A reachability test for systems over polynomial rings usingGröbner basesCitation 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). Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science

Memorandum CaSaR 92-38

A Reachability Test for Systems overPolynomial Rings using Grobner Bases

L.C.G.J.M. Habets

r

Eindhoven, September 1992The Netherlands

Eindhoven University of TechnologyDepartment of Mathematics and Computing ScienceProbability theory, statistics, operations research and systems theoryP.O. Box 5135600 MB Eindhoven - The Netherlands

Secretariate: Dommelbuilding 0.03Telephone: 040-47 3130

ISSN 0926 4493

A Reachability Test for Systems over Polynomial Rings usingGrabner Bases 1 2

L.C.G.J.M. HabetsEindhoven University of Technology

Department of Mathematics and Computing ScienceP.O. Box 513

NL-5600 MB EindhovenThe Netherlands

September 8, 1992

1Research supported by the Netherlands Organization for Scientific Research (NWO)2Submitted to 1993 ACC

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 itpossible to carry out this test explicitly. In this paper it is shown how this can be done in anefficient 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 bothreachable 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 oftenrequired for the design of feedback compensators. Finally, a modification of the reachabilitytest is given to speed up the computations in the non-reachable case.

1 Introduction

Systems over (polynomial) rings can be seen as a rather straightforward generalization ofordinary systems over fields. In the last three decades these systems have been investigatedquite extensively (see for example [1], [10] and [15]), not only because they highlight the mostimportant system theoretic properties very clearly, but also because they have very interestingapplications. For example, systems over polynomial rings can be used to model systems withvarying 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 thesystem:

O"iU(t) = u(t - Ti), (1)

where the Ti, (i = 1, ... , f), are f incommensurate time-delays. A time-delay system can thenbe 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),

(2)

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 asreachability and observability, have been generalized to the systems over rings case. In thisway it was possible to derive results on various problems such as pole-placement, stabilizabilityand input-output decoupling (see [12], [6] and [5] respectively), which are quite similar to thewell-known results for systems over fields. Although from the theoretical point of view thetheory of systems over rings is well established, there remains one shortcoming. In the existingliterature almost no attention is paid to the computational aspects of systems over rings. Inthis article we fill in a part of this gap. We propose a method to solve one of the problems inthis 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 ringis reachable. We give a description which is well suited for the construction of an algorithmictest. In the next part a short introduction to Grabner Bases is given. With this tool fromconstructive 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 veryinteresting from the control point of view because this right-inverse is often needed in thedesign 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 ageneral non-square polynomial matrix. In Section 5 we show how the reachability test can bespeeded up by applying it recursively. After this, the effectiveness of the proposed methodsis illustrated with some examples. Also timing-statistics are given. Certainly in comparisonwith another, rather straightforward method to test reachability, the derived methods behavevery well. Finally some conclusions are drawn.

1

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 algebraicclosure of /(. We denote the ring of all polynomials in f indeterminates 0'1,."'0'1. withcoefficients 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 thering 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 ofview 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 andapplying 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 bythe columns of the matrix

(B I AB I ... IAn - 1 B)

is the free module Rn. Based on the above interpretation, but nevertheless independent ofit, 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. Thenthe system I; = (A, B) is called reachable if the columns of the matrix

(B I AB I '" IAn-

1B)

span the free module R n .

(4)

The condition given in Definition 2.1 is rather difficult to check. Especially for systemsover polynomial rings there are alternative characterizations of reachability which are moresuitable for testing.

Theorem 2.2 Let R = 1C[0"1,'" ,0'1.] and suppose that A E Rnxn and B E Rnxm. Considerthe 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 /(- nZZZ vO'I, ••• ,O'l,Z vQl"'Qn :

ProofSeveral 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 befound in the existing literature.

2

(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

(6)

Because (zI - A I B) is right-invertible over n[z], say with right-inverse M(z), we canalso substitute (u}, ... ,Ul, z) in M(z) to arrive at a right-inverse of (Zl - A(u}, ... ,Ul) IB(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£) IB(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) IB(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) IB(U}, ... ,Ui» = n.

