Some Generic Results on Algebraic Observability and Connections with
Realization Theory
Conference Version (Accepted for ECC '93)
Krister Forsman
Department of Electrical Engineering Linkoping University
S-581 83 Linkoping Sweden
email: krister@isy.liu.se
1993-03-15
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-1458.ps.Z
Some Generic Results on Algebraic Observability and Connections with Realization Theory
Krister Forsman
Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden
email: krister@isy.liu.se
Abstract. We analyze Glad/Fliess algebraic observability for polynomial control systems from a commutative algebraic/algebro-geometric point of view, using some results from the theory of Grobner bases. Furthermore, we discuss some topics in realization theory for polynomial dierential equations. Most issues are treated in a constructive framework.
Keywords: nonlinear control systems, observability, state space realization, Grobner bases, elimination the- ory, commutative algebra, dierential algebra, genericity
1 Introduction
Algebraic observability has been introduced in a dierential algebraic setting by Glad, Fliess and Diop 5, 6, 13]. It was also discussed for discrete time systems, from a slightly dierent perspective, in 10]. It should be noted that the term has also been used by Sontag et al. in a dierent meaning 20]. The concept applies for nonlinear systems in which all nonlinearities are polynomial and, as explained in 6], it basically agrees with the standard denitions of observability given in e.g. 16]. Related work for the C1case can be found in 21].
It is the purpose of this paper to
give some additional aspects of algebraic observability in the light of Grobner base theory.
indicate how the observer relations can be used to obtain information on alternative state space realizations of the same input-output equation and inequations that may have to be added to the i/o-equation.
Theorems 3.4 and 3.3 seem to be new, as do the proofs of theorems 3.1 and 3.2. The methods suggested section 4 also appear to be new, though the results of that section are still rather preliminary. Some of the results are not new, merely restated in a language dierent from the one of dierential algebra. The author believes that dierential algebra is an appropriate framework for dealing with polynomial control systems, both from a theoretical and constructive point of view. However, according to the golden rule of science known as Occam's razor, one should never use more machinery than necessary to obtain desired results and in this spirit the reformulation might have an interest of its own.
We suppose that the reader is familiar with some basic concepts from commutative and dierential algebra, such as polynomial ring, ideal, prime ideal, height and dimension of an
ideal, quotient ring, eld of fractions, localization, transcendence degree and dierential eld.
Some references are 1, 14, 17, 19].
The following notation is used:
hF i is the ideal generated by the set F. We use the abbreviations
x = x1:::xn y = y0:::yn;1 (1) so that e.g. k(y) x] denotesk(y0:::yn;1) x1:::xn].
For the variable y we use subscript to denote time-derivative, as is done in e.g. 17]:
y
i = di
dt i
y(t) (2)
For a prime ideal p k X1:::Xn] the dimension of p is the transcendence degree of the eld of fractions of k X1:::Xn]=p over k. Some refer to this as the coheight of p. This number is equal to the length of the longest ascending chain of prime ideals starting withp 14].
As in the commutative algebra literature capital letters, e.g. XiYi, denote free variables whereas lowercase letters, e.g. xiyi, denote variables subject to relations. For example we could have that k x] = k X1:::Xn]=I for some idealI.
2 Elimination Theory
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] whereXY are sets of variables. One way of doing elimination is to compute a so called Grobner base, or standard base, of an ideal w.r.t. the purely lexicographic term-ordering (plex). For an introduction to GB see e.g. 3, 11]. The computer algebra programs Maple, Axiom, Reduce and Macsyma all have packages for computing GB:s.
The GB for an ideal is a unique nite subset of it, under some natural assumptions 11], so we writeGB(a) for the GB of a. The following two theorems on GB are of special interest in this paper. Recall that a polynomial in k X1:::Xn] is regular w.r.t. Xi if the leading term is in k Xi] for a plex term-ordering rankingXi highest.
Theorem 2.1
Let abe an ideal ink X1:::Xn]. IfG
=GB(a)w.r.t. plex thendima= 0 i for alli there is a p2G
such that p is regular w.r.t. Xi.Proof.
See 11] or 3, page 209]. 2Theorem 2.2
Let k be any eld of characteristic zero. A plex GB for a prime zero- dimensional ideal in k X1:::Xn] has nelements.Proof.
