Descriptor system techniques and software tools

Descriptor system techniques and software tools Andreas V arga Gilching , Germany var ga.an dr eas@gmail.com Abstract The role of the descr iptor system rep resentation as basis for reliab le nu merical comp u tations for system analysis and synthesis, and in particu lar , fo r th e ma- nipulation of rational matr ices, is discussed and av ail- able ro bust n umerical software too ls are described. K eywords Modelling; Differential-algeb raic systems; Ratio- nal matric e s; Numerical analysis; Software tools AMS subject classifications: 34A09, 9 3C, 93 B20, 93B40, 93C05 , 93D20 1 Introduction A linear time-invariant (L TI) continuou s-time de- scriptor sy stem is described by the e q uations E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) + Du ( t ) , (1) where x ( t ) ∈ R n is the state vector, u ( t ) ∈ R m is the input vector, y ( t ) ∈ R p is the outpu t vector , and A , E ∈ R n × n , B ∈ R n × m , C ∈ R p × n , D ∈ R p × m . The square matr ix E is possibly sing ular, but we assume that th e lin e ar matrix pencil A − λ E , with λ a complex parameter, is regular (i. e ., det ( A − λ E ) 6≡ 0 ) . A L TI discrete-time descrip tor system has the form E x ( k + 1 ) = Ax ( k ) + Bu ( k ) , y ( k ) = Cx ( k ) + Du ( k ) . (2) Descriptor system representatio ns of the f orms ( 1 ) and ( 2 ) ar e the mo st g eneral descriptio ns o f L TI sys- tems. Standard L TI st ate-space sy stem s cor r espond to the case E = I n . W e will alternatively deno te the L TI descriptor sy stems ( 1 ) and ( 2 ) w ith th e quadru- ples ( A − λ E , B , C , D ) o r ( A , B , C , D ) if E = I n . Continuou s-time descriptor systems freq uently arise when modeling inter connected systems inv olv- ing linear differential equations and alg ebraic re - lations, and are also co mmon in mo deling con- strained mecha n ical systems (e.g. , contact pro blems). Discrete-time descriptor representations are en c o un- tered in the mo d eling of some econ omic processes. The d escriptor system representation is instrumen tal in de vising general com putationa l procedures (even for standard L TI system s), who se interm ediary step s in volve oper ations leading to d escriptor repr esenta- tions (e.g., system inversion or system c o njugatio n in the discrete- time case). The inp u t-outpu t b ehavior of the L TI systems ( 1 ) and ( 2 ) can b e describ ed in the fo r m y ( λ ) = G ( λ ) u ( λ ) , (3) where u ( λ ) and y ( λ ) are the transformed input and output vectors, using, in the continuou s-time case, the Laplace transform with λ = s , and, in the discrete- time case, the Z -transfor m with λ = z , and whe re G ( λ ) = C ( A − λ E ) − 1 B + D (4) is the tr ansfer fun ction matrix (TFM) of th e system. The tran sfer fun ction matrix G ( λ ) is a rational ma- trix having en tries which are ration al f u nctions in the complex variable λ ( i.e., ratio s of two po lynomials in λ ). G ( λ ) is pr o per ( strictly pr o per ) if each entry of G ( λ ) has the degree of its den ominator larger than or equal to (larger than) the degre e of its n umerator . If E is singular , th en G ( λ ) could h av e entr ies with the degrees of nu m erators exceeding the degrees o f the correspo n ding denominato r s, in which case G ( λ ) is impr oper . W e will use the alternative notation G ( λ ) : =  A − λ E B C D  , (5) to relate the TFM G ( λ ) in ( 4 ) to a particular q uadrup le ( A − λ E , B , C , D ) . An impo rtant application of L T I descriptor sys- tems is to allow the nu merically r e liable manipula- tion of rational matrices (in particular, also of poly- nomial matrices). This is possible, bec a use for any rational matrix G ( λ ) , there exists a quadr uple ( A − λ E , B , C , D ) with A − λ E regular, such th at ( 4 ) is fulfilled. Determ ining th e matrices A , E , B , C and D for a given rational matrix G ( λ ) is kn own as the re- alization problem an d th e q uadrup le ( A − λ E , B , C , D ) is called a descriptor rea liza tio n of G ( λ ) . The solu - tion of the realization problem is not u nique. For ex- ample, if U and V are invertible matrices of the same order as A , th en two realizations ( A − λ E , B , C , D ) and ( e A − λ e E , e B , e C , D ) related as e A − λ e E = U ( A − λ E ) V , e B = U B , e C = CV , (6) have the same transfe r fu nction matrix. The rela- tions ( 6 ) define a (restric ted ) s imilarity transforma- tion between the two de scr iptor system rep resenta- tions. Perfo rming similarity tran sformation s is a basic tool to m anipulate descrip to r system represen ta tio ns. An importan t asp ect is the existence of m inimal re- alizations, which are descriptor r ealizations with the smallest p ossible state dimension n . The characteri- zation of minimal d escriptor re alizations in terms of relev ant systems pr o perties is don e in Section 3 . T o simplify the presentation , we w ill assume in most of the cases d iscussed that the employed r ealizations of rational m atrices are minimal. Complex synthesis appro aches of controller s a nd filters for plants mo deled as L TI systems are of- ten described as conce ptual co mputation al procedu res in terms of input- output representatio ns via TFMs. Since the manipulatio n of ratio nal matric e s is nu mer- ically not advisab le becau se of the potential high sen- siti vity of polyno mial based repre sen tations, it is gen- erally accep ted that the man ipulation of rational ma - trices is best pe rformed via their equ iv alent descrip- tor realizations. In what follo ws, we focus on d is- cussing a selection of descrip tor system technique s which are frequ e ntly encountered as computatio nal blocks o f the syn thesis proce d ures. Whene ver pos- sible, we will indicate the best av ailable n umerical a l- gorithms, but refrain from discussing computational details, which can be found in th e cited references. W e conclud e with the presentatio n of a short overview of available software tools fo r descriptor system s. 2 Basics of manipulating rational matrices In this section, we present the b asic manipula- tions o f ra tional