pp. 72-76In this paper we describe a procedure to visit all feedback classes of locally Brunovsky linear system over fixed R=C(S1) the ring of real continuos functions defined on the unit circle. Furthermore, we give the exact number of such classes throughout partitions of integers, binary strings and colored Ferrers diagrams.S

Carriegos Vieira, Miguel

Castro García, Noemí De

López Cabeceira, María Montserrat

Cybernetics and Physics

English

Castro García, Noemí de

Leon University (Spain)

CYBERNETICS AND PHYSICS, VOL. 2, NO. 2, 2013 , 72–76ENUMERATION OF LOCALLY BRUNOVSKY LINEARSYSTEMS OVER C(S1)-MODULES. A PROCEDUREMiguel V. CarriegosDepartment of MathematicsUniversity of LeónSpainmiguel.carriegos@unileon.esNoeḿı DeCastro-GarćıaDepartment of MathematicsUniversity of LeónSpainncasg@unileon.esMontserrat M. Cb. L ópezDepartment of MathematicsUniversity of LeónSpainmmlopc@unileon.esAbstractIn this paper we describe a procedure to visit all feed-back classes of locally Brunovsky linear system overfixed R = C(S1) the ring of real continuos functionsdefined on the unit circle. Furthermore, we give theexact number of such classes throughout partitions ofintegers, binary strings and colored Ferrers diagrams.Key wordsBrunovsky linear system, feedback classification, Fer-rers diagrams, partition of integers, projective module.1 IntroductionLetR be a commutative ring with unit element1 6= 0.A linear system overR is given by a linear rule (orright hand side) on the formx+ = Ax + Bu wherex ∈ X are states,u ∈ U are inputs, andx+ is the time-derivative or time-shift in the sequential case. Sets ofstatesX and of inputsU areR-modules while mapsA andB areR-linear maps. In this way, we say thata linear systemΣ (see figure 1) and other analogouslinear systemΣ′ are said to be Feedback Equivalentif we can bring one of them into the another by a fi-nite composition of the following Basic Feedback Ac-tions: IsomorphismsQ : U → U ′ in the input mod-ule, isomorphismsP : X → X ′ in the state moduleand feedback actionsF : X → U which transforms(A,B) to system(P (A + BF )P−1, PBQ). In gen-eral, the theory of linear control systems over a com-mutative ringR goes back to the models of [Morse,1976] for delay models. See [Brewer, Bunce and Van-Vleck, 1986], [Carriegos and Sánchez-Giralda, 2001]and [Hermida-Alonso, López-Cabeceira and Trobajo,2005] to do general reading about equivalent linear sys-tems over commutative rings.On the other hand, it is a known that partial reachabil-ity linear map given byϕΣi =(B AB · · · Ai−1B): U⊕i −→ X (1)Σ :UցBX →A XFigure 1. Σ(A,B) Linear Systemis a feedback invariant, up to equivalence, associated toΣ (see [Carriegos, 2003] and [Hermida-Alonso, Pérezand Sánchez-Giralda, 1996]). So, we have a main setof feedback invariants, up to up to isomorphism, asso-ciated to systemΣ, it is, quotient modulesNΣi+1/NΣi =Im(B,AB, ..., AiB)/Im(B,AB, ..., Ai−1B). (2)Furthermore, in the case of reachable linear systemsover a field, or in the more general framework ofprojective-free rings, we known that if allR-modulesNΣi+1/NΣi are free, then there is a complete set of in-variants verifyingX = NΣ1 ⊕NΣ2 /NΣ1 ⊕ · · · ⊕NΣs /NΣs−1. (3)Thus, once we have fixed a projective-free ringR andthe dimensionsm = dimU andn = dimX , all feed-back classes ofm-inputn-dimensional linear systemsare in one to one correspondence with the set of par-titions of integern in decreasing sequences, equiva-lently, all