Implicitization, Graph Ideals and Control Systems

Implicitization, Graph Ideals and Control Systems

Krister Forsman

Department of Electrical Engineering Linkoping University

S-581 83 Linkoping Sweden

email: krister@isy.liu.se

1993-06-24

REGLERTEKNIK

AUTOMATIC CONTROL

LINKÖPING

Technical reports from the automatic control group in Linkoping are available by anonymous ftp at the address130.236.24.1 (joakim.isy.liu.se). This report is contained in the compressed Postscript le named/pub/reports/LiTH-ISY-R-1508.ps.Z

Implicitization, Graph Ideals and Control Systems

Krister Forsman

Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden

email: krister@isy.liu.se

1993-06-24

Abstract. We discuss how the external behavior of a polynomial automatic control system can be determined, i.e. how to nd the dierential equation relating the input and the output of the system given a state space description, focussing on algorithmic aspects. This problem is equivalent to what is known as implicitization in computational algebraic geometry and one way of doing this is to perform elimination in so called graph ideals. We compare dierent methods for implicitization with regard to computational complexity. Moreover, a bound for the degree of the input-output equation in terms of the degrees of the state equations is derived.

Keywords: control systems, state space theory, algebraic geometry, implicitization, B ezout's theorem, com- puter algebra, symbolic computation, elimination, Grobner bases, computational complexity

1 Introduction

In this paper we consider control systems represented as a number of rst order o.d.e:s _

x(^t) = ^f(^x(^t)^u(^t))^{ y}(^t) = ^h(^x(^t)^u(^t)) (1) where^uis a (scalar) input signal,^y is theoutputand ^x = (^x1:::xn) areinternal variables, orstate variables. ^f is thus a function^{Rn+1 !Rn}and ^ha map^Rn+1!R. (1) is referred to as a state space description of the system. Such a description is often quite natural and for many problems with a physical background it can be automatically derived using so called bond graphs 17]. Many times it is interesting to know the d.e. relating the input and the output of a control system. For example, if two systems have the same input-output behavior, they are known to be isomorphic 8, 23]. If a state space realization is minimal, something which can be checked by determining the i/o-equation, some statements can be made about observability and controllability 10, 15, 16].

The problem of determining the input-output equation has been addressed earlier by several authors see e.g. 6, 14, 24] and their references. To the knowledge of the author the connection to implicitization has not been made earlier, nor any investigation of the computational complexity of implicitization using graph ideals.

The most common mathematical tool in nonlinear control theory of today is, no doubt, dierential geometry. However, if the nonlinearities involved are all polynomial there are methods from algebra that can be used instead. This was noted early by several authors e.g. 2, 22], and lately algebraic methods have attracted a lot of attention 7, 14, etc]. In the present work we show how this problem is related to a well-known problem in algebraic geometry that has been solved constructively in several dierent ways. We will also discuss some complexity issues when solving the problem in question.

As mentioned, we only consider^fhwhere all nonlinearities are polynomial. The restric- tion to polynomial nonlinearities is not as severe as one might rst believe. Systems where the nonlinearities are not originally polynomial may be embedded in polynomial ones if the nonlinearities themselves are solutions to algebraic dierential equations 20]. This procedure could have some serious disadvantages, though, since the dimension of the solution space is increased. E.g. the solutions of the d.e. _^x=^exare also solutions to the polynomial d.e. ^x= _^x2 but the latter has a family of solutions ^x(^t) ^{C 2R} that are not solutions to the original equation.

The restriction to polynomial systems suggests that an appropriate mathematical frame- work is algebraic geometry and eld theory, and we will suppose that the reader is acquainted with some of the basic concepts in these branches of mathematics. Some acquaintance with modern elimination theory (Grobner bases and characteristic sets) will also help.

The following notational conventions will be used:

hF i is the ideal generated by the set ^F. We use the abbreviations



x = ^{x1:::xn y} = ^y0:::yn;1 (2) so that e.g. ^K(^y)^xyn] = ^K(^y0:::yn;1)^x1:::xnyn].

For the variables^uy (but not for^x, of course) we use subscript to denote time-derivative:

8i: ^ui = ^di

dt i

u y

i = ^di

dt i

y (3)

It should be noted that even though the systems in this paper are all continuous time systems (described by d.e:s) all the methods suggested work for discrete time systems (dierence equations). This is also a dierence from most of the previous work on the topic.