matrice s, wh ich rep r esent the build- ing blo cks of m o re inv olved m anipulation s. Basic operations W e con sider some oper a tions inv olving a sing le TFM G ( λ ) with the d escriptor realization ( A − λ E , B , C , D ) . The transposed T FM G T ( λ ) cor respond s to th e dual descriptor system with the realiza tio n G T ( λ ) = " A T − λ E T C T B T D T # . If G ( λ ) is invertible, then an in version free real- ization o f the in verse TFM G − 1 ( λ ) is given b y G − 1 ( λ ) =   A − λ E B 0 C D I 0 − I 0   . This r ealization is n o t min imal, e ven if the o riginal realization is minimal. Howe ver , if D is inv ertible, then an alternative r e alization of the inverse is G − 1 ( λ ) =  A − BD − 1 C − λ E − BD − 1 D − 1 C D − 1  , which is minimal if th e origin al realization is mini- mal. Notice that this operatio n may genera lly lea d to an im proper inverse even for standard state-space re- alizations ( A , B , C , D ) with singu lar D . The co njugate (or ad joint ) T FM G ∼ ( λ ) is d efined in th e con tinuous-tim e case as G ∼ ( s ) = G T ( − s ) a nd has the realization G ∼ ( s ) = " − A T − sE T C T − B T D T # , while in th e d iscrete-time ca se G ∼ ( z ) = G T ( z − 1 ) and has the realization G ∼ ( z ) =   E T − zA T 0 − C T zB T I D T 0 I 0   . If G ( z ) has a standard state-space realization ( A , B , C , D ) with A invertible, then an alternative re- alization o f G ∼ ( z ) is G ∼ ( z ) = " A − T − zI − A − T C T B T A − T D T − B T A − T C T # . This operation may lead to a conjugate system with improp e r G ∼ ( z ) , for a stand ard discrete-time state- space realizatio n ( A , B , C , D ) with singular A . Basic couplings Consider no w t wo L TI systems with th e ratio- nal TFMs G 1 ( λ ) and G 2 ( λ ) , having the de- scriptor realization s ( A 1 − λ E 1 , B 1 , C 1 , D 1 ) and ( A 2 − λ E 2 , B 2 , C 2 , D 2 ) , respectively . The product G 1 ( λ ) G 2 ( λ ) re p resents the series cou pling of th e two systems an d has the descrip tor realization G 1 ( λ ) G 2 ( λ ) : =   A 1 − λ E 1 B 1 C 2 B 1 D 2 0 A 2 − λ E 2 B 2 C 1 D 1 C 2 D 1 D 2   . The parallel coupling correspon ds to the sum G 1 ( λ ) + G 2 ( λ ) an d has the realizatio n G 1 ( λ ) + G 2 ( λ ) : =   A 1 − λ E 1 0 B 1 0 A 2 − λ E 2 B 2 C 1 C 2 D 1 + D 2   . The colu mn co ncatena tion of the two sy stem s corr e- sponds to building h G 1 ( λ ) G 2 ( λ ) i and h as the r e alization  G 1 ( λ ) G 2 ( λ )  =    A 1 − λ E 1 0 B 1 0 A 2 − λ E 2 B 2 C 1 0 D 1 0 C 2 D 2    . The r ow co ncatena tion o f the two system s corre- sponds to building  G 1 ( λ ) G 2 ( λ )  and has the real- ization  G 1 ( λ ) G 2 ( λ )  =   A 1 − λ E 1 0 B 1 0 0 A 2 − λ E 2 0 B 2 C 1 C 2 D 1 D 2   . The diagona l stacking of the two systems cor r esponds to building h G 1 ( λ ) 0 0 G 2 ( λ ) i and has the re alization  G 1 ( λ ) 0 0 G 2 ( λ )  =    A 1 − λ E 1 0 B 1 0 0 A 2 − λ E 2 0 B 2 C 1 0 D 1 0 0 C 2 0 D 2    . 3 Minimal Realization The man ipulation of ration al m a tr ices via the ir de- scriptor r e p resentations relies on the fact tha t fo r any rational matrix G ( λ ) ∈ R ( λ ) p × m , there exist n ≥ 0 and the r eal matrices E , A ∈ R n × n , B ∈ R n × m , C ∈ R p × n and D ∈ R p × m , with A − λ E regular, such that ( 4 ) holds an d n has least possible value. If G ( λ ) is prop er , this fact is a well-known r e sult of the realiza tio n the- ory o f standa r d state-space system s for w h ich nu mer- ically reliable min imal realizatio n meth o ds exist. Us- ing this result, a simple realization techniq ue allows to obtain a minimal de scr iptor realization of a ( generally improp e r) ration al matr ix G ( λ ) b y using the additive decomp o sition G ( λ ) = G p ( λ ) + G pol ( λ ) , where G p ( λ ) is the proper p art of G ( λ ) and G pol ( λ ) is its strict poly nomial p art (i.e., withou t constant term). The pro per part G p ( λ ) has a standard state-sp ace re - alization ( A p , B p , C p , D p ) and for the strictly proper TFM λ − 1 G pol ( λ − 1 ) we can build another standard state-space realization ( A pol , B pol , C pol , 0 ) . T hen, we obtain G ( λ ) =   A p − λ I 0 B p 0 I − λ A pol B pol C p C pol D p   . A minimal descriptor system realization ( A − λ E , B , C , D ) is ch aracterized by th e follow- ing five conditions. Theorem 1 ([ V erghese et al , 1 981 ]) . A d escriptor system r ealization ( A − λ E , B , C , D ) of or der n is min- imal if the follo wing con ditions ar e fulfilled : ( i ) rank  A − λ E B  = n , ∀ λ ∈ C , ( ii ) ran k  E B  = n , ( iii ) rank  A − λ E C  = n , ∀ λ ∈ C , ( iv ) rank  E C  = n , ( v ) A N ( E ) ⊆ R ( E ) . Here, N ( E ) den otes the (righ t) nu llspace of E , while R ( E ) denotes the ran ge space of E . The cond itions ( i ) and ( ii ) ar e known as finite and infinite co ntr ollability , respe c ti vely . A system which fulfills both ( i ) and ( ii ) is called contr o llable . Sim- ilarly , the cond itions ( iii ) and ( iv ) are known as fi- nite and infinite observa bility , respec tively . A system which fu lfills b oth ( iii ) and ( iv ) is called observable . Condition ( v ) expresses the ab sence of non-d ynamic modes ( see th e ir definition in Section 5 ). A descrip tor realization which satisfies on ly ( i ) − ( iv ) is called irr e- ducible ( also weak ly m inimal). The numerica l com- putation of min imal re alizations is add r essed, f or ex- ample, in [ V arga , 2017b , Section 1 0.3.1 ]. 4 Canonical Forms of Linea r P encils The main appeal of descr iptor system tech niques lies in their ability to address various an alysis and synthesis problems o f L TI systems in the mo st g e n - eral setting, b oth from theor etical and c omputatio nal standpoin ts. The basic math ematical ingre d ients for addressing analysis a nd synthesis pro blems of de- scriptor systems ar e two canon ical for ms of linear ma- trix pen cils: th e W eierst rass canonica l form of a regu- lar pencil and th e Kr onec ker cano n ical form o f a sin- gular pen cil. For a given a linear pe n cil M − λ N (reg- ular or singular), th e corresponding c a nonical form e M − λ e N can be o b tained u sin g a pencil similarity transform ation of the f orm e M − λ e N = U ( M − λ N ) V , where U and V are suitable in vertible m atrices. If the p encil M − λ N is regu la r an d M , N ∈ C n × n , then, th ere exist invertible matrices U ∈ C n × n and V ∈ C n × n such that U ( M − λ N ) V =  J f − λ I 0 0 I − λ J ∞  , (7) where J f is in a ( complex) Jordan can onical form J f = d iag  J s 1 ( λ 1 ) , J s 2 ( λ 2 ) , . . . , J s k ( λ k )  , (8) with J s i ( λ i ) an e lementary s i × s i Jordan b lock o f the form J s i ( λ i ) =       λ i 1 λ i . . . . . . 