A reachability test for systems over polynomial rings using Gröbner bases Habets, L.C.G.J.M.

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1 A reachability test for systems over polynomial rings using Gröbner bases Habets, L.C.G.J.M. Published: 01/01/1992 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Habets, L. C. G. J. M. (1992). A reachability test for systems over polynomial rings using Gröbner bases. (Memorandum COSOR; Vol. 9238). Eindhoven: Technische Universiteit Eindhoven. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 30. Jun. 2018

2 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science Memorandum CaSaR A Reachability Test for Systems over Polynomial Rings using Grobner Bases L.C.G.J.M. Habets r Eindhoven, September 1992 The Netherlands

3 Eindhoven University of Technology Department of Mathematics and Computing Science Probability theory, statistics, operations research and systems theory P.O. Box MB Eindhoven - The Netherlands Secretariate: Dommelbuilding 0.03 Telephone: ISSN

4 A Reachability Test for Systems over Polynomial Rings using Grabner Bases 1 2 L.C.G.J.M. Habets Eindhoven University of Technology Department of Mathematics and Computing Science P.O. Box 513 NL-5600 MB Eindhoven The Netherlands September 8, Research supported by the Netherlands Organization for Scientific Research (NWO) 2Submitted to 1993 ACC

5 Abstract Conditions for the reachability of a system over a polynomial ring are well known in the literature. However, the verification of these conditions remained a difficult problem in general. Application of the Grobner Basis method from constructive commutative algebra makes it possible to carry out this test explicitly. In this paper it is shown how this can be done in an efficient way. In comparison with a very simple and rather straightforward method, the algorithm proposed in this paper has an enormous advantage: it has a good performance for both reachable and non-reachable systems. Moreover, the method can be used to obtain a rightor left-inverse of a general non-square polynomial matrix. Such inverse matrices are often required for the design of feedback compensators. Finally, a modification of the reachability test is given to speed up the computations in the non-reachable case.

6 1 Introduction Systems over (polynomial) rings can be seen as a rather straightforward generalization of ordinary systems over fields. In the last three decades these systems have been investigated quite extensively (see for example [1], [10] and [15]), not only because they highlight the most important system theoretic properties very clearly, but also because they have very interesting applications. For example, systems over polynomial rings can be used to model systems with varying parameters and time-delay systems. In the last case, some delay operators 0"1,, 0"( are introduced which act on the state trajectory x(t) and the input trajectory u(t) of the system: O"iU(t) = u(t - Ti), (1) where the Ti, (i = 1,..., f), are f incommensurate time-delays. A time-delay system can then be written as: { x(t) = A(O"},,O"()x(t)+B(O"},,O"()u(t), y(t) = C(O"},,O"()x(t) + D(O"},, O"e)u(t), where the matrices..4 = A(O"I,...,O"e), ij = B(O"},...,O"e), C = C(O"I,...,O"e) and ij = D(O"},..., O"e) are all matrices over the polynomial ring R[O"I,...,O"e]. So the quadruple ~ = (..4, ij, C, ij) can be seen as a system over the polynomial ring R[0"1,, O"(]. Several system theoretic concepts, which are well-known for systems over fields, such as reachability and observability, have been generalized to the systems over rings case. In this way it was possible to derive results on various problems such as pole-placement, stabilizability and input-output decoupling (see [12], [6] and [5] respectively), which are quite similar to the well-known results for systems over fields. Although from the theoretical point of view the theory ofsystems over rings is well established, there remains one shortcoming. In the existing literature almost no attention is paid to the computational aspects of systems over rings. In this article we fill in a part of this gap. We propose a method to solve one of the problems in this area explicitly: how to test the reachability of a system over a polynomial ring? The paper is organized as follows. After this introduction, we recall the concept of reachability in Section 2 and state some conditions under which a system over a polynomial ring is reachable. We give a description which is well suited for the construction of an algorithmic test. In the next part a short introduction to Grabner Bases is given. With this tool from constructive commutative algebra, it is possible to solve the reachability problem explicitly. Our algorithm has one very important by-product: given a system (A, B) over R[O"},..., O"(], the algorithm also comes up with a right-inverse of the matrix (zi - A I B). This is very interesting from the control point of view because this right-inverse is often needed in the design of feedback compensators, for example to solve the stabilization problem (see [14]). Moreover, this shows that the algorithm can also be used to find a right- or left-inverse of a general non-square polynomial matrix. In Section 5 we show how the reachability test can be speeded up by applying it recursively. After this, the effectiveness of the proposed methods is illustrated with some examples. Also timing-statistics are given. Certainly in comparison with another, rather straightforward method to test reachability, the derived methods behave very well. Finally some conclusions are drawn. (2) 1

