- Email: [email protected]
Minimum lag descriptions and minimal Gröbner bases
Systems & Control Letters 34 (1998) 289–293
Minimum lag descriptions and minimal Grobner bases Je rey Wood a; ∗; 1 , Eric Rogers a , David H. Owens b a ISIS
Group, Department of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK b Centre for Systems and Control Engineering, School of Engineering, North Park Rd, University of Exeter, Exeter, Devon, EX4 4QF, UK Received 30 November 1997; received in revised form 25 March 1998
Abstract We generalize the concept of a minimum lag description to multidimensional autoregressive discrete systems. We show that our de nition is equivalent to the property that the rows of the representation matrix form a minimal Grobner basis. In c 1998 Elsevier the 1D case, the new de nition is strictly stronger than that of Willems, but yields the same minimum lags. Science B.V. All rights reserved. Keywords: Multidimensional system; Minimum lag description; Grobner basis; Behavioural approach; Module theory
1. Introduction Multidimensional systems theory is the study of systems whose trajectories are functions of more than one independent variable, such as systems described by sets of partial di erential or di erence equations. Interest in such systems has recently increased, following links to multidimensional signal processing and to the theory of convolutional codes. This paper is concerned exclusively with autoregressive (AR) discrete systems. It has been recognized for some time that Grobner bases are of great importance in multidimensional systems theory [2], both as a central computational tool and as part of the solution to the canonical Cauchy problem for discrete linear systems [6]. One of the purposes of this paper is to provide another interpretation of a (minimal) Grobner basis.
∗
Corresponding author. Tel.: +44 1703 595776; fax: +44 1703 594498; e-mail: [email protected]. 1 This work has been sponsored by EPSRC grant no. GR=K 18504.
Our second aim is to extend the behavioural approach to multidimensional systems theory by incorporating the notion of a minimum lag description. The natural de nition we propose is equivalent to the property of being a minimal Grobner basis. We also show that, in the 1D case, a matrix satisfying our de nition is a minimum lag description in the sense of Willems, and where appropriate a minimal basis in the sense of Forney. 2. Monomial orderings and Grobner bases We begin by de ning the standard concepts of a monomial ordering and initial term module [1–3]. Throughout we write R for the polynomial ring k[z1 ; : : : ; zn ]. For clarity, we distinguish between a monomial in R and a monomial vector, often termed a monomial in R q . De nition 1. A monomial of R is an element of the form z1a1 z2a2 : : : znan , where a1 ; : : : ; an ∈ N. A monomial vector is an element of R q for some q of the form pei , where p is a monomial of R, and ei is the ith natural
c 1998 Elsevier Science B.V. All rights reserved. 0167-6911/98/$19.00 PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 4 0 - 1
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basis vector. A monomial ordering on R q is a total ordering of the monomial vectors of R q such that if m1 and m2 are monomial vectors of R q and n ∈= k is a monomial of R then m1 ¿m2 implies nm1 ¿nm2 ¿m2 . Given an arbitrary element x of R q and a monomial ordering ¿ on R q , we denote by in¿ x the initial term of x, i.e. the term of x which is greatest under the given monomial ordering. If N ⊆ R q is a submodule, then in¿ N denotes the submodule of R q generated by elements of the form in¿ x; x ∈ N . We refer to in¿ N as the initial term module of N . The de nition of a Grobner basis [1–3] is now straightforward: De nition 2. Let N be a submodule of R q , and let ¿ denote a monomial ordering on R q . Then a Grobner basis of N is a generating set for N whose initial terms are a generating set for in¿ N . We will call a matrix E with q columns a Grobner basis matrix for N if its rows form a Grobner basis of their R-span, N . If a Grobner basis has the additional property that no initial term of any element is a multiple of an initial term of another element, then it is called a minimal Grobner basis. A minimal Grobner basis matrix must have the property that the initial terms of its rows are precisely the monomial vectors minimal with respect to the property of being in in¿ N . There is also the further restricted notion of a reduced Grobner basis, which for certain monomial orderings gives the Hermite form of a 1D polynomial matrix [6]. Every submodule of R q has a minimal Grobner basis under any monomial ordering. 