1 λ i       and J ∞ is nilpotent and has th e (nilpo tent) Jordan f orm J ∞ = d iag  J s ∞ 1 ( 0 ) , J s ∞ 2 ( 0 ) , . . . , J s ∞ h ( 0 )  . (9 ) The W eier strass ca n onical f o rm ( 7 ) exhibits the fi- nite and infinite eigenv alues of the pen cil M − λ N . Overall, b y in cluding all multiplicities, there are n f = ∑ k i = 1 s i finite eigenvalues and n ∞ = ∑ h i = 1 s ∞ i infinite eigen values . Infinite eigenv alues with s ∞ i = 1 are called simple infi nite e igenvalues . If M a nd N are r eal matrices, then the re exist real matr ic e s U a n d V such that the pencil U ( M − λ N ) V is in a r eal W eierstr ass canon ica l form , where the on ly difference is that J f is in a real Jordan form. In th is form, the elemen - tary real Jordan b locks correspo nd to pairs of co mplex conjuga te eigenv alues. If N = I , then all eigenv alues are finite a n d J f in the W eierstrass f o rm is simp ly th e (real) Jorda n form of M . The transfor mation m atrices can be chosen such th at U = V − 1 . If M − λ N is an arbitrary (singular) p encil with M , N ∈ C m × n , then, the r e exist inv ertible matrices U ∈ C m × m and V ∈ C n × n such that U ( M − λ N ) V =   K r ( λ ) K re g ( λ ) K l ( λ )   , (10) where: 1) The full row rank p encil K r ( λ ) h a s the form K r ( λ ) = d iag  L ε 1 ( λ ) , L ε 2 ( λ ) , · · · , L ε ν r ( λ )  , with L i ( λ ) ( i ≥ 0) an i × ( i + 1 ) bidia g onal p encil of fo rm L i ( λ ) =    − λ 1 . . . . . . − λ 1    ; (1 1 ) 2) Th e regular pe ncil K re g ( λ ) is in a W eierstrass canonical fo rm K re g ( λ ) =  e J f − λ I I − λ e J ∞  , with e J f in a (c o mplex) Jordan c anonical f orm as in ( 8 ) and with e J ∞ in a n ilpotent Jord a n f orm as in ( 9 ); 3) Th e full column ra n k K l ( λ ) h a s the form K l ( λ ) = d iag  L T η 1 ( λ ) , L T η 2 ( λ ) , · · · , L T η ν l ( λ )  . As it is apparent f rom ( 10 ), the Kronecker can oni- cal f orm exhib its the righ t and left singu lar structur es of the pencil M − λ N v ia the full row r a nk block K r ( λ ) and full colum n rank block K l ( λ ) , re spectiv ely , and the eigenv alue structure via th e r egu la r penc il K re g ( λ ) . The full row rank pencil K r ( λ ) is n r × ( n r + ν r ) , wh e r e n r = ∑ ν r i = 1 ε i , th e full column rank pen cil K l ( λ ) is ( n l + ν l ) × n l , wh e re n l = ∑ ν l j = 1 η j , wh ile the regu- lar pencil K re g ( λ ) is n re g × n re g , with n re g = ˜ n f + ˜ n ∞ , where ˜ n f is the nu mber of eigenvalues of J f and ˜ n ∞ is the number of in finite eigenv alues of I − λ e J ∞ (or equiv alently the number of null eigen values of e J ∞ ). The no rmal rank r of the pen cil M − λ N results as r : = rank ( M − λ N ) = n r + ˜ n f + ˜ n ∞ + n l . If M − λ N is r egular , then the re are no left- an d right-Kro necker struc tures and the Kron ecker canon- ical f o rm is simp ly the W eierstrass can o nical form . The red uction of matrix pencils to the W eierstrass or Kronecker ca n onical fo rms gen e r ally inv olves the use o f non-o rthogo nal, po ssibly ill-cond itioned, tran s- formation matrices. Theref o re, the compu tation of these form s must be av oided when devising numer- ically reliable algorithms for de scr iptor systems. A l- ternative cond ensed forms, as th e (o rdered) gen eral- ized real Schur form of a regular pe n cil or various Kronecker-like f orms of sin gular pencils, can b e de- termined by u sing exclusively perf ectly condition ed orthog onal transfor mations an d can be alw ays used instead of the W eierstrass or Kron ecker canonical forms, respectively , in addressing th e computatio n al issues of d escriptor system s. Numerically stable al- gorithms to deter m ine Kronecker-like f orms are de- scribed in [ V arga , 2017b ] and in the literature cited therein. 5 Advanced Descriptor T echniques In this section we discuss a selection of pr oblems in volving rational matrices, whose solutio ns in v olve the use of advanced descriptor sy stem manipu lation technique s. These techniqu es are instrumen ta l in ad- dressing c o ntroller and filter synthesis pro b lems in the most general setting by using numerica lly r eliable al- gorithms for the reductio n of linear matrix p encils to approp riate condensed form s. Normal rank The no rmal rank o f a p × m r ational matr ix G ( λ ) , which we d enote by r ank G ( λ ) , is the maxim al nu m- ber of linea rly independent rows ( o r columns) over the field of ratio nal f unctions R ( λ ) . It can be sh own that the nor mal rank of G ( λ ) is the maxima lly pos- sible rank of the com plex matrix G ( λ ) for all values of λ ∈ C such that G ( λ ) h as finite norm. F or the cal- culation o f the no rmal rank r of G ( λ ) in terms of its descriptor rea liza tion ( A − λ E , B , C , D ) , we use the re- lation r = rank S ( λ ) − n , where r ank S ( λ ) is th e nor mal ran k of the system ma- trix pen cil S ( λ ) defined a s S ( λ ) : =  A − λ E B C D  (12) and n is the ord er o f the descr iptor state-space real- ization. The normal rank r can be easily determined from the Kro n ecker f orm of the pe n cil S ( λ ) as r : = n r + n re g + n l − n , where n r , n re g and n l defines the norm a l ranks of K r ( λ ) , K re g ( λ ) , and K l ( λ ) , respectiv ely , in th e Kro- necker form ( 10 ) o f S ( λ ) . For numerical computation s, the Kro n ecker-like form of the sy stem ma trix pencil provides the same structural inform a tion by using pencil reductio n algo - rithms ba sed on orthog onal transforma tio ns. An even more e fficient way to de termine the nor mal ran k is to determine th e maximu m o f the