized dif ferential R.iccati  equation,they do not provide a full generalization of Proposition 4.1 to t,he time-varyingcase. To f ind a complete characterization of the stabi l i ty radius re(A;B,C) fortime-varying systems is an open problem.5 Robust Lyapunov functions and nonlinear per-turbationsThe following proposition shows how solutions of the parametrized differenl,ial Ric-cati equation DREp can be used to construct robusl LyaprLnov functions for thesys tem (1 . i ) .Proposition 5.1 Suppose 0 < p < l lL?"ll-r. tf pp(.) solves DREo r,henV ( t , x )  : =  - (  x , P o ( t ) r ) ,  t l t o ,  r e C nis a common Lyapunov function for all perturbed systemsi ( t )  =  lA+  BDCI ( t ) x ( t ) ,  t )  t s ,  r ( 16 )  =  aow i r h l l r ( . ) l l ? * < p .l f  (4, B, c) are constant mat.rices and a(,4) C c - a similar resurt to proposition5.1 holds t rue for  l l l ( . ) l l r_  < ra( .4;8,  C) ,  see [8] .Using the above Lyapunov function it is possible to extend our robustness anal-ysis to nonfinear perturbations of the form Ä(t) = B(t)N(C(t)r,t) so that theperturbed system isi ( t )  = A( t )x( t )  + ß( t )N(C(t ) . r ( t ) , r ) ,  r ( ro)  -  x6,  t  )  t11 (b l )where (,4, B, C) satisfy (3.3) and N : IRp x IR,.. + IR- is continuously differcnt.ia.ble.\{e assume N(0,1) = 0 so that 0 is an equil ibrium state of (5.1). The followingre-sult shows that no nonlinear perturbation with global gain smaller than ll1,[l l-1can destabil ize the system.Proposition 5.2 Suppose 7 < l lI, l" l l-1 andl lN(s, t ) l l  S r l lv l l  for  a l l  r  )  to ,  y  eCpThen the origin is globally exponentially stable for the system (5.1).It is not clear whether an analogous tatement holds (over a suitable tinre interval(ro' oo)) if the gain of the nonlinear perturbation is strictly less than ra(A: B,c\.289Y290 D. Il inrichsen et alReferences[1] Bohl, P. (1913) Ueber Differentialgleichungen; J. Reine Angew. IUat,h. 144,284-3 13[2] Coppel, W.A. (1978) Dichotomies in Stahil itg Theory; Lect. Notes l\tath. 629,Springer-Verlag, Berlin et al.[3] Dafeckii, Ju.L. and Krein, M.G. (1974) Stabil ity of Solutions of Di[JerentialEqualions in Banach Spaces; AMS, Providence, Rhode Island[4] Francis, B. and Zames, G. (1983) Feedback m.inimar sensil. i tt i t1 and optimalrohuslness; IEEE Trans. Autom. Control 28, 585-601[5] I lahn, W. (1969) Stabil ity of lrtotion; Die Grundlehren der math. Wis-senschaften Bd. 138, Springer-Verlag, New York[6] I l inrichsen, D.,I lchmann, A. and Pritchard, A.J. (1989) Robaslness of stabil itgof t ime-uarying syslems; J. Diff. Eqns Vol 82.219-250.[7] I l inrichsen, D. and Pritchard, A.J. (1936) Stabitity radii of l inearsystcms; Syst.Contro l  Let t .  7 ,  1-10[8] I l inrichsen, D. and Pritchard, A.J. (1986) Stabil ity radius for structured per-lurbal ions and lhe a lgebraic  Riccat i  equal ionl  Syst .  Contro l  Lct , t , .  8 ,  105-113[9]  Kal rnan,  R.E.  (1960) Conlr ibut ions to the theory of  opt imal  contro l ;  Bul l .  Soc.Math.  Mexico 5,  102-119[10] Perron, O. (1930) Die Stabil it.älsfrage d r Differenlialgleichnngcn; IUath. Z. 32,703-728[ll] Reghis, M. and Megan, M. (1977) Riccati equalion, eract conlrollahil i ty andslabil i ly; Scm. Ecuat. Funct., Univ. Timisoara 42[12] Zames, G. (1981) Feedback and oplimal sensitiuity: moilcl re.ference transforma-tion, muhiytl icaliue seminorms and approrimate inuerses; IEEE Trans. Autom.Contro l  16,  301-320

A stability radius for time-varying linear systems

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A stability radius for time-varying linear systems

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