See 11]. 2This means that a plex-GB for a prime zero-dimensional ideal w.r.t. the variable rank- ingXn<:::<X1 looks as follows:
q ( ) (3)
where deg(piXi) < qi for all i. This is particularly interesting in the case when k is not a number eld and the ideal in question is prime, because in this case we can use lemma 3.2 below to adjoin variables to the coecient ring until the ideal becomes zero-dimensional. We can compare the GB to a Ritt characteristic set and easily see that the GB is actually a characteristic set after such simple manipulations.
We say that a property is genericon a space if it holds on a dense subset of the space in question.
Theorem 2.3
LetG
be a plex GB for a prime zero-dimensional ideal in k X1:::Xn]. Generically the n;1 rst elements ofG
are linear in the leading variable.Proof.
Proved in 11]. 2Here the term generic refers to the coecients of the polynomials that generate the ideal.
Thus the generic look of a plex-GB for a zero-dimensional prime ideal w.r.t. the variable ranking Xn<:::<X1 is
fX
1
;p
1
::: X
n;1
;p
n;1
p
n
g (4)
wherepi2k Xn] for all i, and degpn>degpi fori= 1:::n;1. An ideal with such a GB is said to be in generic position (g.p.)
Another formulation is: if a prime zero-dimensional ideal is in g.p. its GB provides a primitive element for the corresponding eld extension.
3 Algebraic Observability
Consider a polynomial system in state space form:
_
x(t) = f(x(t)) y(t) = h(x(t)) (5) wheref is a polynomial mappingRn!Rn and ha polynomial output map Rn!R i.e.
f
1
:::f
n
h2k X1:::Xn] (6) Thus there is no input signal so far. Most of the analysis below goes through in the case of an input signal as well, though just change the coecient eldk to includeuu _ :::u() for some large enough. Some of the additional diculties arising when inputs are added are treated in 6].
The Lie derivativew.r.t. f is the dierential operator Lf = Pfi@xi@ . One of our main objects of study in this paper will be the ideal
L
jfh = hy0;hy1;Lfh:::yj;Ljfhi (7) for dierent values of j. The generators of the ideal are thus some j polynomials Lifh, each one associated to a \tag-variable" yi. The technique of tag-variables is commonly used in constructive commutative algebra see e.g. 11, 18] or even 15]. In the current context, we of course think of thesubscript of y as denoting time-derivative: yi = dtdiiy(t).Let us now state some important properties of the ideal
L
jfh recalling the abbrevia- tions (1).Lemma 3.1
We have that (L
jfh)\k x] = 0whereL
jfh is dened by formula (7).Proof.
Write Ti =yi;Lifh. Suppose thats2k x]\L
jfh. By denitions = X
i
i T
i
i
2k x] (8)
If s 6= 0 then we x a value 2 kn of x such that s( ) 6= 0. Then, for each i, we choose
y
i =fi( ). Thus the rhs of (8) is zero, which is a contradiction. 2
Theorem 3.1 L
jfh is prime for all j.Proof.
We prove that A = k xy0:::yj]=(L
jfh) is an integral domain. Suppose thatpq= 0 inA. According to lemma 3.1 it is impossible that bothpand q are ink x]. Suppose
pcontains someyi. Then we use the relationTi = 0 to replaceyi with an expression in x. In this way we can eliminate all yi from p, and get a contradiction. 2 The above theorem can also be proved using Ritt's characterization of prime ideals via characteristic sets 17, page 89], as is done in 12].
Now we only need the following theorem in order to state some facts about algebraic observability:
Lemma 3.2
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.Proof.
See 1, proposition 3.11, iv]. 2In our applications R=k xy] andS =k A]nf0g for a subsetAfy0:::yng.
Lemma 3.3
We havedim(L
n;1f h) = n.Proof.
Since, for any prime ideal p ink xy] we havedimp+ htp = dimk xy] = 2n (9) where htp denotes the height of p it is necessary and sucient that ht(
L
n;1f h) =n. NowL
n;1f h :::L
1fhL
0fh 0 (10)is a strictly descending chain of prime ideals according to theorem 3.1. Thus ht(
L
n;1f h)n. The generalized Krull's Hauptidealsatz 14, theorem 13.5] states that the height of a prime ideal is less than or equal to the minimal number of generators of the ideal, so ht(L
n;1f h)nand we have equality. 2
Denition 3.1
The system (5) is algebraically observable if each xi is algebraic over thedierential eldkhyi. 2
This just means that the system is algebraically observable if for each xi there is an r such that there is a polynomial
] deg( ) 0 (11)
We call this ai, if it exists, an observer relation forxi. The lemma below serves to simplify the concept of algebraic observability a little. To prove it we need that
trdegL=k = trdegL=K+trdegK =k (12)
for a towerL K k of eld extensions 19, theorem 12.56].