rank of S ( λ ) for a few random values of the frequen cy variable λ by using singular values based rank ev aluations. Poles and zer os The poles of G ( λ ) ar e re lated to Λ ( A − λ E ) , the eig en- values of the pole pen cil A − λ E (also known as the generalized eigenv alues of the pair ( A , E ) ). For a minimal realization ( A − λ E , B , C , D ) of G ( λ ) , the fi- nite poles of G ( λ ) are the n p , f finite eigenv alues in the W eierstrass canonical fo rm of the regular pencil A − λ E , while the nu mber of infinite poles is given by n p , ∞ = ∑ h i = 1 ( s ∞ i − 1 ) , where s ∞ i is th e mu ltiplicity of the i -th infinite eigenvalue. Th e infinite eigenv al- ues o f multiplicity ones are the so-called no n-dyn amic modes . Th e McMillan degree of G ( λ ) , denoted b y δ  G ( λ )  , is the total numbe r of po les n p : = n p , f + n p , ∞ of G ( λ ) δ  G ( λ )  : = n p and satisfi es δ  G ( λ )  ≤ n . A pr oper G ( λ ) h as o nly finite poles. A proper G ( λ ) is stable if a ll its poles be long to the approp riate stable region C s ⊂ C , where C s is the open lef t half plane of C , for a co ntinuou s-time sys- tem, and the interior of the un it circle centered in the origin, for a d iscr ete-time system. G ( λ ) is unstable if it has at lea st one p ole (finite or in finite) ou tside of the stability dom ain C s . The zero s of G ( λ ) are tho se comp lex values of λ (includin g also infinity), wh e re the r ank of the sys- tem matrix pe ncil ( 12 ) dro p s below its normal rank n + r . T herefor e, the zer os can be de fin ed o n the ba- sis o f the eig en values of the regular part K re g ( λ ) of the Kronecker form ( 10 ) of S ( λ ) . The finite zeros of G ( λ ) are the n z , f finite eigenvalues o f the r egular pencil K re g ( λ ) , while n z , ∞ , the total numb er of infi- nite zeros, is the sum o f multiplicities of infinite ze- ros, which are defined by the multiplicities of infinite eigenv alues of K re g ( λ ) minus one. The total n umber of zer o s is n z : = n z , f + n z , ∞ . A pro per and stable G ( z ) is minimu m-phase if all its zeros are finite and stable. The numb er of poles and zeros of G ( λ ) satisfy the relation n p = n z + n l + n r , where n r and n l are the normal ranks o f K r ( λ ) and K l ( λ ) , respectively , in the Kron ecker form ( 10 ) of S ( λ ) . Numerically stable algor ithms for the com puta- tion of poles emp loy orthogonal transformation s to reduce the pole pencil A − λ E to a quasi-upper tri- angular f orm (i.e., with th e pair ( A , E ) in a general- ized Sch ur fo rm), wh ile for the comp utation of z e ros use or thogon al tran sformation s to re d uce the system matrix pe ncil S ( λ ) to special Kr onecker-like fo rms [ Misra et al , 19 94 ]. Rational nullspace bases Let G ( λ ) be a p × m rational matrix o f normal rank r and let ( A − λ E , B , C , D ) be a minimal descripto r r e - alization of G ( λ ) . The set of 1 × p ration al (r ow) vec- tors { v ( λ ) } satisfyin g v ( λ ) G ( λ ) = 0 is a linear space, called th e left nu llspace of G ( λ ) , and has dimension p − r . Analogou sly , th e set of m × 1 r ational (c o lumn) vectors { w ( λ ) } satisfying G ( λ ) w ( λ ) = 0 is a linear space, called the right nullspace of G ( λ ) , a n d has d i- mension m − r . The p − r rows of a ( p − r ) × p rationa l matr ix N l ( λ ) satisfying N l ( λ ) G ( λ ) = 0 is a basis of the left nullspace of G ( λ ) , provided N l ( λ ) has f ull row rank . Analogou sly , the m − r columns o f a m × ( m − r ) ra - tional ma tr ix N r ( λ ) satisfyin g G ( λ ) N r ( λ ) = 0 is a ba- sis o f the rig ht nullsp a c e of G ( λ ) , provided N r ( λ ) has full column rank. The determina tio n of a rational le f t nullspace basis N l ( λ ) of G ( λ ) can be easil y turned into the pr oblem of determ in ing a rational basis of the system matrix S ( λ ) . Let M l ( λ ) be a suitable ratio nal matrix su c h that Y l ( λ ) : = [ M l ( λ ) N l ( λ ) ] (13) is a left n ullspace ba sis of the associated system ma- trix pencil S ( λ ) ( 12 ). T hus, to deter mine N l ( λ ) we can determine first Y l ( λ ) , a left n ullspace basis o f S ( λ ) , and th en N l ( λ ) re su lts as N l ( λ ) = Y l ( λ )  0 I p  . By du ality , if Y r ( λ ) is a r ig ht nullspace b asis of S ( λ ) , then a righ t nullspace basis of G ( λ ) is given by N r ( λ ) = [ 0 I m ] Y r ( λ ) . The Kro necker canon ica l fo rm ( 10 ) o f the system pencil S ( λ ) in ( 12 ) allows to ea sily determin e lef t an d right nullspac e b ases of G ( λ ) . Let S ( λ ) = U S ( λ ) V be the Kron ecker canon ical form ( 10 ) o f S ( λ ) , whe r e U and V are the respe cti ve lef t and right tran sf o rmation matrices. If Y l ( λ ) is a left n ullspace basis o f S ( λ ) , then N l ( λ ) = Y l ( λ ) U  0 I p  . (14) Similarly , if Y r ( λ ) is a right n ullspace basis of S ( λ ) then N r ( λ ) = [ 0 I m ] V Y r ( λ ) . (15) W e choose Y l ( λ ) o f the f orm Y l ( λ ) =  0 0 Y l , 3 ( λ )  , (16) where Y l , 3 ( λ ) satisfies Y l , 3 ( λ ) K l ( λ ) = 0. Similarly , we ch oose Y r ( λ ) o f the f orm Y r ( λ ) =   Y r , 1 ( λ ) 0 0   , (17) where Y r , 1 ( λ ) satisfies K r ( λ ) Y r , 1 ( λ ) = 0 . Both Y l , 3 ( λ ) and Y r , 1 ( λ ) can be determin ed as p olynom ial or ratio- nal m atrices a nd the re su lting bases are polynom ial o r rational a s well. Numerically reliable compu tational appro aches to compute prop er nullspace bases of rational matri- ces rely on using Kron ecker-like f o rms (instead of the Kronecker f o rm), which can b e determined by using exclusively or thogon al similarity transf o rma- tions. Moreover , these m ethods a r e ab le to d etermine nullspace bases o f minim al McMillan degree and with arbitrary assign ed poles [ V arga , 20 08 ]. Additiv e decompositions Let G ( λ ) be a rational TFM with a descriptor system realization G ( λ ) = ( A − λ E , B , C , D ) . Consider a dis- junct p a rtition of the com p lex plan e C as C = C g ∪ C b , C g ∩ C b = / 0 , (18) where b oth C g and C b are symmetrically lo cated with respect to th e real axis, and C g has at le a st o ne point on the real ax is. Since C g and