Observability at an initial state for polynomial systems

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Observability at an initial state for polynomial systems✩ Yu Kawano 1 , Toshiyuki Ohtsuka Graduate School of Engineering Science, Osaka University, Machikaneyama 1-3, Toyonaka, Osaka, 560-8531, Japan

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Article history: Received 6 July 2011 Received in revised form 16 October 2012 Accepted 1 November 2012 Available online xxxx Keywords: Nonlinear systems Polynomial models Observability Algebraic approaches

abstract We consider observability at an initial state for polynomial systems. When testing for local observability for nonlinear systems, the observability rank condition based on the inverse function theorem is commonly used. However, the rank condition is a sufficient condition, and we cannot test for global observability using the rank condition. In this paper, first we derive necessary and sufficient conditions for global observability at an initial state for continuous-time polynomial systems. Then, necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring. Using the same procedure, we derive both global and local observability conditions for discrete-time polynomial systems. Each condition is characterized by a finite set of equations, since polynomial rings are Noetherian. Finally, examples demonstrate the proposed criteria for testing for observability. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we consider global and local observability at an initial state for continuous-time and discrete-time polynomial systems. Observability is one of the most fundamental properties, along with controllability and stability, in modern control theory. If sensors are arranged such that a system is observable, in principle, we may uniquely determine the initial states of the system without detecting them. Observability indicates how to arrange sensors to determine states with a smaller number of sensors than the number of states, and plays an important role in control theory. Thus, observability is an important criterion when implementing sensors for physical systems. For nonlinear systems, there are various definitions of observability. Most of observability are defined in terms of the distinguishability of a pair of initial states. Definitions of observability of nonlinear systems can be broadly classified into three points of view (Hermann & Krener, 1977; Isidori, 1995; Nijmeijer, 1982; Nijmeijer & van der Schaft, 1991; Sontag, 1990). First, there are definitions of distinguishability depending on the properties of inputs. Second, using one of these definitions of distinguishability,

✩ This work has been partly supported by a Grant-in-Aid for Exploratory Research (No. 24656263) from the Japan Society for the Promotion of Science. The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milan, Italy. This paper was recommended for publication in revised form by Associate Editor Henri Huijberts under the direction of Editor Andrew R. Teel. E-mail addresses: [email protected] (Y. Kawano), [email protected] (T. Ohtsuka). 1 Tel.: +81 06 6850 6358; fax: +81 06 6850 6341.

0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2013.01.020

global and local observability have been defined. Third, two types of global and local observability have been defined. One is called global or local observability at an initial state. The other is simply called global or local observability and means that a system is globally or locally observable for all initial states. In addition, there are some slightly different definitions of global and local observability (Hermann & Krener, 1977; Nijmeijer, 1982). In this paper, distinguishability depending on input is defined on the basis of multiple-experiment observability (Sontag, 1979). The definitions of observability and weak observability (Hermann & Krener, 1977) are adopted as the definitions of global and local observability, respectively. Note that, the definition of weak observability is sometimes adopted as the definition of local observability (Nijmeijer, 1982). In this paper, we focus on the global/local observability at an initial state. Although, algebraic observability (Forsman, 1993) has been introduced in a differential algebraic setting, we adopt a definition of observability slightly different. Local observability is worth researching in its own right. Global observability is a too strong property when we control nonlinear systems in practice. Sometimes, local observability for possible initial states is sufficient to control dynamical systems even if they are not globally observable. When testing for local observability at an initial state, the observability rank condition (Nijmeijer, 1982; Nijmeijer & van der Schaft, 1991) based on the inverse function theorem is well known. However, this rank condition is a sufficient condition for local observability, and to determine whether this condition is satisfied or not, we may need to compute output sequences as functions of the initial state over many time steps. Moreover, we cannot verify global observability at an initial state using the rank condition.

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Y. Kawano, T. Ohtsuka / Automatica (

In this paper, we restrict our considerations to polynomial systems. Polynomial systems are one of the important subclasses of nonlinear systems. Various types of real system can be represented as polynomial systems, and by exploiting the Taylor series expansion, most of the nonlinear functions can be approximated by polynomial functions. Furthermore, we can convert exactly a wide range of nonlinearities into polynomials by augmenting a state (Ohtsuka, 2005, 2008). It is worth mentioning that, even for polynomial systems, the observability rank condition is still only sufficient for the local observability. Thus, there is no universal criterion for the local observability of polynomial systems. For the global observability of polynomial systems on the basis of commutative algebra, it has been shown that a finite set of equations characterizes the observability (Ne˘sić, 1998; Sontag, 1979; Tibken, 2004). In particular, Tibken (2004) gives necessary and sufficient conditions for global observability that are described only in terms of ideals and corresponding varieties for continuous-time autonomous polynomial systems. In this paper, employing the same procedure as that used for global observability, we derive global observability conditions at an initial state. Moreover, using the localization of a polynomial ring, we derive local observability conditions at an initial state. Whereas the observability rank condition guarantees only sufficiency, our condition also guarantees necessity. The obtained conditions are characterized by a finite set of equations, as in the case for the global observability condition for all initial states. Conditions for both continuous-time and discrete-time polynomial systems are obtained with similar forms. The original feature of this paper is that, we exploit the localization of a ring to obtain a necessary and sufficient condition for the local observability of polynomial systems. The remainder of this paper is organized as follows. In Section 2, we derive observability conditions at an initial state for a continuous-time polynomial system. First, the distinguishability and global/local observability are defined. Then, the set of initial states that cannot be distinguished from the initial state is characterized by a finite set of equations based on Hilbert’s basis theorem (Atiyah & Macdonald, 1969; Cox, Little, & O’Shea, 1992, 2005; Kunz, 1984) to test for global and local observability. Using the set of indistinguishable initial states, we derive necessary and sufficient conditions for global observability at the initial state for continuous-time polynomial systems. Then, we derive a necessary and sufficient condition for local observability at the initial state. The local observability condition is derived from the condition for global observability at the initial state using the localization of a polynomial ring. In Section 3, for discrete-time polynomial systems, distinguishability is defined, and global and local observability conditions are obtained. Concluding remarks are given in Section 4, and some lemmas are proved in the Appendices. Notation 1.1. Throughout the paper, R and N denote the field of real numbers and the set of non-negative integers, respectively. A commutative ring with an identity is denoted by R, and the polynomial ring over R with variables xi (i = 1, 2, . . . , n) is denoted by R[x](:= R[x1 , . . . , xn ]). Polynomial rings over R with variables ξi (i = 1, 2, . . . , n) and ξi , ηi (i = 1, 2, . . . , n) are denoted by R[ξ ](:= R[ξ1 , . . . , ξn ]) and R[ξ , η](:= R[ξ1 , . . . , ξn , η1 , . . . , ηn ]), respectively. The ideal generated by a1 , a2 , . . . , as ∈ R[ξ , η] is denoted as ⟨a1 , . . . , as ⟩ ⊂ R[ξ , η]. The sets V(I ) ⊂ Rn and V(J ) ⊂ Rn × Rn represent affine algebraic varieties, which are the set of common zeros of all elements of the ideals I ⊂ R[ξ ] and J ⊂ R[ξ , η], respectively. Refer to Atiyah and Macdonald (1969), Cox et al. (1992, 2005) and Kunz (1984) for technical terms in commutative algebra and algebraic geometry.

