In this paper the new concept of 2D structured system is defined and a characterization of global reachability is obtained. This extends a well known result for 1D structured  systems, according to which (A ,B ) is (generically) reachable if and only if the matrix [A  B] is full generically row rank and irreducible

Pereira, Ricardo

Rocha, Paula

Simões, Rita

English

Repositório Institucional da Universidade de Aveiro

GLOBAL REACHABILITY OF 2D STRUCTURED SYSTEMSRicardo Pereira, ∗ Paula Rocha, ∗∗ Rita Simo˜es ∗∗ Department of Mathematics, University of AveiroCampus de Santiago, 3810-193, Aveiro, Portugal{ricardopereira,ritasimoes}@ua.pt∗∗ Department of Electrical and Computer EngineeringFaculty of Engineering, University of OportoRua Dr. Roberto Frias 4200-465, Porto, Portugalmprocha@fe.up.ptSubmitted to theSPECIAL SESSION ON GEOMETRIC CONTROL THEORYAbstract: In this paper the new concept of 2D structured system is defined and a characteri-zation of global reachability is obtained. This extends a well known result for 1D structuredsystems, according to which (Aλ, Bλ) is (generically) reachable if and only if the matrix[Aλ Bλ]is full generically row rank and irreducible.Keywords: Structured System, 2D Discrete System, Global ReachabilityFOREWORDThis paper is dedicated to Professor Fa´tima Leite, towhom we wish to pay our tribute for the outstandingrole in the development of Mathematical Systemsand Control Theory in Portugal. The second authoris particularly thankful for the encouragement andhelp she received from her at several stages of her(academic) life.1. INTRODUCTIONStructured systems have been introduced in (Lin,1974) in order to model phenomena where the onlyavailable information is the existence or absence ofrelations between the relevant variables. Since then avast literature has been produced on this subject, inparticular for structured linear systems in state spaceform (Glover and Silverman, 1976; Shields and Pear-son, 1976; Mayeda, 1981; Dion et al., 2003)1 This work was partially supported by FCT–Fundac¸a˜o para aCieˆncia e Tecnologia through CIDMA–Centro de Investigac¸a˜o eDesenvolvimento em Matema´tica e Aplicac¸o˜es of the University ofAveiro, Portugal.In such systems, the system matrices are supposed tohave entries that are zero or then assume arbitraryvalues, and each nonzero entry is identified with aparameter. In this setting, several system theoreticproperties have been defined in a generic way, i.e.,as holding for almost all the concretizations of thevalues of the parameters. One of these propertiesis reachability, that is defined as the possibility ofattaining an arbitrary state starting from the origin, byusing a suitable control sequence.In this paper we study the property of reachabilityfor 2D structured systems. Whereas dynamical sys-tems evolve over time (a one-dimensional variable),2D systems evolve over a two-dimensional domain(for instance (1D) space-time or 2D space) (Fornasiniand Marchesini, 1978). To our knowledge up to nowno research has been done on 2D structured systems.The results of this paper concern the characterizationof global reachability for 2D state space systems de-scribed by a Fornasini-Marchesini model (Fornasiniand Marchesini, 1978) and constitute a first step tobuild a full theory of 2D structured systems.2. STRUCTURED SYSTEMSA matrix M ∈ Rn×m is said to be a structured matrixif its entries are either fixed zeros or independentparameters, in which case they are referred to as thenonzero entries. In this paper, we assume that theactual