2.1. Grobner bases and the Cauchy problem Recall that the behaviour B of a discrete nD system is the set of all its trajectories, which are functions from N n (or Z n ) to k q , where k is some eld (typically R or C) and q is the number of components (e.g. inputs plus outputs). Any autoregressive n behaviour B ⊆ (k q )(N ) has a kernel representation E, i.e. can be written as the kernel of a polynomial matrix E under the action of indeterminates on trajectories by backward shifts [6]. We can construct the “orthogonal module” B⊥ , which is the submodule of R q spanned by the rows of E, and is independent of the representation [6]. Then for any monomial ordering ¿ on R q , a complete ini-
tial condition set for B is given by the set of lattice points in {1; : : : ; q} × N n corresponding to monomial vectors of R q not in in¿ B⊥ [6]. A Grobner basis for B⊥ provides an algorithm for computing a trajectory from its initial conditions; this is demonstrated in a more general setting in [7]. These results on the Cauchy problem have also been exn tended to behaviours in (k q )(Z ) [7, 8, 10]. 3. Minimum lags and Grobner bases We are going to introduce an ordering on the representations of any given behaviour. So that there is a least element under this ordering, it is necessary to exclude from consideration those representations which have obvious redundancy. The acceptable representations will be those satisfying the following new de nition: De nition 3. A polynomial matrix E with q columns is said to be row-irreducible (with respect to a monomial ordering on R q ) if no row has an initial term which is a multiple of the initial term of another row. n
Consider an arbitrary AR behaviour B ⊆ (k q )(N ) . By de nition, a minimal Grobner basis matrix for the orthogonal module B⊥ is row-irreducible. Since every submodule of R q has a minimal Grobner basis, every nD AR behaviour has a row-irreducible representation. Representations which are not row-irreducible can be reduced: if v1 and v2 are two rows of E such that r(in¿ v1 ) = in¿ v2 , then we can replace v2 by rv1 − v2 to obtain a new matrix which is also a representation of the kernel of E but which has rows with lower initial terms. In the 1D case, minimum lag descriptions have previously been de ned simply by minimizing the lag (degree) of the equations. This is ambiguous when dealing with more than one indeterminate – is z12 z2 larger or smaller than z1 z22 ? To avoid con ict with existing nomenclature in the 1D case, we use the term “irreducible minimum lag description”. De nition 4. Let q ∈ N be xed, and let ¿ denote a monomial ordering on R q . For any matrix E with q columns, and any monomial vector x in R q , let x (E) denote the number of rows of E with initial term x under ¿, and we construct a list (E) of these indices in ascending order of the monomial vectors x. We now pre-order the matrices E with q columns,
J. Wood et al. / Systems & Control Letters 34 (1998) 289–293
