CONSTRUCTION OF ALGEBRAIC AND DIFFERENCE EQUATIONS WITH A PRESCRIBED SOLUTION SPACE

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1 Int J Appl Math omput Sci 7 Vol 7 No 9 3 DOI: /amcs-7- ONSTRUTION OF ALGEBRAI AND DIFFERENE EQUATIONS WITH A PRESRIBED SOLUTION SPAE LAZAROS MOYSIS a NIHOLAS P KARAMPETAKIS a a Department of Mathematics Faculty of Sciences Aristotle University of Thessaloniki 44 Thessaloniki Greece {lazarosmkarampet}@mathauthgr This paper studies the solution space of systems of algebraic and difference equations given as auto-regressive AR representations Aσβk = where σ denotes the shift forward operator and Aσ is a regular polynomial matrix The solution space of such systems consists of forward and backward propagating solutions over a finite time horizon This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of Aσ This work deals with the inverse problem of constructing a family of polynomial matrices Aσ such that the system Aσβk = satisfies some given forward and backward behavior Initially the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased This result is used to construct a system satisfying a certain backward behavior By combining this result with the method provided by Gohberg et al 9 for constructing a system with a forward behavior an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior Keywords: algebraic and difference equations behavior exact modeling auto-regressive representation discrete time system higher order system Introduction Let R be the field of reals R [s] the ring of polynomials with coefficients from R and Rs the field of rational functions By R[s] p m Rs p m R pr s p m we denote the sets of p m polynomial rational and proper rational matrices with real coefficients respectively We consider systems of higher order discrete time algebraic and difference equations that are described by the matrix equation A q βk +q+a q βk +q + +A βk = or equivalently Aσβk = where k = N q A A q R r r βk : [N] R r is the state of the system σ denotes the forward shift operator ie σ βk = βk +and Aσ =A q σ q + + A σ + A R[σ] r r 3 is a regular polynomial matrix with det [Aσ] and A q A The number q is often called the lag of orresponding author the system Markovsky et al 6 Systems described by are called auto-regressive AR representations Such higher-order systems often appear in systems theory since they can accurately model many natural or artificial systems like economic or biological discrete time phenomena Such examples include the population growth model in biology and the Leontief multisector economy model in economics ampbell 98 as well as other applications in engineering social sciences and medicine for positive systems Kaczorek 4 or D systems Kaczorek The solution space of consists of both forward and backward solutions and is denoted as B:={βk is satisfied k [N q]} 4 Forward solutions are defined in the sense that the initial conditions are given and βk is to be determined in a forward fashion from its previous values Backward solutions are defined in the sense that the final conditions are given and βk is to be determined in a backward fashion from its future values The solution of such systems has been previously 7 L Moysis and NP Karampetakis This is an open access article distributed under the reative ommons Attribution-Nonommercial-NoDerivs license Download Date /3/7 :9 PM

2 L Moysis and NP Karampetakis studied by various authors Antoniou et al 998; Antsaklis and Michel 6; Gohberg et al 9; Karampetakis 4 A method for constructing the forward resp backward solution space of based on the finite resp infinite elementary divisor structure of Aσ was presented by Gohberg et al 9 resp Antoniou et al 998 whereas the results were extended for non-regular systems by Karampetakis 4 The algebraic structure of polynomial matrices was studied by Bernstein 9 Gantmacher 99 Gohberg et al 9 Hayton et al 988 Kaczorek 7 Karampetakis and Vologiannidis 3 Karampetakis et al 4 Praagman 99 Zaballa and Tisseur and in the references cited therein An interesting problem that we face in this paper is the following inverse problem: Given a certain forward/backward solution space find a system of algebraic and difference equations with the prescribed solution space This problem was initially presented and solved by Gohberg et al 9 but only for the case of the forward solution space More specifically Gohberg et al 9 proposed a formula that connects the form of the unknown polynomial matrix Aσ and a finite Jordan pair that can be constructed from the prescribed forward solution space The main aim of this work is to extend the results presented by Gohberg et al 9 for the case where except from a given