7 2 Reachability of systems over polynomial rings Let /( be an arbitrary field of characteristic zero (think of R or Q), and let K be the algebraic closure of /(. We denote the ring of all polynomials in f indeterminates 0'1,."'0'1. with coefficients in /( by R := /([0'1,,0'1.]. Let A and B be n x nand n x m matrices respectively, which have all their entries in R. Then we can consider the discrete time system over the ring R given by the equations { x(t + 1) = Ax(t) + Bu(t), xed) = xo. (3) Now clearly x(t) E R n (t E Z+) and u(t) E Rm (t E Z+). From a system theoretic point of view we call the system (3) reachable if for all xo, x E R n there exists a time instant T E Z+ and an input sequence u(d), u(i),...,u(t-i) such that, starting the system in xed) = Xo and applying this input sequence, it reaches the state x at time T, i.e. x(t) = x. It is easily shown (see for example [1, Section 2.1.]), that this property is satisfied iff the module generated by the columns of the matrix (B I AB I... I A n - 1 B) is the free module Rn. Based on the above interpretation, but nevertheless independent of it, reachability of systems over rings is defined as follows (compare [15, p. 16]). Definition 2.1 Let R be an arbitrary commutative ring, and A E Rnxn, B E Rnxm. Then the system I; = (A, B) is called reachable if the columns of the matrix (B I AB I '" I A n - 1 B) (4) span the free module R n. The condition given in Definition 2.1 is rather difficult to check. Especially for systems over polynomial rings there are alternative characterizations of reachability which are more suitable for testing. Theorem 2.2 Let R = 1C[0"1,'",0'1.] and suppose that A E Rnxn and B E Rnxm. Consider the system I; = (A, B). Then the following four conditions are equivalent: (i) I; = (A, B) is reachable, (ii) (zi - A I B) is right-invertible over R[z], (""'J \.I( - - -) E /(-HI \.I( ) E /(- n ZZZ vo'i,,o'l,z vql"'qn : Proof Several of these conditions are already well-known. From the implication scheme (i) => (ii) => (iii) => (iv) => (i), we only prove (ii) => (iii) => (iv). The other implications can be found in the existing literature. 2

8 (i) ::} (ii): In [14, Theoreme 2.1.], the equivalence of (i) and (ii) is proved. (ii) ::} (iii): Let (u},...,ul, z) E Kl+I, and suppose that (ql... qn) E K,n is such that Because (zi - A I B) is right-invertible over n[z], say with right-inverse M(z), we can also substitute (u},...,ul, z) in M(z) to arrive at a right-inverse of (Zl - A(u},...,Ul) I B(U},...,Ul»' Right-multiplication of (6) with M(UI,...,Ul,Z) yields: (ql qn).1= (0.. 0) M(UI,".,Ui, z) = (0.. 0). Hence (ql qn) = (0.. 0). (iii) ::} (iv): Let (u},...,ul, z) E Kl+ 1, and consider the matrix (Zl - A(u},...,u ) I B(u},...,ud). From (iii) it follows that for all (ql... qn) E Kn for which (ql... qn) i- (0 0) we have (ql'" qn)(zl- A(UI,".,Ui) I B(u},...,ud) i- (0.. 0). So the n rows of (Zl - A(u},...,Ui) I B(u},...,Ui» are linearly independent. This immediately implies that rank(zl- A(u},...,ue) I B(U},...,Ui» = n. (iv) ::} (i): This is a generalization of the well-known PBH-test (see for example [9]). A complete proof of the exact statement can be found in [8, Theorem ]. Condition (ii) in Theorem 2.2 is a very important characterization of reachability from the control point of view. In several control problems, such as the stabilization problem, the right-inverse of (zi - A I B) can be used to design a compensator. The computation of such a right-inverse is therefore a very interesting question. With help of condition (iii), Olbrot and Lee ([11]) derived genericity conditions for the reachability of systems over polynomial rings. They showed that such a system is generically reachable if the number of inputs m to the system is strictly larger than the number of indeterminates in the polynomial ring K[al,...,ai]. Moreover, if m :5, so when the number of inputs is smaller or equal to the number of indeterminates, a system is generically not reachable. 3 A reachability test using Grobner Bases In this section it is shown how Grobner Bases can be used to test the reachability of a system over a polynomial ring. To do so, we translate one of the reachability conditions in Theorem 2.2 into terms of polynomial ideals. These ideals can then be manipulated using the Grobner Basis method. Therefore we start with a short introduction on the theory of Grobner Bases. The Grobner Basis method is a technique from constructive commutative algebra to solve various questions on polynomial ideals such as the membership problem, or to find a solution of a system of algebraic equations. The method was introduced by B. Buchberger in 1965 and nowadays most computer-algebra packages contain software for the computation of Grabner Bases. Good references are [3] for the algorithmic part of the problem, and [13] for a more (6) 3

