Minimal state space realization of MIMO systems in the max algebra

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1 KULeuven Department of Electrical Engineering (ESAT) SISTA Technical report Minimal tate pace realization of MIMO ytem in the max algebra B De Schutter and B De Moor If you want to cite thi report pleae ue the following reference intead: B De Schutter and B De Moor Minimal tate pace realization of MIMO ytem in the max algebra Proceeding of the 3rd European Control Conference (ECC 95) Rome Italy pp Sept 1995 ESATSISTA KULeuven Leuven Belgium phone: (ecretary) fax: URL: Thi report can alo be downloaded viahttp://pubdechutterinfo/ab/94_54html
2 MINIMAL STATE SPACE REALIZATION OF MIMO SYSTEMS IN THE MAX ALGEBRA Bart De Schutter and Bart De Moor ESATSISTA KULeuven Kardinaal Mercierlaan 94 B3001 Leuven Belgium tel: fax: Keyword: dicrete event ytem max algebra tate pace model minimal realization extended linear complementarity problem Abtract The topic of thi paper i the (partial) minimal realization problem in the max algebra which i one of the modeling framework that can be ued to model dicrete event ytem We ue the fact that a ytem of multivariate maxalgebraic polynomial equalitie can be tranformed into an Extended Linear Complementarity Problem to find all equivalent minimal tate pace realization of a multiple input multiple output (MIMO) maxlinear dicrete event ytem tarting from it impule repone matrice We alo give a geometrical decription of the et of all minimal tate pace realization 1 Introduction 11 Overview In thi paper we conider dicrete event ytem example of which are flexible manufacturing ytem ubway traffic network parallel proceing ytem telecommunication network There exit a wide range of framework to model and analyze dicrete event ytem: Petri net generalized emimarkov procee formal language perturbation analyi computer imulation and o on We concentrate on a ubcla of dicrete event ytem that can be decribed with the max algebra [1 2 3 Although the decription of thee ytem i nonlinear in linear algebra the model become linear when we formulate it in the max algebra In thi paper we only conider ytem that can be decribed by Thi paper preent reearch reult of the Belgian program on interuniverity attraction pole (IUAP50) initiated by the Belgian State Prime Miniter Office for Science Technology and Culture The cientific reponibility i aumed by it author Reearch aitant with the NFWO (Belgian National Fund for Scientific Reearch) Senior reearch aociate with the NFWO a timeinvariant tate pace model Therefore we limit ourelve to determinitic ytem ie ytem in which the equence and the duration of the activitie are fixed or can be determined in advance In order to analyze ytem it i advantageou to have a compact decription ie a decription with a few parameter a poible For a ytem that can be decribed by a maxlinear tate pace model thi give rie to the minimal tate pace realization problem In thi paper we addre the (partial) minimal tate pace realization problem for maxalgebraic multiple input multiple output (MIMO) ytem Firt we dicu the problem of olving a ytem of multivariate maxalgebraic polynomial equalitie and inequalitie and then we ue the reult to olve the minimal tate pace realization problem for MIMO maxlinear dicrete event ytem We alo give a geometric characterization of the et of all minimal tate pace realization and illutrate the procedure with an example 12 The max algebra In thi ection we give a hort introduction to the max algebra and tate the definition theorem and propertie thatweneedintheremainderofthipaper Amorecomplete overview of the max algebra can be found in [1 3 The baic maxalgebraic operation are defined a follow: a b = max(ab) a b = a+b where ab R { } The reulting tructure R max = (R { } ) i called the max algebra The zero elementfoiε def = : a R {ε} : a ε = a = ε a Define R ε = R {ε} Let r R The rth maxalgebraic power of a R i repreented by a r and correpond to ra in linear algebra Hence a 0 = 0 and a 1 = a i the invere element of a wrt in R ε If r > 0 then ε r = ε; if r 0 then ε r i not defined The maxalgebraic operation are extended to matrice