Lemma 3.4
The system (5) is algebraically observable i each xi is algebraic over k(y).Proof.
It is easy to prove that the extension k(y0:::yr)=k(y) is algebraic for an arbitraryr 2 N. As a direct consequence of formula (12) the extension k(xiy0:::yr)=k(y0:::yr) is algebraic ik(xiy0:::yn;1)=k(y) is. 2 The following theorem was rst proved by Glad in 13]. The proof given here diers substantially from the original one.
Theorem 3.2
The system (5) is algebraically observable i(
L
n;1f h) \ k y] = 0Proof.
y0:::yn;1 are algebraically independent i (L
n;1f h) \ k y] = 0 18]. But the extensionk(y)k(x) is algebraic i y0:::yn;1 are algebraically independent. 2 Thus we have a simple test for algebraic observability: we just check if there is a relation betweeny0:::yn;1 using elimination theory.To be able to use the results on zero-dimensional GB:s we will now change the ground
eld using theorem 3.2. In other words we will make embeddings k x y] ,! k(y) x]. The extension of a prime ideal p k x y] to k(y) x] is pe = k(y) x]p. The extension pe is not always prime 1], but the ideals that are of interest to us do have prime extensions:
Theorem 3.3
If the system (5) is algebraically observable, then(L
n;1f h)ea zero-dimensional prime ideal in k(y) x]and (L
nfh)e is zero-dimensional prime in k(y) xyn].Proof.
According to theorem 3.1L
n;1f h is a prime ideal in k x y]. Theorem 3.2 assures thatL
n;1f h contracts to zero ink y], so lemma 3.2 implies that (L
n;1f h)eis prime in k(y) x].Lemma 3.3 implies that the dimension is zero, since we have decreased the dimension of the ring to n by adjoining variables to the ground eld. The situation is completely analogous
for
L
nfh and (L
nfh)e. 2Theorem 3.3 is not only of theoretical interest, it also has practical consequences if we wish to compute with
L
nfh: computations are more ecient when made in a ring of fewer variables (K xyn] compared tok y0:::ynx]). This observation was used by the author in the Maple package Polycon 9]. See also 8].It now follows from theorem 2.2 that for an algebraically observable system there is only one observer relation for each variable, in principle.
Theorem 3.4
Suppose that (5) is a generic algebraically observable system. Then, for alli,x
i is a rational function of y0:::yn.
Proof.
According to theorem 3.3 the result follows as we apply theorem 2.3 to (L
nfh)e. 2 So, even ifL
n;1f h always contains observer relations for all state variables if the system is algebraically observable, the chance of getting a relation linear inxi is greater if we compute inL
nfh instead.We may summarize this section in the words: if we are only interested in the observability of (5) we may study the non-dierential ideal
L
nfh instead of the dierential ideal_x1;f1:::x_n;fny;h] (13) It may seem that there is no reason to consider
L
n;1f h when we have access toL
nfh, but this is not true, as we will see in the next section.Note that not all of the analysis is trivial to repeat in the discrete time case.
4 The Inverse Problem: Recreating Dynamics
This section contains a discussion on how the observer relations described above can provide information on
alternative state space realizations of the same input-output equation and observer relations.
an inequation that may have to be added to the i/o-equation in order to get equivalence between solutions 4, 21].
More formally we wish to nd the rhs vector eld f, given the observability morphism
H : Cn ! Cn
H: x 7! (hLfh:::Ln;1f h) (14) and the external d.e. p(y) = 0. Also, we seek an inequation q(y) 6= 0 such that there is a one-to-one correspondence between solutions ofp= 0q6= 0 and those of _x=f.
The problem addressed is thus a particularly simple, but not trivial, instance of the state space realization problem.
The mathematical rigor of this section is not yet as satisfactory as that of the preceding one, but the results seem interesting enough to be mentioned in this context.
