Strongly stable algorithm for computing periodic system zeros

Transcription

1 Stongly stable algoithm fo computing peiodic system zeos Andas Vaga Geman Aeospace Cente DLR - bepfaffenhofen Institute of Robotics and System Dynamics D Wessling, Gemany AndasVaga@dlde Abstact We popose a computationally efficient and numeically eliable algoithm to compute the finite zeos of a linea discete-time peiodic system The zeos ae defined in tems of the tansfe-function matix coesponding to an equivalent lifted time-invaiant state-space system The poposed method elies on stuctue peseving manipulations of the associated system pencil to extact successively lowe complexity subpencils which contains the finite zeos of the peiodic system The new algoithm uses exclusively stuctue peseving othogonal tansfomations and fo the oveall computation of zeos the stong numeical stability can be poved I INTRDUCTIN We conside the poblem of computing zeos of peiodic time-vaying descipto systems of the fom E x( + 1) = A x() + B u() y() = C x() + D u() whee the matices E R ν +1 n +1, A R ν +1 n, B R ν +1 m, C R p n, D R p m ae peiodic with peiod K 1 Fo solvability of these equations we will assume that the dimensions of A and E fulfil the condition K =1 ν = K =1 n A geneal, efficient and numeically eliable algoithm to compute the zeos of such a system epesents a univesal analysis tool of peiodic systems Besides chaacteizing when the system is minimum-phase o not, the zeos povide pactically infomation on all stuctual popeties of a system Fo instance, eachability/stabilizability and obsevability/detectability can be easily studied by computing the zeos of paticula peiodic systems without outputs o inputs, espectively Even the poles of a peiodic system can be seen as a paticula type of zeos fo a system with no inputs and no outputs Fo the computation of zeos it is impotant to conside the moe geneal case of time-vaying dimensions Since the tansmission zeos of a standad system ae defined in tems of a minimal ealization, a simila definition is appopiate also fo the zeos of a peiodic system (see fo example 10) Howeve, the minimal ealization theoy of standad peiodic systems (ie, E = I n+1 ) evealed (see fo example 3, 5) that minimal ode (ie, eachable and obsevable) state-space ealizations of peiodic systems have, in geneal, time-vaying state dimensions It follows immediately that the minimal ealization of a peiodic descipto system computed, fo example, via a fowad-bacwad decomposition (1) 14, leads in geneal to ectangula descipto matices E as well Time-vaying input and output vecto dimensions occu when evaluating zeos of paticula system as those appeaing in an algoithm to evaluate the tansfe-function matix of a peiodic system 17 Finally, the development of geneal algoithms able to addess the case of time-vaying dimensions, is one of the equiements fomulated fo a satisfactoy numeical algoithm fo peiodic systems 18 The definition of zeos of a peiodic descipto system can be intoduced stating fom two input-output equivalent time-invaiant lifted efomulations 11, 8 While the diect application of the numeically stable methods of 4, 12 to these epesentations is cetainly possible, the esulting algoithms ae completely inappopiate Povided all E ae identity matices (ie, we have a standad peiodic system), constucting explicitly the lifted epesentation of 11 involves foming poducts of up to K matices Thus, applying the algoithm of 4 to this lifted system can lead to sevee numeical difficulties When applying the method of 12 to the lage ode lifted descipto epesentation of 8, the computational complexity fo lage ode systems o lage peiods is vey high Assuming constant dimensions µ i = n i = n, such an algoithm has a complexity of (K 3 n 3 ), instead of an expected complexity of (Kn 3 ) fo a satisfactoy algoithm 18 Specific equiements fo satisfactoy numeical algoithms fo peiodic systems have been fomulated in 18 Besides low computational complexity, the numeical stability of algoithms is a main equiement A fist geneal method to compute the zeos of peiodic systems, belonging to the family of fast, stuctue exploiting algoithms, has been poposed in 19 This algoithm elies exclusively on using