(iv) ::} (i): This is a generalization of the well-known PBH-test (see for example [9]). Acomplete proof of the exact statement can be found in [8, Theorem 2.2.4.]. •

Condition (ii) in Theorem 2.2 is a very important characterization of reachability fromthe control point of view. In several control problems, such as the stabilization problem, theright-inverse of (zI - A IB) can be used to design a compensator. The computation of sucha right-inverse is therefore a very interesting question.

With help of condition (iii), Olbrot and Lee ([11]) derived genericity conditions for thereachability of systems over polynomial rings. They showed that such a system is genericallyreachable if the number of inputs m to the system is strictly larger than the number ofindeterminates £ in the polynomial ring K[al, ... ,ai]. Moreover, if m :5 £, so when thenumber of inputs is smaller or equal to the number of indeterminates, a system is genericallynot 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 systemover a polynomial ring. To do so, we translate one of the reachability conditions in Theorem2.2 into terms of polynomial ideals. These ideals can then be manipulated using the GrobnerBasis 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 solvevarious questions on polynomial ideals such as the membership problem, or to find a solutionof a system of algebraic equations. The method was introduced by B. Buchberger in 1965 andnowadays most computer-algebra packages contain software for the computation of GrabnerBases. Good references are [3] for the algorithmic part of the problem, and [13] for a more

3

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 statethe results we need in the sequel. To do so, we first have a short glimpse at the theory. Forthe 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 whichgenerate the ideal. To find these polynomials of low complexity a sort of reduction processis carried out. For polynomials in one indeterminate the degree is a good measure for thecomplexity. Moreover, the degree induces a ranking on these polynomials. The EuclidianDivision Algorithm uses this ranking to compute remainders. Such a remainder has a lowercomplexity (lower degree) than the original polynomial. In this way it is possible to findpolynomials of lower complexity in the one indeterminate case.

When confronted with polynomials in more than one indeterminate, the situation is muchmore 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 inducesa ranking on the polynomials. Then one has to find a generalized remainder algorithm. Thisremainder algorithm is used to simplify polynomials with respect to each other in order tofind polynomials of lower 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 havealready introduced a generalized degree on K[Xl' ... ,Xl]. Then the initial term of p, denotedby in(p), is the term in p of highest degree. A Grabner Basis of an ideal I can then be definedin 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 aGrobner Basis of I if the set

in(G) = {in(9) 19 E G}

generates the ideal in(I) = {in(J) I f E I}.

As a direct consequence, a finite set of monomials (Le. polynomials consisting of onlyone term) {ml, ... ,mk} is a Grabner Basis of the ideal (ml, ... ,mk) generated by thesemonomials (see [13, p. 218, Remark 1.5.6]).

Grabner Bases can be calculated using the algorithm of Buchberger (see [3]). It canbe seen as an explicit implementation of the reduction process, rather intuitively describedabove. 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 Gsuch 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 multiplicationby 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 thereachability of a system over a polynomial ring. First we translate condition (iii) in Theorem2.2 into terms of polynomial ideals.

Consider a system ~ = (A, B) over K[al, ... , ae] and let qT = (ql··· qn) be an n­dimensional row-vector consisting of the n indeterminates ql, ... , qn. Define the row-vectorpT as

pT := qT . (zI _ A I B).

4

(7)

Then the entries P}, ... ,Pn+m of pT = (PI'" Pn+m) are polynomials in the indeterminatesUl, ... ,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 notationK[u, z, q] for this ring. The next proposition shows how the reachability of ~ = (A, B) dependson 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 inK [u}, ... , Ui, Z, q}, ... , qn] genemted by the polynomials Pi (i = 1, ... , n +m) defined by

(PI'" Pn+m) = pT = qT . (zI - A IB).