2 Implicitization and Elimination

We will now see how the problem stated in the introduction is related to a problem in algebraic geometry, regarding polynomial mappings. Let ^f be as in equation (1). The extended Lie derivative operator w.r.t.^f is dened by

L

f = ^Xn

i=1 f

i

@

@x

i

+^X1

i=0 u

i+1

@

@u

i

This means that taking the Lie derivative of a function w.r.t. ^f is the same thing as dier- entiating it w.r.t. ^tand then replacing all _^xi with their corresponding right hand side in the state space equations. If we take successive Lie-derivatives of the output map ^h w.r.t. ^f we thus get expressions for the ^yi in terms of ^x and^u:

y

0=^{h y1} =^{Lfh ::: yn}=^Lnfh (4) Let^K =^k(^u0u1:::) i.e.^K is the transcendental extension of ^k containing all derivatives of the input.

Lemma 2.1

^y0:::yn are algebraically dependent over ^K.

Proof.

All ^Lifh2K(^x) and the transcendence degree of ^K(^x) over^K isⁿsoⁿ+1 elements

of ^K(^x) are algebraically dependent. ²

In a geometric language the lemma states that the functions ^Lifh parametrize a variety over ^K. Determining the input-output relation is now the same thing as nding the implicit equation for the variety. This is known asimplicitization in algebraic geometry 5, etc]. The most straightforward way of doing implicitization is probably to perform elimination in the ideal

G = ^{hy0;h y1;Lfh:::yn;Lnfhi} (5) The generators of the ideal areⁿ+ 1 polynomials of the type ^Lifh, each one associated to a

\tag-variable"^yi. We call^Gthegraph-idealof the mapping (^x) ^7! (^y0:::yn). The technique of tag-variables is commonly used in constructive algebraic geometry see e.g. 5, 12, 13, 21].

In 21] it is attributed to D. Spear, but it appears to be much older 18].

An alternative way of determining dependency relations is suggested in 19].

Elimination consists in nding the elements of an ideal that belong to a certain subring.

For example we may wish to nd ^a\k^X] for an ideal ^{a  k}^XY] where^XY are sets of variables. One way of doing elimination is to compute a Grobner base (GB), or standard base, of an ideal w.r.t. the purely lexicographic term-ordering (plex). For an introduction to GB see e.g. 4, 5, 13]. The computer algebra programs Maple, Axiom, Reduce and Macsyma all have GB packages.

There is also a method based on GB for nding univariate contractions, that does not require an entire plex GB to be computed. This method was rst presented in 3], so we call it the Boege-Gebauer-Kredel algorithm (BGK). It has been implemented in the Maple GB package under the name^finduni. The BGK algorithm is in general more ecient than computation of a plex GB, but to use it we have to know that the contraction in question is non-zero. We return to this in the next Section.

Yet another method for elimination is characteristic sets (CS), see e.g. 5, 12, 15] and their references.

The rest of the paper will be spent discussing computational aspects of the elimination of the ^x-variables in the graph ideal ^G.

3 Complexity

The complexity of GB computations has been studied thoroughly by many authors see the work cited in e.g. 4]. For some polynomials of total degree ^d in ⁿ variables the worst case complexity is ^O(^d2n). Thus, if we are able to decrease the number of variables involved without changing the problem too much, we drastically improve computation times. We will now use some commutative algebra to perform such a reduction of the number of variables.

The following theorem is very useful:

Lemma 3.1

Let ^R be a commutative ring and ^S a multiplicative system in ^R. There is a one-to-one correspondence between the prime ideals of ^R that do not intersect ^S and the prime ideals of ^S;1R (the localization of ^R at ^S).

Proof.

See 1, proposition 3.11, iv]. ²

In our application^R =^K^xy ^yn] and^S=^K^y]^nf0^g, so to change the ground eld using lemma 3.1 we make an extension^K^xy ^yn]^!K(^y)^xyn]. In 10] it is proved that^Gis prime and the extension of the prime ideal^GK^xy ^yn] to^K(^y)^{x yn}] is^Ge=^K(^y)^xyn]^G. The

Example no Method 1 Method 2 Method 3 Method 4

(6) 61 ^>2000 544 *

(7) 1.8 1.0 1.1 0.25

(8) 72 ^>2000 ^>2000 *

(9) 13 1440 187 36

Table 1: Computation time in CPU-seconds for dierent methods of implicitization using the graph ideal^G. The symbol *means that the computation failed because a Maple object occurring during the course of computation was too large.

extension of an arbitrary prime ideal is not necessarily prime 1], but ^Gdoes have a prime extension:

We may assume that the variables ^y are not algebraically dependent (if they are, the system is not observable 10, 15]), so lemma 3.1 ensures that ^Ge is prime and that ^Gec=^G. In other words we are free to work with^Geinstead of ^Gand then contract the nal result to the original ring.

From the complexity bound stated in the beginning of this Section we could expect that computing is more ecient with^Ge than with ^G, and this is the case in the examples tried below. But there is one more advantage with ^Ge compared to ^G namely that ^Ge is zero- dimensional in^K(^y)^{x yn}] so that the BGK algorithm could be used!