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2. Observability at an initial state for continuous-time polynomial systems 2.1. Indistinguishable initial states Distinguishability is one of the most important concepts in nonlinear control system theory. For continuous-time and discrete-time nonlinear systems, most of the observability are defined in terms of distinguishability. Here, the set of initial states indistinguishable from an initial state is characterized for continuous-time polynomial systems on the basis of Hilbert’s basis theorem (Atiyah & Macdonald, 1969; Cox et al., 1992, 2005; Kunz, 1984). Let us consider a continuous-time polynomial system. The state-space representation of the system is described by

ΣC

x˙ (t ) = f (x(t ), u(t )), x(0) = x0 , y(t ) = h(x(t ), u(t )),



where x ∈ Rn , u ∈ Rm and y ∈ Rq denote the state, input and output vectors of the system ΣC , respectively. The elements of f and h are polynomial functions of x and u. Since the output of the system ΣC depends on time t, the initial state x0 and input u, we denote the output at time t as y(t ; x0 , u). The distinguishability of continuous-time nonlinear systems and the global and local observability of nonlinear systems are defined as follows (Hermann & Krener, 1977; Isidori, 1995; Nijmeijer & van der Schaft, 1991; Sontag, 1979). Definition 2.1. A pair of initial states (ξ , η) ∈ Rn × Rn of a system ΣC is distinguishable if there exist an admissible piecewise constant input u and a time t ≥ 0 that yield y(t ; ξ , u) ̸= y(t ; η, u). Definition 2.2. A system is globally observable at an initial state σ ∈ Rn if for all ξ ∈ Rn \ {σ }, pairs of (σ , ξ ) are distinguishable. Definition 2.3. A system is locally observable at an initial state σ ∈ Rn if there exists a neighborhood of σ denoted as U (σ ) ⊂ Rn such that for all ξ ∈ U (σ ) \ {σ }, pairs of (σ , ξ ) are distinguishable. Sontag (1979) gives some definitions of observability depending on inputs for discrete-time systems. The definitions can be readily extended to continuous-time systems. Definition 2.1 is based on the definition of multiple-experiment observability: each pair of states can be distinguished by some input sequence that may depend on the initial states (Sontag, 1979). In this paper, the definitions of observability and weak observability (Hermann & Krener, 1977) are adopted as the definitions of global and local observability, respectively. Note that, the definition of weak observability is sometimes adopted as the definition of local observability (Nijmeijer, 1982). Furthermore, there are some slightly different definitions of global and local observability (Hermann & Krener, 1977; Nijmeijer, 1982). Remark 2.1. Multiple-experiment observability does not imply that the same input distinguishes all pairs of initial states, i.e., single-experiment observability (Sontag, 1979). For the polynomial system ΣC , f and hi (i = 1, 2, . . . , q) are represented as f (x, u) =

r 

gk (x)qk (u),

k=0

h i ( x , u) =

si 

pi,j (x)ˆqi,j (u),

j=0

where gk ∈ (R[x])n (k = 0, 1, . . . , r ), pi,j ∈ R[x] (i = 1, 2, . . . , q; j = 0, 1, . . . , si ), and qk (k = 0, . . . , r ) and qˆ i,j (i = 1, . . . , q; j =

Y. Kawano, T. Ohtsuka / Automatica (

0, . . . , si ) are monomial functions of elements of u and distinct from each other. All positive integers r and si (i = 1, . . . , q) are uniquely defined for the given f and h of the system ΣC . The distinguishability of the system ΣC is characterized by the Lie derivatives of output functions. Lemma 2.1. A pair of initial states (ξ , η) ∈ Rn × Rn of the system ΣC cannot be distinguished in terms of outputs if and only if, for all (τ1 , τ2 , . . . , τℓ ) ∈ {g0 , g1 , . . . , gr }ℓ (ℓ = 0, 1, . . .), Lτ1 Lτ2 · · · Lτℓ pi,j (ξ ) = Lτ1 Lτ2 · · · Lτℓ pi,j (η),