othogonal tansfomations and it can be shown that it is numeically stable in the following esticted sense: the computed zeos in the pesence of oundoff eos ae exact fo a slightly petubed lifted system pencil Howeve, by pefoming ow compessions of the system pencil which destoy its cyclic stuctue, this algoithm is not stongly numeically stable This means that it is not possible to demonstate fo it that the computed zeos ae exact fo an oiginal system with slightly petubed system matices In this pape we popose a numeical appoach to compute the finite zeos of the peiodic system (1) which meets the equiements fomulated in 18 fo a satisfactoy numeical

2 algoithm egading geneality, speed and accuacy The poposed method elies on stuctue peseving manipulations of the associated system pencil to extact successively lowe complexity subpencils which contains the finite zeos of the peiodic system The new algoithm uses exclusively stuctue peseving othogonal tansfomations and fo the oveall computation of zeos the stong numeical stability can be poved Besides the finite zeos, the eduction algoithm also povides infomation to deduce the infinite zeos stuctue as well as the complete Konece stuctue of the lifted system pencil Fo this eason, the poposed new algoithm can be seen as a genealization of the method of 12 Notation Fo a K-peiodic matix X i R µ +1 n we use altenatively the scipt notation X := diag (X, X +1,, X +K 1 ), which associates the bloc-diagonal matix X to the cyclic matix sequence X i, i =,, + K 1 stating at time moment We will use the notation n := {n,, n +K 1 } to denote the time-vaying dimensions of peiodic matices By using the scipt notation, the peiodic system (1) will be altenatively denoted by the quintuple (E, A, B, C, D ) The dimensional infomation on state-, input- and outputvectos of this system is povided by the tiple (n, m, p ) To simplify the notation fo the case = 1, we dop the index used fo the sampling time in the system matices and dimensions II ZERS AND PLES F PERIDIC SYSTEMS We define the zeos and poles of the peiodic system (1), using the tansfe-function matix (TFM) coesponding to the associated staced lifted epesentation 8 This lifting technique uses the input-output behavio of the system ove time intevals of length K, athe then 1 Fo a given sampling time, the coesponding input-, output- and statevectos u S (h) = ut ( + hk) u T ( + hk + K 1) T y S (h) = yt ( + hk) y T ( + hk + K 1) T x S (h) = xt ( + hk) x T ( + hk + K 1) T have dimensions M = K =1 m, P = K =1 p and N = K =1 n, espectively The lifted system has a time-invaiant descipto system epesentation of the fom L S xs (h + 1) = F S xs (h) + GS us (h) y S (h) = HS xs (h) + J S us (h) (2) whee G S = B, H S = C, J S = D, and A E F S zl S = E+K 3 A +K 2 E +K 2 ze +K 1 A +K 1 (3) Assuming the squae pencil (3) is egula (ie det(f S zls ) is not identically 0), the TFM of the staced lifted system is W (z) = H S (zl S F S ) 1 G S + J S (4) and the associated system pencil is defined as S S F S (z) = zl S G S, (5) H S which both depend on the sampling time bviously W +K (z) = W (z) and the TFMs at two successive values of ae elated by the following elation 6 W +1 (z) = 0 IP p zi p 0 W (z) J S 0 z 1 I m I M m 0 It follows fom this elation that poles and zeos of the TFMs fo diffeent sampling time, can only diffe at z = 0 and z = The nomal an L of the TFM W (z) (ie, the an ove ationals) is the numbe of non-zeo diagonal elements in the Smith-McMillan fom of W (z) In ode to define poles and zeos of the peiodic system (1), we need the minimality of the system and of the ealization (2) This is equivalent to the notion of eachability and obsevability at finite and infinite eigenvalues of the pencil (5), as intoduced in 20 If we assume that the system (1) is minimal in that sense (this implies time-vaying state dimensions and ectangula descipto matices) then we have the following definitions of poles, zeos and minimal indices of the TFM (4) based on the system matix (5) of the staced lifted system (2) Definition 1: The tansmission zeos of the TFM W (z) of the minimal ode peiodic system (E, A, B, C, D ) ae the invaiant zeos of the associated system pencil (5) Definition 2: The