value of each of the nonzero entries is unknown,but can take any real value (including zero). Thereforea structured matrixM having r nonzero entries can beparameterized by means of a parameter vector λ ∈ Rrand is denoted by Mλ.Example 1. Let λi, i = 1, 2, 3, be free parameters.The matrixMλ =[λ1 λ2λ3 0]is a structured matrix where λ = (λ1, λ2, λ3) ∈ R3.However, neither[λ1 λ2λ1 0]nor[λ1 λ21 0]are structured matrices.Let us consider a discrete time-invariant system of theformx(t+ 1) = Ax(t) +Bu(t), (2.1)where A ∈ Rn×n, B ∈ Rn×m, x(·) ∈ Rn denotes thestate of the system and u(·) ∈ Rm the input.If in the system (2.1) we assume that the matricesA and B are structured matrices having together rnonzero entries, the system can be parameterized bymeans of a parameter vector λ ∈ Rr. The set ofparameterized systems thus obtained is called a struc-tured system and is denoted byx(t+ 1) = Aλx(t) +Bλu(t) (2.2)with λ ∈ Rr, or simply by (Aλ, Bλ).By choosing λ, system (2.2) becomes completelyknown and can be written as a system of the form(2.1). Thus, for each value of λ, its system theoreticproperties can be studied in the usual way. It is clearthat these properties may depend on the parameter val-ues and hold for some of them while for others not. Inthis context, for structured systems, the relevant issueis not whether a property holds for some particularparameter values, but rather whether it is a genericproperty, in the sense that it holds “for almost all pa-rameter values”, i.e., it holds for all parameter valuesexcept for those in some proper algebraic variety inthe parameter space (which is a set with Lebesguemeasure zero) (Davison and Wang, 1973). Hence weshall say that the structured system (2.2) has a certainproperty P if P is a generic property of the system.In this paper we shall focus in the study of reachabil-ity. As is well-known, the system (2.1) is said to bereachable if for every x∗ ∈ Rn there exist t∗ > 0 andan input sequence u(t), t = 0, 1, . . . , t∗−1, that steersthe state from x(0) = 0 to x(t∗) = x∗.Characterizations of reachability for completely spec-ified systems of type (2.1) are given by the followingresults (Kucˇera, 1992).Theorem 2. The system (2.1) is reachable if and onlyif rankRn = n, where Rn is the reachability matrixof the system, i.e.,Rn := [B AB · · · An−1B] .Theorem 3. (PBH test). The system (2.1) is reachableif and only ifrank[zI −A | B] = n,∀z ∈ C.By Theorem 2, system (2.2) is reachable if and only ifthe reachability matrixRn = [Bλ AλBλ · · · An−1λ Bλ]has rank n for almost all λ ∈ Rr. But, noting thatRn is a polynomial matrix in r indeterminates, we canshow that this is equivalent to say that rankRn = n,for some λ∗ ∈ R. This means that the structured sys-tem (2.2) is reachable if and only if it is reachable forone choice of λ. However, neither this characterizationnor, equivalently, the study of the rank of the polyno-mial matrix Rn yield useful tests. This suggests thata different approach should be adopted, for instancebased only on the structure of the relevant matrices.Unfortunately, given two structured matrices Aλ andBλ the reachability matrixRn is not necessarily struc-tured, as is illustrated by the next example.Example 4. LetAλ =[λ1 λ20 λ3]and Bλ =[λ40]be two structured matrices. ThenR2 = [Bλ AλBλ] = [λ4 λ1λ40 0]is not a structured matrix since its nonzero entries arenot independent.Moreover, define a structured polynomial matrixMλ(z) asMλ(z) =MλNzN + · · ·+Mλ1 z +Mλ0for some nonnegative integer N , where the coeffi-cients Mλj , j = 0, 1, . . . , N , are