3 in the uual way If AB Rε m n then (A B) ij = a ij b ij If A Rε m p and B R p n ε then (A B) ij = p a ik b kj The matrix E n i the n by n identity k=1 matrix in the max algebra: (E n ) ij = 0 if i = j and (E n ) ij = ε if i j The m by n maxalgebraic zero matrix i repreented byε m n : (ε m n ) ij = ε for all ij In contrat to linear algebra there exit no invere element wrt in R ε To overcome thi problem we hall ue the extended max algebra S max [8 10 which i a kind of ymmetrization of the max algebra We hall retrict ourelve to an intuitive introduction to the mot important feature of S max For a formal derivation and for the proof of the propertie and theorem of thi ection the intereted reader i referred to [ We introduce two new element for each element x R ε : x and x Thi give rie to an extenion S of R ε that contain three clae of element: the maxpoitive or zero element: S R ε the maxnegative or zero element: S = { a a R ε } the balanced element: S = {a a R ε } where S = S S S By definition we have ε = ε = ε The element of S and S are called igned The operation between an element of S and an element of S i defined a follow: a ( b) = a if a > b a ( b) = b if a < b a ( a) = a where a b R ε The ign correpond to the ign in linear algebra By analogy we write a b intead of a ( b) We have ( a) = a ( a) ( b) = (a b) ( a) b = a ( b) = (a b) for a b S If one of the operand i balanced we evaluate the expreion a follow: a b = a (b ( b)) a b = a (b ( b)) and we ue the fact that both and are aociative and commutative in S and that i ditributive wrt in S The reulting tructure S max = (S ) i called the extended max algebra Let a S The maxpoitive part a + and the maxnegative part a of a are defined a follow: if a R ε then a + = a and a = ε if a S then a + = ε and a = a if a S then there exit an element b R ε uch that a = b and then a + = a = b So a + a R ε and a = a + a In linear algebra we have a a = 0 for all a R but in S max we have a a = a ε (unle a = ε) Therefore we introduce a new relation the balance relation repreented by Definition 11 (Balance) Conider ab S We ay that a balance b denoted by a b if and only if a + b = b + a Property 12 a S : a = a a ε We could ay that the balance relation i the S max counterpart of the equality relation However the balance relation i not an equivalence relation ince it i not tranitive A ign in a balance mean that the element hould be at the other ide: Property 13 abc S : a b c if and only if a b c If both ide of a balance are igned we can replace the balance by an equality: Property 14 ab S S : a b a = b Definition 15 (Determinant) Let A S n n The maxalgebraic determinant of A i defined a deta = σ P n gn(σ) n i=1 a iσ(i) where P n i the et of all permutation of {1n} and gn(σ) = 0 if the permutation σ i even and gn(σ) = 0 if the permutation i odd Definition 16 (Minor rank) Conider A S m n The maxalgebraic minor rank of A i the dimenion of the larget quare ubmatrix A ub of A uch that deta ub / ε Definition 17 (Characteritic equation) Let A S n n The maxalgebraic characteritic equation of A i defined a det(a λ E n ) ε If we work thi out we get λ n n a p λ n p ε Theorem 18 (CayleyHamilton) In S max every quare matrix atifie it characteritic equation
4 2 Sytem of multivariate maxalgebraic polynomial equalitie and inequalitie In thi ection we conider ytem of multivariate maxalgebraic polynomial equalitie and inequalitie which can be een a a generalized framework for many important maxalgebraic problem uch a matrix decompoition tranformation of tate pace model tate pace realization of impule repone contruction of matrice with a given characteritic polynomial and o on [6 7 Conider the following problem: Given a et of integer {m k } and three et of real number {a ki } {b k } and {c kij } with i = 1m k j = 1n and k = 1p 1 +p 2 find x R n uch that m k i=1 m k i=1 a ki a ki n j=1 n j=1 x j c kij x j c kij or how that no uch x exit = b k for k = 1p 1 (1) b k for k =p 1 +1 p 1 +p 2 (2) We