left and ight minimal indices of the TFM W (z) of the minimal peiodic system (E, A, B, C, D ) ae those of the associated system pencil (5) Definition 3: The poles of the TFM W (z) of the minimal peiodic system (E, A, B, C, D ) ae the zeos of the associated pole pencil F S zls defined in (3) The above definitions of zeos and poles of a peiodic system ae consistent with definitions based on the lifting technique intoduced in 11 applicable to systems with E squae and invetible In this case, the tansfe-function matices of the two lifted systems ae the same, thus the coesponding definitions of poles and zeos coincide Fom the definition of zeos follows that the tansmission zeos of the peiodic system (1) (finite and infinite) ae those values of z whee the an of the lifted system matix S S(z) dops below its nomal an N + L The infinite zeos and thei multiplicities can be defined in tems of the infinite eigenvalues of the pencil S S (z) (see the elationship between the null zeos of W (1/λ) and the Konece fom of S S (z) 20) To each Jodan bloc of size j at the eigenvalue

3 coesponds an elementay diviso λ j 1 in the Smith- McMillan fom of S S (1/λ) Because of this diffeence of one fo the stuctue at infinity, the simple eigenvalues at of the pencil F S zls (at most N an E +K 1) play no ole when counting the infinite zeos III CMPUTATINAL APPRACH In this section we popose an efficient computational appoach to detemine the zeos of the staced lifted system (2) at = 1 The zeos fo othe time moments = 2,, K can be computed in a simila manne by just pemuting the ode of the undelying matices Instead of S S (z) in (5), we conside an equivalent pencil S(z) with pemuted bloc ows and columns S 1 T 1 S 2 T 2 S(z) = (6) S K 1 T K 1 zt K S K whee fo = 1,, K A B S := E, T C D := To educe this pencil by peseving its stuctue, we will use othogonal tansfomations of the fom S = Q S Z, T = Q T Z +1 (7) which coesponds to apply to S(z) fom left and ight, the bloc-diagonal matices Q and Z, espectively The poposed algoithm to compute zeos can be applied to compute the system poles as well by defining In a simila way, with o S := A, T := E S := A B, T := E S := A C E, T := the zeos algoithm can be used to compute the input decoupling zeos and output decoupling zeos, espectively 7 We also discuss some computational enhancements which aise in the case of standad peiodic systems (ie, E = I n+1 ) The algoithm we popose has thee main steps, which we discuss in the subsequent thee subsections A Computation of the compessed system In the fist step we educe the poblem to an equivalent one, but fo squae and non-singula peiodic descipto matices Let U and V be othogonal peiodic matices such that A U A B V,11 A,12 B,1 = A I C D I,21 A,22 B,2 C,1 C,2 D E U E V+1,11 = I I whee E,11 R fo = 1,, K, ae squae, non-singula matices The compession of each E to a nonsingula E,11 can be done by computing a full othogonal decomposition U E V +1 = diag (E,11, ) using eithe the singula-value decomposition (SVD) o a an-evealing QRdecomposition followed by an RQ-decomposition In both cases, we can assume that each E,11 esults uppe tiangula If we constuct the new system matices E = E,11, A = A,11, B = A,12 B,1 A,21 A,22 B C =, D C =,2,1 C,2 D then the pencils S(z) and the tansfomed pencil S(z) coesponding to the educed matices have the same Konececanonical fom, thus we have the following staightfowad esult Theoem 1: The oiginal system (E, A, B, C, D) and the compessed system (E, A, B, C, D) have the same tansmission zeos, left and ight minimal indices Note that in geneal the compessed system has timevaying dimensions not only fo the state vecto but also fo the input and output vectos, even when the oiginal system has constant input and output dimensions B Isolation of the finite pat In the second step we isolate a peiodic descipto system (E c, A c, B c, C c, D c ) whee E c and D c ae squae invetible matices nce educed to this fom, the tansmission zeos of the system ae the chaacteistic multiplies of the peiodic pai (E c, A c B c (D c ) 1 C c ) It will be shown late that the chaacteistic multiplies can be obtained without inveting D c The goal of the eduction in this step is to isolate a egula pat of the tansfomed pencil S(z) which contains the finite zeos Fo this pupose we edefine the dimensions m := n + m, p := ν p and n := Fo convenience, we will also euse the oiginal notation by edefining (E, A, B, C, D) := (E, A, B, C, D) The isolation of the pat containing the finite zeos is done in two steps In the fist step, we isolate a pat which coesponds to a peiodic descipto system (E, A, B, C, D ), whee E is squae invetible and D is full ow an This is pefomed by employing the following pocedue genealizing the Algoithm S-REDUCE of 12:

4 Algoithm PS-REDUCE input (E, A, B, C, D, n, m, p), output (E, A, B, C, D, n, m, p ) step i 1 Fo each = 1,, K, compess the ows of D with othogonal X (i) and tansfom C ; C,1 D,1 := X (i) C D, C,2 whee D,1 R (p τ (i) ) m has full ow an and C,2 R τ (i) n ; if τ (i) = 0 fo = 1,, K, then go to exit 1 2 Fo each = 1,, K, compess the columns of C,2 with othogonal V (i) such that C,2 V (i) = C,22, with C,22 R τ (i) µ(i) full column an; 3 Fo each = 1,, K, detemine othogonal U (i) such that U (i) E V (i) +1 is uppe tiangula; tansfom the system and patition as: U (i) (i) A B V X (i) C D I m A,11 A,12 B,1 = A,21 A,22 B,2 C,11 C,12 D,1 C,22 U (i) E V (i) E,11 E +1 =,12 E,22 whee A,22 R µ(i) +1 µ(i) 4 Set A := A,11, E := E,11, B := B,1, C := A,21, D C :=,11 5 Update n := n µ (i) B,2 D,1, p := p (τ (i), = 1,, K 6 If n = 0 fo = 1,, K, then go to exit 2 7 If µ (i) = 0 fo = 1,, K, then go to exit 1 8 i := i + 1 go to step i; exit 1 comment Full an matix D found (E, A, B, C, D ) := (E, A, B, C, D); n := n; m := m; p := p exit 2 comment System has no finite zeos µ (i) +1 ), fo Rema The compession of C,2 to a full column an matix can be done simultaneously with maintaining E uppe tiangula by using an algoithm simila to that of 15 fo standad descipto systems Details fo achieving this ae given in the next section bvious simplifications aise when E = I In this case, it possible to devise PS-REDUCE such that the educed E is also the identity matix This amounts to educe C,2 by pefoming Lyapunov similaity tansfomations, thus ensuing that U (i) = V (i) +1 The PS-REDUCE algoithm detemines implicitly the nomal an of the TFM W (z) If ρ K is the an defect of the oiginal descipto matix E K, then the nomal an of the TFM W (z) is K L = p ρ K =1 At the end of PS-REDUCE algoithm we obtain globally the educed matices S and T in (7) in the fom A B E S = C D, T = S T whee each S has full column an and the leading nonzeo ows of T fom a full ow an matix The oveall educed system pencil can be put, afte obvious ow and column pemutations, in the fom S Ŝ(z) = (z) S, (z) with S x (z) (x = o ) of the fom S1 x T1 x S2 x T2 x S x (z) = SK 1 x TK 1 x ztk x SK x whee fo = 1,, K S A := B C D, T E := A ;l,l A ;l,l 1 A ;l,2 A ;l,1 A ;l 1,l 1 A ;l,2 A ;l,1 S = A ;2,2 A ;2,1 A ;1,1 E;l,l 1 E;l,2 E ;l,1 E;l,2 E ;l,1 T = E;2,1 whee l is the numbe of steps pefomed in the Algoithm PS-REDUCE, A (i) ;i,i Rτ µ(i) ) is full column an and E;i,i 1 (i+1) Rτ µ (i) +1 is full ow an The subpencils S (z) and S (z) have the same stuctue as the oiginal system pencil S(z) S (z) contains the finite zeos of the peiodic system and the infomation on the ight Konece stuctue (eg, ight nullspace) of S(z) The tailing subpencil S (z), has full column an fo all finite values of z, an thus contains infomation on the odes of infinite zeos and

5 the left Konece stuctue (eg, left nullspace) of S(z) These facts can be poven similaly as done in 13 fo standad systems This leads to the following esult Theoem 2: The odes of the infinite elementay divisos of S (z) ae equal to the odes of the infinite zeos of the system (E, A, B, C, D) A dual algoithm to PS-REDUCE can be devised to compute a educed system (E c, A c, B c, C c, D c ), whee E c is squae invetible and D c is full column an In this case, the oveall educed system pencil can be put, afte ow and column pemutations, in the fom S Ŝ(z) = c (z) S c (z) whee both S c (z) and S c (z) have the same stuctue as S(z) This time S c (z) contains the finite and left Konece stuctue, while S c (z), having full ow an fo all finite z, contains the infinite and ight Konece stuctue By pefoming these two algoithms successively, we get finally (E c, A