(i = 1, . . . , q; j = 0, . . . , si )

(1)

hold, where Lτ p(x) := (∂ p/∂ x)τ . Note that, in Lemma 2.1, Lτ p(x) ∈ R[x] holds if τ is an element of (R[x]n ) and p is an element of R[x]. Lemma 2.1 can be proved using the same procedure as that for an input-affine analytic system (Sontag, 1990). The following theorem (Hermann & Krener, 1977; Isidori, 1995; Nijmeijer & van der Schaft, 1991) based on the inverse function theorem is commonly used when testing for the local observability of continuous-time nonlinear systems. Proposition 2.1 (Observability Rank Condition). The system ΣC is locally observable at an initial state σ ∈ Rn if dim dO(σ ) = n, dO(x) = spanR {dH (x) : H ∈ O}, O(x) = {Lτ1 · · · Lτℓ pi,j (x) : (τ1 , . . . , τℓ ) ∈ {g0 , . . . , gr }ℓ , (ℓ = 0, 1 · · · ; i = 1, . . . , q; j = 0, . . . , si )}

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From (3) and (5), we define an ideal J1 ⊂ R[ξ , η] as J1 = J0 +

q 

J1,i ,

i=1

J1,i = ⟨Lg0 pi,0 (ξ ) − Lg0 pi,0 (η), Lg0 pi,1 (ξ ) − Lg0 pi,1 (η), . . . , Lgr pi,si (ξ ) − Lgr pi,si (η)⟩.

(6)

The algebraic variety V(J1 ) ⊂ Rn × Rn denotes the set of (ξ , η) satisfying (3) and (5). From condition (1), if a pair of initial states (ξ , η) ∈ Rn × Rn cannot be distinguished in terms of the output, then Lgl Lgk pi,j (ξ ) = Lgl Lgk pi,j (η)

(i = 1, . . . , q; j = 0, . . . , si ; k, l = 0, . . . , r ). From (3), (5) and (7), we define an ideal J2 ⊂ R[ξ , η] as q  J2 = J1 + J2,i ,

(7)

i=1

J2,i = ⟨Lg0 Lg0 pi,0 (ξ ) − Lg0 Lg0 pi,0 (η), Lg0 Lg0 pi,1 (ξ ) − Lg0 Lg0 pi,1 (η), . . . , Lgr Lgr pi,si (ξ ) − Lgr Lgr pi,si (η)⟩.

(8)

J 0 ⊂ J 1 ⊂ · · · ⊂ J N = J N +1 = · · · = J

(9)

The algebraic variety V(J2 ) ⊂ Rn × Rn denotes the set of (ξ , η) satisfying (3), (5) and (7). From (4), (6) and (8), ideals satisfy the inclusion property J0 ⊂ J1 ⊂ J2 . Continuing the same procedure, we obtain ideals Ji ⊂ R[ξ , η] (i = 0, 1, 2, . . .) and an ascending chain of ideals J0 ⊂ J1 ⊂ · · ·. According to Hilbert’s basis theorem (Atiyah & Macdonald, 1969; Cox et al., 1992, 2005; Kunz, 1984), the chain will stabilize. In other words, there exists a certain ideal J ⊂ R[ξ , η] that satisfies

(2)

for a certain natural number N. In particular, the chain (9) has the following strict inclusion property:

This is an extension of the observability condition for linear systems. On the basis of this rank condition, the decomposition and canonical form of input-affine nonlinear systems are studied (Hermann & Krener, 1977; Isidori, 1995; Nijmeijer & van der Schaft, 1991). However, it is only a sufficient condition for local observability, and we cannot necessarily verify global observability using this rank condition. In this paper, we derive necessary and sufficient conditions for global and local observability on the basis of Lemma 2.1. To test for observability at an initial state σ ∈ Rn of the polynomial system ΣC , the set of initial states satisfying condition (1) is needed. However, it is difficult to determine how many Lie derivatives are sufficient to test for observability. Here, we characterize the set of initial states indistinguishable from σ ∈ Rn for the system ΣC using a finite set of equations on the basis of Hilbert’s basis theorem. The finite set can be obtained uniquely and explicitly for the system ΣC . From condition (1), if a pair of initial states (ξ , η) ∈ Rn × Rn cannot be distinguished in terms of the output, then

as stated in Lemma A.1. Let ϕσ : R[ξ , η] → R[ξ ] be a mapping that substitutes η = σ ∈ Rn for an element of R[ξ , η], which is a surjective ring homomorphism over R[ξ , η]. Suppose the ideal J ⊂ R[ξ ] denotes the image ϕσ (J ) of J in (10). The ideal J characterizes the set of initial states that are indistinguishable from σ , i.e., V(J ) ⊂ Rn represents the set of indistinguishable initial states. Since a finite set of polynomials generate the ideal J , V(J ) consists of the common zeros of a finite set of polynomials. Thus, the set of initial states that cannot be distinguished from σ ∈ Rn is obtained as the common zeros of a finite set of equations for continuous-time polynomial systems.

holds.

pi,j (ξ ) = pi,j (η)

J 0 ( J 1 ( · · · ( J N = J N +1 = J ,

(i = 1, . . . , q; j = 0, . . . , si ).

(3)

From (3), we define an ideal J0 ⊂ R[ξ , η] as J0 = ⟨p1,0 (ξ ) − p1,0 (η), p1,1 (ξ )

− p1,1 (η), . . . , pq,sq (ξ ) − pq,sq (η)⟩.