call (1) (2) a ytem of multivariate maxalgebraic polynomial equalitie and inequalitie Note that the exponent can be negative or real In[7wehavehownthatthiproblemiequivalenttoan Extended Linear Complementarity Problem (ELCP) [5 Thi lead to an algorithm that yield the entire olution et of problem (1) (2) In general thi olution et conit of the union of face of a polyhedron P and i defined by three et of vector X cen X inf X fin and a et Λ Thee et can be characterized a follow: X cen i a et of central ray of P It i a bai for the larget linear ubpace of P Let u call P red the polyhedron obtained by ubtracting thi larget linear ubpace from P X inf i a et of extreme ray or vertice at infinity of the polyhedron P red X fin i the et of the finite vertice of the polyhedron P red Λ i a et of pair { } X inf X fin with X inf X inf X fin X fin and X fin Each pair determine a face F of the polyhedron P that belong to the olution et: X inf contain the extreme ray of F and X fin contain the finite vertice of F The olution et of problem (1) (2) i characterized by the following theorem: Theorem 21 When X cen X inf X fin and Λ are given then x i a finite olution of the ytem of multivariate maxalgebraic polynomial equalitie and inequalitie if and only if there exit a pair { } X inf X fin Λ uch that x = x k X cen λ k x k + x k X inf with λ k R κ k µ k 0 and k κ k x k + µ k = 1 x k X fin µ k x k Theorem 22 In general the et of the (finite) olution of a ytem of multivariate maxalgebraic polynomial equalitie and inequalitie conit of the union of face of a polyhedron Remark: Solution for which ome of the component are equal to ε can be obtained by a limit or a threhold procedure Thee olution would correpond to point at infinity of the polyhedron P See[7 for more information on thi ubject 3 Minimal tate pace realization Conider a dicrete event ytem that can be decribed by the following nth order tate pace model with m input and l output: x(k +1) = A x(k) B u(k) (3) y(k) = C x(k) (4) where A Rε n n B Rε n m and C R l n ε The vector x repreent the tate u i the input vector and y i the output vector of the ytem Ifweapplyaunitimpule: e(k) = 0ifk = 0ande(k) = ε if k 0 to the ith input of the ytem and if we aume that the initial tate x(0) atifie x(0) =ε n 1 we get y(k) = C A k 1 B i for k = 12 a the output of the ytem where B i i the ith column of B We repeat thi experiment for all input i = 12m and tore the output in l by m matrice G k = C A k B for k = 01 The G k are called the impule repone matrice or Markov parameter Suppoe that A B and C are unknown and that we only know the Markov parameter(eg from experiment where we aume that the ytem i maxlinear and timeinvariant and that there i no noie preent) How can we contruct A B and C from the G k? Thi problem i called tate pace realization If we make the dimenion of A minimal we have a minimal tate pace realization problem In order to olve thi problem we firt need a lower bound r for the minimal ytem order A a direct conequence of the CayleyHamilton theorem the Markov parameter atify the maxalgebraic characteritic equation of A So we could try to find a table relationhip
5 of the form a p G k+r p ε l m for k = 01 (5) p=0 with a few term a poible where the a p hould correpond to the coefficient of the characteritic equation of a matrix with entrie in R ε Expreion (5) i a ytem of linear balance with the a p a unknown In [6 4 we have applied thi procedure to obtain a lower bound r for the minimal ytem order of a ingle input ingle output ytem If we decompoe the a p a a p = a + p a p and if we ue Propertie 13 and 14 we can tranform (5) into a ytem of multivariate maxalgebraic polynomial equalitie with the maxpoitive and the maxnegative part of the a p a variable We could alo ue the following theorem [8 9 to obtain a lower bound for the minimal ytem order: Theorem 31 Let G k = C A k B for k = 01 be the Markov parameter of a timeinvariant maxlinear ytem with ytem matrice A B and C Then the maxalgebraic minor rank of the block Hankel matrix H = G 0 G 1 G 2 G 1 G 2 G 3 G 2 G 3 G 4 