c, B c, C c, D c ), the educed system with both E c and D c squae invetible matices veall we can then show the following esult Theoem 3: The system (E, A, B, C, D) and the educed system (E c, A c, B c, C c, D c ) have the same finite tansmission zeos As aleady mentioned, the finite zeos can be computed as the chaacteistic multiplies of the peiodic pai (E c, A c B c (D c ) 1 C c ), whee E c is invetible To avoid the invesion of D c, it is possible to deflate a pat of simple infinite (non-dynamic) eigenvalues by pefoming a final compession of the system matices To do that, we detemine V fo = 1,, K by compessing the columns of C c D c system matices as to Df and tansfoming the educed A f E f D f A c := C c E c := B c D c V, V +1 This eduction can be always pefomed such that the esulting non-singula E f ae uppe tiangula (see 1, pp 33-34) By this final eduction we succeeded to isolate the egula pat S f (z) of the system pencil S(z) which contains the finite tansmission zeos This pat has the same stuctue as the oiginal system pencil, thus we can feely associate this pencil to a peiodic eigenvalue poblem defined by the peiodic pai (E f, A f ) The chaacteistic multiplies of this pai ae the finite tansmission zeos of the peiodic system C Computation of the finite zeos The thid step of zeos computations consists in solving the peiodic eigenvalue poblem fo the esulting peiodic pai (E f, A f ) Fo constant dimension, the peiodic QZ-algoithm 2, 9 can be employed fo this pupose Fo time-vaying dimensions the extended peiodic Schu fom based eduction 16 can be applied to the n f +1 nf peiodic matices (E f ) 1 A f, = 1,, K This is always possible, since E f is non-singula (and also uppe tiangula) Howeve, a stongly numeically stable appoach must avoid any invesion, and theefoe we can easily extend the appoach of 16 to compute two peiodic tansfomation matices U and V such that à f := U A f V A f,11 A f,12 = 0 à f,,22 (8) Ẽ f := U E f V E f,11 E f,12 +1 = 0 Ẽ f,,22 whee A f,11, Ef,11 n f Rnf fo n f = min {n f }, Af,22, R (nf +1 nf ) (n f nf ), E f,22, R(nf +1 nf ) (n f +1 nf ) fo = 1,, K Moeove, one of matices in the leading position, say A f K,11, is in Hessenbeg fom, Af,11 fo = 1, K 1, E f,11 and Ef,22 fo = 1, K ae uppe tiangula, and A,22 fo = 1, K ae uppe tapezoidal Thus, the pai (E f 11, Af 11 ) is in a genealized peiodic Hessenbeg fom as equied by the application of the peiodic QZalgoithm By applying this algoithm we compute n f finite zeos epesenting the so-called coe set of zeos, which ae independent of the time moment Additionally, thee ae also n f 1 nf null zeos, whose numbe vaies accoding to the chosen time moment IV NUMERICAL ASPECTS Fo the eduction of S(z) we employed exclusively stuctue peseving othogonal tansfomations of the fom (7) Thus it possible to pove that the computed finite zeos ae exact fo slightly petubed initial matices S, T, which satisfy X X ε X X, X = S, T whee, in each case, ε X is a modest multiple of the elative machine pecision ε M It follows that the poposed algoithm is stongly bacwad stable Regading the computational complexity of the poposed algoithm, we note that all eductions ae pefomed K- times on low ode matices, thus the oveall computational complexity is popotional with K To estimate the wostcase computational complexity, we assume constant dimensions n, m and p fo state-, input- and output vectos, espectively, and E ae invetible The system compession pefomed by using eithe SVD-based o an-evealing QRdecomposition based eductions equies (Kn 3 ) floating point opeations (flops) The compessions of D, = 1,, K can be done by computing successively K anevealing QR-decompositions of p m matices and applying the tansfomation to a p n sub-bloc This eduction step, although pefomed moe than once, has a computational