Then

(zI - A I B) is right-invertible over n[z],

The auto-reduced Grobner Basis ofP consists precisely of the 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 amatrix 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+mqi = 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)'

On the other hand it is clear from (7) that also

(P}, .. ·,Pn+m) C (q}, ... ,qn)'

Hence, in the ring K[u, z, q] we have

(8)

(9)

(10)

Because the polynomials q}, ... , qn are clearly monomials in K[u, z, q], it follows fromDefinition 3.3 and the subsequent remark that the polynomials ql,"" qn form a GrobnerBasis of P, independent of the generalized notion of degree. Moreover, this Grobner Basis isauto-reduced because none of these polynomials divides an other one. This implies that thisBasis is unique upto multiplication with constants in K.

5

"¢:" Now suppose that the auto-reduced Grobner Basis of P consists of the polynomialsq1,' .. ,qn' Define the ideal Q in K[O', z, q] as Q = (q}, ... ,qn). Then P = Q. But then also thevarieties of P and Q are equal: V(P) = V( Q). Because Q is generated by a set of polynomialsof 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 ~ 0Pn+m ~ 0 qn = 0

Let now (0"1, ... ,O"e, i) E ,(:£+1, and consider the matrix

(12)

(13)(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) ofTheorem 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 reachabilitytest for systems over the polynomial ring n = K[O'l, ... ,O't]. Let E = (A,B). First computethe polynomials PI, .. . ,Pn+m in K[O'I, , O't, Z, q1,' .. , qn] with help offormula (7). Computean auto-reduced Grobner Basis of (P1, ,Pn+m), using Buchberger's algorithm. Such analgorithm is available in most computer-algebra packages. If the Grobner Basis consistsprecisely 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 IB)

In the last section it was shown how the reachability of a system E = (A, B) can be tested, byverifying the right-invertibility of the matrix (zI - A IB) using Grobner Bases. However, thecomputation of a right-inverse of (zI - A I B) over R[z] is also very interesting for its own sakebecause this inverse is needed in the design of compensators for various control problems. Inthis section we show that the Grobner Basis construction of the last section implicitly carriesall 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" ·qn) of indeterminates anddefine:

pT = (Pl'" Pn+m) = qT. (zI - A I B).

Because (zI - A I B) is right-invertible, the Grobner Basis of (PI,' .. ,Pn+m) is the set ofpolynomials {q1,"" qn}. This set is obtained after application of Buchberger's algorithm.

6

This algorithm consists of successive steps from one polynomial set to an other, startingwith {PI, ... ,Pn+m} and ending with {qI, ... , qn}, such that these sets, and all the sets inbetween, generate the same ideal. Each polynomial in a new polynomial set is constructed asa 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 coefficients0ji in K[O', z, q] (j = 1, ... , n +mj i = 1, ... , n), such that

n+mVi E {I, ... ,n}: qi = 2: QjiPj·

j=I

(14)

Let A(q) denote the (n +m) X n matrix over K[O',zHq] such that the (j,i)th entry of A(q) is0ji. Then, according to (14):

qT = pT . A(q).

Substitution of pT = qT . (zI - A IB) in (15) yields for all q:

qT = qT . (zI - A IB) . A(q).

(15)

(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 of indeterminates. Suppose that A(q) is an k Xn polynomialmatrix over K[O'I, ... ,O'i, z][qI, ... ,qn] such that

(17)

Then the matrix A(O), obtained by substituting (qI'" qn) = (0···0) in A(q),is a right-inverseof 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 theindeterminate>' with coefficients in K[O', z, q]. For all >. # 0, these polynomials are zero, sothey must be identically zero. Thus for>. = 0 we have:

qT = qT . M . A(O). (18)

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 thematrix A(q) over K[O', z,q] defined in (14) and (15). From formula (16) and Theorem 4.5we immediately see that substitution of (qI ... qn) =(0·· ·0) in A(q) yields a right-inverse of(zI - A I B) over n[z].

7

Remark 4.6 The method described above to compute a right-inverse of a polynomial matrixusing Grobner Bases, is not only valid in the case where the polynomial matrix has the specialform (zI - A I B). In the construction this special structure was never used. Therefore thismethod is also applicable in the general case. In this way Grobner Bases can be used to testthe right- and left-invertibility of non-square polynomial matrices and to compute left- andright-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 canalso be used in a slightly different way. It is possible to reformulate the condition in order toget 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 inthe next lemma.