i a lower bound for the minimal ytem order In practice we only conider a truncated verion of the emiinfinite block Hankel matrix H: G 0 G 1 G p G 1 G 2 G p+1 H pq = G q G q+1 G p+q The maxalgebraic minor rank of H pq i a lower bound for the minimal ytem order WeaumethattheentrieofalltheG k arefiniteand that the ytem exhibit a periodic teady tate behavior of the following kind: n 0 d N and c R uch that n n 0 : G n+d = c d G n (6) It can be hown [1 8 that a ufficient condition for (6) to hold i that the ytem matrix A i irreducible ie (A A 2 A n ) ij ε for all ij Thi will for example be the cae for a dicrete event ytem without eparate independent ubytem and with a cyclic behavior or with feedback from the output to the input like flexible production ytem in which the part are carried around on a limited number of pallet that circulate in the ytem We tart with r equal to the lower bound Now we try to find an rth order partial tate pace realization of the given impule repone: we have to find A Rε r r B Rε r m and C R l r ε uch that C A k B = G k for k = 01N 1 (7) for N large enough If we work out the equation of the form (7) we get for k = 0: c ip b pj = (G 0 ) ij (8) for i = 12l and j = 12m For k > 0 we obtain c ip (A k ) pq b qj = (G k ) ij (9) q=1 for i = 12l and j = 12m Since (A k ) pq = i 1=1 i k 1 =1 equation (9) can be rewritten a r k 1 q=1 =1 c ip r u=1 v=1 a pi1 a i1i 2 a ik 1 q r a uv γ kpquv b qj = (G k ) ij (10) where γ kpquv i the number of time that a uv appear in the th term of (A k ) pq If a uv doe not appear in that term we take γ kpquv = 0 ince a 0 = 0 a = 0 the identity element for If we ue the fact that x R ε : x x = x and xy R ε : x y x x y y we can remove many redundant term There are then w kij term in (10) where w kij r k+1 If we put all unknown in one large vector x of length r(r + m + l) we have to olve a ytem of multivariate maxalgebraic polynomial equation of the following form: w kij r(r+m+l) q=1 r(r+m+l) q=1 x q δ0ijpq = (G 0 ) ij (11) x q δ kijpq = (G k ) ij (12) for i = 12l; j = 12m and k = 12N 1 Ifwefindaolutionxof(11) (12) weextracttheentrie of the ytem matrice A B and C from x If we do not get any olution thi mean that r i le than the minimal ytem order ie the lower bound i not tight Then we have to augment our etimate of the minimal ytem order and repeat the above procedure
6 but with r +1 intead of r We continue until we find a olution of (11) (12) Thi yield a minimal tate pace realization of the firt N impule repone matrice If N i large enough we can obtain the et of all realization of the given impule repone by putting all component that are maller than ome threhold equal to ε if neceary If the ytem doe not exhibit the teady tate behavior of (6) then the procedure preented in thi paper will in general only yield partial tate pace realization However in ome cae the ELCP technique can till be applied if an analogou but more complicated threhold procedure i ued Now we can characterize the et of all(partial) minimal tate pace realization of a given impule repone: Theorem 32 In general the et of all (partial) minimal tate pace realization of the impule repone of a maxlinear timeinvariant dicrete event ytem conit of the union of face of a polyhedron in the xpace where x i the vector obtained by putting the component of the ytem matrice in one large vector 4 Example We hall now illutrate the preceding procedure with an example Example 41 We tart from a ytem with ytem matrice: A = ε 14 ε ε B = ε 9 and ε [ 2 ε 9 C = 0 4 ε Now we are going to contruct all equivalent minimal tate pace realization of thi ytem tarting from it impule repone matrice which are given by {G k } k=0 = [ [ [ [ [ [ [ Note that the G k exhibit the behavior of (6) with n 0 = 1 d = 2 and c = 95 The relation of the form (5) with a few term a poible i given by G k+2 8 G k+1 19 G k ε 2 2 for k = 01 or equivalently G k+2 = 8 G k+1 19 G k for k = 01 by Propertie 13 and 14 So the minimal ytem order i greater than or equal to r = 2 The maxalgebraic minor rank of the truncated Hankel matrix H 66 i alo equal