6 complexity of (K(n + p)pn) A wost-case computational complexity of (Kn 3 ) is also guaanteed fo the last steps, to compute the finite zeos via the extended QZ-algoithm The only citical computation is to maintain efficiently the uppe tiangula fom of E at successive steps of the PS-REDUCE algoithm Note that by just computing U such that U (i) E V (i) +1 is uppe tiangula is an opeation of complexity (n 3 ) This would mae the oveall wost-case complexity to maintain E uppe tiangula fo = 1,, K to become (Kn 4 ) To avoid this, we can pefom the compession of C,2 with V (i) and estoing the uppe-tiangula fom of E 1 V (i) simultaneously, by employing Givens otations The technique is entiely simila to that poposed in 15 and also employed in 12 veall, this equies to pefom the eduction fo values of in a evese ode, fo = K, K 1,, 1 Using this appoach, this computation has pe iteation step a complexity at most (Kηn 2 ), whee η is small compaed to n Thus, the oveall complexity of the compession-estoing algoithm is (Kn 3 ) Summing up, the computation of finite zeos has a wost-case complexity which can be bounded by K(p + n)(m + n)n, which coesponds to what is expected fo a satisfactoy algoithm fo peiodic systems 18 V CNCLUSIN In this pape we developed a stongly numeically bacwad stable algoithm to compute the finite zeos of a staced system matix of a peiodic system This algoithm can be applied to find the finite zeos, finite poles and finite decoupling zeos of the system matix and povides infomation to detemine the odes of infinite zeos as well as the left and ight nullspace stuctues of the coesponding lifted tansfe function These last aspects will be addessed in a sepaate pape The algoithm wos fo matices of vaying dimension, and peseves the bloc cyclic stuctue of the coesponding lifted system pencil This leads to two main benefits: 1) a satisfactoy wost-case computational complexity, which is linea in the peiod K and cubic in the maximum dimension of the blocs; and 2) stong numeical stability achieved by employing exclusively stuctue peseving othogonal tansfomations Accoding to 18, this algoithm is well-suited fo obust softwae implementations VI REFERENCES 1 T Beelen and P Van Dooen An impoved algoithm fo the computation of Konece s canonical fom of a singula pencil Lin Alg & Appl, 105:9 65, A W Bojanczy, G Golub, and P Van Dooen The peiodic Schu decomposition Algoithms and applications In F T Lu (Ed), Poceedings SPIE Confeence, vol 1770, pp 31 42, July P Colanei and S Longhi The ealization poblem fo linea peiodic systems Automatica, 31: , A Emami-Naeini and P M Van Dooen Computation of zeos of linea multivaiable systems Automatica, 18: , I Gohbeg, M A Kaashoe, and J Kos Classification of linea peiodic diffeence equations unde peiodic o inematic similaity SIAM J Matix Anal Appl, 21: , M Gasselli and S Longhi Zeos and poles of linea peiodic discete-time systems Cicuits, Systems and Signal Pocessing, 7: , M Gasselli and S Longhi Finite zeo stuctue of linea peiodic discete-time systems Int J Systems Sci, 22: , M Gasselli and S Longhi Pole-placement fo noneachable peiodic discete-time systems Math Contol Signals Syst, 4: , J J Hench and A J Laub Numeical solution of the discete-time peiodic Riccati equation IEEE Tans Autom Contol, 39: , A MacFalane and N Kacanias Poles and zeos of linea multivaiable systems: a suvey of the algebaic, geometic, and complex-vaiable theoy Int J Contol, 24:33 74, R A Meye and C S Buus A unified analysis of multiate and peiodically time-vaying digital filtes IEEE Tans Cicuits and Systems, 22: , P Misa, P Van Dooen, and A Vaga Computation of stuctual invaiants of genealized state-space systems Automatica, 30: , F Svaice Computation of the stuctual invaiants of linea multivaiable systems with an extended vesion of the pogam ZERS Systems & Contol Lett, 6: , P Van Dooen Two point bounday value and peiodic eigenvalue poblems Poc CACSD 99 Symposium, Kohala Coast, Hawaii, A Vaga Computation of ieducible genealized statespace ealizations Kybenetia, 26:89 106, A Vaga Balancing elated methods fo minimal ealization of peiodic systems Systems & Contol Lett, 36: , A Vaga Computation of tansfe functions matices of peiodic systems Poc of CDC 2002, Las Vegas, Nevada, A Vaga and P Van Dooen Computational methods fo peiodic systems - an oveview Poc of IFAC Woshop on Peiodic Contol Systems, Como, Italy, pp , A Vaga and P Van Dooen Computing the zeos of peiodic descipto systems Systems & Contol Lett, 50, 2003 (to appea) 20 G Veghese, P Van Dooen, and T Kailath Popeties of the system matix of a genealized state-space system Int J Contol, 30: , 1979