forward behavior a backward behavior is also provided In order to achieve this we connect the backward solution space of with the forward solution space of its dual system A βk + q+a βk + q + + A q βk = Structural properties of regular polynomial matrices In this section we provide some background on the finite and infinite zero structure of polynomial matrices Definition Gantmacher 99; Vardulakis 99 A square polynomial matrix Aσ R[σ] r r is called unimodular if det Aσ =c R c A rational matrix Aσ R pr σ r r is called biproper if lim σ Aσ = E R r r with ranke = r Theorem Gantmacher 99; Vardulakis 99 Let Aσ be as in 3 There exist unimodular matrices U L σu R σ R[σ] r r such that SAσ σ = U L σaσu R σ = diag f z σf z+ σf r σ 6 with z r and f j σ f j+ σ forj = zz + r SAσ σ is called the Smith form of Aσ where f j σ R [σ] are the invariant polynomials of Aσ The zeros λ i of f j σ j = zz +rare called finite zeros of Aσ Assume that Aσ has l finite distinct zeros The partial multiplicities n ij of each zero λ i i =lsatisfy n iz n iz+ n ir with f j σ =σ λ i nij ˆfj σ j = zr and ˆf j λ i Thetermsσ λ i nij are called finite elementary divisors of Aσ at λ i Themultiplicity of each zero is defined as n i = r j=z n ij We denote by n the sum of the degrees of the finite elementary divisors of Aσ and combine this with the results given by Gohberg et al 9 This methodology was used for the first time by Karampetakis for the case of continuous time systems where the finite and infinite zero structure of Aσ is connected with the smooth and impulsive behavior of respectively This paper is organized as follows In Section some background is provided on the algebraic structure of polynomial matrices In Section 3 the finite and infinite zero structure of a polynomial matrix is connected to the forward and backward solution space of its corresponding system Section 4 deals with the inverse problem of modeling the forward and backward solution space of a system and Section combines the results of the previous sections into a single algorithm Lastly in Section 6 following the behavioral framework of Antoulas and Willems 993 Markovsky et al 6 Willems 986; 99; 7 or Zerz 8a; 8b; we study the conditions under which the constructed system is most powerful Section 7 concludes the paper n := deg detaσ r l =deg f j σ = j=z i= j=z r n ij Similarly we can find U L σ Rσ r r U R σ Rσ r r having no poles and zeros at σ = λ such that S λ Aσ σ = U L σaσu R σ 7 = diag σ λ nz σ λ nr 8 S λ Aσ σ is called the Smith form of Aσ at the local point λ Theorem Vardulakis et al 98 Let Aσ be as in 3 There exist biproper matrices U L σu R σ R pr σ r r such that S Aσ σ Download Date /3/7 :9 PM

3 onstruction of algebraic and difference equations with a prescribed solution space with = U L σaσu R σ r u {}}{ = diag σ q σ qu σ ˆqu+ σ ˆqu+ σ ˆqr u q q u ˆq r ˆq r ˆq u+ > 9 and u r SAσ σ is called the Smith form of Aσ at infinity If p is the number of q i s in 9 with q i > then we say that Aσ has p poles at infinity each one of order q i > Also if z is the number of ˆq i s in 9 then we say that Aσ has z zeros at infinity each one of order ˆq i > Lemma Vardulakis 99 orollary 34 For Aσ in 3 we have q = q Definition Vardulakis 99 Section 4 The dual polynomial matrix of Aσ is defined as Ãσ :=σ q A = A σ q +A σ q + +A q σ Theorem 3 Vardulakis 99 Section 4 Let Ãσ be as in There exist matrices ŨLσ Rσ r r Ũ R σ Rσ r r having no poles or zeros at σ = such that S σ =ŨLσÃσŨRσ Ãσ =diagσ μ σ μr S σ being the local Smith form of Ãσ at σ = Ãσ The terms σ μj are the finite elementary divisors of Ãσ at zero and are called the infinite elementary divisors ied of Aσ The connection between the Smith form at the infinity of Aσ andthesmith format the zeroofthedual matrix is given by Hayton et al 988 and Vardulakis 99: S Ãσ σ = diag σ μ σ μr = diag σ q q σ q qu σ q+ˆqu+ σ q+ˆqr iped ized where by iped and ized we denote the infinite pole and infinite zero elementary divisors respectively From the above formula it can be seen that the the orders of the infinite elementary divisors of Aσ are given by q=q μ = q q = μ j = q q j j = 3u μ j = q +ˆq j j = u +r 3 We denote by μ the sum of the degrees of the infinite elementary divisors of Aσ ie r μ := μ j 4 j= Lemma Antoniou et al 998; Gohberg et al 9 Let Aσ be as in 3 Let also n and μ be the sums of the degrees of the finite and infinite elementary divisors of Aσ respectively as defined in 7 and 4 Then n + μ = r q The above relation is of fundamental importance since it will help us determine whether or not the construction of a system with a prescribed behavior is possible It should be noted that in the case where the matrix Aσ is non-regular that is Aσ R[σ] r m and