the Ferrers diagrams of integern (see [Knuth,2004] to get a complete reading about partitions of in-teger subject).This paper is organized as it follows: In section2, our study is focused over continuos real functionsR = C(K) defined in a topological spaceK (see[Brunovsky, 1970]), in particular is given necessaryand sufficient conditions for classifying linear systemsoverR = C(S1) by Ferrers diagrams (see [Carriegosand Sánchez-Giralda, 2001] and [Ferrer, Garcı́a-Planasand Puerta, 1997] to do a previous reading). In sec-tion 3, we obtain the enumeration and the number ofCYBERNETICS AND PHYSICS, VOL. 2, NO. 2, 2013 73all feedback classes of reachable linear systems overthe freeR = C(S1)-module of rankn. In section 4, wedesign a procedure to obtain and enumerate all suchclasses. In section 5, we extend feedback classes studyof reachable linear systems overR = C(S1)-modules.2 The Unit Circle and Colored Ferrer’s DiagramsIn this section, we strongly use next result (see [Car-riegos and Sánchez-Giralda, 2001]):The classificationproblem (in the case of projective invariants) is actuallyequivalent to the problem of characterization of all pos-sible decompositions of finitely generatedR-modulesU andX on the formU = P0 ⊕ P1X = P1 ⊕ P2 ⊕ · · · ⊕ Ps(4)whereP0 represents a solution forKer(B) andPi rep-resents a solution forNΣi /NΣi−1. Thus, the only restric-tion to solve the system of equations is thatPi+1 mustbe a direct summand ofPi for all i.In order to above theorem, to give the complete clas-sification of locally Brunovsky systems is needed toknow exactly the monoid(Proj(R),⊕) of isomor-phism classes of finitely generatedR-modules withthe direct sum as internal operation. The full descrip-tion of the monoid(Proj(R),⊕) is a great task. Ofcourse, if finitely generated projective are free, then(Proj(R),⊕) is isomorphic to(N∪{0},+), but in gen-eral this is not the case. IfR = C(K) is the ring ofcontinuos functions defined on a compact topologicalspaceK, then(Proj(R),⊕) ≡ (Vect(K),⊕) depend,of course, on the topology ofK (see [Swan, 1962]).Our paper is devoted to study ofK = S1 the real unitcircumference. In this case,(Proj(R = C(S1)),⊕) isthe commutative monoid generated by the symbolsR(representing trivial vector bundles) andP (represent-ing the Möbius Strip) modulo the relationP ⊕ P = R⊕R = R2. (5)Consequently, there is only two isomorphism classesof rankr projectiveR-modules:Rr (the free one) andRr−1 ⊕ P . Thus, we may characterize the feedbackclass of a locally Brunovsky linear system overR bya colored Ferrer’s diagram: Because(Proj(R),⊕) isthe commutative monoid generated by the symbolsRandP , then every building block is a rank1 projectivemodule, and there are two classes depicted byfor R andP respectively. Observe that we have the rule(5), it is figure 2.So, locally Brunovsky linear systems over the finitelygenerated moduleX of rankn would be describe bya colored Ferrer’s diagram with exactlyn buildingblocks (white or grey) where the following four=Figure 2. Equation 5Figure 3. Example 2.1restrictions apply:i) There is at most one grey block on each row (byequation (5)).ii ) Parity condition: ifX = Rn, then there are an evennumber of grey blocks in the whole diagram.iii ) Theith row is at most as long as the (i− 1)th row(by decreasing ranks in the sequence (4)).iv) If two rows have the same lengthr, then they areequal (byRr is not a direct summand ofRr−1 ⊕P northe