by saying E1 6E2 if and only if (E1 )¿(E2 ) under the lexicographic ordering, i.e. if and only if either (E1 ) = (E2 ) or, for the lowest monomial vector y ∈ R q for which y (E1 ) and y (E2 ) di er, we must have y (E1 )¿y (E2 ). We write E1 ¡E2 to 6 E1 . mean E1 6E2 but E2 6 n For an AR behaviour B ⊆ (k q )(N ) , a matrix E is called an irreducible minimum lag description of B if it is minimal under the ordering above with respect to the property of being a row-irreducible kernel representation of B. The minimum lag vectors are the initial terms of the rows of any irreducible minimum lag description, the minimum lag monomials are the non-zero components of these vectors, and the minimum lags themselves are the total degrees of the minimum lag monomials. Note that everything above is de ned with respect to a monomial ordering. Intuitively, an irreducible minimum lag description is determined by maximizing the number of “independent restrictions” with lowest possible initial term, then by maximizing the number of independent restrictions with next-lowest possible initial term, and so on until no further independent restrictions can be added. Looking at it this way, it is clear that an irreducible minimum lag description of a behaviour always exists. An irreducible minimum lag description of a given behaviour is therefore given by restrictions “with minimum possible degree”. The matrix pre-ordering of De nition 4 is not a partial ordering, since there will exist distinct matrix representations with the same list (·). It is however clearly transitive, and any two matrices can be compared under the order. Any two matrices E1 and E2 satisfying E1 6E2 and E2 6E1 under the ordering clearly have not only the same minimum lags, but also the same minimum lag vectors. The minimum lags are therefore well-de ned (with respect to the monomial ordering). Note that row-irreducibility of a representation E can be characterized by the law that if x (E)¿1 for some monomial vector x, then x (E) = 1 and y (E) = 0 for all proper multiples y of x. The usefulness of irreducible minimum lag descriptions as de ned above arises from the following new result: Theorem 5. Let ¿ be an arbitrary monomial ordering of R q . A polynomial matrix is an irreducible minimum lag description of the corresponding AR
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behaviour B if and only if it is a minimal Grobner basis matrix for B⊥ . Proof. ⇐: Suppose that E is a minimal Grobner basis matrix for B⊥ . Let C be an irreducible minimum lag description of B, which exists due to the remarks following De nition 4. Assume that (E) 6= (C), and let x be the smallest monomial vector of R q under ¿ such that x (E) 6= x (C). Assume x (E) = 0 and hence x (C) = 1. Since we must have x ∈ in¿ B⊥ , let y be a minimal monomial vector of in¿ B⊥ which divides x. Then y (E) = 1 as E is a minimal Grobner basis matrix, and so x 6= y. By de nition of x, y (C) = 1, which contradicts the row-irreducibility of C. Hence x (E)¿1, and so x (E) = 1 as E is row-irreducible, from which x (C) = 0. But now (C)¡(E), which contradicts the fact that C is an irreducible minimum lag description of B. The assumption that (E) 6= (C) must therefore have been wrong, and we see that E is also an irreducible minimum lag description of B. ⇒: Now suppose that E is an irreducible minimum lag description of B. Then by the rst part of the proof, the rows of E must have the same initial terms as any minimal Grobner basis, and therefore E is a minimal Grobner basis matrix. Theorem 5 can also be viewed as providing a new interpretation of a minimal Grobner basis. 3.1. Minimum lag descriptions in the 1D case In the 1D case, we also have the concepts of minimality de ned by Willems and by Forney [4]: De nition 6. Let E be a kernel representation of a 1D behaviour B in (k q )N . Suppose E has g = g(E) rows. Then the lag structure of E is the g-tuple (E) = (1 ; 2 ; : : : ; g ), where i is the degree of the row with the ith highest degree. We order the lag structures as follows: (E1 )6(E2 ) i (E1 ) = (E2 ), or g(E1 )¡g(E2 ), or g(E1 ) = g(E2 ) and, for the lowest i such that i (E1 ) and i (E2 ) di er, i (E1 )¡i (E2 ). A representation E ∗ of B is said to be a minimum lag description of B if its lag structure is minimal among all lag structures of kernel representations of B [9]. A representation E is said to be a minimal basis in the sense of Forney if it has minimal total lag among all polynomial matrix representations for the rational function space generated by its rows [4].