r m or Aσ R[σ] r r and det Aσ = the algebraic structure of Aσ and by extension the solution space of are connected with additional invariants due to the left and right null space of Aσ see Karampetakis 4; Praagman 99 3 Jordan pairs and solutions of Aσβk = In this section we will connect the finite and infinite zero structure of the polynomial matrix Aσ with the forward and backward behavior of the system 3 Finite Jordan pairs and the forward solution space Definition 3 Antoniou et al 998; Gohberg et al 9 Let i r ni J i ni ni be a matrix pair where J i is in Jordan form corresponding to the zero of λ i of Aσ with multiplicity n i That is J i consists of Jordan blocks with sizes equal to the partial multiplicities of λ i Thisiscalledafinite Jordan pair of Aσ or a finite eigenpair corresponding to λ i iff det Aσ has a zero at λ i of multiplicity n i rankcol i Ji k q = n k= i q 3 A k i Ji k = k= Taking an eigenpair for each distinct finite zero λ i i =lofaσ we can create the finite spectral pair of Aσ ie F r n J F n n where F = l J F = blockdiag J J l 6 Download Date /3/7 :9 PM

4 L Moysis and NP Karampetakis The finite spectral pair satisfies similar properties ie degdet Aσ = n rankcol F J k q F = n k= 7 q A k F J k F = 8 k= Theorem 4 Antoniou et al 998; Gohberg et al 9 The forward solution space B F of is given by the column span of the matrix k Ψ F = F J F 9 and has dimension dim B F = n 3 Infinite Jordan pairs and the backward solution space Definition 4 Antoniou et al 998; Gohberg et al 9 An eigenpair of the dual matrix Ãσ corresponding to the eigenvalue λ = is called an infinite eigenpair of Aσ or an infinite Jordan pair Taking an eigenpair for each finite zero λ =of Ãσ we construct the infinite spectral pair of Aσ = μ μr J = blockdiag J μ J μr which satisfies the following: det Ãσ has a zero at λ =ofmultiplicity μ rankcol q k J = μ k= q 3 A k J q k = k= Theorem Antoniou et al 998 The backward solution space B B of is given by the column span of the matrix N k Ψ B = J and has dimension dim B B =μ In the following we shall provide an analytic formula for the backward solution space 4 onstruction of a system with a given backward behavior In the previous section we have provided a method for constructing the complete forward and backward solution space of when knowledge of the finite and infinite Jordan pairs of the matrix Aσ is available If the Jordan pairs of Aσ are not given beforehand a method for constructing them can be found in the works of Gohberg et al 9 and Karampetakis 4 In the following sections we study the inverse problem that is: Given a specific forward or backward behavior how to construct a polynomial matrix Aσ and its corresponding homogenous system Aσβk =that will satisfy the given behavior An answer to this modeling problem was first proposed by Gohberg et al 9 for the case of the forward solution space We first present these results and then extend them in order to include the case of the backward solution space as well In the next section we will combine these results into a single algorithm for modelling both the forward and the backward solution space of a system Suppose that a finite number of vector valued functions β i k: [N] R r of the form β i k :=λ k i β ini + kλ k i β ini k + + λ k ni i β i n i for λ i i= lor 3 β i k =δkβ ini + + δ k n i β i 4 for λ i = are given where δk denotes the discrete Kronecker delta function and β i β ini r We want to construct a system of difference equations with 3 4 as its solutions By analogy to the continuous time case studied by Gohberg et al 9 Proposition 9 if 3 4 are solutions of the system then the vectors β i β ini are generalized eigenvectors of Aσ corresponding to λ i and the matrices i J i with i = β i β ini λ i J i = λ i λ i constitute a Jordan pair of Aσ functions for λ i and β i λ k i Thus the vector β ikλ k i + β i λ k i 6 β i δk β i δk + β i δk 7 respectively for λ i =are also solutions of the system Gohberg et al 9 Section 83 The above set of vector valued functions can be written in matrix form as Download Date /3/7 :9 PM