(4)

The algebraic variety V(J0 ) ⊂ Rn × Rn denotes the set of (ξ , η) satisfying (3). From condition (1), if a pair of initial states (ξ , η) ∈ Rn × Rn cannot be distinguished in terms of the output, then Lgk pi,j (ξ ) = Lgk pi,j (η)

(i = 1, . . . , q; j = 0, . . . , si ; k = 0, . . . , r ).

(5)

(10)

Remark 2.2. To obtain the ideal J in (10), the Gröbner basis of ideals plays an important role. We can determine whether two ideals are equivalent using the Gröbner basis. The Gröbner basis is uniquely determined if we give a monomial ordering for the indeterminate elements of the polynomial ring R[ξ , η], for example, ξ1 ≺ · · · ≺ ξn ≺ η1 ≺ · · · ≺ ηn . Remark 2.3. Depending on the context, (ξ , η) is considered as an element of (R[ξ , η])2n or Rn × Rn . In (4), (6) and (8), (ξ , η) is considered as an element of (R[ξ , η])2n . In (1), (3), (5) and (7), (ξ , η) is considered as an element of Rn × Rn . Remark 2.4. Since the polynomial ring R[ξ ] is also Noetherian (Atiyah & Macdonald, 1969; Cox et al., 2005; Kunz, 1984), the ideals of R[ξ ] satisfy an ascending chain condition. To obtain an ideal characterizing the initial states that are indistinguishable from σ , in fact, we can generate ideals that are subsets of R[ξ ]. However, once we generate ideals of R[ξ , η], we do not need to regenerate ideals even if we test for global observability at another initial state σ0 ∈ Rn . We can obtain the set of initial states indistinguishable from σ0 by only substituting σ0 ∈ Rn into η ∈ (R[ξ , η])n .

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We characterized the set of initial states that are indistinguishable from an initial state by an ideal of the polynomial ring and its algebraic variety for continuous-time polynomial systems. Here, by using these ideal and algebraic variety, we derive global observability conditions at an initial state for such a system. To test for global observability at σ ∈ Rn for the system ΣC , we define an ideal I ⊂ R[ξ ] as (11)

Then, V(I) ⊂ Rn represents {σ }. We now obtain the following theorem. Theorem 2.1. The polynomial system ΣC is globally observable at an initial state σ ∈ Rn if and only if one of the following equivalent conditions, (i) V(√ J ) = V(I), (ii) V( J : I) ⊂ V(I), holds. Proof. First, we show (i) is a necessary and sufficient condition for the global observability at σ ∈ Rn . V(J ) denotes the set of initial states that are indistinguishable from σ , and V(I) = {σ }. Thus, from Definition 2.2, (i) is a necessary and sufficient condition for global observability at σ . Next, we show that condition (i) is equivalent to condition (ii). We show that condition (i) implies condition (ii). The following properties, V( J : I ) ⊂ V( J ),

(12)

V( J ) = V(J ),

(13)

  V( J ) \ V(I ) ⊂ V( J : I )

(14)







hold for any ideals I and J ⊂ R[ξ ] (Cox et al., 1992, 2005). From (12) and (13), we obtain V( J : I) ⊂ V( J ) = V(J ).



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√

2.2. Global observability at an initial state

I = ⟨ξ1 − σ1 , ξ2 − σ2 , . . . , ξn − σn ⟩.

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system. More specifically, the algebraic variety V( J : I) is the smallest variety that contains the set V(J )\ V(I). That is, condition (ii) of Theorem 2.1 does not consider most of the set satisfying condition (i) of Theorem 2.1 out of variety V(J ). Theorem 2.1 leads to the following simple sufficient conditions. Corollary 2.1. The polynomial system ΣC is globally observable at σ ∈ Rn if one of the following conditions,

√

(i) V( J : I) = ∅, (ii) J = I, √ (iii) J : I = R[ξ ], holds. If condition (ii) or (iii) of Corollary √2.1 is satisfied, then we need not compute the variety V(J ) or V( J : I ), because condition (i) or (ii) of Theorem 2.1 is trivially satisfied. Conditions (ii) and (iii) of Corollary 2.1 are easily checked using the Gro¨ bner basis. In condition (ii) of Theorem 2.1 and condition (i) of Corollary 2.1, we consider not the field of complex numbers but the field of real numbers. Since we do not assume the field to be algebraically closed, i.e., the field of complex numbers in this case, condition (ii) of Theorem 2.1 and condition (i) of Corollary 2.1 are more general than the similar condition for global observability for all initial states (Tibken, 2004). 2.3. Local observability at an initial state On the basis of Theorem 2.1, we derive local observability conditions at an initial state for continuous-time systems. To this end, the properties of a localization of the polynomial ring (Atiyah & Macdonald, 1969; Cox et al., 2005; Kunz, 1984) play important roles. Suppose that, p ⊂ R[x] is a prime ideal. Then a localization R[x]p of R[x] at p is defined as R[x]p = {a/b : a, b ∈ R[x], b ̸∈ p},



and for an ideal I ⊂ R[x], an ideal Ip ⊂ R[x]p is defined as

Therefore, condition (ii) holds if (i) holds. Then, we show that condition (ii) implies condition (i). From (13) and (14), we obtain V(J ) \ V(I) = V( J ) \ V(I) ⊂ V( J : I).





Thus, the following equation holds if (ii) holds.

Ip = {a/b : a ∈ I , b ∈ R[x], b ̸∈ p}. Suppose that I is an ideal of R[x] and p is a prime ideal of R[x], then Ip ∩ R[x] is also an ideal of R[x]. Then, we define an affine algebraic variety V(Ip ) ⊂ V(I ) ⊂ Rn corresponding to the ideal Ip ⊂ R[x]p as V(Ip ) := V(Ip ∩ R[x]).