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Note that not every behaviour B has a representation which is minimal in the sense of Forney. For such a representation to exist, it is necessary that the k[z1 ]span of a generating set of B⊥ is equal to the set of polynomial vectors in its k(z1 )-span. We now clarify the relationships between the de nitions above and those presented earlier in the paper. Lemma 7. Let B be a 1D AR behaviour in (k q )N . (1) The following are equivalent for any kernel representation E of B: (a) E is row-irreducible. (b) E is an irreducible minimum lag description of B with respect to ¿, in the sense of Definition 4. (c) E is a minimal Grobner basis matrix. (2) Suppose that ¿ is a re nement of the partial ordering by degree. If a kernel representation E is row-irreducible, then it satis es the following equivalent conditions: (a) E is a minimum lag description of B in the sense of De nition 6. (b) E is row proper, i.e. the matrix of coecients in the highest degree for each row has full row rank. (c) E is minimal with respect to the total lag and the number of rows. (3) If B has a minimal basis in the sense of Forney, then the following are equivalent for any kernel representation E: (a) E is a minimal basis in the sense of Forney. (b) E is row proper and left prime. (c) E is minimal with respect to the total lag. Proof. The equivalence of (1b) and (1c) has been proved in Theorem 5, and these conditions are obviously sucient for (1a). Conversely if E is rowirreducible, then each row of E must have its initial term in a distinct column, and now Buchberger’s criterion [3] states that E must be a Grobner basis matrix, and also minimal. Now suppose that E is row-irreducible. Without loss of generality assume that the natural basis vectors are ordered e1 ¿e2 ¿ · · · ¿eq , and consider the k-matrix [E]h containing the coecients of highest degree in each row. By the choice of monomial ordering, there must be a permutation of the rows and columns of [E]h which brings it into the form (E1 E2 ), where E1 is upper triangular and invertible. Therefore [E]h has full row rank, i.e. (2b) holds. The equivalence of (2a)
and (2b) is given in [9]. The equivalence of (2b) and (2c) is given in [5]. Equivalence of (3a) and (3b) is given in [4]. Equivalence of (3a) and (3c) is given in [5]. Not all 1D minimum lag descriptions are rowirreducible; consider the following example: ! ! 1 0 1 1 0 1 0 E= ; E = : 2z 1 z z 1 0 It is not hard to see that E is a minimum lag description in the sense of Willems, but it is not row-irreducible for any monomial ordering. E can easily be reduced to E 0 , which has the same row span represents the same behaviour. The matrix E 0 is row-irreducible and hence an irreducible minimum lag description (under the ordering re ning the partial order by degree and with e1 ¡e2 ¡e3 ). 3.2. The minimum lags We have not used the quali er “irreducible” in de ning the minimum lags. This is because, in the 1D case, the minimum lags as de ned in De nition 4 are equal to the minimum lags as de ned by Willems: if E is an irreducible minimum lag description of B and C is a minimum lag description of B in the sense of Willems, then by Lemma 7 E is also a minimum lag description in the sense of Willems, and so E and C have the same minimum lags. Theorem 5 gives us an interesting interpretation for the minimum lag vectors in the nD case. They are a minimal generating set for in¿ B⊥ . The minimum lag vectors can therefore be determined easily from the initial condition set ; a monomial vector x ∈ R q is a minimum lag vector if x ∈= but contains any proper divisor of x. This follows from the results of Oberst [6] and is illustrated for a 2D scalar system in Fig. 1. In particular, we see that knowledge of the minimum lag vectors is equivalent to knowledge of the initial condition set. 4. Conclusions We have introduced the concept of an irreducible minimum lag description, which applies to multidimensional systems and is equivalent to the notion of a minimal Grobner basis. We have summarized the relationship between these concepts and the de nitions
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References
Fig. 1. The minimum lag vectors.
of Willems and Forney, in the 1D case. In the nD case, the identi cation with a minimal Grobner basis provides an interpretation of the minimum lag vectors in terms of the Cauchy problem. The theory as described applies only to autoregressive discrete systems de ned on the lattice N n . Using the tools recently developed which extend Grobner basis theory to the Laurent polynomial ring [7, 8, 10], it should be possible to produce corresponding results for discrete systems on Z n . Acknowledgements The authors would like to thank the reviewers for their many helpful comments, corrections and suggestions.