structured matrices.Then the matrix[zI −Aλ | Bλ]=[I 0]z +[−Aλ Bλ]associated to the pair (Aλ, Bλ) of structured matricesis also not structured.Thus, if we wish to make a study of reachability of astructured system by analyzing structured matrices wemust use different tools.The next two concepts are fundamental for thisstudy (Dion et al., 2003)Let Aλ ∈ Rn×n and Bλ ∈ Rn×m be structuredmatrices. The pair (Aλ, Bλ) is said to be:• reducible, or to be in form I, if there exists apermutation matrix P ∈ Rn×n such thatP−1AλP =[Aλ11 0Aλ21 Aλ22]and PBλ =[0Bλ2]where Aλij is an ni × nj structured matrix fori, j = 1, 2, with 0 < n1 ≤ n and n1 + n2 = n,and where Bλ2 is an n2 ×m structured matrix.• not of full generic row rank, or to be in form II,if the generic rank of[Aλ Bλ]is less than n.Recall that the generic rank of a structured matrix Mλis ρ if it is equal to ρ for almost all λ ∈ Rr. Thiscoincides with the maximal rank that Mλ achieves asa function of the parameter λ.A necessary and sufficient condition for a pair(Aλ, Bλ) to be in form II is that[Aλ Bλ]has a zerosubmatrix of order k × l where k + l ≥ n + m + 1(Shields and Pearson, 1976).For structured systems of type (2.2), the followingresult has been proved (see (Glover and Silverman,1976; Lin, 1974; Shields and Pearson, 1976)).Theorem 5. The structured system (2.2) is (generi-cally) reachable if and only if the pair (Aλ, Bλ) isneither in form I nor in form II.The main goal of this paper is to generalize thistheorem for 2D structured systems.3. 2D SYSTEMSOne of the most frequent representations of 2D sys-tems is the Fornasini-Marchesini state space model(Fornasini and Marchesini, 1978) which is describedby the following 2D first order state updating equationx(i+ 1, j + 1) = A1x(i, j + 1) +A2x(i+ 1, j)+B1u(i, j + 1) +B2u(i+ 1, j),(3.1)with local states x(·, ·) ∈ Rn, inputs u(·, ·) ∈ Rm,state matrices A1, A2 ∈ Rn×n and input matricesB1, B2 ∈ Rn×m. In the sequel this 2D system willbe denoted by (A1, A2, B1, B2). The correspondingupdating scheme is illustrated in Figure 1.• ••(i, j + 1) (i+ 1, j + 1)(i+ 1, j)Figure 1Introducing the shift operatorsσ1x(i, j) := x(i+ 1, j),σ2x(i, j) := x(i, j + 1),the equation (3.1) can be written asσ1σ2x = A1σ2x+A2σ1x+B1σ2u+B2σ1u,or, equivalently,σ1x =(A1 +A2σ1σ−12)x+(B1 +B2σ1σ−12)u.Defining a new operator σ := σ1σ−12 equation (3.1)can be written asσ1x = (A1 +A2σ)x+ (B1 +B2σ)u. (3.2)The initial conditions for this equation may be as-signed by specifying the values of the state on theseparation set C0, whereCk = {(i, j) ∈ Z2 : i+ j = k},see Figure 2.• • • • • • • • • • • • • C0• • • • • • • • • Ck(0,k)(k,0)Figure 2Defining the global state on the separation set Ck asXk(t) :=(x(k + t,−t))t∈Zand the global input asUk(t) :=(u(k + t,−t))t∈Z,by (3.2), the global state evolution is given byXk+1 = A(σ)Xk +B(σ)Uk, (3.3)where A(σ) = A1 + A2σ, B(σ) = B1 + B2σ, andthe action of σ on Xk is given byσXk(t) =(x(k + (t+ 1),−(t+ 1)))t∈Z= Xk(t+ 1).The action of σ on Uk is analogous.Denote by R[[z, z−1]]the set of bilateral Laurentformal power series in the indeterminate z with coef-ficients in R and define the z−transform Z : (R)Z →R[[z, z−1]]byZ[Wk] :=+∞∑t=−∞Wk(t)z−twhich will be denotet by Wk(z), with k ∈ Z. Forvector signals in(Rl)Zthe z−transform is definedcomponentwise.ThenZ[σXk] =+∞∑t=−∞σXk(t)z−t=+∞∑t=−∞Xk(t+ 1)z−t= z+∞∑t=−∞Xk(t+ 1)z−(t+1)= z+∞∑t=−∞Xk(t)z−t= zXk(t)and hence, by (3.3), we obtainXk+1(z) = A(z)Xk(z) +B(z)Uk(z), (3.4)where A(z) = A1 + A2z, B(z) = B1 + B2z andUk(z) := Z[Uk].4. LOCAL AND GLOBAL REACHABILITYWhen dealing