converse).Example 2.1. In figure 3, we can see the feed-back class of locally Brunovsky linear systems overX = (R⊕R⊕ P )⊕ (R⊕ P )⊕ (R) freeR = C(S1)-module of rankn = 6.3 Number of Locally Brunovsky Linear SystemsOver the FreeR = C(S1)-module of RanknLet’s denote bypR(n) the number of non-isomorphicdecompositions ofRn, while p̃R(n) denotes the num-ber of non-isomorphic decompositionsRn ∼= P1 ⊕· · · ⊕Ps with Pi+1 direct summand ofPi. Note that, ifR is projectively trivial, theñpR(n) = pR(n) = p(n)is the number of partitions of integern, but in generalp̃R(n) ≤ pR(n).So, the number of feedback classes of locallyBrunovsky linear systems over the freeR = C(S1)-module of rankn is p̃R(n). Thus, in order to give allfeedback classes equivalence we have to visit all par-titions x of n and to determinate that colored Ferrer’sdiagrams verifying the above four conditions on par-tition x. If we denote byp̃R(x) the number of suchdiagrams, thenp̃R(n) =∑xp̃R(x). (6)It is known a procedure to obtain all partitions of agiven integern in inverse lexicographic order. Thus, letx be a fixed partition ofn. First, conditioni) is directbecause inverse lexicographic order. Second, in orderto control conditioniv) we writex = xc11 , xc22 , . . . , xchhwheren = c1x1 + c2x2 + . . . + chxh andxi > xi+1for all i. In this way, by conditioniv), we can definetheith-row-block, of the Ferrers diagram, as the blockof ci rows (each row of lengthxi) associated toxcii74 CYBERNETICS AND PHYSICS, VOL. 2, NO. 2, 2013The key is denote the colored Ferrer’s diagrams, asso-ciated to a given sequence (4) with partitionx of n, asa binary stringw = w1w2 . . . wh, wherewi = 0 if andonly if the ith-row-block associated toxcii is white.Finally, by conditioni), the ith-row-block associatedto xcii affect conditionii ) if and only if ci is an odd ex-ponent. Letcr be first odd exponent in the wayxr ≥ xjfor all xj such thatcj is an odd number, then followingcomputing sentences are equivalent to conditionii ):1. (sum,w[i]c[i],i,1,h) = 0 mod(2)2. if (c[i] odd number) then(sum,w[i]c[i],i,1,h) = 0 mod(2)3. if (i not r and c[i] odd number) then(sum,w[i]c[i],i,1,h) = 1 mod(2)4. if (i not r and c[i] odd number) then(sum,w[i],i,1,h) = 1 mod(2)In this way, in order to verify above four conditions,observe thatci exponents are free for alli 6= r andonly cr control the parity conditionii ), i.e. parity iscontrolled by the bitwr throughout the equalitywr +∑i6=rwi = 0 mod2, (7)wherewi bits are free for alli 6= r. So, we have nextresult:Theorem 3.1. LetR = C(S1) be the ring of real con-tinuos functions defined on the unit circle. The numberof all feedback classes of locally Brunovsky linear sys-tems over the freeR = C(S1)-module of rankn is givenbyp̃R(n) =∑xp̃R(x) =∑x2k,where, if x is a partition denoted byx =xc11 , xc22 , · · · , xchh , thenk = h if not exists an odd ex-ponentci in partition x, andk = h− 1 in other case.Proof. It is known that the set of all feedback classesof locally Brunovsky linear systems over the freeR =C(S1)-module of rankn is the disjoint union, in par-titions of n, of sets of all feedback classes of locallyBrunovsky linear systems over the freeR = C(S1)-module of rankn throughout a given partitionx. Thus,p̃R(n) =∑x p̃R(x).On the other hand, in particular casex = xc11 withc1 odd number we have only a colored Ferrers di-agram (all building blocks are white) and it verifiesp̃R(x) = 2h−1 = 1. In other cases, by sentences 