[1] T. Becker, V. Weispfenning, Grobner Bases: A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics, vol. 141, Springer, Berlin, 1993. [2] B. Buchberger, Grobner bases: an algorithmic method in polynomial ideal theory, in: N.K. Bose (Ed.), Multidimensional Systems Theory, Reidel, Dordrecht, 1985, pp. 184 –232. [3] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer, Berlin, 1995. [4] G.D. Forney, Minimal bases of rational vector spaces, with applications to multivariable linear systems, SIAM J. Control 13 (1975) 493 –520. [5] H.F. Munzner, D. Pratzel-Wolters, Minimal bases of polynomial modules, structural indices and Brunovsky transformations, Internat. J. Control 30 (2) (1979) 291–318. [6] U. Oberst, Multidimensional constant linear systems, Acta Appl. Math. 20 (1990) 1–175. [7] F. Pauer, A. Unterkircher, Grobner bases for ideals in Laurent polynomial rings and their application to systems of di erence equations, Preprint, 1997. [8] F. Pauer, S. Zampieri, Grobner bases with respect to generalized term orders and their application to the modelling problem, J. Symbolic Comput. 21 (1996) 155 –168. [9] J.C. Willems, From time series to linear system, Part I: Finitedimensional linear time invariant systems, Automatica 22 (5) (1986) 561–580. [10] E. Zerz, U. Oberst, The canonical Cauchy problem for linear systems of partial di erence equations with constant coecients over the complete r-dimensional integral lattice Z r , Acta Appl. Math. 31 (3) (1993) 249 –273.
Minimum lag descriptions and minimal Grobner bases Je rey Wood a; ∗; 1 , Eric Rogers a , David H. Owens b a ISIS
Group, Department of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK b Centre for Systems and Control Engineering, School of Engineering, North Park Rd, University of Exeter, Exeter, Devon, EX4 4QF, UK Received 30 November 1997; received in revised form 25 March 1998
Abstract We generalize the concept of a minimum lag description to multidimensional autoregressive discrete systems. We show that our de nition is equivalent to the property that the rows of the representation matrix form a minimal Grobner basis. In c 1998 Elsevier the 1D case, the new de nition is strictly stronger than that of Willems, but yields the same minimum lags. Science B.V. All rights reserved. Keywords: Multidimensional system; Minimum lag description; Grobner basis; Behavioural approach; Module theory
1. Introduction Multidimensional systems theory is the study of systems whose trajectories are functions of more than one independent variable, such as systems described by sets of partial di erential or di erence equations. Interest in such systems has recently increased, following links to multidimensional signal processing and to the theory of convolutional codes. This paper is concerned exclusively with autoregressive (AR) discrete systems. It has been recognized for some time that Grobner bases are of great importance in multidimensional systems theory [2], both as a central computational tool and as part of the solution to the canonical Cauchy problem for discrete linear systems [6]. One of the purposes of this paper is to provide another interpretation of a (minimal) Grobner basis.
∗
Corresponding author. Tel.: +44 1703 595776; fax: +44 1703 594498; e-mail: [email protected]. 1 This work has been sponsored by EPSRC grant no. GR=K 18504.
Our second aim is to extend the behavioural approach to multidimensional systems theory by incorporating the notion of a minimum lag description. The natural de nition we propose is equivalent to the property of being a minimal Grobner basis. We also show that, in the 1D case, a matrix satisfying our de nition is a minimum lag description in the sense of Willems, and where appropriate a minimal basis in the sense of Forney. 2. Monomial orderings and Grobner bases We begin by de ning the standard concepts of a monomial ordering and initial term module [1–3]. Throughout we write R for the polynomial ring k[z1 ; : : : ; zn ]. For clarity, we distinguish between a monomial in R and a monomial vector, often termed a monomial in R q . De nition 1. A monomial of R is an element of the form z1a1 z2a2 : : : znan , where a1 ; : : : ; an ∈ N. A monomial vector is an element of R q for some q of the form pei , where p is a monomial of R, and ei is the ith natural
c 1998 Elsevier Science B.V. All rights reserved. 0167-6911/98/$19.00 PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 4 0 - 1