C b are d isjo in t, each pole of G ( λ ) lies eith er in C g or in C b . Using a sim - ilarity transformation of the form ( 6 ), we can d eter- mine an eq uiv alent representation of G ( λ ) with parti- tioned system matrices of the form G ( λ ) =  U A V − λ U E V U B C V D  =   A g − λ E g 0 B g 0 A b − λ E b B b C g C b D   , (19) where Λ ( A g − λ E g ) ⊂ C g and Λ ( A b − λ E b ) ⊂ C b . It follows that G ( λ ) can b e additively deco mposed as G ( λ ) = G g ( λ ) + G b ( λ ) , (20) where G g ( λ ) =  A g − λ E g B g C g D  , G b ( λ ) =  A b − λ E b B b C b 0  , and G g ( λ ) has on ly po le s in C g , while G b ( λ ) has on ly poles in C b . The spectral separ ation in ( 1 9 ) is auto- matically p r ovided b y the W eierstrass c anonical form of the pole pencil A − λ E , where the diagonal J or- dan block s are suitab ly permuted to correspo nd to th e desired eig en value sp litting. T h is appro ach automati- cally leads to par tial fraction expansion s of G g ( λ ) an d G b ( λ ) . A numer ic a lly reliable app roach to compute spectral separations as in ( 19 ) has been p roposed in [ K ˚ agstr ¨ om and V an Dooren , 1990 ]. Coprime factorizations Consider a disjunct partition ( 18 ) of the complex plane C , wh ere both C g and C b are symmetrically lo - cated with respect to the real axis, and such th at C g has at least one po int on the real axis. Any rational matrix G ( λ ) can be expressed in a left fr actional fo rm G ( λ ) = M − 1 ( λ ) N ( λ ) , (21) or in a right fraction al form G ( λ ) = N ( λ ) M − 1 ( λ ) , (22 ) where both the den o minator factor M ( λ ) and the nu- merator factor N ( λ ) have on ly poles in C g . These fractional factorizations over a “good ” domain of poles C g are im portant in v arious observer, fault de- tection filter, or controller syn thesis method s, becau se they allow to achieve the placem ent of all p oles o f a TFM G ( λ ) in the domain C g simply , by a premu lti- plication o r postmu ltiplication of G ( λ ) with a suitable M ( λ ) . Of special in terest are th e so-called co prime fac- torizations, where th e factor s satisfy addition al co- primeness con ditions. A fractional r e p resentation of the for m ( 21 ) is a left co p rime factorization (LCF) of G ( λ ) with respect to C g , if th ere exist U ( λ ) and V ( λ ) with poles on ly in C g which satisfy the Bezo ut iden tity M ( λ ) U ( λ ) + N ( λ ) V ( λ ) = I . A fraction al re presentation o f th e form ( 22 ) is a right coprime facto rization (RCF) of G ( λ ) with respect to C g , if there exist U ( λ ) and V ( λ ) with po les o nly in C g which satisfy U ( λ ) M ( λ ) + V ( λ ) N ( λ ) = I . For th e comp u tation of a right coprim e factoriza- tion of G ( λ ) with a minimal descriptor realization ( A − λ E , B , C , D ) it is sufficient to determine a state- feedback matr ix F such th at all finite eigen values in Λ ( A + BF − λ E ) b elong to C g and all infinite eigen- values in Λ ( A + BF − λ E ) are simple. The descrip tor realizations o f the factors are giv en by  N ( λ ) M ( λ )  =   A + BF − λ E B C + DF D F I m   . Similarly , to determ ine a left copr ime factorization, it is suf ficient to determin e an output- injection matrix K such that all finite eigenv alues in Λ ( A + KC − λ E ) be- long to C g and all infin ite eigen values in Λ ( A + KC − λ E ) are simple. Th e descripto r realizations o f the fac- tors are given by [ N ( λ ) M ( λ ) ] =  A + KC − λ E B + K D K C D I p  . An impo rtant class o f coprime factorizations is the class of co prime factorizations with minimum- degree denomin ators. Th e McMillan degree of G ( λ ) satisfies δ ( G ( λ )) = n g + n b , where n g and n b are the nu mber o f poles of G ( λ ) in C g and C b , r e spectiv ely . The d e nom- inator factor M ( λ ) has th e min imum-d egree proper ty if δ ( M ( λ )) = n b . Special classes of coprime factor- izations, as the c o prime factorization s with inner de- nominato rs or the n ormalized co prime factorizations, have importan t applications in solvin g v arious anal- ysis and synthesis problem s. For the comp utation o f coprime factoriza tions with minim um degree den om- inators, descripto r system repr e sentation based meth - ods have been d eveloped, which rely o n iterativ e pole dislocation techn iques [ V arga , 1998 , 201 7a ]. Full rank compr essions Row comp ressions o f a p × m rational matrix G ( λ ) of normal rank r < p to a full row ra n k matr ix can be d e- termined by p re-multip ly ing G ( λ ) with an inv ertible rational matrix U ( λ ) to obtain U ( λ ) G ( λ ) =  R ( λ ) 0  , where R ( λ ) has full ro w rank r . Of par ticular im- portance for solvin g m odel-match ing pro blems is the case whe n U ( λ ) has the for m U ( λ ) = Q ∼ ( λ ) , where Q ( λ ) is a square inner matrix, that is, Q ( λ ) is sta- ble an d Q ∼ ( λ ) Q ( λ ) = I . I f we p artition Q ( λ ) as Q ( λ ) = [ Q 1 ( λ ) Q 2 ( λ ) ] , with Q 1 ( λ ) h aving r column s, then we have G ( λ ) = Q ( λ )  R ( λ ) 0  = Q 1 ( λ ) R ( λ ) . (2 3) The full colum n ran k m atrix Q 1 ( λ ) is an inner basis of the image space of G ( λ ) , while Q 2 ( λ ) is called its inner orth o gonal comp lem ent. W e call ( 23 ) th e inner– full-r ow-rank factorization of G ( λ ) and it ca n be in- terpreted as the generalizatio n of the o rthogo nal rank- revealing QR factor ization of a constan t matrix. The column compr ession o f G ( λ ) to a full co l- umn rank matrix can be obtained in a similar way , by post-multiplyin g G ( λ ) with Q ∼ ( λ ) , where Q ( λ ) is a square inner matrix . W ith Q ( λ ) partitioned as Q ( λ ) = h Q 1 ( λ ) Q 2 ( λ ) i , with Q 1 ( λ ) having r rows , the n we can write G ( λ ) = [ R ( λ ) 0 ] Q ( λ ) = R ( λ ) Q 1 ( λ ) . (24) The full r ow ran k matrix Q 1 ( λ ) is co -inner (i. e ., Q 1 ( λ ) Q ∼ 1 ( λ ) = I ) and is a basis of th e co -image space of G ( λ ) . The factorization ( 24 ) is called the full- column-rank– co-inner factorizatio n and can be seen as a g eneralization of th e o rthogo nal ran k-revealing RQ factorizatio n of a constant matrix. The primary role of the inner matrix Q ( λ ) is to achieve the row or