V(J ) \ V(I) ⊂ V(I).

We show some properties of the ideal I defined by (11).

That is, V(J ) ⊂ V(I)

(15)

holds if (ii) holds. Eq. (15) is equivalent to condition (i) because V(J ) ⊃ V(I) always holds.  To verify condition (i) or (ii) of Theorem 2.1, it suffices to verify n that every common √ zero ξ ∈ R of a finite set of generators of the ideal J or J : I belongs to V(I) = {σ }. Thus, the global observability of the system ΣC is tested by checking the common zeros of a finite set of equations that are explicitly given √ by generators of the ideal J or J : I. In contrast, using the condition of Lemma 2.1 to directly derive a set of initial states indistinguishable from a given initial state, it is difficult to determine how many Lie derivatives are sufficient. Thus, it is difficult, without using Hilbert’s basis theorem, to find the set of initial states indistinguishable from a given initial state. The difference between conditions (i) and (ii) of Theorem 2.1 is that condition (ii) considers a smaller variety than (i) for the same

Lemma 2.2. The ideal I is a maximal ideal of R[x]. Since we can choose a basis of the ideal I as {ξ1 − σ1 , . . . , ξn −

σn }, Lemma 2.2 is trivial, and I(V(I)) = I

(16)

holds. Since a maximal ideal is prime, we can localize a polynomial ring R[ξ ] by I. From Lemma B.7, II ∩ R[ξ ] = I holds. Thus, we obtain V(II ) = V(I).

(17)

From the definition of a local ring R[ξ ]p , if I ⊂ J, then Ip ⊂ Jp holds for ideals I ⊂ R[ξ ] and J ⊂ R[ξ ]. However, the converse is not necessarily true. The inclusion property (10) holds even after the ring is localized. The ideal JI characterizes the initial states that are indistinguishable from σ ∈ Rn in any neighborhood of σ . Now, we obtain one of the main results of this paper.

Y. Kawano, T. Ohtsuka / Automatica (

Theorem 2.2. The polynomial system ΣC is locally observable at an initial state σ ∈ Rn if and only if V(JI ) = V(I)

(18)

holds. Proof. Suppose the minimal decomposition of the variety V(J ) is X0 ∪ · · · ∪ Xt , where Xi satisfies V(I) ⊂ Xi (i = 0, 1, . . . , s) and V(I) ̸⊂ Xi (i = s+1, s+2, . . . , t ) for some s ≤ t. Then, Lemma B.12 implies V(JI ) = X0 ∪ · · · ∪ Xs .

(19)

Since V(J ) ⊃ V(I) always holds, and I satisfies (16), the proof of Lemma B.13 implies that there exists X0 such that X0 ⊃ V(I), then (19) implies that V(JI ) ⊃ V(I) always holds. Thus, we need to show that V(JI ) ⊂ V(I) is necessary and sufficient for local observability. (Necessity) From Definition 2.3, the system ΣC is locally observable at the initial state σ if and only if there exists a neighborhood of σ such that σ is indistinguishable only from σ . That is, σ is an isolated point of V(J ). Since {σ } = V(I) is an irreducible variety, the irreducible variety X0 ⊃ V(I) in (19) must be identical to V(I). That is, V(I) = X0 and V(I) ̸⊂ Xi (i = 1, . . . , t ) are satisfied. Thus, V(JI ) = X0 = V(I). (Sufficiency) We prove this by contraposition. If system ΣC is not locally observable at σ , that is, σ is not an isolated point of V(J ), then σ is also not an isolated point of Xi (i = 0, . . . , s), and consequently Xi ) {σ } = V(I) (i = 0, . . . , s) is satisfied. Thus, (19) implies

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Theorem 2.2 is derived from Theorem 2.1 by localizing the polynomial ring R[ξ ] at I. To check condition (18), it suffices to check whether V(JI ), which is the common zeros of a finite set of polynomials, equals {σ }. To test for local observability at an initial state for nonlinear systems, the observability rank condition (2) is commonly used. However, it is a sufficient condition. Condition (2) guarantees necessity if an input-affine continuous-time analytic system is accessible (Hermann & Krener, 1977; Nijmeijer & van der Schaft, 1991). In contrast, the obtained condition in Theorem 2.2 guarantees necessity in addition to sufficiency for all systems described by ΣC . The original feature of this paper is that, we exploit the localization of a ring to obtain a necessary and sufficient condition for the local observability of the polynomial system ΣC . In a similar manner to condition (ii) of Theorem 2.1, another local observability condition is derived from condition (18). Theorem 2.3. The polynomial system ΣC is locally observable at an initial state σ ∈ Rn if and only if V(( J : I)I ) ⊂ V(I)



(20)

holds. Proof. We show that, condition (18) is equivalent to condition (20). First, we show that condition (18) implies condition (20). For any ideals I , J ⊂ R[ξ ] and any prime ideal p ⊂ R[ξ ], the following property holds (Atiyah & Macdonald, 1969).

(I : J )p = Ip : Jp .

(21)

From (21), (12), (B.1) and (B.18), we obtain V(( J : I)I ) = V( J I : II ) ⊂ V( J I ) = V(JI ).







V(JI ) = X0 ∪ · · · ∪ Xs ) V(I).

Therefore, condition (20) holds if (18) holds. Then we show that, condition (20) implies condition (18). From (B.18) and (17), we obtain

This completes the proof.

V(JI ) \ V(I) = V( J I ) \ V(I) = V( J I ) \ V(II ).