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basis vector. A monomial ordering on R q is a total ordering of the monomial vectors of R q such that if m1 and m2 are monomial vectors of R q and n ∈= k is a monomial of R then m1 ¿m2 implies nm1 ¿nm2 ¿m2 . Given an arbitrary element x of R q and a monomial ordering ¿ on R q , we denote by in¿ x the initial term of x, i.e. the term of x which is greatest under the given monomial ordering. If N ⊆ R q is a submodule, then in¿ N denotes the submodule of R q generated by elements of the form in¿ x; x ∈ N . We refer to in¿ N as the initial term module of N . The de nition of a Grobner basis [1–3] is now straightforward: De nition 2. Let N be a submodule of R q , and let ¿ denote a monomial ordering on R q . Then a Grobner basis of N is a generating set for N whose initial terms are a generating set for in¿ N . We will call a matrix E with q columns a Grobner basis matrix for N if its rows form a Grobner basis of their R-span, N . If a Grobner basis has the additional property that no initial term of any element is a multiple of an initial term of another element, then it is called a minimal Grobner basis. A minimal Grobner basis matrix must have the property that the initial terms of its rows are precisely the monomial vectors minimal with respect to the property of being in in¿ N . There is also the further restricted notion of a reduced Grobner basis, which for certain monomial orderings gives the Hermite form of a 1D polynomial matrix [6]. Every submodule of R q has a minimal Grobner basis under any monomial ordering. 2.1. Grobner bases and the Cauchy problem Recall that the behaviour B of a discrete nD system is the set of all its trajectories, which are functions from N n (or Z n ) to k q , where k is some eld (typically R or C) and q is the number of components (e.g. inputs plus outputs). Any autoregressive n behaviour B ⊆ (k q )(N ) has a kernel representation E, i.e. can be written as the kernel of a polynomial matrix E under the action of indeterminates on trajectories by backward shifts [6]. We can construct the “orthogonal module” B⊥ , which is the submodule of R q spanned by the rows of E, and is independent of the representation [6]. Then for any monomial ordering ¿ on R q , a complete ini-
tial condition set for B is given by the set of lattice points in {1; : : : ; q} × N n corresponding to monomial vectors of R q not in in¿ B⊥ [6]. A Grobner basis for B⊥ provides an algorithm for computing a trajectory from its initial conditions; this is demonstrated in a more general setting in [7]. These results on the Cauchy problem have also been exn tended to behaviours in (k q )(Z ) [7, 8, 10]. 3. Minimum lags and Grobner bases We are going to introduce an ordering on the representations of any given behaviour. So that there is a least element under this ordering, it is necessary to exclude from consideration those representations which have obvious redundancy. The acceptable representations will be those satisfying the following new de nition: De nition 3. A polynomial matrix E with q columns is said to be row-irreducible (with respect to a monomial ordering on R q ) if no row has an initial term which is a multiple of the initial term of another row. n
Consider an arbitrary AR behaviour B ⊆ (k q )(N ) . By de nition, a minimal Grobner basis matrix for the orthogonal module B⊥ is row-irreducible. Since every submodule of R q has a minimal Grobner basis, every nD AR behaviour has a row-irreducible representation. Representations which are not row-irreducible can be reduced: if v1 and v2 are two rows of E such that r(in¿ v1 ) = in¿ v2 , then we can replace v2 by rv1 − v2 to obtain a new matrix which is also a representation of the kernel of E but which has rows with lower initial terms. In the 1D case, minimum lag descriptions have previously been de ned simply by minimizing the lag (degree) of the equations. This is ambiguous when dealing with more than one indeterminate – is z12 z2 larger or smaller than z1 z22 ? To avoid con ict with existing nomenclature in the 1D case, we use the term “irreducible minimum lag description”. De nition 4. Let q ∈ N be xed, and let ¿ denote a monomial ordering on R q . For any matrix E with q columns, and any monomial vector x in R q , let x (E) denote the number of rows of E with initial term x under ¿, and we construct a list (E) of these indices in ascending order of the monomial vectors x. We now pre-order the matrices E with q columns,
J. Wood et al. / Systems & Control Letters 34 (1998) 289–293