5 onstruction of algebraic and difference equations with a prescribed solution space 3 i Ji k where we make use of the formulas λ k k i λ k i k k n n i λ i i Ji k = λ k i k k n n i λ i i λ i λ k i δk δk δk n i Ji k δk δk n i = λ i = δk with i r ni J i ni ni Define := l r n 8 J := blockdiag J J l n n 9 where n := l i= n i Then we have the following theorem Theorem 6 Gohberg et al 9 Let a be a complex number different from λ i and define Aσ =I r J ai n q σ av q + +σ a q V 3 where q = ind J is the least integer such that the matrix J S q = rq n 3 J q has full column rank and V V q is the generalized inverse of J ai n S q = rq n 3 J ai n q Then β i k are solutions of Furthermore q is the minimal possible lag of any polynomial matrix with this property Remark Different choices of a λ i will lead to the construction of matrices that are left unimodular equivalent That is if A σ and A σ are any two matrices that are constructed for different values of the parameter a there exists a unimodular matrix Uσ such that A σ =UσA σ 33 Thus since multiplication by Uσ does not alter the algebraic structure of Aσ the behavior of the corresponding system remains the same In consequence the choice of the parameter a does not impose any limitations Notice that since the time sequences in the form of 3 constitute solutions to they satisfy for all k regardless of how we consider time propagation This means that solutions in the form of 3 can either be regarded as solutions propagating forward in time for initial conditions βββq or solutions moving backward for the final conditions βnβn βn q + Remark In the case where a backward propagating time sequence is given in the form of β i k = i Ji N k x N corresponding to a zero λ i with the final conditions β i N β i N q + it can be rewritten as = i i J q i β i k = i J i k J N i x N x x N 34 and it can be clearly seen that it corresponds to the finite Jordan pair J of Aσ for the initial conditions β i β i q = i i J i q x 3 Therefore in order to construct the system with β i k as its solution one can follow Theorem 6 using the pair J On the other hand time sequences in the form of 4 can only be considered forward solutions for the system Overall the finite elementary divisors are connected to either forward/backward propagating solutions of the form 3 for a non-zero λ i or to strictly forward propagating solutions of the form 4 for λ i = In the following we will show how the infinite elementary divisors are connected to the backward solutions of the system What we need to note here is that although we have created an auto-regressive representation for the given forward solution space if the equation n+μ = r q is not satisfied for μ = then the above algorithm will give rise to an AR the representation which will include some extra forward/backward behavior An example showcasing this is given in the last section Now we will provide a theorem for the backward solution space Theorem 7 Karampetakis 4 Let Ãσ be the dual matrix of Aσ as defined in From let Ũ R σ = ũ σ ũ r σ and ũ i j σ Ãi σ be Download Date /3/7 :9 PM

6 4 L Moysis and NP Karampetakis the i-th derivatives of ũ j σ and Ãσ for j =r and i = μ j with μ j defined in 3 Define x ji = i!ũi j σ 36 Then the discrete time vector valued functions β j k =x jμj δn k+ + x j δn k μ j + 37 are linearly independent backward solutions of Aσβk = Lemma 3 Karampetakis ; Vardulakis 99 The vectors x ji defined in 36 for i = μ j and j =r form Jordan chains for Ãσ corresponding to the eigenvalue λ i =with lengths μ j asin3 Thus they satisfy the following systems of equations: A q x j = 38 A qj + A q x jq qj for the case of infinite pole elementary divisors ie μ j = q q j j =u and A q A q A x j A = 39 x jq+ˆqj } A {{ A q } Q for the case of infinite zero elementary divisors ie μ j = q +ˆq j j = u +r The proof of 39 that corresponds to the infinite zero elementary divisors was provided by Vardulakis 99 The procedure can be easily extended with no significant changes to include the case of infinite pole elementary divisors 38 The following theorem showcases the connection between the backward solutions of and the forward solutions of the dual system Theorem 8 Let x ji be the vectors defined in 36 for i = μ j and j =r The vector valued function 37 is a solution of the AR representation iff the vector β j k =x j δk μ j ++ + x jμj δk 4 is a solution of the dual system ie Ãσ β j k = Proof With no loss of generality we shall study the case of μ j = q +ˆq j The proof of μ j = q q j proceeds in a similar manner Necessity First we will prove that if β j k is a solution of then β j k is a solution of N Let Bz = Z[ βk] = k= z k βk be the Z-transform of βk Taking Z-transforms on the dual system we have ] Z [Ãσ βk = Ãz Bz = β in 4 where β in := z q I r zi r A β A q A βq The Z transform of β j k in 4 is 4 B j z =x j z q ˆqj+ + + x jq+ˆqj 43 Substituting βz in 4 we obtain Ãz B j z =z q A + + A q x j z q ˆqj+ + + x jq+ˆqj = z q I r I r z I r z q ˆqj+ I r A x jq+ˆqj A q A 44 x j A q Thus Ãz B j z = A z q I r zi r A q A + I r z q ˆqj+ I r Q T x jq+ˆqj x j x jq+ˆqj x jˆqj } {{ } = see Lemma 3 Hence Ãz B j z = z q I r zi r A A q A x jˆqj x jq+ˆqj 4 46 Download Date /3/7 :9 PM