1,2, 3, 4 and equation (7), it is clear that all feedbackclasses of locally Brunovsky linear systems over thefreeR = C(S1)-module of rankn throughout a givenpartitionx = xc11 , xc22 , · · · , xchh are in one-to-one cor-respondence with the set of all binary strings of lengthx=61 w=0p̃R(x)=2h−1=1x=51,11 w=00 w=11p̃R(x)=2h−1=2x=41,21 w=00 w=11p̃R(x)=2h−1=2x=41,12 w=00 w=01p̃R(x)=2h−1=2x=32 w=0 w=1p̃R(x)=2h=2x=31,21,11 w=000 w=101w=110 w=011p̃R(x)=2h−1=4x=31,13 w=00 w=11p̃R(x)=2h−1=2x=23 w=0p̃R(x)=2h−1=1x=22,12 w=00 w=01w=10 w=11p̃R(x)=2h=4x=21,14 w=00 w=01p̃R(x)=2h−1=2x=16 w=0 w=1p̃R(x)=2h=2Figure 4. Feedback classes of Example 3.2k, wherek = h if not exists an odd exponentci in par-tition x, andk = h− 1 in other case. So,̃pR(x) = 2k.Example 3.2. All p̃R(n) feedback classes of locallyBrunovsky linear systems over the freeR = C(S1)-module of rankn = 6, are listed in figure 4. We con-CYBERNETICS AND PHYSICS, VOL. 2, NO. 2, 2013 75clude that there exist̃pR(n) =∑x p̃R(x) = 24 feed-back classes of locally Brunovsky linear systems overthe freeR = C(S1)-module of rankn = 6. Observethat, inside each binary stringw of each partitionxwe have mark (if possible) parity control bitwr withdouble underline.4 ProcedureIt is known an algorithm to obtain all partitions of agiven integern (see [Knuth, 2004]). We include it fora complete study of our subject:input(n){m=1, h=1, x[1]=n;for i=2 to n do x[i]=1;output(x[1]);while(x[1] not 1)do{if(x[h]==2)then{m=m+1, x[h]=1, x[m]=1, h=h-1}else{r=x[h]+1, t=m-h+1, x[h]=r,while(t>=r)do{h=h+1, x[h]=r, t=t-r}if(t==0)then{m=h}else{m=h+1, if(t>1)then{ h=h+1, x[h]=t}}}output(x[m])}}Next, we give our procedure to give all col-ored Ferrers diagramsD, associated to a partitionx = xc11 , xc22 , · · · , xchh , verifying conditionsi), ii ), iii )andiv).input(x[1],x[2],· · · ,x[h],c[1],c[2],· · · ,c[h]){r=0, k=h, boolean cont=true;for i=1 to h do if(c[i]mod2==1∧cont)then{r=i, k=h-1, cont=false}for i=0 to pow(2,k)-1 do{aux=integer[i].toBinaryString.ofLength[k]for j=1 to h doif(j<r)then w[j]=aux[j]elseif(j>r) then w[j]=aux[j-1]w[r]=sum(aux[j]c[j],j,1,h,j not r)mod2output(w[1],w[2],· · · ,w[h])}}input(w[1],w[2],· · · ,w[h]){D = ∅;for i=1 to h do{if(w[i]=0)then {add toD a building white block with c[i]rows and x[i] columns}else{add toD a building grey block with c[i]rows and x[i] columns}}output(D)}Example 4.1. Enumerate allp̃R(n) feedback classesof locally Brunovsky linear systems over the freeR =C(S1)-module of rankn = 50 associated to partitionx = 92, 63, 34, 21 of n. We haveh = 4 row-blocks in0→000w=00001→001w=01012→010w=01103→011w=00114→100w=11005→101w=10016→110w=10107→111w=1111Figure 5. Feedback classes of Example 4.1each diagramD, r = 2 position of control bit andk =h− 1 = 3 free bits, then there exist23 feedback classesof locally Brunovsky linear systems over the freeR =C(S1)-module of rankn = 380 associated to partitionx = 92, 63, 34, 21 of n:0 1 2 3 4 5 6 792 w1 0 0 0 0 1 1 1 163 w2 0 1 1 0 1 0 0 1 pcb34 w3 0 0 1 1 0 0 1 121 w4 0 1 0 1 0 1 0 1pcb:parity control bitwith colored Ferrers diagrams of figure 55 Number of Locally Brunovsky Linear SystemsOver X = Rn−1 ⊕ P an R = C(S1)-moduleAnalogously to section 3, locally Brunovsky lin-ear systems over the finitely generated moduleX =Rn−1 ⊕ P would be describe by a colored Ferrer’s di-agram with exactlyn building blocks (white or grey)76 CYBERNETICS AND PHYSICS, VOL. 2, NO. 2, 2013x=61 w=0p̃R(x)=2h−1=1x=51,11 