colu mn com pression o f G ( λ ) to a full rank m a trix. If G ( λ ) has n o zeros on the bou nd- ary of th e stability doma in C s , th en it is possible to ac h iev e simultaneously tha t all zeros o f R ( λ ) re- sult in the stable region C s . Addition ally , if G ( λ ) is stable, then R ( λ ) results stable to o, and thu s, minimum- phase. I n this case, the factorization ( 23 ) is called the inner-outer factorizatio n of G ( λ ) , with R ( λ ) outer (i.e., minimu m-phase and full row r a nk), and the factoriza tio n ( 24 ) is called the co- outer–co- inner fa ctorization of G ( λ ) , with R ( λ ) co-o ute r (i.e., minimum- phase an d f ull column r ank). The inner- outer and co-oute r–co-in ner factorizations are in stru- mental in solving ap proxim ate con tr oller an d fault de- tection filter synthe sis p roblems, which in volve the minimization o f H ∞ -norm or H 2 -norm pe r forman ce criteria. General methods to d etermine inner-outer factorizations are based o n t he computation o f spe - cial Kron ecker-lik e for ms o f th e system matrix pen c il [ Oar ˘ a and V arga , 2000 ; Oar ˘ a , 2 005 ]. Linear rational matrix equations The solution of mod el-matching p roblems encou n- tered in the synthesis of con trollers or filters inv olves the solutio n of the linear rationa l m a tr ix equ ation G ( λ ) X ( λ ) = F ( λ ) , (25) with G ( λ ) a p × m ratio nal m atrix and F ( λ ) a p × q rational matrix. This equa tio n has a solution p rovided the com patibility cond ition rank G ( λ ) = rank [ G ( λ ) F ( λ ) ] (26) is fulfilled. Assum e G ( λ ) and F ( λ ) have descripto r realizations o f the form G ( λ ) =  A − λ E B G C D G  , F ( λ ) =  A − λ E B F C D F  , which share the system pair ( A − λ E , C ) . It is easy to observe that any solutio n X ( λ ) of ( 25 ) is also part of the solution Y ( λ ) = h W ( λ ) X ( λ ) i of the linear (polyno m ial) equation S G ( λ ) Y ( λ ) =  B F D F  , (27) where S G ( λ ) is the associa te d system m atrix pe ncil S G ( λ ) =  A − λ E B G C D G  . (28) Therefo re, an alternativ e to solving ( 25 ), is to solve ( 27 ) fo r Y ( λ ) in stead and compu te X ( λ ) as X ( λ ) = [ 0 I p ] Y ( λ ) . (29) The co mpatibility con dition ( 26 ) beco mes rank  A − λ E B G C D G  = r ank  A − λ E B G B F C D G D F  . If G ( λ ) is in vertible, a d escriptor system realiza- tion of X ( λ ) can b e explicitly o btained as X ( λ ) =   A − λ E B G B F C D G D F 0 − I p 0   . (30) The ge neral solution o f ( 25 ) can be expressed as X ( λ ) = X 0 ( λ ) + X r ( λ ) Y ( λ ) , where X 0 ( λ ) is any particular solution of ( 25 ), X r ( λ ) is a rational basis matr ix for the right nullspace of G ( λ ) , an d Y ( λ ) is an arbitrary rational matr ix with suitable dimension s. Gene r al methods to determin e both X 0 ( λ ) and X r ( λ ) can be devised by using th e Kronecker canonical form of the ass ociated system matrix pe ncil S G ( λ ) in ( 28 ). It is also p ossible to choose Y ( λ ) to obtain special solutions, as, f o r exam- ple, with least Mc M illan d egree. A num e rically so und computatio nal appro ach to determine least Mc M illan degree solutions is based on the redu ction of th e sys- tem m atrix pe n cil S G ( λ ) to a Kronecker-like form an d has been prop osed in [ V arga , 2004 ]. Ap pr oximate model matching W e consider the following standard formu lation of the approx imate mod el-matching p r oblem ( MMP): deter- mine fo r a given stable G ( λ ) and a stable F ( λ ) , a sta- ble ration al matrix X ( λ ) such that k F ( λ ) − G ( λ ) X ( λ ) k = min , where either the L 2 -norm or L ∞ -norm of the ap p roxi- mation erro r E ( λ ) : = F ( λ ) − G ( λ ) X ( λ ) are u sed. The correspo n ding problems a re called L 2 -MMP and L ∞ - MMP , respectively . In the absence of gene r al n ecessary and sufficient condition s fo r the existence of an op timal so lution of the MMPs, an often em ployed sufficient co ndition is to assum e that G ( λ ) h a s no zeros on th e bo undar y of C s . Fur thermore , in the ca se of th e L 2 -norm and for a con tinuous-tim e system, it is assumed that F ( s ) is strictly p r oper . Th ese cond itio ns ar e clearly no t nec- essary (e.g ., if an exact solu tion exists). The inner -outer factorizatio n ( 23 ) o f G ( λ ) , with R ( λ ) ou ter , can b e employed to r educe the MMPs to simpler ones, the so-called least distance pr oblems (LDPs). The factor ization ( 23 ) allows to express the error no rm as k E ( λ ) k =      e F 1 ( λ ) − Y ( λ ) e F 2 ( λ )      , (31) where Y ( λ ) : = R ( λ ) X ( λ ) and Q ∼ ( λ ) F ( λ ) =  Q ∼ 1 ( λ ) F ( λ ) Q ∼ 2 ( λ ) F ( λ )  : =  e F 1 ( λ ) e F 2 ( λ )  . The terms e F 1 ( λ ) and e F 2 ( λ ) are gener ally un stable, and may even be impro per in the d iscrete-time case (i.e., if Q ( z ) has p o les in the o rigin). The pr o blem o f computin g a stable solution X ( λ ) which minimizes the error norm k E ( λ ) k has been thus redu ced to a L DP to comp ute the stable solution Y ( λ ) which m inimizes the norm in ( 31 ). The solution of the origin a l MMP is given by X ( λ ) = R † ( λ ) Y ( λ ) , where R † ( λ ) is a stable right inv erse of R ( λ ) (i.e., R ( λ ) R † ( λ ) = I ). The solu tion of the LDP in the case of L 2 -norm is straightfor ward . Let L s ( λ ) be the stable part and let L u ( λ ) be the unstable part in th e ad ditive decompo si- tion e F 1 ( λ ) = L s ( λ ) + L u ( λ ) , (32) where, in the continuous-time case, we take the un- stable projection L u ( λ ) strictly p roper . The solution of the LDP is Y ( λ ) = L s ( λ ) and th e achieved minimum error no rm of E ( λ ) is k E ( λ ) k 2 =    L u ( λ ) e F 2 ( λ )    2 . The so lu tion of th e LDP in the c a se of L ∞ -norm is more complicated, a n d follows the approach described in [ Francis , 1987 ] for co n tinuous- tim e sys- tems. The s olution procedur e in volv es th e solution of a Neh a ri problem and, if e F 2 ( λ ) is pre sen t (i.e., G ( λ ) h as no f ull r ow rank), the repea ted solutio n of a special spe ctral factorization p roblem in a so-called γ -iteration appro x imation p rocess. Details can be found in [ Francis , 1987 ] fo r the continuo us-time case, o r in [ V arga , 2017 b , Chapter 9 and 10 ]. 