Remark 2.5. We can compute the ideal JI . The minimal primary decomposition of an ideal of a polynomial ring can be computed. Using the Gröbner basis, we can check weather or not an ideal contains other ideals. Thus, Lemma B.8 implies that JI is easily computed from J using the minimal primary decomposition and Gröbner basis. Remark 2.6. We can prove Theorem 2.2 from Lemma B.11. If a field is algebraically closed, i.e., the field of complex numbers in this case, we √ can prove Theorem 2.2 without Lemma B.11 by using the property I = I(V(I )) (Cox et al., 1992, 2005; Kunz, 1984). The lemma is complicated. However, owing to Lemma B.11, we do not need to assume that the field is algebraically closed. Thus, we can derive the local observability condition for more general fields, i.e., the field of real numbers in this case. Remark 2.7. Since the localization Rp [ξ ] of polynomial ring R[ξ ] is also Noetherian (Atiyah & Macdonald, 1969; Cox et al., 2005; Kunz, 1984), the ideals of Rp [ξ ] satisfy an ascending chain condition. In the case that we only check the local observability at σ ∈ Rn , we do not need to generate ideals of R[ξ , η]. We can generate ideals of the local ring Rp [ξ ] that characterize the initial states indistinguishable from σ ∈ Rn in its neighborhood. Remark 2.8. The local observability condition in Theorem 2.2 can be extended to suitable classes of rational systems because the ascending chain condition of a Noetherian ring can be exploited. However, it is not trivial to extend the result of global observability to rational systems because the ascending chain condition may not be exploited.





From (14), (21) and (B.1), we also obtain V( J I ) \ V(II ) ⊂ V(( J I ∩ R[ξ ]) : (II ∩ R[ξ ]))





 = V(( J : I)I ). Thus, V(JI ) \ V(I) ⊂ V(I) holds if (20) holds. That is, V(JI ) ⊂ V(I)

(22)

holds if (20) holds. Eq. (22) is equivalent to condition (18) because V(JI ) ⊃ V(I) always holds.  Theorems 2.2 and 2.3 lead to the following simple sufficient conditions. Corollary 2.2. The polynomial system ΣC is locally observable at σ ∈ Rn if one of the following conditions, (i) (ii) (iii)

√

V(( J : I)I ) = ∅, J√ I = II , ( J : I)I = R[ξ ]I ,

holds. The difference between conditions (18) and (20) is that condition (20) considers a smaller variety than (18) for the same system. Condition (18) or (20) is trivially satisfied if condition (ii) or (iii) of Corollary 2.2 is satisfied, and conditions (ii) and (iii) of Corollary 2.2 are easily checked. Compared to the global observability conditions, the local observability conditions are obtained, from the global observability conditions, by only localizing the polynomial ring R[ξ ] at σ .

6

Y. Kawano, T. Ohtsuka / Automatica (

Example 2.1. We test the global and local observability at an initial state of the following continuous-time polynomial system.

 −x1 (t ) x˙ (t ) =  x1 (t )x3 (t )  , x1 (t ) + x22 (t ) 

y(t ) = 2x2 (t ). Since it is easily checked that the observability rank condition for continuous-time systems (2) does not hold at the origin, the observability at the origin of the system cannot be determined from the conventional result. That is, we cannot determine that the system is not locally observable at the origin simply because the observability rank condition, i.e., the sufficient condition, does not hold. The observability of the system is thus tested using the main results of this work. First, we test for global observability at the origin. We generate an ideal I ⊂ R[ξ ](:= R[ξ1 , ξ2 , ξ3 ]) as

)

–

Next, we give counter examples to the necessity of Corollaries 2.1 and 2.2. That is, we demonstrate that Corollaries 2.1 and 2.2 are only sufficient for the global and local observability, respectively. Example 2.2. This example presents a globally observable polynomial system not satisfying conditions (ii) and (iii) of Corollary 2.1. Let us consider the following one-dimensional polynomial system: x˙ (t ) = 0, y(t ) = x(t )(1 + x2 (t )). We test the global observability at the origin. The ideals J ⊂ R[ξ ] and I ⊂ R[ξ ] are generated as

J = ⟨ξ (1 + ξ 2 )⟩, I = ⟨ξ ⟩.

I = ⟨ξ1 , ξ2 , ξ3 ⟩,

To check condition (i) of Corollary 2.1, we compute the ideal I ⊂ R[ξ ]. This ideal is obtained as

and the following ideals, which are subsets of R[ξ , η](:= R[ξ1 , ξ2 , ξ3 , η1 , η2 , η3 ]):



J:

J : I = ⟨1 + ξ 2 ⟩.

√

Since 1 + ξ 2 has no real zero, the algebraic variety V( J : I) ⊂ R is the empty set. That is, the system satisfies condition (i) of Corollary 2.1, and consequently, the system is globally observable at the origin. However, the system does not satisfy conditions (ii) and (iii) of Corollary 2.1.