with 2D systems, the concept of reach-ability is naturally introduced in two different forms:a weak (local) and a strong (global) form which refer,respectively, to single local states and to global states.These notions are defined next as in (Fornasini andMarchesini, 1978).Definition 6. The 2D state space model (3.1) is• locally reachable if, upon assuming X0 ≡ 0, forevery x∗ ∈ Rn there exists (i, j) ∈ Z2, withi + j > 0, and an input sequence u(·, ·) suchthat x(i, j) = x∗. In this case, we say that x∗ isreachable in i+ j steps.• globally reachable if, upon assuming X0 ≡ 0,for every global state sequence X ∗ with valuesin Rn there exists k ∈ Z+ and an input sequenceU0,U1, . . . ,Uk−1 such that the global state Xkcoincides with X ∗. In this case, we say that X ∗is reachable in k steps.Clearly, global reachability implies local reachability.In this paper we shall focus on this global property.Bearing in mind that, if X0 ≡ 0, thenXk(z) =k−1∑l=0Ak−1−l(z)B(z)Ul(z)=[B(z) A(z)B(z) · · · Ak−1(z)B(z)]Uk−1(z)Uk−2(z)...U0(z) ,it is easy to see that the global state X ∗ is reachable ink steps if and only ifX∗(z) = Z[X ∗] ∈ ImRk(z)withRk(z) := [B(z) A(z)B(z) · · · Ak−1(z)B(z)].The matrixRn(z) = [B(z) A(z)B(z) · · · An−1(z)B(z)] ,where n is the dimension of the local state and thepolynomial matrices A(z) and B(z) are defined as in(3.4), is called global reachability matrix of the 2Dsystem (A1, A2, B1, B2).In the following theorem (Fornasini and Marchesini,1978), global reachability is characterized in terms ofthe global reachability matrix.Theorem 7. The 2D system (A1, A2, B1, B2) is globalreachable if and only if the polynomial matrix Rn(z)has rank n, i.e., rankRn(z) = n.5. 2D STRUCTURED SYSTEMSIn the sequel we consider 2D systems of the form (3.1),where the matrices A1, A2, B1 and B2 are structured,i.e., their entries are either fixed zeros or independentfree parameters. In this case, the polynomial matricesAλ(z) = Aλ1 + Aλ2z and Bλ(z) = Bλ1 + Bλ2 z arestructured matrices too. Moreover, their evaluationsfor any ν∗ ∈ C, yield matrices Aλ (ν∗) and Bλ (ν∗)that are also structured.Similar to the 1D case, we say that a 2D structured sys-tem (Aλ1 , Aλ2 , Bλ1 , Bλ2 ) is (globally / locally) reach-able if it is generically (globally / locally) reachable,i.e., if it is reachable for almost all λ ∈ Rr. Againthis is equivalent to say that (Aλ∗1 , Aλ∗2 , Bλ∗1 , Bλ∗2 ) isreachable for at least one value λ∗ ∈ Rr.As in the 1D case, the notions of matrix pairs inform I and in form II play an important role in thecharacterization of 2D reachability, now applied tothe polynomial matrix pair (Aλ(z), Bλ(z)). In thepolynomial case the definitions remain the same as inthe constant case, with the difference that the (generic)rank of[Aλ(z) Bλ(z)]is to be understood as its rankas a polynomial matrix.Lemma 8. Let ν∗ ∈ C \ {0}. Then the pair of struc-tured matrices (Aλ(z), Bλ(z)) is neither in form I norin form II if and only if the pair of structured matrices(Aλ (ν∗) , Bλ (ν∗)) is not in form I nor in form II,where Aλ(z) = Aλ1 +Aλ2z and Bλ(z) = Bλ1 +Bλ2 z.PROOF. If ν∗ ∈ C\{0} both implications are obvi-ous since the pairs of structured matrices(Aλ(z), Bλ(z)) and (Aλ (ν∗) , Bλ (ν∗)) have the samezero structure. 