by saying E1 6E2 if and only if (E1 )¿(E2 ) under the lexicographic ordering, i.e. if and only if either (E1 ) = (E2 ) or, for the lowest monomial vector y ∈ R q for which y (E1 ) and y (E2 ) di er, we must have y (E1 )¿y (E2 ). We write E1 ¡E2 to 6 E1 . mean E1 6E2 but E2 6 n For an AR behaviour B ⊆ (k q )(N ) , a matrix E is called an irreducible minimum lag description of B if it is minimal under the ordering above with respect to the property of being a row-irreducible kernel representation of B. The minimum lag vectors are the initial terms of the rows of any irreducible minimum lag description, the minimum lag monomials are the non-zero components of these vectors, and the minimum lags themselves are the total degrees of the minimum lag monomials. Note that everything above is de ned with respect to a monomial ordering. Intuitively, an irreducible minimum lag description is determined by maximizing the number of “independent restrictions” with lowest possible initial term, then by maximizing the number of independent restrictions with next-lowest possible initial term, and so on until no further independent restrictions can be added. Looking at it this way, it is clear that an irreducible minimum lag description of a behaviour always exists. An irreducible minimum lag description of a given behaviour is therefore given by restrictions “with minimum possible degree”. The matrix pre-ordering of De nition 4 is not a partial ordering, since there will exist distinct matrix representations with the same list (·). It is however clearly transitive, and any two matrices can be compared under the order. Any two matrices E1 and E2 satisfying E1 6E2 and E2 6E1 under the ordering clearly have not only the same minimum lags, but also the same minimum lag vectors. The minimum lags are therefore well-de ned (with respect to the monomial ordering). Note that row-irreducibility of a representation E can be characterized by the law that if x (E)¿1 for some monomial vector x, then x (E) = 1 and y (E) = 0 for all proper multiples y of x. The usefulness of irreducible minimum lag descriptions as de ned above arises from the following new result: Theorem 5. Let ¿ be an arbitrary monomial ordering of R q . A polynomial matrix is an irreducible minimum lag description of the corresponding AR
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behaviour B if and only if it is a minimal Grobner basis matrix for B⊥ . Proof. ⇐: Suppose that E is a minimal Grobner basis matrix for B⊥ . Let C be an irreducible minimum lag description of B, which exists due to the remarks following De nition 4. Assume that (E) 6= (C), and let x be the smallest monomial vector of R q under ¿ such that x (E) 6= x (C). Assume x (E) = 0 and hence x (C) = 1. Since we must have x ∈ in¿ B⊥ , let y be a minimal monomial vector of in¿ B⊥ which divides x. Then y (E) = 1 as E is a minimal Grobner basis matrix, and so x 6= y. By de nition of x, y (C) = 1, which contradicts the row-irreducibility of C. Hence x (E)¿1, and so x (E) = 1 as E is row-irreducible, from which x (C) = 0. But now (C)¡(E), which contradicts the fact that C is an irreducible minimum lag description of B. The assumption that (E) 6= (C) must therefore have been wrong, and we see that E is also an irreducible minimum lag description of B. ⇒: Now suppose that E is an irreducible minimum lag description of B. Then by the rst part of the proof, the rows of E must have the same initial terms as any minimal Grobner basis, and therefore E is a minimal Grobner basis matrix. Theorem 5 can also be viewed as providing a new interpretation of a minimal Grobner basis. 3.1. Minimum lag descriptions in the 1D case In the 1D case, we also have the concepts of minimality de ned by Willems and by Forney [4]: De nition 6. Let E be a kernel representation of a 1D behaviour B in (k q )N . Suppose E has g = g(E) rows. Then the lag structure of E is the g-tuple (E) = (1 ; 2 ; : : : ; g ), where i is the degree of the row with the ith highest degree. We order the lag structures as follows: (E1 )6(E2 ) i (E1 ) = (E2 ), or g(E1 )¡g(E2 ), or g(E1 ) = g(E2 ) and, for the lowest i such that i (E1 ) and i (E2 ) di er, i (E1 )¡i (E2 ). A representation E ∗ of B is said to be a minimum lag description of B if its lag structure is minimal among all lag structures of kernel representations of B [9]. A representation E is said to be a minimal basis in the sense of Forney if it has minimal total lag among all polynomial matrix representations for the rational function space generated by its rows [4].