9 theoretical point of view. The application of Grabner Bases in the field of systems theory, especially for nonlinear systems, was investigated by Forsman in [7]. Because it is not our goal to give a detailed treatment of Grabner Bases, we only state the results we need in the sequel. To do so, we first have a short glimpse at the theory. For the details we refer to [3] and [13] as already mentioned above. A Grabner Basis of a polynomial ideal is a set of polynomials of low complexity which generate the ideal. To find these polynomials of low complexity a sort of reduction process is carried out. For polynomials in one indeterminate the degree is a good measure for the complexity. Moreover, the degree induces a ranking on these polynomials. The Euclidian Division Algorithm uses this ranking to compute remainders. Such a remainder has a lower complexity (lower degree) than the original polynomial. In this way it is possible to find polynomials of lower complexity in the one indeterminate case. When confronted with polynomials in more than one indeterminate, the situation is much more complicated. To mimic the process described above, one first has to introduce a generalized notion of degree, which incorporates the well-known properties of degrees and induces a ranking on the polynomials. Then one has to find a generalized remainder algorithm. This remainder algorithm is used to simplify polynomials with respect to each other in order to find polynomials oflower complexity. This finally leads to the concept of Grabner Bases. Let p E K[Xl,., Xl] be a polynomial in f indeterminates, and suppose that we have already introduced a generalized degree on K[Xl'...,Xl]. Then the initial term of p, denoted by in(p), is the term in p of highest degree. A Grabner Basis of an ideal I can then be defined in the following way (compare [13, DeL 1.5.]): Definition 3.3 Let I be an ideal in K[xl,..., Xl], I i- {O}. A finite subset G of I is a Grobner Basis of I if the set in(g) = {in(9) 19 E G} generates the ideal in(i) = {in(j) If E I}. As a direct consequence, a finite set of monomials (Le. polynomials consisting of only one term) {ml,...,mk} is a Grabner Basis of the ideal (ml,...,mk) generated by these monomials (see [13, p. 218, Remark 1.5.6]). Grabner Bases can be calculated using the algorithm of Buchberger (see [3]). It can be seen as an explicit implementation of the reduction process, rather intuitively described above. The algorithm yields a so called auto-reduced Grabner Basis. For a Grabner Basis G, consisting only of monomials, this means that there do not exist monomials 91 and 92 in G such that 91 is divisible by 92. One can prove that an auto-reduced Grabner Basis is unique (in the given ordering induced by the definition of the generalized degree) up to multiplication by non-zero constants from the field K (see [2]). After this short introduction on Grabner Bases, we show how they can be used to test the reachability of a system over a polynomial ring. First we translate condition (iii) in Theorem 2.2 into terms of polynomial ideals. Consider a system ~ = (A, B) over K[al,..., ae] and let qt = (ql qn) be an n dimensional row-vector consisting of the n indeterminates ql,..., qn. Define the row-vector pt as pt := qt. (zi _ A I B). (7) 4

10 Then the entries P},...,Pn+m of p T = (PI'" Pn+m) are polynomials in the indeterminates Ul,...,Ui,z,q},...,qn with coefficients in K. So P = (Pl,...,Pn+m) is an ideal in the polynomial ring K[Ul,'.., Ui, Z, qi,...,qn]. In the sequel we will often use the shorthand notation K[u, z, q] for this ring. The next proposition shows how the reachability of ~ = (A, B) depends on the ideal P. Proposition 3.4 Let A E nnxn and B E nnxm where n = K[u},..., Ui]. Let qt = (qi... qn) be an n-vector of indeterminates and P = (p},...,pn+m) the polynomial ideal in K [u},..., Ui, Z, q},..., qn] genemted by the polynomials Pi (i = 1,..., n +m) defined by Then (PI'" Pn+m) = p T = qt. (zi - A I B). (zi - A I B) is right-invertible over n[z], The auto-reduced Grobner Basis ofp consists precisely ofthe polynomials ql,..., qn (independent of the choice of the genemlized degree). Proof ":::>" Assume that (zi - A I B) is right-invertible over n[z]. Then there exists a matrix M(u},...,Ui, z) over n[z] such that (zi - A I B). M(u},...,Ui, z) = I. Right multiplication of (7) with M( Ul,.,UR., z) yields (Pl'.. Pn+m)M(z, UI,, Ui) = qt. (zi - A I B). M(z, u},...,ur.) = (qi... qn). Let i E {I,..., n}. The i th column of M (z, U},..., Ud consists of polynomials a},..., a n + m in K[Ul,...,Ui, z] C K[u, z, q]. So n+m qi = L ajpj E (PI,...,Pn+m)' j=l Since this holds for all polynomials qi (i = 1,..., n), this shows that in the ring K[u, z, q]: (ql,...,qn) C (P},,Pn+m)' (8) On the other hand it is clear from (7) that also (P},..,Pn+m) C (q},...,qn)' (9) Hence, in the ring K[u, z, q] we have Because the polynomials q},..., qn are clearly monomials in K[u, z, q], it follows from Definition 3.3 and the subsequent remark that the polynomials ql,"" qn form a Grobner Basis of P, independent of the generalized notion of degree. Moreover, this Grobner Basis is auto-reduced because none of these polynomials divides an other one. This implies that this Basis is unique upto multiplication with constants in K. (10) 5