J0 = ⟨2ξ2 − 2η2 ⟩, J1 = J0 + ⟨2ξ1 ξ3 − 2η1 η3 ⟩, J2 = J1 + ⟨2ξ1 (ξ1 + ξ22 ) − 2ξ1 ξ3 − (2η1 (η1 + η22 ) − 2η1 η3 )⟩, J3 = J2 + · · · . Here, J7 = J6 is obtained. Thus, we define J ⊂ R[ξ , η] as J6 . By substituting η = 0 ∈ R3 , we obtain J = ϕ0 (J ) ⊂ R[ξ ] as

J := ϕ0 (J ) = ⟨ξ12 , ξ1 ξ3 , ξ2 ⟩,

√

(23)

and

   ξ1 3 ξ2 ∈ R : ξ1 = 0, ξ2 = 0 . V(J ) = ξ3 Thus, V(J ) ̸= V(I) is obtained. From Theorem 2.1, the system is not globally observable at the origin. Next, we test for local observability at the origin. The minimal primary decomposition of J is computed as

J = q1 ∩ q2 ,

Example 2.3. This example gives a globally observable system not satisfying the sufficient conditions of Corollaries 2.1 and 2.2 for the global and local observability. We test the global observability at the origin of the following two-dimensional polynomial system: x˙ (t ) =



x1 ( t ) , x2 ( t )



y(t ) = x21 (t ) + x22 (t ). First, we show that the system does not satisfy the sufficient conditions of Corollary 2.1. The ideals J ⊂ R[ξ1 , ξ2 ] and I ⊂ R[ξ1 , ξ2 ] are generated as

J = ⟨ξ12 + ξ22 ⟩,

q1 = ⟨ξ1 , ξ2 ⟩,

I = ⟨ξ1 , ξ2 ⟩,

q2 = ⟨ξ , ξ2 , ξ3 ⟩.

and

Since both q1 and q2 are subsets of the ideal I, we have JI ∩ R[ξ ] = J . Therefore, Theorem 2.2 implies that the system is not locally



observable at the origin. Finally, we test for global observability at σ := [1, 0, 0]T , where observability rank condition (2) holds. It cannot be determined that the system is globally observable at σ simply because observability rank condition (2) holds at σ . We generate an ideal I ⊂ R[ξ ](:= R[ξ1 , ξ2 , ξ3 ]) as

Thus, neither condition (i) nor (ii) nor (iii) of Corollary 2.1 holds. Furthermore, neither condition (i) nor (ii) nor (iii) of Corollary 2.1 holds, because we have

2 1

I = ⟨ξ1 − 1, ξ2 , ξ3 ⟩, and by substituting η = σ ∈ R3 , we obtain J = ϕσ (J ) ⊂ R[ξ ] as

J := ϕσ (J ) = ⟨ξ1 − 1, ξ2 , ξ3 ⟩, where J ⊂ R[ξ , η] is the same ideal as J in (23). Since J = I is obtained, from condition (ii) of Corollary 2.1, the system is globally observable at σ . For linear systems, because of the Cayley–Hamilton theorem, it is guaranteed that we generate fewer ideals than the dimension of the system. However, in this example, we generated more ideals than the dimension of the system.

√

J : I ⊂ R[ξ1 , ξ2 ] is obtained as

J : I = J.

JI ( I I , and

 ( J : I)I = JI ( R[ξ1 , ξ2 ]I . However, it is possible to show that, the system is globally observable at the origin by exploiting condition (i) of Theorem 2.1. By computing V(J ) ⊂ R2 , we obtain V(J ) = {(0, 0)} = V(I). Therefore, condition (i) of Theorem 2.1 implies that the system is globally observable at the origin and consequently locally observable at the origin, where the conditions of Corollaries 2.1 and 2.2 do not hold.

Y. Kawano, T. Ohtsuka / Automatica (

3. Observability at an initial state for discrete-time polynomial systems In this section, we show global and local observability conditions at an initial state for discrete-time polynomial systems. First, for discrete-time polynomial systems, the set of pairs of indistinguishable initial states are derived and then employed to obtain the observability conditions. Let us consider a discrete-time polynomial system. The state equation of the system is described by

ΣD

x[t + 1] = f (x[t ], u[t ]), x[0] = x0 , y[t ] = h(x[t ], u[t ]),

where t ∈ N denotes the time step, and x ∈ Rn , u ∈ Rm and y ∈ Rq denote the state, input and output vectors of the system ΣD , respectively. The elements of f and h are polynomial functions of x and u. Since the output of the system ΣD depends on the time step t, the initial state x0 and the input sequence UN = (u[0], u[1], . . . , u[N ]), we denote the output at time step N as y(N , x0 , UN ). The distinguishability of discrete-time nonlinear systems is defined as follows. Definition 3.1. A pair of initial states (ξ , η) ∈ R × R of a system ΣD is distinguishable if there exists an input sequence UN that yields y(N , ξ , UN ) ̸= y(N , η, UN ). n

n

Definition 3.1 is based on multiple-experiment observability (Sontag, 1979). Definitions 2.2 and 2.3 are also adopted as definitions of global and local observability at an initial state for the system ΣD , respectively. Since the elements of f and h are polynomial functions of x and u, we can write the ith element of output ΣD as yi [t ] =

ri,t 

hi,t ,j (x0 )pi,t ,j (Ut ),

(25)

j =0

where hi,t ,j ∈ R[x] (i = 1, 2, . . . , q; t = 0, 1, . . . ; j = 0, 1, . . . , ri,t ) and pi,t ,j (i = 1, 2, . . . , q; t = 0, 1, . . . ; j = 0, 1, . . . , ri,t ) are monomial functions of the elements of Ut and are distinct from each other, and the positive integer rt ,j is uniquely defined for the given f and h of the system ΣD . From (25), a pair of initial states (ξ , η) ∈ Rn × Rn is indistinguishable if and only if hi,t ,j (ξ ) = hi,t ,j (η),

(i = 1, . . . , q; t = 0, 1, . . . ; j = 0, . . . , ri,t )

(26)

holds. The following theorem (Nijmeijer, 1982) based on the inverse function theorem is commonly used when testing for the local observability of discrete-time nonlinear systems.