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Note that not every behaviour B has a representation which is minimal in the sense of Forney. For such a representation to exist, it is necessary that the k[z1 ]span of a generating set of B⊥ is equal to the set of polynomial vectors in its k(z1 )-span. We now clarify the relationships between the de nitions above and those presented earlier in the paper. Lemma 7. Let B be a 1D AR behaviour in (k q )N . (1) The following are equivalent for any kernel representation E of B: (a) E is row-irreducible. (b) E is an irreducible minimum lag description of B with respect to ¿, in the sense of Definition 4. (c) E is a minimal Grobner basis matrix. (2) Suppose that ¿ is a re nement of the partial ordering by degree. If a kernel representation E is row-irreducible, then it satis es the following equivalent conditions: (a) E is a minimum lag description of B in the sense of De nition 6. (b) E is row proper, i.e. the matrix of coecients in the highest degree for each row has full row rank. (c) E is minimal with respect to the total lag and the number of rows. (3) If B has a minimal basis in the sense of Forney, then the following are equivalent for any kernel representation E: (a) E is a minimal basis in the sense of Forney. (b) E is row proper and left prime. (c) E is minimal with respect to the total lag. Proof. The equivalence of (1b) and (1c) has been proved in Theorem 5, and these conditions are obviously sucient for (1a). Conversely if E is rowirreducible, then each row of E must have its initial term in a distinct column, and now Buchberger’s criterion [3] states that E must be a Grobner basis matrix, and also minimal. Now suppose that E is row-irreducible. Without loss of generality assume that the natural basis vectors are ordered e1 ¿e2 ¿ · · · ¿eq , and consider the k-matrix [E]h containing the coecients of highest degree in each row. By the choice of monomial ordering, there must be a permutation of the rows and columns of [E]h which brings it into the form (E1 E2 ), where E1 is upper triangular and invertible. Therefore [E]h has full row rank, i.e. (2b) holds. The equivalence of (2a)
and (2b) is given in [9]. The equivalence of (2b) and (2c) is given in [5]. Equivalence of (3a) and (3b) is given in [4]. Equivalence of (3a) and (3c) is given in [5]. Not all 1D minimum lag descriptions are rowirreducible; consider the following example: ! ! 1 0 1 1 0 1 0 E= ; E = : 2z 1 z z 1 0 It is not hard to see that E is a minimum lag description in the sense of Willems, but it is not row-irreducible for any monomial ordering. E can easily be reduced to E 0 , which has the same row span represents the same behaviour. The matrix E 0 is row-irreducible and hence an irreducible minimum lag description (under the ordering re ning the partial order by degree and with e1 ¡e2 ¡e3 ). 3.2. The minimum lags We have not used the quali er “irreducible” in de ning the minimum lags. This is because, in the 1D case, the minimum lags as de ned in De nition 4 are equal to the minimum lags as de ned by Willems: if E is an irreducible minimum lag description of B and C is a minimum lag description of B in the sense of Willems, then by Lemma 7 E is also a minimum lag description in the sense of Willems, and so E and C have the same minimum lags. Theorem 5 gives us an interesting interpretation for the minimum lag vectors in the nD case. They are a minimal generating set for in¿ B⊥ . The minimum lag vectors can therefore be determined easily from the initial condition set ; a monomial vector x ∈ R q is a minimum lag vector if x ∈= but contains any proper divisor of x. This follows from the results of Oberst [6] and is illustrated for a 2D scalar system in Fig. 1. In particular, we see that knowledge of the minimum lag vectors is equivalent to knowledge of the initial condition set. 4. Conclusions We have introduced the concept of an irreducible minimum lag description, which applies to multidimensional systems and is equivalent to the notion of a minimal Grobner basis. We have summarized the relationship between these concepts and the de nitions
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References
Fig. 1. The minimum lag vectors.
of Willems and Forney, in the 1D case. In the nD case, the identi cation with a minimal Grobner basis provides an interpretation of the minimum lag vectors in terms of the Cauchy problem. The theory as described applies only to autoregressive discrete systems de ned on the lattice N n . Using the tools recently developed which extend Grobner basis theory to the Laurent polynomial ring [7, 8, 10], it should be possible to produce corresponding results for discrete systems on Z n . Acknowledgements The authors would like to thank the reviewers for their many helpful comments, corrections and suggestions.
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