11 " :" Now suppose that the auto-reduced Grobner Basis of P consists of the polynomials q1,'..,qn' Define the ideal Q in K[O', z, q] as Q = (q},...,qn). Then P = Q. But then also the varieties of P and Q are equal: V(P) = V( Q). Because Q is generated by a set of polynomials of very low complexity, the variety V( Q) of Q is easy to find: V(Q) = {(O'},...,O'e,z,q},...,qn) E,(:n+i'+I I q1 = q2 =... = qn = O}. (11) Since V(P) = V( Q) this implies that PI ~ O} <==> {q1 ~ 0 Pn+m ~ 0 qn = 0 Let now (0"1,...,O"e, i) E,(: +1, and consider the matrix (ZI - A(O"},...,O"e) I B(O"},...,O"e)). Pre-multiplication of (13) with qt = (q1... qn) yields (Pl'" Pn+m) = (q1'" qn)(zi - A(O"},...,O"t) I B(O"},...,O"t)), where Pi (i = 1,..., n) is equal to the polynomial Pi after substitution of (0"1,...,0", i) for (O'l,...,O't,z). Now suppose that PI = O'''',Pn+m = O. After substitution of (O"},...,o-,i) for (O'},...,0', z), it follows from (12) that q1 = q2 =... = qn = O. Hence condition (iii) of Theorem 2.2 is satisfied. This implies that (zi - A I B) is right-invertible over R[z]. Combining the results of Theorem 2.2 and Proposition 3.4 it is easy to derive a reachability test for systems over the polynomial ring n = K[O'l,...,O't]. Let E = (A,B). First compute the polynomials PI,...,Pn+m in K[O'I,, O't, Z, q1,'.., qn] with help offormula (7). Compute an auto-reduced Grobner Basis of (P1,,Pn+m), using Buchberger's algorithm. Such an algorithm is available in most computer-algebra packages. If the Grobner Basis consists precisely of the polynomials q},...,qn, the matrix (zi - A I B) is right-invertible over R[z], so E = (A, B) is reachable. Otherwise, when the Grobner Basis contains other polynomials, E = (A, B) is not reachable. 4 Computation of the right-inverse of (zi - A I B) In the last section it was shown how the reachability of a system E = (A, B) can be tested, by verifying the right-invertibility of the matrix (zi - A I B) using Grobner Bases. However, the computation of a right-inverse of(zi - A I B) over R[z] is also very interesting for its own sake because this inverse is needed in the design of compensators for various control problems. In this section we show that the Grobner Basis construction of the last section implicitly carries all the information needed to write down a right-inverse of (zi - A I B) immediately. Suppose that the system E = (A, B) over n is reachable. Then (zi - A I B) is rightinvertible over R[z]. Introduce again the row-vector qt = (q1" define: p T = (Pl'" Pn+m) = qt. (zi - A I B). (12) (13) qn) of indeterminates and Because (zi - A I B) is right-invertible, the Grobner Basis of (PI,'..,Pn+m) is the set of polynomials {q1,"" qn}. This set is obtained after application of Buchberger's algorithm. 6

12 This algorithm consists of successive steps from one polynomial set to an other, starting with {PI,...,Pn+m} and ending with {qi,..., qn}, such that these sets, and all the sets in between, generate the same ideal. Each polynomial in a new polynomial set is constructed as a linear combination of polynomials in the former set. Because this is done successively from {PI,...,Pn+m} to {qi,...,qn}, Buchberger's algorithm does not only yield a Grobner Basis {qi,'..,qn} of (PI,...,Pn+m), but, by an accurate bookkeeping of coefficients, also coefficients 0ji in K[O', z, q] (j = 1,..., n +mj i = 1,..., n), such that n+m Vi E {I,...,n}: qi = 2: QjiPj j=i Let A(q) denote the (n +m) X n matrix over K[O',zHq] such that the (j,i)th entry of A(q) is 0ji. Then, according to (14): (14) qt = p T. A(q). (15) Substitution of p T = qt. (zi - A I B) in (15) yields for all q: qt = qt. (zi - A I B). A(q). (16) Theorem 4.5 Let M be an n X k matrix (k ~ n) with all entries in n[z] = K[O'I,"',O'i,Z]. Let qt = (qi... qn) be a row-vector ofindeterminates. Suppose that A(q) is an k Xn polynomial matrix over K[O'I,...,O'i, z][qi,...,qn] such that Then the matrix A(O), obtained by substituting (qi'" qn) = (0 0) in A(q),is a right-inverse of Mover n[z]. Proof Suppose that (17) holds truej then it also holds while replacing q by >.q, where >. E K\{O}, Le. V>. E K\{O} : >.qt = >.qt. M A(>.q). Dividing both right- and left-hand side by >. yields: V>. E K\{O} : qt = qt. M A(>.q). Now qt. M. A(>.q) - qt can be seen as an n-dimensional row-vector of polynomials in the indeterminate>' with coefficients in K[O', z, q]. For all >. # 0, these polynomials are zero, so they must be identically zero. Thus for>. = 0 we have: qt = qt. M. A(O). Clearly A(O) is an k x n matrix over n[z], while qt is a row-vector of indeterminates. So, according to (18), A(O) is a right-inverse of Mover n[z]. Theorem 4.5 can be used to derive a right-inverse of (zi - A I B) over n[z] from the matrix A(q) over K[O', z,q] defined in (14) and (15). From formula (16) and Theorem 4.5 we immediately see that substitution of (qi... qn) = (0 0) in A(q) yields a right-inverse of (zi - A I B) over n[z]. (17) (18) 7