7 onstruction of algebraic and difference equations with a prescribed solution space We can easily check that 4 and 46 coincide in the case where J j = β j x jq+ˆqj Rμj μj = 47 4 β j q x jˆqj where j = llet Therefore β j k is a solution of for the above initial conditions Sufficiency Now we will show the opposite that is if β j k is a solution of then β j k is a solution of Since β j k is a solution of we shall have that Ãz B j z =z q A + + A q x j z q ˆqj+ + + x jq+ˆqj = A β j z q I r zi r 48 A q A β j q The above equation holds true if and only if the proper part on the left-hand side of the above equation cf 4 x j z q ˆq j+ I r z q I r I r Q 49 x jq+ˆqj is equal to zero This holds true if and only if the condition 39 is satisfied However according to Lemma 3 if 39 is satisfied then β j k is a solution of The previous theorem proves that the problem of finding an AR representation of the form with the following backward behavior: β j k =x jμj δn k+ + x j δn k μ j + is equivalent to that of finding an AR representation of the form satisfying the forward behavior β j k =x j δk μ j ++ + x jμj δk However this problem can easily be solved using Theorem 6 This is demonstrated in the following Theorem 9 functions: β j k = Suppose that the following l vector valued μ j w= x jμj wδn w k are given where x j x jμj R r j =l Define j = x j x jμj R r μ j 3 with μ = = l R r μ J = blockdiag J J l R μ μ 6 l j= μ j Leta and define Ãσ =I r J ai r q σ av q + +σ a q V 7 where q = ind J is the least integer such that the matrix J S q = 8 J q has full column rank and V = V V q is the generalized inverse of J ai n S q = 9 J ai n q Then β j k are solutions of where Aσ =σ q à σ Furthermore q is the minimal possible degree of any r r polynomial matrix with this property Example We want to find a polynomial matrix Aσ such that the system Aσβk = has the backward solution β k = δn k+ δn k x 3 x + δn k + δn k 3 6 x x We begin by forming the matrices = x x x x 3 = 6 J = 6 Download Date /3/7 :9 PM

8 6 L Moysis and NP Karampetakis Then we start assuming values for q Starting from q = the matrix S = S = does not have full column rank For q =the matrix S = S = = J 63 has full column rank and thus A σ has lag q =Let a =and V = V V = J I = 6 Thus Ãσ =I J I 4 σ V +σ V σ σ = σ σ + σ 3 σ with Smith form at zero 66 S Ãσ σ = σ 4 67 Therefore the matrix that we are looking for is Aσ =σ Ã σ σ σ = σ + σ σ with det Aσ =and Smith form at infinity SAσ σ =σ S σ = Ãσ 69 σ σ onstruction of a system with given forward and backward behaviors In this section we combine all the results presented in previous sections in order to create an algorithm for constructing a system that satisfies both given forward and backward behaviors We have already presented methods for constructing a system with a given either forward or backward behavior In the last case we have constructed a dual polynomial matrix Ãσ with a forward behavior resulting from a given backward behavior Our aim is to work in a similar way for the forward behavior as well ie to connect it to the behavior in the dual system We can either connect the forward behavior of to a forward or a backward behavior of the dual system We shall work with the first result so that the problem of finding a system with a given forward-backward behavior is reduced to that of finding its dual system which exhibits a certain forward behavior First we outline the connection between the forward behavior of and a corresponding forward behavior of the dual system Theorem Let Aσ be defined as in 3 with rank Rσ Aσ = r If β j k = j Jj kx is a solution of where j r nj J j nj nj is a finite Jordan pair corresponding to the zero λ j of Aσ then β j k = j Jj J kx j is a solution of Proof Since j J j is a finite Jordan pair of the polynomial matrix Aσ by Definition 3 it satisfies the equation A q j J q j + + A j J j + A j = 7 Now substituting β j k into we obtain Ãσ β j k = Ãσ jj j J k j x = A j J k+q+ j x + + A q j J j = A q j J q j + + A j J j + A j J j 7 = k+ x k+q+ x Thus β j k is a solution of the dual system In the above theorem we managed to match a certain forward solution of to a forward solution of a result that we will utilize in the following Since the matrix J j is not in Jordan form we can find a non-singular constant matrix U R nj nj such that Jj = U J j U where J j is in Jordan form With this change the solution of Ãσ βk =can also be written as follows: βk = j J j J kx j = j U J j U U Jj ku x = j Jj k U x 7 where j = j U J j Therefore we see that instead of using the matrix pair j J j r nj J j nj nj Download Date /3/7 :9 PM