Let us now compare the following four methods for elimination in the graph-ideal ^G: 1. Find the contraction of the ideal ^Ge to ^K(^y)^yn] using the BGK algorithm.

2. Compute a plex GB for^Gwith a term-ordering of the type ^{fx g} ^{> yn >::: > y0}. 3. Compute a plex GB for^Ge using a term-ordering of the type ^fxg ^{> yn}.

4. Compute a CS for ^Gwith a ranking of the type^fxg ^{> fu0u1:::g > fy} ^yng. A detailed, theoretical analysis of the complexity of these methods seems to be very dicult to perform. Instead the three dierent methods were compared on four examples, using Maple version V Release 2 running on a Sun Sparc 2 with 32 Mbytes of memory. The execution times are showed in table 1. The characteristic set package used was the one described in 25].

It is the experience of the author, supported by the computations above, that the dier- ence between GB and CS is not as large as sometimes claimed, when the localization trick is used in connection with GB. However GB:s have one advantage compared to CS, namely that there is no correspondence to the BGK algorithm for CS, since these always use a plex term-ordering.

The following are the four test examples used:

f = (^u;x1x2x21;2^x2)^{ h} = ^x1x2 (6)

f = (^ux2ux1)^{ h} = ^x2+^ux1 (7)

f = (^x1x2;ux3+ 3^ux21;x2)^{ h} = ^x1+^x2 (8)

f = (^;x2;ux21u)^{ h} = ^x21+^x2 (9) For the systems (6){(9) the sizes of the i/o-relation as a Maple object were 2 kbytes, 100 bytes, 8 kbytes and 2 kbytes, respectively.

4 The Degree of the Input-Output Relation

Even though a theoretical investigation of the computational complexity of the implicitization problem seems dicult, there is one thing we can do in order to get an estimate of the amount of work involved in nding the dependency relation in the graph ideal ^G. This is to derive an upper bound for the degree of an input-output relation in terms of the degrees of the right-hand sides in the state equations. In this Section we will derive such a bound, which is sharp. Our main tool in doing this will be the ane version of B#ezout's theorem, which reads as follows:

Lemma 4.1 (Ane version of B ezout's theorem)

A system of ⁿ equations ^f1 = 0^{ :::fn} = 0^{ fi 2 k}^X1:::Xn] with nitely many solutions has at most ^Qni=1 deg(^fi) solutions in

A

^{n. Here} deg refers to the total degree.

Proof.

See 11, page 223] (This proof is rather inaccessible for the non-expert, unfortunately.

But, surprisingly, no other reference is known to the author.) ² What we need is a bound for the degree of the relation between algebraically dependent polynomials. The following lemma will do the job.

Lemma 4.2

Consider ⁿ+ 1 polynomials ^{f0:::fn 2 k}^X1:::Xn] and write ^di = deg^fi. Suppose w.l.o.g. that ⁸ⁱ : ^{d0 di} and let ^p be a polynomial of lowest possible total degree such that the ^fi satisfy a relation ^p(^f0:::fn) ^ 0. Then

deg^{p Yn}

i=1 d

i

In other words: the degree of the relation is bounded by the product of theⁿhighest degrees of the polynomials.

Proof.

The original system is ^z0 = ^f0:::zn = ^fn but since we are interested in the degree of the dependency relation we make the system zero-dimensional by considering

z = ^{f0 ::: z} = ^fn (10)

for a single variable ^zinstead. A consequence of (10) is that

f

1

;f

0 = 0^{ f2;f0} = 0^{ ::: fn;f0} = 0 (11) This places us in a position where we may use theorem 4.1, for the ⁿ equations in (11) are still of degrees^di since ^{d0 di}. The ane version of B#ezout's theorem states that there are at most ^d1:::dn solutions to (11). This implies that elimination of the ^x in (10) renders a univariate polynomial in^z of degree less than or equal to ^d1:::dn. ²

Theorem 4.1

^Ifdi= deg(^Lifhx)then the degree of the input-output relation^pin the output variable ^y satises

deg^{p Yn}

i=1 d

i

Proof.

Apply lemma 4.2 to the system corresponding to ^G. ² Equality can occur in the bound of the theorem, as shown by the following example:

Example 4.1

As is proved in 9] any choice of algebraically independent ^h0:::hn;1 corresponds to a rational state space realization. If we choose

h

0 =^Yn

i=1 x

i

 h

1 =^{xd11  h2} =^{xd22  ::: hn}=^xdnn (12) then obviously ^h0:::hn;1 are algebraically independent, and a relation of lowest possible degree is provided by

h



0

; n

Y

i=1 h



i

i = 0 (13)

where^ = ^{d1:::dn i} = ^Qj6=idj. ²

Acknowledgement

This work was nancially supported by the Swedish Council for Technical Research (TFR).

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