13 Remark 4.6 The method described above to compute a right-inverse of a polynomial matrix using Grobner Bases, is not only valid in the case where the polynomial matrix has the special form (zi - A I B). In the construction this special structure was never used. Therefore this method is also applicable in the general case. In this way Grobner Bases can be used to test the right- and left-invertibility of non-square polynomial matrices and to compute left- and right-inverses of such matrices. 5 A recursive method to test reachability Condition (iii) in Theorem 2.2 for the reachability of a system over a polynomial ring can also be used in a slightly different way. It is possible to reformulate the condition in order to get a sort of recursive method. This observation was already made by Olbrot and Lee in [11], where they used it to prove their result on genericity. This alternative condition is stated in the next lemma. Lemma 5.7 Let n = K[(il,'..,(ii], and suppose that A E nnxn and B E nnxm. Consider the system 'E = (A, B) over n. Then 'E = (A, B) is reachable, \-I( - - -) 0i+! \-I' {I } \-I( ) 0n-j. v (il,...,(ii, z E ~ vj E,...,n v qj+l... qn E ~. (~11Iqj+!'" qn)' (H - A(Ul,...,Ui) I B(Ul,'",Ui)) =I (0 0). (19) j-l Proof "=>" Suppose that 'E = (A, B) is reachable. Then, according to condition (iii) of Theorem 2.2, the equality implies that (0.. 0lllqj+!... qn) = (0.. 0). Trivially this can not happen, hence the necessity of (19) is obvious. "- ::" Suppose that condition (19) is satisfied and assume that 'E = (A,B) is not reachable. Then there exist (Ul,...,Ui, z) E j(l+l and a row-vector il = (iil... iin) E j(n, (iii'" iin) =I (0 0), such that (iii'" iin). (H - A(Ul,...,Ui) I B(Ul,...,Ui)) = (0 0) (use condition (iii) of Theorem 2.2). Define j := min{i I iii =I O}. Then the n-vector ;j.ii T is of the form ~ ii T = ~ (iii...iin) = ( iij_ ~n ). qj qj ~l qj qj J- But of course we still have 1 1 -=-. ii T. (H- A(ul,...,Ui) IB(ul,...,Ui)) = -=-(0 0) = (0 0). ~ ~ This contradicts (19), and therefore 'E = (A, B) must be reachable. This proves the claim. _ 8

14 The condition of Lemma 5.7 is easily tested for each j E {I,..., n} separately, using the Grabner Basis method. Let j E {I,..., n} and introduce the n-dimensional row-vector qt = (L:j!IIlqi+!'.. qn) i-i where qi+i,"" qn are considered as indeterminates. Define p T = (PI'" Pn+m) := qt. (zi - A I B). The entries PI,...,Pn+m of p T can be seen as polynomials in K[uI,...,Ut,z,qi+I,..,qn]' When (A, B) is reachable, it follows from Lemma 5.7 that these polynomials do not have a common zero. According to the Hilbert-Nullstellensatz this implies that the ideal P = (PI,...,Pn+m) is the whole ring K[UI,...,Ul,z,qj+I,...,qn]' Therefore it follows from Definition 3.3 that the auto-reduced Grabner Basis of P consists of only one polynomial: the constant polynomial!. (Here we made the assumption that the Grabner Basis is normalized.) Now the reachability of a system ~ = (A, B) can be investigated by carrying out the test described above for each j E {I,..., n}. This leads to the following, in a sense recursive, algorithm. Algorithm 5.8 Let A E nnxn and B E nnxm. Then the algorithm below is a test for the reachablity of the system ~ = (A, B) over n. j:= I;G:= {I}; while j ~ nand G = {I} do qt:= (0 Olllqj+I qn); ~ j-i p T = (PI'" Pn+m) := qt. (zi - A I B); G ;= GrobnerBasis((PI,...,Pn+m}); j := j + 1; od; if G = {I} then ~ = (A, B) is reachable; else ~ == (A, B) is not reachable; fi; Algorithm 5.8 has one important advantage in comparison with the method of Section 3 due to the following fact. The complexity of the computation of Grabner Bases is highly dependent on the number of indeterminates in the polynomial ring. In the method of Section 3, a Grabner Basis over a ring with n+f+ 1 indeterminates has to be calculated. In Algorithm 5.8 n Grabner Bases have to be computed, with each n+f+1-j indeterminates (j = 1,..., n). Because the number of indeterminates is lower in this case, it is possible that this method is faster, despite the fact that more Grabner Bases have to be calculated. Note that these computations become in each step less involved because the number of indeterminates is strictly decreasing. Moreover, if the system ~ = (A, B) is not reachable, it is very likely that this is detected in the first step, after the computation of only one Grabner Basis. From the proofon genericity in [11] it follows that generically this will be the case. Therefore we expect Algorithm 5.8 to be a faster method for the detection of the non-reachability of a 9