6 Software T ools Sev eral basic r equireme n ts ar e desirable wh en im- plementing robust software to ols for num erical com - putations: • employing exclusiv ely numerically stable o r numerically reliable algorith ms; • ensurin g h igh co mputation al efficiency; • enfo r cing ro bustness aga inst nume r ical excep - tions (overflows, under flows) and po orly scaled data; • ensurin g ease of use, high portability an d h igh reusability . The above requ irements have b een enf orced in the development o f high-p erform a nce linear algebra soft- ware lib raries, such as BLAS [ Dongarr a et al , 1 990 ], a collection of basic linear algeb ra subroutines, an d LAP A CK [ Anderson et al , 19 99 ], a compre h ensive linear algebra p ackage b ased on BLAS. These re- quiremen ts have be e n also ad opted to im plement SLI- CO T [ Benner et al , 1999 ; V an Huffel e t al , 2004 ], a subroutin e libr ary for control theory , based primar- ily on BLAS an d LAP A CK. The gen eral-pur pose li- brary LAP A CK contain s over 1 300 subro utines and covers most of the basic linear alg ebra comp u tations for solving systems of linear equations and eigen - value p r oblems. The release 4.5 of the specialized library SLI CO T 1 is a free software distributed under the GNU General Pub lic Lic e nse ( GPL). The substan- tially enrich ed current release 5.7 is freely distributed under a BSD 3 -Clause License via GitHu b 2 and con- tains over 500 su broutin e s. SLICO T covers the b a- sic computation a l p roblems for the analysis and d e- sign of linea r control sy stem s, su c h as linear system analysis and synthesis, filtering, iden tification, solu- tion of matrix equation s, model reduction, and sys- tem tran sformation s. Of special interest is th e com- prehen sive collection of routines for h andling descrip- tor systems and for solving gen eralized linear matrix equations, as well as, the routines for co mputing var - ious Kronecker-like f o rms. The subroutin e libraries BLAS, LAP ACK and SLI CO T ha ve been originally implemented in th e g eneral-p u rpose languag e Fortran 77 and, therefo re, pr ovide a high level o f reusab ility , which allows their easy incorpo ration in user-friendly software environments, f or example, MA TL A B. In the c ase of MA TLAB, selected L AP ACK rou tines un- derlie the linear algeb ra f unctionalities, while th e in- corpor ation of selected SLICOT r outines was po ssible via suitable gateways, as the provided mex -func tio n interface. The Control System T o olbox (CST) of MA TLAB supports b oth descripto r system state-space models and in put-ou tput representation s with impr oper ratio- nal TFMs, and provides a rich function ality cov er- ing the basic system op erations and couplin gs, m odel conv ersions, as well as some advanced fu nctionality such as p ole and zer o computatio ns, minimal r ealiza- tions, and the solution of gen eralized L yapunov and Riccati equation s. Howe ver , most of the functions of th e CST can on ly han dle descripto r system s with proper T FMs and impor tant function ality is curren tly lacking for ha n dling the mor e genera l descripto r sys- tems with improp er TFMs, notab ly for determining the comp lete pole an d zero structures or for compu t- ing min imal ord er realizations, to mention o nly a few limitations. T o facilitate the implem e n tation of the synthesis proced u res o f fault de tection and isolation filters de- scribed in the book [ V arga , 2 017b ], a n ew collection of freely av ailable m - files, called the Descripto r Sys- tem T ools (DSTOOLS), has been implemen ted for MA TLAB. DS TOOLS is pr imarily intended to p ro- 1 http://www.sl icot.org/ 2 https://githu b.com/SLICOT/ SLICOT- Reference/ vide an extended function ality f or both M A TLAB (e.g., with matrix pencil manipulation meth o ds for the comp utation o f Krone c ker -like form s), and fo r the CST by providing f unctions f or minim al realiza- tion of descripto r systems, co m putation o f pole and zero structures, computation of nullspa c e and range space bases, additive decom positions, several fac- torizations o f ratio nal matrices (e.g., coprime, fu ll rank, inner-outer), ev aluation of ν -gap distance , ex- act and approximate solution of linear rationa l ma- trix equations, eigenv alue assignmen t and stabiliza- tion via state feedb ack, etc. The appro ach used to de- velop DSTOOLS exploits MA TLAB’ s matrix a n d ob - ject man ipulation features by means of a flexible an d function ally rich collectio n of m - files, intended for noncritical com putations, wh ile simultaneously en- forcing hig hly efficient and n umerically soun d co m- putations via mex -function s (calling Fortran routines from SLI COT), to solve critical n umerical pro b lems requirin g th e u se of structure-exploiting algorithms. An im portant aspect of im plementing DSTOOLS was to en sure that standa r d systems are fully suppor ted, using specific algor ithms. In the same vein, all alg o- rithms are available for bo th c ontinuo us- and discrete- time sy stems. A prec ursor of DSTOOLS was the Descrip- tor Systems T oolbox f or MA TLAB, a pr oprietary software of th e Ger m an Aerospace Center ( DLR), developed between 1996 and 2 006 ( f or the status of this toolbox around 2 000 see [ V arga , 20 00 ]). Some descriptor system fu nctionality covering the basic manipulatio n o f ration al matrices is a lso av ailable in the free, open -source software Scilab 3 and Octave 4 . A n otable recent d ev elopmen t is the free sof t- ware package DescriptorSystems 5 , wh ich im- plements the complete fu nctionality of DSTOOLS in the Ju lia lang uage. Julia is a po werful and flexible dynamic lang uage, su itab le fo r scientific and nu mer- ical compu ting, with perfo r mance comparab le to tra- ditional statically- typed lan guages such as Fortran or C. As a pr ogramm ing languag e, Julia features op- tional typing , multiple dispatch, and goo d per for- mance, achieved using typ e inferen ce and ju st-in-time compilation [ Bezanson et al. , 2017 ]. The un derlying Julia pack ages MatrixEqua tions 6 , for solv ing various control related matrix equations (L yapunov , Sylvester , Riccati), and Mat rixPencils 7 , fo r ma- nipulation of m atrix pencils, p r ovide the req uired b a- sic computationa l fu nctionality for the implementa- 3 http:scilab.o rg 4 http://www.gn u.org/softwar e/octave/ 5 https://githu b.com/andreas varga/DescriptorSystems.jl 6 https://githu b.com/andreas varga/MatrixEquations.jl 7 https://githu b.com/andreas varga/MatrixPencils.jl tion of DescriptorSystems (e.g ., such as pro- vided b y SLICOT fo r DSTOOLS). A new featu re of this pac k age is the f u ll suppo r t fo r mode ls with bo th real and complex data (Note: DSTOOLS suppo rts only mod els with re al data). 