The most obvious approach to the problem is to dene a mapping
b
H : x 7! (Lfh:::Lnfh) (15) which is compatible withH andp. Then clearly
f = J(H);1Hb (16)
where J denotes jacobian matrix w.r.t. x. The matrix J(H) is identically singular i the system is not algebraically observable 7]. However, if the jacobian conjecture is true, detJ(H) 62 k i H is bijective 2]. So if the observability morphism is not one-to-one, detJ(H) = 0 denes a non-trivial variety. If there is a solution to the state equations that lies on this variety we cannot expect system equivalence. In other words, the inequation that has to be added is somehow related to det ( )= 0. Alternative realizations of may arise
if p is nonlinear in yn, since Hb is not uniquely determined in this case. The main diculty with this approach is that it might be hard to determine all possible, or even one, Hb. No algorithm is known to the author.
An algebraic formulation of the problem could be: determine the contraction of the dierential ideal A =
L
n;1f hp] to the ringRi =k _xix] for each i. This is not interesting in case the elimination only renders the equations _xi=fi. It becomes more interesting when the ideal considered is non-prime, for then we may get several suggestions for state equations, one for each factor of the generator of the contraction. We may also get factors in xonly that give conditions on the states. Unfortunately, non-prime dierential ideals are quite dicult to handle constructively: Ritt's suggestions for calculating characteristic sets in this case seem hard to implement 17].Thus it is tempting to truncate the dierential ideal again and then use Grobner bases, which are applicable to all types of ideals. Let us rst describe this method in general terms and then consider some examples. Suppose that the system considered is algebraically observable.
1. Find the d.e. p= 0 fory, for example by elimination in (
L
nfh)e. 2. Find an observer relationai(xiy) = 0 for xi in (L
n;1f h)e.3. Dierentiate ai w.r.t. time, to get dai 2 k yynxiFi] where Fi represents the time derivative of xi.
4. Repeat steps 2 and 3 for each iin order to geta1:::anda1:::dan 5. Compute Grobner bases for the ideal
a = (
L
n;1f h) + hpda1:::dani (17) ink yynx F1:::Fn] w.r.t. rankings such that theyi and all Fi except one are elim- inated. This gives one or several expressions forFi in terms of x.The above procedure gives part of the contraction A \Ri for each i, but the reader should keep in mind that it has not yet been proved that a is a proper truncation ofA.
It is easy to see that for rst order systems the procedures described above give the same result, ifHb can be determined. For higher order systems they are not equivalent, though, as shown by example 4.2.
Example 4.1
(Compare 4, example 2].) Consider the system_
x = x y = x2+x (18)
Elimination in the ideal
L
1f(x2+x) givesp = y0+ 4y20;(4y0+1)y1+y21 (19) So p= 0 is a dierential equation for y but, as noted in 4], whereasy(t) ;1=4 is a zero of
p there is no corresponding solution to (18). Thus we have to add the inequation y6 ;1=4 to get equivalence between solutions. In 4] an algorithm due to Seidenberg is used to keep track of inequations. The problem is also discussed in 21].
Now, the method described above will work as follows in this example: the observer relation for x in
L
n;1f h is simply the generator y0;x2;x. Dierentiate it and add its derivative and p toL
0fh to get ideal a. The GB for a w.r.t. the ranking y1 > y0 > F > x contains the polynomial(1 +2x)2(x+ 1;F)(x;F) (20) This shows that a is not prime, and each of the factors tells us something: we have that
x=;1=2 or _x=x or _x=x+ 1
The interpretation is clear: if x(t) 6 ;1=2 (i.e. y(t) 6 ;1=4) and y = x2 +x then the equation p= 0 can be realized either as _x=xor as _x=x+1. 2
Example 4.2
Consider the system _x
1 = x2 x_2 = 1 y = x21 (21)
Proceeding as above we nd that e.g.
a
1 = y0;x21 a2 = 4x22y0;y12 (22) etc. The GB fora w.r.t. the rankingy2>y1>y0>F2 >F1>x2>x1 contains 6 elements, among which are
x 6
1 x
2(F2;1)(F2+1) (23)
and x1(F1;x2). This suggests that the inequation to add isx1x2 6= 0, but detJ(H) = 4x21. Comparing this with the d.e. for y (which we do not display for lack of space) we nd that
the proper inequation isx16= 0. 2
Acknowledgements
This work was nancially supported by the Swedish Council for Technical Research (TFR).
The author wishes to thank an anonymous referee for helpful suggestions.
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