Lemma 5.7 Let n = K[(il,' .. ,(ii], and suppose that A E nnxn and B E nnxm. Considerthe 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) ofTheorem 2.2, the equality

implies that (0· .. 0lllqj+! ... qn) = (0· .. 0). Trivially this can not happen, hence the necessityof (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) IB(Ul, ... ,Ui)) = (0···0)

(use condition (iii) of Theorem 2.2). Define j := min{i I iii =I O}. Then the n-vector ;j .iiT isof the form

~ iiT = ~ (iiI ... iin) = (0 ... 0 111 iij_+1 ... ~n ).qj qj ~l qj qj

J-

But of course we still have

1 1-=- . iiT . (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

The condition of Lemma 5.7 is easily tested for each j E {I, ... , n} separately, using theGrabner 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

pT = (PI'" Pn+m) := qT . (zI - A IB).

The entries PI, ... ,Pn+m of pT 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 havea 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: theconstant 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 testdescribed 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 thereachablity of the system ~ = (A, B) over n.

j:= I;G:= {I};while j ~ nand G = {I} do

qT:= (0· ··Olllqj+I ···qn);~

j-I

pT = (PI'" Pn+m) := qT . (zI - A IB);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 Section3 due to the following fact. The complexity of the computation of Grabner Bases is highlydependent on the number of indeterminates in the polynomial ring. In the method of Section3, a Grabner Basis over a ring with n+f+ 1 indeterminates has to be calculated. In Algorithm5.8 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 methodis faster, despite the fact that more Grabner Bases have to be calculated. Note that thesecomputations become in each step less involved because the number of indeterminates isstrictly decreasing. Moreover, if the system ~ = (A, B) is not reachable, it is very likely thatthis is detected in the first step, after the computation of only one Grabner Basis. Fromthe proof on genericity in [11] it follows that generically this will be the case. Therefore weexpect Algorithm 5.8 to be a faster method for the detection of the non-reachability of a

9

system. This claim is illustrated in Section 6, where the performance of both the algorithmsis 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 thecomputations do not give any clue for the construction of a right-inverse of (zI - A I B). Sowhen this inverse is really needed, the method of Section 3 is of course the most favorableone.

Remark 5.9 Algorithm 5.8 can be seen as a special case of the method of Section 3 (onlythe conclusions are derived in a different way) in the sense that in each step a number of theindeterminates qt, ... ,qn is substituted by some zeros and a one. In Algorithm 5.8 this is donein a special order (from qt to qn), but this order does not make any difference for the problemunder consideration. Therefore one can obtain alternative algorithms by changing the orderof 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 orderof substitution.

6 Examples

The purpose of this section is to show the effectiveness of the methods proposed in thispaper. This is illustrated with help of some examples. Also a comparison of the performancesof the various methods is made. Moreover, to illuminate the advantages and drawbacks ofthe algorithms derived in this paper very clearly, the performances are compared with a verysimple 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 nandn 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-invertibleiff these minors do not have a common zero. According to the Hilbert-Nullstellensatz thisimplies that E = (A, B) is reachable iff (Tt, ... , TN) = n[z], Le. iff the ideal generated bythe n x n minors Tt, ... , TN of (zI - A I B) is the whole ring R[z]. This last conditionis easily verified with help of Grobner Bases. According to Definition 3.3, the normalizedauto-reduced Grobner Basis of (Tt, •.. ,TN) consists of only one polynomial in this case: theconstant polynomial 1.

The rest of this section contains three different examples. These experiments were madein the computer-algebra package MAPLE V, running on a Sun/Spare Workstation with a25MHz processor. To compute Grobner Bases, we used the function gbasis from the grobnerpackage, with the total-degree ordering and an automatic ordering of the indeterminates (forany details, see [4, pp. 469-478]). All the timings are given in CPU seconds without excludingthe time for garbage collection. This garbage collection took place every 1Mb. Statistics onthe use of memory are not given because the required amount of memory was not critical inthese examples.