9 onstruction of algebraic and difference equations with a prescribed solution space where the matrix Jj is not in Jordan form we can use β k = δn k+ the matrix pair 3 x j = j U J j r nj J j = U J j U nj nj 7 Overall from Remark as well as Theorems 8 and the connection between the finite and infinite elementary divisors of Aσ as well as the solutions of and its dual system are summarized in Table To sum up our results in order to construct an AR representation for a certain forward/backward behavior one must follow Algorithm Algorithm onstruction of an AR representation with a given forward and backward behavior for λ j Step Transform the finite Jordan pairs j r nj J j nj nj that correspond to solutions of the form βk = j Jj kx for λ j to the finite Jordan pairs 7 that correspond to solutions of the form βk = k U j Jj x of the dual system that we are looking for Step onstruct infinite Jordan pairs of the matrix Aσ as in Theorem 9 These correspond to the finite Jordan pairs of the dual system at σ = Step 3 onstruct the polynomial matrix Ãσ using the method presented in Theorem 6 Step 4 Obtain the polynomial matrix Aσ =σ q Ã/σ and thus the AR representation that we are looking for It should be mentioned again that Aσ is not the only polynomial matrix satisfying the given behavior Different choices of a will result in different matrices that are left unimodular equivalent as mentioned in Remark Example We want to construct an AR representation with the following forward and backward solutions: β k = k + k k 73 β β Table onnection between the solutions of and Aσβk = Ãσβk = forward/backward forward/backward solutions solutions λ i λ i = λ i strictly forward solutions strictly backward solu- λ i = tions λ i = strictly backward solutions λ i = strictly forward solutions λ i = 7 δn k 74 x First define the matrix pairs = β β = J = J = 4 = 4 4 U J U = U J = = x x = J 3 = The complete matrix pair is = = 7 3 J J = = J 76 For q = the matrix S = does not have full column rank For q = the matrix 3 S = = J 77 3 has full column rank Let a =Then where Ãσ =I J ai 4 { σ av +σ a V } 78 V V = J ai = 8 The resulting matrix is 3 4 Ãσ 6 3σ +σ σ + σ = 3 3σ +σ σ +4σ 79 Download Date /3/7 :9 PM

10 8 L Moysis and NP Karampetakis Therefore the matrix that we are looking for is β k = δn k+ δn k 84 Aσ =σ Ã x x σ 6σ 3σ + σ 8 First define the matrix pairs = 3 σ 3σ + σ +4 = β β = Indeed the vector functions β kβ k are solutions of the system ie they satisfy Aσβ i k =fori = J = 8 For a different choice of the parameter a for example a = the resulting matrix is A σ = 6 7σ = J = 86 +σ 4 + σ σ + σ 6+σ 8 We observe that the matrix J is not invertible Hence in order to move on we set b =and use the pairs and as expected the two matrices are left unimodular equivalent that is = = β β = 87 4 Aσ = 8 3 A σ 8 J = J + bi = with U unimodular } {{ } U The above algorithm may suffer from computational difficulties in the case where the forward behavior has polynomial vector valued functions since these are connected with the finite elementary divisors of Aσ at zero In that case the Jordan matrix J will have the zero determinant and thus will not be invertible This problem can easily be surpassed by replacing σ with σ + b in Aσ whereb does not correspond to a zero of the polynomial matrix This replacement moves all possible zeros of Aσ to non-zero places The following remark indicates this solution Remark 3 Gohberg et al 9 Proof of Theorem 73 Let Aσ be a polynomial matrix a If J is a finite Jordan pair of Aσ then J + bi n is a finite Jordan pair of Aσ b b If J is an infinite Jordan pair of Aσ then J I μ + bj is an infinite Jordan pair of Aσ b The use of the above lemma is illustrated with the following example Example 3 We want to construct an AR representation of the form with the following forward and backward behavior: β k = δk+ δk 83 β β = = 88 J = J I + bj = = J 89 Working with the above pairs we construct the matrix Aσ b =Aσ Let also J = 4 = U J U = U J = 4 = 9 The complete matrix pair is = = 9 J J = = J 93 Now start assuming values for q For q = the matrix S = does not have full column rank For q = the matrix S = = J 4 94 Download Date /3/7 :9 PM

11 onstruction of algebraic and difference equations with a prescribed solution space 9 has dets =/4 so it has full rank Let a = Then à σ =I J ai 4 σ av +σ a V 9 where 8 V V = J ai 4 = 4 The resulting matrix is σ +σ à σ = +3σ σ σ + σ +σ + σ 96 The matrix Aσ is the dual of the above matrix Aσ = σ à σ 97 σ σ = +3σ σ +σ σ and the matrix that we were initially searching for is given by substituting σ by σ +and is equal to σ σ Aσ = σ σ 3σ σ 98 with SAσ σ = σ 99 SAσ σ =σ S Ãσ σ = σ = σ As expected Aσβ i k =for i = 6 Notes on the power of a model σ The notion of power in modeling was introduced by Willems 986 and later studied by Willems 7 and Zerz 8a The power of a model is defined as the ability of the constructed model to describe the given behavior ie the given data but as