15 system. This claim is illustrated in Section 6, where the performance of both the algorithms is compared based on some examples for both the reachable and the non-reachable case. The application of Algorithm 5.8 also has a drawback. Suppose that a system E = (A, B) is reachable and we apply Algorithm 5.8. Then we end up with a correct conclusion, but the computations do not give any clue for the construction of a right-inverse of (zi - A I B). So when this inverse is really needed, the method of Section 3 is of course the most favorable one. Remark 5.9 Algorithm 5.8 can be seen as a special case of the method of Section 3 (only the conclusions are derived in a different way) in the sense that in each step a number of the indeterminates qt,...,qn is substituted by some zeros and a one. In Algorithm 5.8 this is done in a special order (from qt to qn), but this order does not make any difference for the problem under consideration. Therefore one can obtain alternative algorithms by changing the order of substitution. In fact, the result is the same as when the order of the rows of (zi - A I B) is permuted. In this way it is possible to influence the computing-time by changing the order of substitution. 6 Examples The purpose of this section is to show the effectiveness of the methods proposed in this paper. This is illustrated with help of some examples. Also a comparison of the performances of the various methods is made. Moreover, to illuminate the advantages and drawbacks of the algorithms derived in this paper very clearly, the performances are compared with a very simple method to test reachablity. This rather straightforward method is introduced first. Let A and B be matrices over the polynomial ring n = K[O"t,...,0" ] of size n x nand n x m respectively. Then an alternative method to test the right-invertibility of the matrix (zi - A I B) over n[z] is the following. First compute all the n x n minors Tt,..., TN of (zi - A I B). Then it is clear (see for example [11, p. 111]) that (zi - A I B) is right-invertible iff these minors do not have a common zero. According to the Hilbert-Nullstellensatz this implies that E = (A, B) is reachable iff (Tt,..., TN) = n[z], Le. iff the ideal generated by the n x n minors Tt,..., TN of (zi - A I B) is the whole ring R[z]. This last condition is easily verified with help of Grobner Bases. According to Definition 3.3, the normalized auto-reduced Grobner Basis of (Tt,..,TN) consists of only one polynomial in this case: the constant polynomial 1. The rest of this section contains three different examples. These experiments were made in the computer-algebra package MAPLE V, running on a Sun/Spare Workstation with a 25MHz processor. To compute Grobner Bases, we used the function gbasis from the grobner package, with the total-degree ordering and an automatic ordering of the indeterminates (for any details, see [4, pp ]). All the timings are given in CPU seconds without excluding the time for garbage collection. This garbage collection took place every 1Mb. Statistics on the use of memory are not given because the required amount of memory was not critical in these examples. 10

16 Example 6.10 Consider the matrices A, Band B 1 over the polynomial ring R[O'], given by _0'2 + 30' - 8 0' ' +7-20' + 6 ) 20'2-50' + 4 0'+4 (20) 0'2 + 30' ) B = 0 50' + 1 ( 40' + 7-0' +2 0'2 + 30' - 2 ) B 1 = 0. ( 40' + 7 (21) So B 1 consists of the first column of B. Based on the genericity conditions in [11], we expect ~ = (A, B) to be reachable, but ~l = (A, Bd not to be reachable. The reachability of both ~ = (A, B) and ~l = (A, B 1 ) is now tested with four different methods: Method 1 The method described in Section 3, based on the computation of a Grabner Basis of qt. (zi - A I B). Method 2 Algorithm 5.8 in Section 5. Method 3 A modification of Algorithm 5.8 as mentioned in Remark 5.9. The substitution order (from ql to qn in Algorithm 5.8) is changed into the reverse direction (from qn to ql). Method 4 The method based on the computation of a Grabner Basis of the ideal generated by all the minors of (zi - A I B), as explained at the beginning of this section. The results of the application of these methods on this particular example are given in Table 1. First the conclusion (reachable/not reachable) is given, then the computing time (in CPU seconds) needed to arrive at the result. The computer time needed by Method 4 to verify the reachability of ~ = (A, B) was highly variable. The indicated value is the mean of four samples. I Table 1 ~ ~ = (A,B) I ~l = (A,B1 ) I Method 1 reachable not reachable Method 2 reachable not reachable Method 3 reachable not reachable Method 4 reachable not reachable From Table 1 it is clear that the results obtained with Method 4 are rather extreme. Although this method is the fastest in the non-reachable case, it is very slow for the reachable system ~ = (A, B). On the contrary, the Methods 1 to 3, as proposed in this paper, behave very well in both cases. The recursive Methods 2 and 3 are a little bit faster than Method 1, especially in the non-reachable case. This is exactly as expected. Finally there is a little difference between Method 2 and Method 3, probably due to the structure of the example under consideration. 11