10

Example 6.10 Consider the matrices A, Band B1 over the polynomial ring R[O'], given by

_0'2 + 30' - 80'2 + 4

-80' +7

-20' +6 )20'2 - 50' +4

0'+4

(20)

(

0'2 +30' - 2 2 )B = 0 50' + 1

40' +7 -0' +2 (

0'2 + 30' - 2 )B1 = 0 .

40' +7

(21)

So B1 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, B1 ) is now tested with four differentmethods:

Method 1 The method described in Section 3, based on the computation of a Grabner Basisof qT . (zI - A IB).

Method 2 Algorithm 5.8 in Section 5.

Method 3 A modification of Algorithm 5.8 as mentioned in Remark 5.9. The substitutionorder (from ql to qn in Algorithm 5.8) is changed into the reverse direction (from qn toql).

Method 4 The method based on the computation of a Grabner Basis of the ideal generatedby 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 inTable 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 toverify the reachability of ~ = (A, B) was highly variable. The indicated value is the mean offour samples.

Method 1 reachable not reachable17.8 98.4

Method 2 reachable not reachable8.8 27.1

Method 3 reachable not reachable9.0 19.9

Method 4 reachable not reachable56.1.102 3.3

I Table 1 ~ ~ = (A,B) I ~l = (A,B1 ) I

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 reachablesystem ~ = (A, B). On the contrary, the Methods 1 to 3, as proposed in this paper, behavevery well in both cases. The recursive Methods 2 and 3 are a little bit faster than Method1, especially in the non-reachable case. This is exactly as expected. Finally there is a littledifference between Method 2 and Method 3, probably due to the structure of the exampleunder consideration.

11

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 B1 over the polynomial ring R[0'1' 0'2] givenby

( "dl 0'2 + 0'1 '" -'" +3 )A= 0'2 - 1 0'2 + 1 0'1 - 50'1

2 + 1 1 0'2 + 1

( "d", 0'1 - 1

'" ~ 3 )( "d", ",-1 )

B= 1 0'22 + 1 B1 = 1 0'22 + 1 .0 0'1 - 0'2 0'2 +2 0 0'1 - 0'2

(22)

(23)

Again, B1 consists of the first two columns of B. After application of the same methods asmentioned in Example 6.10, Table 2 is obtained. The results confirm our expectations basedon the genericity conditions in [11]: E = (A, B) is reachable, E1 = (A, B1 ) is not.

Method 1 reachable not reachable55.7 362.9

Method 2 reachable not reachable20.3 43.7

Method 3 reachable not reachable19.7 145.3

Method 4 reachable not reachable158.6.103 21.3

A careful study of Table 2 yields almost the same conclusions as in Example 6.10. AgainMethod 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 systemE1 = (A, B1 ), it is clear that in comparison with Method 4, all the algorithms derived inthis paper are overwhelming improvements: they solve the problem in all cases within areasonable amount of time. Nevertheless, the performances of the Methods 1 to 3 are alsodifferent 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-reachablecase. Moreover, this example shows that the performances of these methods (the differencebetween 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 - 20'2 +140'10'2 - 50'2 +2

(24)

12

(25)

and let B1 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, B1 );

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 reachable1266.8 14,182.9

Method 2 reachable not reachable314.3 695.8

Method 3 reachable not reachable174.2 583.6

Method 4 not not reachablecomputed 110.8

This example is a good illustration of the performance of the recursive methods in thenon-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 therecursive methods do very well too, but here the performance is clearly dependent on theorder of substitution.

Remark 6.13 Examples 6.10-6.12 suggest that also in the reachable case, the recursive meth­ods to test reachability are faster than the method of Section 3. This conjecture was falsifiedby an example (not mentioned in this paper), in which the method of Section 3 was fasterthan both the Methods 2 and 3. In the non-reachable case however, the recursive methodsperformed better in all our examples.