anything little else as possible Accordingly this takes place if we define as B the behavior of the system we have constructed that is the complete set of vector valued functions which satisfy it: B = {w :[N] R r Aσwk =} or equivalently B=kerAσ and we do not simply desire this behavior to include the given functions This should obviously be the aim of the modeling procedure but the optimal goal for the constructed model is to have no other behavior linearly independent from the prescribed Thus for any other model with the behavior B we want {B more powerful than B } {B B } 3 Now as have mentioned previously for a given number of vector valued functions the system created by the proposed algorithms may still include some extra forward/backward behavior if the equation n + μ = rq is not satisfied In this case the system model is not the most powerful one and no such model can be created for a square and regular system matrix Theorem Given the following vector valued functions: β j k =λ k j β jq j + + k λ k qj j β j q j β i k =δkβ ipi + + δ k p i β i β p k =x pμp δn k + + x p δn k μ p + for j =m i =m and p =m let m m m n = q i + p i μ = μ p j= i= p= The system Aσβk = constructed by the proposed Algorithm corresponding to the behavior B= ker Aσ is the most powerful model that describes the above vector functions iff there exists q N such that holds An example where the constructed system is not the most powerful model is given below Example 4 We want to construct an AR representation with the following forward and backward solutions: β k = + k β β β k = δn k x + δn k x Download Date /3/7 :9 PM

12 3 L Moysis and NP Karampetakis + 3 δn k 4 = σ + 9σ 6 9σ + 93σ 74 4σ + 48σ σ + 48σ σ 4σ 399 3σ + 874σ x σ 93σ 7 + σ 48σ First define the matrix pairs + 67σ 874σ = β β = with Smith forms SAσ σ = 8 J = σ 77σ 6 J = = S Ãσ σ = σ 3 U J U = U J As expected the final system matrix Aσ has an = extra finite elementary divisor at σ = 6/77 This gives rise to a third solution vector for the system linearly independent of β = k and β k Using the method x x = 3 proposed by Karampetakis 4 we find that the third solution is 3 96 J = 949 k β 3 k = 9 The complete matrix pair is = J = blockdiag J J Before we start assuming values for q we see that Hence the above system Aσβk =is not the most for the functions given above we have n = μ =3and powerful model describing the vectors β k and β k since we want to create a square regular system r =3 A construction of a non-regular system that satisfies However we easily observethat there is no q that satisfies β k β k is still possible for example the system with matrix n + μ = rq =3q Aσ = 7 Hence we expect the final system to include undesired 9 σ + σ4 σ σ solutions For q = the matrix S = does not have full column rank For q = the matrix S = T J T T R 6 has full column rank Thus let a =and Ãσ =I 3 J ai { σ av +σ a V } 6 where V V is the generalized inverse of V = V V = J ai 7 omputing Ãσ and after some simplifications we obtain the matrix we are looking for: Aσ =σ Ã σ is non-regular and satisfies Aσβ i k = What must be noted though is that solutions of non-regular systems can either be connected to their fed s and ied s or the structure of the right null space since non-regular systems have an infinite number of forward and backward propagating solutions due to the right null space of Aσ as evidenced by Karampetakis 4 Therefore in this case too the constructed system includes an additional undesired behavior 7 onclusions We have proposed an algorithm for constructing a regular AR representation which satisfies given forward and backward behaviors This is a discrete time analog of the work by Karampetakis where the problem was formulated for continuous time AR representations as well as the modeling of the smooth and impulsive system behavior Our further aim is to extend this theory to the Download Date /3/7 :9 PM

13 onstruction of algebraic and difference equations with a prescribed solution space 3 case of non-regular AR representations since we have noticed that for a given forward-backward behavior we can always achieve a non-regular AR representation that satisfies this behavior Acknowledgment The authors wish to thank the anonymous reviewers for their insightful comments that significantly improved the quality of this paper References Antoniou E Vardulakis A and Karampetakis N 998 A spectral characterization of the behavior of discrete time AR-representations over a finite time interval Kybernetika 34: 64 Antoulas A and Willems J 993 A behavioral approach to linear exact modeling IEEE Transactions on Automatic ontrol 38: Antsaklis PJ and Michel AN 6 Linear Systems nd Edn Birkhäuser Boston MA Bernstein DS 9 