17 The next example shows that the proposed methods can easily handle systems over polynomial rings with more than one indeterminate. Example 6.11 Consider the matrices A, Band B 1 over the polynomial ring R[0'1' 0'2] given by 0'2 ( "dl + 0'1 '" -'" +3 ) A= 0'2-1 0' '1-5 0' ' '1-1 ( "d", ( "d", ",-1 ) B= 1 0' B 1 = 1 0' '" ~ 3 ) 0 0'1-0'2 0' '1-0'2 (22) (23) Again, B 1 consists of the first two columns of B. After application of the same methods as mentioned in Example 6.10, Table 2 is obtained. The results confirm our expectations based on the genericity conditions in [11]: E = (A, B) is reachable, E 1 = (A, B 1 ) is not. Method 1 reachable not reachable Method 2 reachable not reachable Method 3 reachable not reachable Method 4 reachable not reachable A careful study of Table 2 yields almost the same conclusions as in Example Again Method 4 is extremely slow in the reachable case. (Therefore it was only applied once). Although this algorithm remains the fastest option to verify the non-reachability ofthe system E 1 = (A, B 1 ), it is clear that in comparison with Method 4, all the algorithms derived in this paper are overwhelming improvements: they solve the problem in all cases within a reasonable amount of time. Nevertheless, the performances of the Methods 1 to 3 are also different from each other. In the reachable case, the recursive methods (Methods 2 and 3) are somewhat faster than Method 1, but the advantage becomes evident in the non-reachable case. Moreover, this example shows that the performances of these methods (the difference between Method 2 and Method 3) depend on the order of substitution. To illustrate the performance of the proposed methods for a more complicated problem, we consider the following experiment. Example 6.12 Let A and B be matrices over the polynomial ring R[O'I, 0'2] given by 0'1 + 0'2 0'2 2-20' '10'2-50'2 + 2 (24) 12

18 (25) and let B 1 be the submatrix of B, consisting of the first two columns of B. On the systems ~ = (A, B) and ~l = (A, Bt), the Methods 1 to 3 are applied to test reachability. Method 4, based on the minors of (zi - A I B), is only applied to ~l = (A, B 1 ); the computation for ~ = (A, B) would take too much time. The results are given in Table 3. 1 Table 3 ~ ~ = (A,B) I ~l = (A,Bt) I Method 1 reachable not reachable ,182.9 Method 2 reachable not reachable Method 3 reachable not reachable Method 4 not not reachable computed This example is a good illustration of the performance of the recursive methods in the non-reachable case: they are about 20 times faster than the method proposed in Section 3. Nevertheless, Method 4 is also in this experiment still much faster. In the reachable case the recursive methods do very well too, but here the performance is clearly dependent on the order of substitution. Remark 6.13 Examples suggest that also in the reachable case, the recursive methods to test reachability are faster than the method of Section 3. This conjecture was falsified by an example (not mentioned in this paper), in which the method of Section 3 was faster than both the Methods 2 and 3. In the non-reachable case however, the recursive methods performed better in all our examples. A review on the results of this section leads to the following conclusion. The methods proposed in this paper to test the reachability of a system ~ = (A,B) over a polynomial ring are far more effective than the simple method based on the computation of a Grobner Basis of an ideal generated by the minors of the matrix (zi - A I B). Although this simple method is the fastest option to detect the non-reachability of a system, it is extremely slow in the reachable case. This makes this method unappropriate as a general purpose reachability test. On the other hand, the methods of Sections 3 and 5 behave very well in both the reachable and the non-reachable case. In this respect they are clearly improvements of the method, based on the minors of (zi - A I B). When a system is reachable, the method of Section 3 and the recursive Algorithm 5.8 perform almost the same: the question which method is faster depends on the problem under consideration. However, when a system is not reachable, the recursive method of Algorithm 5.8 is clearly the most favorable one. Moreover, when using a recursive method, the computing time is dependent on the order of substitution (recall Remark 5.9). The exact relationship is not very clear, but it is likely that it is somewhat related to the structure of the system under consideration. 13

19 7 Conclusions In this paper it was shown how the Grobner Basis technique from the field of constructive commutative algebra can be used to test the reachablity of a system over a polynomial ring explicitly. Moreover, when a system ~ = (A,B) is reachable, the same computations can be used to construct a right-inverse of the matrix (zi - A I B). This right-inverse has several interesting applications from the control point of view. In test-examples the algorithm was very effective, and showed a better performance than a more straightforward method, based on the minors of (zi - A I B). Finally, in the non-reachable case it is possible to speed up the computation by doing the test recursively. References (1] J.W. Brewer, J.W. Bunce and F.S. Van Vleck, Linear systems over commutative rings. Lecture notes in pure and applied mathematics, vol New York, Marcel Dekker, [2] B. Buchberger, Some properties of Grobner Bases for Polynomial Ideals. ACM SIGSAM Bull., vol. 10, No.4, pp , [3] B. Buchberger, Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory. In N.K. Bose (ed.), Multidimensional Systems Theory, pp Dordrecht, Reidel, [4] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan and S.M. Watt, Maple V Library Reference Manual. New York, Springer Verlag, [5] K.B. Datta and M.L.J. Hautus, Decoupling of multivariable control systems over unique factorization domains. SIAM J. Control and Optimization, vol. 22, pp , [6] E. Emre, On necessary and sufficient conditions for regulation of linear systems over rings. SIAM J. Control and Optimization, vol. 20, pp , [7] K. Forsman, Constructive Commutative Algebra in Nonlinear Control Theory. Linkoping Studies in Science and Technology, Dissertations, No Linkoping University, [8] L.C.G.J.M. Habets, Stabilization of time-delay systems: An overview of the algebraic approach. EUT Report 92-WSK-02, Eindhoven University of Technology, [9] M.L.J. Hautus, Controllability and observability conditions of linear autonomous systems. Indag. Math., vol. 31, pp , [10] E.W. Kamen, Lectures on Algebraic System Theory: Linear Systems Over Rings. NASA Contractor Report 3016, [11] E.B. Lee and A.W. Olbrot, On reachability over polynomial rings and a related genericity problem. Int. J. Systems Sci., vol 13, pp , [12] A.S. Morse, Ring models for delay-differential systems. Automatica, vol. 12, pp ,