A review on the results of this section leads to the following conclusion. The methodsproposed in this paper to test the reachability of a system ~ = (A,B) over a polynomial ringare far more effective than the simple method based on the computation of a Grobner Basisof an ideal generated by the minors of the matrix (zI - A I B). Although this simple methodis the fastest option to detect the non-reachability of a system, it is extremely slow in thereachable 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 reachableand 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 3and the recursive Algorithm 5.8 perform almost the same: the question which method is fasterdepends on the problem under consideration. However, when a system is not reachable, therecursive method of Algorithm 5.8 is clearly the most favorable one. Moreover, when usinga recursive method, the computing time is dependent on the order of substitution (recallRemark 5.9). The exact relationship is not very clear, but it is likely that it is somewhatrelated to the structure of the system under consideration.

13

7 Conclusions

In this paper it was shown how the Grobner Basis technique from the field of constructivecommutative algebra can be used to test the reachablity of a system over a polynomial ringexplicitly. Moreover, when a system ~ = (A,B) is reachable, the same computations can beused to construct a right-inverse of the matrix (zI - A I B). This right-inverse has severalinteresting applications from the control point of view. In test-examples the algorithm wasvery effective, and showed a better performance than a more straightforward method, basedon the minors of (zI - A I B). Finally, in the non-reachable case it is possible to speed upthe 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. 104. New York, Marcel Dekker,1986.

[2] B. Buchberger, Some properties of Grobner Bases for Polynomial Ideals. ACM SIGSAMBull., vol. 10, No.4, pp. 19-24, 1976.

[3] B. Buchberger, Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory.In N.K. Bose (ed.), Multidimensional Systems Theory, pp. 184-232. Dordrecht, Reidel,1985.

[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, 1991.

[5] K.B. Datta and M.L.J. Hautus, Decoupling of multivariable control systems over uniquefactorization domains. SIAM J. Control and Optimization, vol. 22, pp. 28-39, 1984.

[6] E. Emre, On necessary and sufficient conditions for regulation of linear systems overrings. SIAM J. Control and Optimization, vol. 20, pp. 155-160, 1982.

[7] K. Forsman, Constructive Commutative Algebra in Nonlinear Control Theory. LinkopingStudies in Science and Technology, Dissertations, No. 261. Linkoping University, 1991.

[8] L.C.G.J.M. Habets, Stabilization of time-delay systems: An overview of the algebraicapproach. EUT Report 92-WSK-02, Eindhoven University of Technology, 1992.

[9] M.L.J. Hautus, Controllability and observability conditions of linear autonomous sys­tems. Indag. Math., vol. 31, pp. 443-448, 1969.

[10] E.W. Kamen, Lectures on Algebraic System Theory: Linear Systems Over Rings. NASAContractor Report 3016, 1978.

[11] E.B. Lee and A.W. Olbrot, On reachability over polynomial rings and a related genericityproblem. Int. J. Systems Sci., vol 13, pp. 109-113, 1982.

[12] A.S. Morse, Ring models for delay-differential systems. Automatica, vol. 12, pp. 529-531,1976.

14

[13] F. Pauer and M. Pfeifhofer, The theory of Grabner Bases. L 'Enseignement Mathematique,vol. 34, pp. 215-232, 1988.

[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 theFifth International Conference on Analysis and Optimization of Systems, Versailles, De­cember 14-17, 1982, pp. 523-547. Lecture Notes in Control and Information Sciences, vol.44. Berlin-Heidelberg-New York, Springer Verlag, 1982.

[15] E.D. Sontag, Linear systems over commutative rings: a survey. Ricerche di Automatica,vol. 7, pp. 1-34, 1976.

15

List of COSOR-memoranda - 1992

Number Month Author Title92-01 January F.W. Steutel On the addition of log-convex functions and sequences

92-02 January P. v.d. Laan Selection constants for Uniform populations

92-03 February E.E.M. v. Berkum Data reduction in statistical inferenceH.N. LinssenD.A.Overdijk

92-04 February H.J.C. Huijberts Strong dynamic input-output decoupling:H. Nijmeijer from linearity to nonlinearity

92-05 March S.J.L. v. Eijndhoven Introduction to a behavioral approachJ.M. Soethoudt of continuous-time systems

92-06 April P.J. Zwietering The minimal number of layers of a perceptron that sortsE.H.L. AartsJ. Wessels