2Remark 9. The “if” part also holds for ν∗ = 0. Infact, if ν∗ = 0 then Aλ (0) = Aλ1 and Bλ (0) =Bλ1 . Since all the zero entries of the matrix Aλ(z)are also zero in Aλ1 and the same happens betweenBλ(z) and Bλ1 , the set of zero entries for the pair(Aλ(z), Bλ(z)) is contained in the set of zero entriesof (Aλ(0), Bλ(0)) = (Aλ1 , Bλ1 ). The result is easilyobtained by the definition of form I and the character-ization of form II.The following example shows that the “only if” partdoes not hold for ν∗ = 0.Example 10. Let Aλ1 =[λ1 00 0], Aλ2 =[0 λ2λ3 λ4],Bλ1 =[00]and Bλ2 =[λ5λ6].Since all the entries of (Aλ1 +Aλ2 , Bλ1 +Bλ2 ) are free,this pair is neither in form I nor in form II and byLemma 8 the same holds for the pair (Aλ(z), Bλ(z)).However, its clear that the pair (Aλ(0), Bλ(0)) =(Aλ1 , Bλ1)is in form I and II.The next theorem characterizes the global reachabilityof 2D structured systemsTheorem 11. A 2D structured system (Aλ1 , Aλ2 , Bλ1 , Bλ2 )is globally reachable if and only if the pair of struc-tured matrices (Aλ(z), Bλ(z)) is neither in form I norin form II, where Aλ(z) = Aλ1 + Aλ2z and Bλ(z) =Bλ1 +Bλ2 z.PROOF. By definition, the 2D structured system(Aλ1 , Aλ2 , Bλ1 , Bλ2 ) is globally reachable if thereexists λ∗ ∈ Rr such that the 2D system(Aλ∗1 , Aλ∗2 , Bλ∗1 , Bλ∗2 ) is globally reachable.Then, by Theorem 7, rankRnλ∗(z) = n, whereRnλ∗(z) is the global reachability matrix of the 2Dsystem (Aλ∗1 , Aλ∗2 , Bλ∗1 , Bλ∗2 ). Note that, in this case,the setL := {η ∈ C : rankRnλ∗(η) < rankRnλ∗(z)}corresponds to the common zeros of the n × nminors of Rnλ∗(z), and is hence a finite set. ThusrankRnλ∗(z) = n means that there exist ν∗ ∈ C \ Lsuch thatrankRnλ∗(ν∗) = n.By Theorem 2, the system corresponding to the pair(Aλ∗ (ν∗) , Bλ∗ (ν∗)) is reachable, for all ν∗ ∈ C \L.Thus, by definition, (Aλ (ν∗) , Bλ (ν∗)) is a structuredsystem which is reachable.By Theorem 5 we have that the pair of structuredmatrices (Aλ (ν∗) , Bλ (ν∗)) is neither in form I nor inform II and, by Lemma 8, (Aλ (z) , Bλ (z)) is neitherin form I nor in form II. The converse implication isanalogous. 26. REFERENCESDavison, Edward J. and Shih-Ho Wang (1973). Prop-erties of linear time-invariant multivariable sys-tems subject to arbitrary output and state feed-back. IEEE Trans. Autom. Control 18, 24–32.Dion, Jean-Michel, Christian Commault and Jacobvan der Woude (2003). Generic properties andcontrol of linear structured systems: A survey.Automatica 39(7), 1125–1144.Fornasini, E. and G. Marchesini (1978). Doubly-indexed dynamical systems: State-space mod-els and structural properties. Math. Syst. Theory12, 59–72.Glover, K. and L.M. Silverman (1976). Characteriza-tion of structural controllability. IEEE Trans. Au-tom. Control 21, 534–537.Kucˇera, Vladimir (1992). Analysis and Design of Dis-crete Linear Control Systems. Prentice Hall Inter-national Series in Systems and Control Engineer-ing. New York: Prentice Hall. viii.Lin, Ching-Tai (1974). Structural controllability. IEEETrans. Autom. Control 19, 201–208.Mayeda, Hirokazu (1981). On structural control-lability theorem. IEEE Trans. Autom. Control26, 795–798.Shields, Robert W. and J.Boyd Pearson (1976). Struc-tural controllability of multiinput linear systems.IEEE Trans. Autom. Control 21, 203–212.

Analysis and Design of Discrete Linear Control Systems.

Characterization of structural controllability.

Doublyindexed dynamical systems: State-space models and structural properties.

Generic properties and control of linear structured systems: A survey.

On structural controllability theorem.

Properties of linear time-invariant multivariable systems subject to arbitrary output and state feedback.

Structural controllability of multiinput linear systems.

Global reachability of 2D structured systems

Global reachability of 2D structured systems

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