w=10 w=01p̃R(x)=2h−1=2x=41,21 w=10 w=11p̃R(x)=2h−1=2x=41,12 w=10 w=11p̃R(x)=2h−1=2x=31,21,11 w=100 w=001w=010 w=111p̃R(x)=2h−1=4x=31,13 w=10 w=01p̃R(x)=2h−1=2x=23 w=1p̃R(x)=2h−1=1x=21,14 w=10 w=11p̃R(x)=2h−1=2Figure 6. Feedback classes of Example 3.2where it changes parity condition:ii ) Parity condition:if X = Rn−1 ⊕ P , then there are an odd number ofgrey blocks in the whole diagram. So, in this case wehaveif (i not r and c[i] odd number) then (sum,w[i],i,1,h) = 0mod(2). Observe that, in order to write a procedure wehave to replace the computing linew[r]=sum(aux[j]c[j],j,1,h,j not r)+1 mod2.Theorem 5.1. LetR = C(S1) be the ring of real con-tinuos functions defined on the unit circle. The num-ber p̃R(n) of all feedback classes of locally Brunovskylinear systems overX = Rn−1 ⊕ P a R = C(S1)-module is given by sum of̃pR(x) on x, where, ifxis a partition denoted byx = xc11 , xc22 , · · · , xchh , thenp̃R(x) = 0 if not exists an odd exponentci in partitionx, andp̃R(x) = 2h−1 in other case.Example 5.2. All p̃R(n) feedback classes of locallyBrunovsky linear systems overX = R5 ⊕ P anR =C(S1)-module, are listed in figure 6. Note that parti-tionsx = 32, x = 22, 12 andx = 16 verify p̃R(x) = 0,so this partitions do not math feedback classes.6 ConclusionIn this paper, we design computing procedure for ob-taining feedback equivalent classes of linear systemsunder determined conditions. In this way, our blow-up relation from integers partitions to feedback classeslead to suppose that it is possible to design computingprocedures over other similar rings.AcknowledgementsThis research has been partially supported by MCyT(Spain) and JCyL (Spain).ReferencesBrewer, J.W., Bunce, J.W., and VanVleck, F.S., (1986).Linear systems over commutative rings. MarcelDekker. New York.Brunovsky, P.A., (1970).A classification of linear con-trollable systems. Kybernetika. 3, pp. 173–187.Carriegos, M.V., (2003).On the local-global decompo-sition of linear control systems. Communications inNonlinear Sci. and Num. Sim. 9, pp. 149–156.Carriegos, M.V., (2007).A study of control systemsfrom the geometry of ring of scalars. IPACS Elec-tronic Library, http://lib.physcon.ru.Carriegos, M.V., and Sánchez-Giralda, T., (2001).Canonical forms for linear dynamical systems overcommutative rings: The local case. Ring Theoryand Algebraic Geometry. Marcel Deker. New York.pp. 113–131.Ferrer, J., Garcı́a-Planas, M.I., and Puerta, F., (1997).Brunowsky local form of a holomorphic family ofpairs of matrices. Linear Algebra and Its Applica-tions. 253, pp. 175–198.Hermida-Alonso, J.A., López-Cabeceira, M.M., andTrobajo, M.T., (2005).When are dynamic and staticfeedback equivalent?Linear Algebra and Its Appli-cations. 405, pp. 74–82.Hermida-Alonso, J.A., Pérez, P., and Sánchez-Giralda, T., (1996).Brunovsky’s canonical form forlinear dynamical systems over commutative rings.Linear Algebra and Its Applications. 233, pp. 131–147.Knuth, D.E., (2004).The art of computer program-ming. Generating all partitions4(Pre-Fascicle 3B).Addison Wesley. Online Previews.Morse, A.S., (1976).Ring models for delay-differentialsystems. 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Enumeration of locally Brunovsky linear systems over C(S1)-modules. A procedure

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Enumeration of locally Brunovsky linear systems over C(S1)-modules. A procedure

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