92-07 April F.P.A. Coolen Maximum Imprecision Related to Intervals of Measuresand Bayesian Inference with Conjugate Imprecise PriorDensities

92-08 May I.J.B.F. Adan A Note on "The effect of varying routing probability inJ. Wessels two parallel queues with dynamic routing under aW.H.M. Zijm threshold-type scheduling"

92-09 May I.J.B.F. Adan Upper and lower bounds for the waiting time in theG.J.J.A.N. v. Houtum symmetric shortest queue systemJ. v.d. Wal

92-10 May P. v.d. Laan Subset Selection: Robustness and Imprecise Selection

92-11 May R.J .M. Vaessens A Local Search TemplateE.H.L. Aarts (Extended Abstract)J.K. Lenstra

92-12 May F.P.A. Coolen Elicitation of Expert Knowledge and Assessment of Im-precise Prior Densities for Lifetime Distributions

92-13 May M.A. Peters Mixed H2I Hoo Control in a Stochastic FrameworkA.A. Stoorvogel

Number92-14

92-15

92-16

92-17

92-18

92-19

92-20

92-21

92-22

92-23

MonthJune

June

June

June

June

June

June

June

June

June

AuthorP.J. ZwieteringE.H.L. AartsJ. Wessels

P. van del' Laan

J.J.A.M. BrandsF.W. SteutelR.J .G. Wilms

S.J.L. v. EijndhovenJ .M. Soethoudt

J .A. HoogeveenH. OosterhoutS.L. van del' Velde

F.P.A. Coolen

J .A. HoogeveenS.L. van de Velde

J .A. HoogeveenS.L. van de Velde

P. van del' Laan

T.J.A. StorckenP.H.M. Ruys

-2-

TitleThe construction of minimal multi-layered perceptrons:a case study for sorting

Experiments: Design, Parametric and NonparametricAnalysis, and Selection

On the number of maxima in a discrete sample

Introduction to a behavioral approach of continuous-timesystems part II

New lower and upper bounds for scheduling around asmall common due date

On Bernoulli Experiments with Imprecise PriorProbabilities

Minimizing Total Inventory Cost on a Single Machinein Just-in-Time Manufacturing

Polynomial-time algorithms for single-machinebicriteria scheduling

The best variety or an almost best one? A comparison ofsubset selection procedures

Extensions of choice behaviour

92-24 July L.C.G.J.M. Habets Characteristic Sets in Commutative Algebra:overview

an

92-25

92-26

July

July

P.J. Zwietering Exact Classification With Two-Layered PerceptronsE.H.L. AartsJ. Wessels

M.W.P. Savelsbergh Preprocessing and Probing Techniques for Mixed IntegerProgramming Problems

-3-

Number Month Author Title92-27 July I.J.B.F. Adan Analysing EklErlc Queues

W.A. van deWaarsenburg

J. Wessels

92-28 July O.J. Boxma The compensation approach applied to a 2 x 2 switchG.J. van Houtum

92-29 July E.H.L. Aarts Job Shop Scheduling by Local SearchP.J .M. van LaarhovenJ.K. LenstraN.L.J. Ulder

92-30 August G.A.P. Kindervater Local Search in Physical Distribution ManagementM.W.P. Savelsbergh

92-31 August M. Makowski MP-DIT Mathematical Program data Interchange ToolM.W.P. Savelsbergh

92-32 August J .A. Hoogeveen Complexity of scheduling multiprocessor tasks withS.L. van de Velde prespecified processor allocationsB. Veltman

92-33 August O.J. Boxma Tandem queues with deterministic service timesJ.A.C. Resing

92-34 September J.H.J. Einmahl A Bahadur-Kiefer theorem beyond the largestobservation

92-35 September F.P.A. Coolen On non-informativeness in a classical Bayesianinference problem

92-36 September M.A. Peters A Mixed H 2 / Hoo Function for a Discrete Time System

92-37 September I.J.B.F. Adan Product forms as a solution base for queueingJ. \Vessels systems

92-38 September L.C.G.J.M. Habets A Reachability Test for Systems over Polynomial llingsusing Grabner Bases