7 Recommended Reading The th eoretical aspects of descriptor systems are discussed in the two textbooks [ Dai , 1989 ; Du an , 2010 ]. For a th oroug h tr e a tment of ration al m a- trices in a system theoretical context two authorita- ti ve references are [ Kailath , 1980 ] and [ V idyasagar , 2011 ]. M ost of th e concepts and techniqu es pre- sented in this article are also discussed in d epth in [ Zhou et al , 1996 ] for stand ard systems. T he book V arga [ 20 1 7b ] illustrates the use of descripto r system technique s to solve the synthesis p roblems of fault de- tection and isola tio n filter s in the most general set- ting. Chapters 9 and 10 o f th is b ook describe in de - tails the presented descriptor system techniques fo r the manipulatio n o f rational matrices and also gi ve details o n a vailable n umerically reliab le algorith ms. These algorithm s form the basis of th e implemen- tation of the function s a vailable in th e DSTOOLS and De scriptorSyst ems collection s. A compre- hensive d o cumentatio n o f DSTOOLS is available in arXiv [ V arga , 2 018 ]. A shorte r version o f th is article appeared in the Encycloped ia o f Systems and Control [ V arga , 2019 ]. REFERENCES Anderson E, Bai Z, Bishop J, Dem mel J, Du Croz J, Greenbau m A, Ham marling S, McKenney A, Os- trouchov S, Sor ensen D ( 1 999) LAP A CK User’ s Guide, T hird Edition . SIAM, Philad elphia Benner P , Mehr mann V , Sima V , V a n H u ffel S, V arga A (199 9) SLICO T – a subroutin e library in sys- tems an d control theor y . I n : Datta BN (ed) Ap- plied and Compu tational Control, Signals and Cir- cuits, vol 1, Birkh ¨ auser, p p 499 –539 , DOI 1 0.100 7/ 978- 1- 4612- 0571- 5 10 Bezanson J, Edelman A, Karp inski S, Sh ah VB (2017 ): Julia: A fr esh approach to nume rical com - puting. SIAM Revie w , v ol 59, 1:65–98 see also: https: //juli alang. org/ Dai L (198 9) Singular Control Systems, Lectu re Notes in Contr ol and Inf ormation Sciences, vol 118. Sprin ger V er lag, New Y or k Dongarr a JJ, Croz JD, Hammar ling S, Duff I (1990) A set o f level 3 basic linear algebra sub progr ams. A CM Trans Math Software 16:1 –17 Duan GR (2010) Analysis an d Design o f Descr ip tor Linear Systems, Ad vances in Mechan ics and Math- ematics, vol 2 3. Spring e r, New Y ork Francis B A (19 87) A Course in H ∞ Theory , Lecture Notes in Co ntrol and Info rmation Sciences, vol 88. Springer-V erlag , New Y o rk K ˚ agstr ¨ om B, V an Doore n P (1 990) Additive d ecom- position of a transfer fu nction with respect to a specified region . In: Kaashoek MA , van Schuppe n JH, Ran ACM (eds) Sign al Processing, Scatter ing and Operato r theory , and Numerical Methods, Pro- ceedings of th e Intern ational Sym posium on Math- ematical Theory o f Network s and Systems 1989, Amsterdam, The Netherland s, Birh h ¨ auser , Boston, vol 3, pp 469– 477 Kailath T (19 80) Lin ear Sy stems. Pren tice Hall, En - glew ood Cliffs Misra P , V an Dooren P , V arga A (199 4) Computatio n of structural in variants of g eneralized state-space systems. Au tomatica 30:1 921– 1 936 Oar ˘ a C ( 2005) Constructive solution s to spectral and inner-outer factorizations with respect to the disk. Automatica 41:18 55–1 8 66, DOI 10. 1016/j. automatica.2 005.0 4.009 Oar ˘ a C, V arga A (200 0 ) Comp utation of general inner-outer and spectral factor izations. IEEE T rans Automat Control 45:2 3 07–2 325 V an Huffel S, Sima V , V arga A, Ham marling S, Dele- becque F (2004 ) High- perfor mance numerica l soft- ware for con trol. IEE E Con trol Syst M ag 24:60–7 6 V arga A (1998) Computation of coprime factor- izations of rationa l matrices. Linear Algebra Appl 271:83 –115 , DOI 10.101 6/S0024 - 3795(97) 00256 - 5 V arga A (2000) A D E S C R I P T O R S Y S T E M S toolbox for M A T L A B . In : Proceedings of the IEEE Inter- national Sy mposium on Computer-Aided Control System Design, An c horage, AK , USA, p p 150–1 55 V arga A ( 2004) Com p utation o f lea st ord er solution s of linear rational equation s. In : Proceed in gs o f the Internatio nal Symposium on Mathem atical T heory of Networks and Systems, Leuven, Belgium V arga A (2008) On computin g nullspace bases – a fault detection p erspective. In: Pr oceeding s of the IF AC W orld Cong ress, Seo ul, K orea, p p 6295 – 6300 V arga A (2017a ) On recursive co mputation o f coprime factorizations of r a tional matrices. https: //arxi v.org/ abs/1703.07307 V arga A ( 2017b ) Solving Fault Diagno sis Prob- lems – Linear Synthesis T echn iques, Stud ies in Systems, Decision and Control, vol 8 4. Springer Internatio n al Publishing , DOI 10 .1007 / 978- 3- 319- 51559- 5 V arga A (2018) Descrip tor Sy stem T ools (DSTOOLS) V0 .71, User’ s Guide. https: //arxi v.org/ abs/1707.07140 V arga A ( 2019) Descriptor system techniqu es and software tools. In: Baillieul J, Samad T (ed s) En- cyclopedia of Systems and Contro l, Spr inger Lon- don, DOI 10.10 07/978 - 1- 44 71- 5102- 9 10005 4- 1, https: //doi. org/10 .1007/978- 1- 4471- 5102- 9_100054- 1 V erghese G, L ´ evy B, Kailath T (1981) A general- ized state-space f or sing ular systems. I EEE T rans Automat Contr ol 26:81 1–83 1 , DOI 10.1109/T A C. 1981. 11027 63 V id y asagar M (2011) Control Sy stem Syn thesis: A Factorization Appro ach. ”Mo rgan & Claypoo l” Zhou K, Doyle JC, Glover K (19 96) Robust and Op- timal Contro l. Prentice Hall, Upp er Saddle River