20 [13] F. Pauer and M. Pfeifhofer, The theory of Grabner Bases. L 'Enseignement Mathematique, vol. 34, pp , [14] Y. Rouchaleau, Regulation statique et dynamique d'un systeme hereditaire. In A. Bensoussan and J.L. Lions (eds.), Analysis and Optimization of systems, Proceedings of the Fifth International Conference on Analysis and Optimization of Systems, Versailles, December 14-17, 1982, pp Lecture Notes in Control and Information Sciences, vol. 44. Berlin-Heidelberg-New York, Springer Verlag, [15] E.D. Sontag, Linear systems over commutative rings: a survey. Ricerche di Automatica, vol. 7, pp. 1-34,

21 List of COSOR-memoranda Number Month Author Title January F.W. Steutel On the addition oflog-convex functions and sequences January P. v.d. Laan Selection constants for Uniform populations February E.E.M. v. Berkum Data reduction in statistical inference H.N. Linssen D.A.Overdijk February H.J.C. Huijberts Strong dynamic input-output decoupling: H. Nijmeijer from linearity to nonlinearity March S.J.L. v. Eijndhoven Introduction to a behavioral approach J.M. Soethoudt of continuous-time systems April P.J. Zwietering The minimal number oflayers of a perceptron that sorts E.H.L. Aarts J. Wessels April F.P.A. Coolen Maximum Imprecision Related to Intervals of Measures and Bayesian Inference with Conjugate Imprecise Prior Densities May I.J.B.F. Adan A Note on "The effect of varying routing probability in J. Wessels two parallel queues with dynamic routing under a W.H.M. Zijm threshold-type scheduling" May I.J.B.F. Adan Upper and lower bounds for the waiting time in the G.J.J.A.N. v. Houtum symmetric shortest queue system J. v.d. Wal May P. v.d. Laan Subset Selection: Robustness and Imprecise Selection May R.J.M. Vaessens A Local Search Template E.H.L. Aarts (Extended Abstract) J.K. Lenstra May F.P.A. Coolen Elicitation of Expert Knowledge and Assessment of Imprecise Prior Densities for Lifetime Distributions May M.A. Peters Mixed H2I H oo Control in a Stochastic Framework A.A. Stoorvogel

22 -2- Number Month June Author P.J. Zwietering E.H.L. Aarts J. Wessels Title The construction of minimal multi-layered perceptrons: a case study for sorting June P. van del' Laan Experiments: Design, Parametric and Nonparametric Analysis, and Selection June J.J.A.M. Brands F.W. Steutel R.J.G. Wilms On the number of maxima in a discrete sample June S.J.L. v. Eijndhoven J.M. Soethoudt Introduction to a behavioral approach of continuous-time systems part II June J.A. Hoogeveen H. Oosterhout S.L. van del' Velde New lower and upper bounds for scheduling around a small common due date June F.P.A. Coolen On Bernoulli Experiments with Imprecise Prior Probabilities June J.A. Hoogeveen S.L. van de Velde Minimizing Total Inventory Cost on a Single Machine in Just-in-Time Manufacturing June J.A. Hoogeveen S.L. van de Velde Polynomial-time algorithms for single-machine bicriteria scheduling June P. van del' Laan The best variety or an almost best one? A comparison of subset selection procedures June T.J.A. Storcken P.H.M. Ruys Extensions of choice behaviour July L.C.G.J.M. Habets Characteristic Sets in Commutative Algebra: overview an July P.J. Zwietering E.H.L. Aarts J. Wessels Exact Classification With Two-Layered Perceptrons July M.W.P. Savelsbergh Preprocessing and Probing Techniques for Mixed Integer Programming Problems

23 Number Month Author Title July I.J.B.F. Adan Analysing EklErlc Queues W.A. van de Waarsenburg J. Wessels July O.J. Boxma The compensation approach applied to a 2 x 2 switch G.J. van Houtum July E.H.L. Aarts Job Shop Scheduling by Local Search P.J.M. van Laarhoven J.K. Lenstra N.L.J. Ulder August G.A.P. Kindervater Local Search in Physical Distribution Management M.W.P. Savelsbergh August M. Makowski MP-DIT Mathematical Program data Interchange Tool M.W.P. Savelsbergh August J.A. Hoogeveen Complexity of scheduling multiprocessor tasks with S.L. van de Velde prespecified processor allocations B. Veltman August O.J. Boxma Tandem queues with deterministic service times J.A.C. Resing September J.H.J. Einmahl A Bahadur-Kiefer theorem beyond the largest observation September F.P.A. Coolen On non-informativeness in a classical Bayesian inference problem September M.A. Peters A Mixed H 2 / H oo Function for a Discrete Time System September I.J.B.F. Adan Product forms as a solution base for queueing J. \Vessels systems September L.C.G.J.M. Habets A Reachability Test for Systems over Polynomial llings using Grabner Bases