to 2 Now we try to find a econd order tate pace realization of the impule repone matrice Let u take N = 3 The ELCP algorithm of [5 yield the ray and vertice oftable1and2andthepairofubetoftable3 Ifwe take N > 3 we get the ame reult but if we take N < 3 ome combination of the ray and the vertice only lead to a partial realization of the given impule repone: they only fit the firt N impule repone matrice Any arbitrary finite minimal realization can now be expreed a x = λ 1 x c 1 +λ 2 x c 2 + k κ k x i k +x f l (13) with λ 1 λ 2 R κ k 0 and x i k Xinf x f l Xfin with {128} and where x i the column vector obtained by putting the entrie of the ytem matrice in one large column vector Expreion (13) how that the et of all equivalent minimal tate pace realization of the given impule repone i the union of 8 face of a polyhedron in the xpace 5 Concluion and future reearch We have hown that the problem of finding a minimal tate pace realization of the impule repone of a multiple input multiple output maxlinear timeinvariant dicrete event ytem (that exhibit a particular kind of periodic teady tate behavior) can be reformulated a a ytem of multivariate polynomial equation in the max algebra Thi mean that we can ue the ELCP algorithm of [5 to olve uch a problem One of the main characteritic of the ELCP algorithm of [5 i that it find all olution Thi provide a geometrical inight in the et of all equivalent minimal tate pace realization of an impule repone On the other hand thi alo lead to large computation time and torage pace requirement if the number of variable and equation i large Therefore it might be intereting to develop algorithm that only find one olution a we have done for the minimal realization problem for ingle input ingleoutputytemin[4 Amongtheetofallpoible realization we could alo try to find certain privileged realization uch a balanced realization Reference [1 F Baccelli G Cohen GJ Older and JP Quadrat Synchronization and Linearity New York: John Wiley & Son 1992 [2 GCohenDDuboiJPQuadratandMViot A linearytemtheoretic view of dicreteevent pro
7 cee and it ue for performance evaluation in manufacturing IEEE Tranaction on Automatic Control vol 30 no 3 pp Mar 1985 [3 RA CuninghameGreen Minimax Algebra vol 166 of Lecture Note in Economic and Mathematical Sytem Berlin Germany: SpringerVerlag 1979 [4 B De Schutter and B De Moor The characteritic equation and minimal tate pace realization of SISO ytem in the max algebra in 11th International Conference on Analyi and Optimization of Sytem (SophiaAntipoli France June 1994) (G Cohen and JP Quadrat ed) vol 199 of Lecture Note in Control and Information Science pp Springer 1994 [5 B De Schutter and B De Moor The extended linear complementarity problem Mathematical Programming vol 71 no 3 pp Dec 1995 [6 B De Schutter and B De Moor Minimal realization in the max algebra i an extended linear complementarity problem Sytem & Control Letter vol 25 no 2 pp May 1995 [7 B De Schutter and B De Moor A method to find all olution of a ytem of multivariate polynomial equalitie and inequalitie in the max algebra Dicrete Event Dynamic Sytem: Theory and Application vol 6 no 2 pp Mar 1996 [8 S Gaubert Théorie de Sytème Linéaire dan le Dioïde PhD thei Ecole Nationale Supérieure de Mine de Pari France July 1992 [9 S Gaubert On rational erie in one variable over certain dioid Tech rep 2162 INRIA Le Chenay France Jan 1994 [10 Max Plu Linear ytem in (max+) algebra in Proceeding of the 29th IEEE Conference on Deciion and Control Honolulu Hawaii pp Dec 1990 Set X cen X fin Ray x c 1 x c 2 x f 1 x f 2 a a a a b b b b c c c c Table 1: The central ray and the finite vertice for Example 41 Set X inf Ray x i 1 x i 2 x i 3 x i 4 x i 5 x i 6 x i 7 x i 8 a a a a b b b b c c c c Table 2: The vertice at infinity for Example 41 X inf X fin 1 {x i 1x i 2x i 4} {x f 1} 2 {x i 1x i 2x i 5} {x f 1} 3 {x i 2x i 3} {x f 2} 4 {x i 2x i 7} {x f 2} X inf X fin 5 {x i 3x i 6x i 8} {x f 2} 6 {x i 4x i 8} {x f 1} 7 {x i 5x i 8} {x f 1} 8 {x i 6x i 7x i 8} {x f 2} Table 3: The pair of ubet for Example 41