Matrix Mathematics Theory Facts and Formulas nd Edn Princeton University Press Princeton NJ ampbell S 98 Singular Systems of Differential Equations Vol Research Notes in Mathematics Pitman London Gantmacher FR 99 The Theory of Matrices Vols helsea Publishing o New York NY Gohberg I Lancaster P and Rodman L 9 Matrix Polynomials Reprint SIAM Philadelphia PA Hayton G Pugh A and Fretwell P 988 Infinite elementary divisors of a matrix polynomial and implications International Journal of ontrol 47: 3 64 Kaczorek T 7 Polynomial and Rational Matrices Applications in Dynamical Systems Theory Springer Dordrecht Kaczorek T 4 Minimum energy control of fractional descriptor positive discrete-time linear systems International Journal of Applied Mathematics and omputer Science 44: DOI: 478/amcs-4-4 Kaczorek T Analysis of the descriptor Roesser model with the use of the Drazin inverse International Journal of Applied Mathematics and omputer Science 3: DOI: /amcs--4 Karampetakis NP 4 On the solution space of discrete time AR-representations over a finite time horizon Linear Algebra and Its Applications 38: 83 6 Karampetakis NP onstruction of algebraic-differential equations with given smooth and impulsive behaviour IMA Journal of Mathematical ontrol and Information 3: 9 4 Karampetakis NP and Vologiannidis S 3 Infinite elementary divisor structure-preserving transformations for polynomial matrices International Journal of Applied Mathematics and omputer Science 34: Karampetakis N Vologiannidis S and Vardulakis A 4 A new notion of equivalence for discrete time AR representations International Journal of ontrol 776: Markovsky I Willems J Van Huffel S and De Moor B 6 Exact and Approximate Modeling of Linear Systems A Behavioral Approach SIAM Philadelphia PA Praagman 99 Invariants of polynomial matrices Proceedings of the st European ontrol onference Grenoble France pp Vardulakis A 99 Linear Multivariable ontrol Algebraic Analysis and Synthesis Methods John Wiley & Sons hichester Vardulakis A Limebeer D and Karcanias N 98 Structure and Smith MacMillan form of a rational matrix at infinity International Journal of ontrol 34: 7 7 Willems J 986 From time series to linear system II: Exact modelling Automatica 6: Willems J 99 Paradigms and puzzles in the theory of dynamical systems IEEE Transansactions on Automatic ontrol 363: 9 94 Willems J 7 Recursive computation of the MPUM in A hiuso et al Eds Modeling Estimation and ontrol Springer Berlin pp Zaballa I and Tisseur F Finite and infinite elementary divisors of matrix polynomials: A global approach MIMS EPrint 78 Manchester Institute for Mathematical Sciences University of Manchester Manchester Zerz E 8a Behavioral systems theory: A survey International Journal of Applied Mathematics and omputer Science 83: 6 7 DOI: 478/v Zerz E 8b The discrete multidimensional MPUM Multidimensional System Signal Processing 93 4: 37 3 Zerz E Levandovskyy V and Schindelar K Exact linear modeling with polynomial coefficients Multidimensional System Signal Processing 3: 6 and LATEX Lazaros Moysis is a PhD student at the Department of Mathematics Aristotle University of Thessaloniki Greece He received a Bachelor s degree in and a Master s degree in 3 His research interests include the theory of control systems linear systems descriptor and higher order systems of differential or difference equations as well as matrix theory He has published works in international journals and conferences as well as some free tutorials on Matlab Download Date /3/7 :9 PM

14 3 L Moysis and NP Karampetakis Nicholas P Karampetakis was born in Drama Greece in 967 He received the Bachelor s and PhD degrees in mathematics from the School of Mathematics Aristotle University of Thessaloniki Greece in 989 and 993 respectively Since August 4 he has been a professor of mathematical theory of control systems with the School of Mathematics Aristotle University of Thessaloniki and since September 3 he has been the head of that school His research interests lie mainly in algebraic methods for computer aided design of control systems ASD numerical and symbolic algorithms for ASD polynomial matrix theory and issues related to mathematical systems theory He is a senior member of the IEEE and a vice-chair of the IEEE Action Group on Symbolic Methods for ASD He is a member of the Technical ommittee on Linear ontrol Systems of the International Federation of Automatic ontrol IFA He is also an associate editor of the Journal of Multidimensional Systems and Signal Processing Systems Science and ontrol Engineering andtheinternational Journal of ircuits Systems and Signal Processing Received: May 6 Revised: 4 September 6 Accepted: October 6 Download Date /3/7 :9 PM