–

7

are indistinguishable from σ ∈ Rn , we need to find all initial states satisfying (26). Here, we characterize the set of initial states indistinguishable from σ for the discrete-time system ΣD using a finite set of equations on the basis of Hilbert’s basis theorem. First, we consider the output of the system at time step 0. From condition (26), if the initial states (ξ , η) ∈ Rn × Rn cannot be distinguished in terms of the output at time step 0, then hi,0,j (ξ ) = hi,0,j (η),

(i = 1, . . . , q; j = 0, . . . , ri,0 ).

(28)

From (28), we define an ideal J0 ⊂ R[ξ , η] as



(24)

)

J0 =

q 

J0,i ,

i=1

J0,i = ⟨hi,0,0 (ξ ) − hi,0,0 (η), hi,0,1 (ξ ) − hi,0,1 (η), . . . , hi,0,ri,0 (ξ ) − hi,0,ri,0 (η)⟩.

(29)

The variety V(J0 ) ⊂ Rn × Rn denotes the set of (ξ , η) satisfying (28). Then we consider the output of the system at time step 1. From condition (26), if the initial states (ξ , η) ∈ Rn × Rn cannot be distinguished in terms of the output at time step 1, then hi,1,j (ξ ) = hi,1,j (η),

(i = 1, . . . , q; j = 0, . . . , ri,1 ).

(30)

From (28) and (30), we define an ideal J1 ⊂ R[ξ , η] as J1 = J0 +

q 

J1,i ,

i=1

J1,i = ⟨hi,1,0 (ξ ) − hi,1,0 (η), hi,1,1 (ξ ) − hi,1,1 (η), . . . , hi,1,ri,1 (ξ ) − hi,1,ri,1 (η)⟩.

(31)

The variety V(J1 ) ⊂ Rn × Rn denotes the set of (ξ , η) satisfying (28) and (30). That is, V(J1 ) ⊂ Rn × Rn is the set of indistinguishable states for the first time step. Similarly, the set of indistinguishable states (ξ , η) for the first two steps is given by V(J2 ), where J2 ⊂ R[ξ , η] is defined as J2 = J1 +

q 

J2,i ,

i =1

J2,i = ⟨hi,2,0 (ξ ) − hi,2,0 (η), hi,2,1 (ξ ) − hi,2,1 (η), . . . , hi,2,ri,2 (ξ ) − hi,2,ri,2 (η)⟩.

(32)

From (29), (31) and (32), the ideals satisfy the inclusion property J0 ⊂ J1 ⊂ J2 . Continuing the same procedure, we obtain ideals Ji ⊂ R[ξ , η], (i = 0, 1, 2, . . .) and an ascending chain of ideals J0 ⊂ J1 ⊂ · · ·. According to Hilbert’s basis theorem, the chain will stabilize. In other words, there exists a certain ideal J ⊂ R[ξ , η] that satisfies J 0 ⊂ J 1 ⊂ · · · ⊂ J N = J N +1 = · · · = J

(33)

Proposition 3.1 (Observability Rank Condition). The system ΣD is locally observable at an initial state σ ∈ Rn if

for a certain natural number N. In particular, the chain (33) has the following strict inclusion property:

dim dO(σ ) = n,

J 0 ( J 1 ( · · · ( J N = J N +1 = J ,

dO(x) = spanR {dH (x) : H ∈ O},

as stated in Lemma A.2. Let ϕσ : R[ξ , η] → R[ξ ] be a mapping that substitutes η = σ ∈ Rn for an element of R[ξ , η], which is a surjective ring homomorphism over R[ξ , η]. Suppose the ideal J D ⊂ R[ξ ] denotes the image ϕσ (J ) of J in (34), which implies that J D characterizes the set of initial states that are indistinguishable from σ . That is, V(J D ) denotes the set of indistinguishable initial states. Since a finite set of polynomials generates the ideal J D , V(J D ) consists of the common zeros of a finite set of polynomials. Thus, the set of initial states that cannot be distinguished from σ is obtained for discrete-time polynomial systems as the common zeros of a finite set of equations.

O(x) = {hi,t ,j (x) (i = 1, . . . , q; t = 0, 1, . . . ; j = 0, . . . , ri,t )}

(27)

holds. The observability rank condition (27) is a sufficient condition for local observability at an initial state, and we cannot verify global observability at an initial state using the condition. Here, we derive necessary and sufficient conditions for global and local observability at an initial state for the system ΣD by a similar procedure to that for ΣC . To obtain the set of initial states that

(34)

8

Y. Kawano, T. Ohtsuka / Automatica (

Remark 3.1. Depending on the context, (ξ , η) is considered as an element of either (R[ξ , η])2n or Rn × Rn . In (29), (31) and (32), (ξ , η) is considered as an element of (R[ξ , η])2n . In (26), (28) and (30), (ξ , η) is considered as an element of Rn × Rn .

)

–

Since q1 ⊂ I and q2 ̸⊂ I, from Lemma B.8, JID ∩ R[ξ ] is obtained as

JID ∩ R[ξ ] = q1 = ⟨ξ1 , ξ2 , ξ32 ⟩,

Theorem 3.1. The polynomial system ΣD is globally observable at an initial state σ ∈ Rn if and only if

and we obtain V(JID ) = V(I). From Theorem 3.2, the system is locally observable at the origin. Note that the observability rank condition is not satisfied at the origin. Next, we test for global observability at the origin in case (ii). We generate the following ideals, which are subsets of R[ξ , η]:

V(J D ) = V(I)

J0 = ⟨ξ2 − η2 ⟩,

In a similar manner to Theorems 2.1 and 2.2, we can prove the following observability conditions.

(35)

holds.

J1 = J0 + ⟨ξ1 + ξ22 − (η1 + η22 ), ξ32 − η32 ⟩, J2 = J1 + ⟨2ξ32 − ξ33 + (ξ1 + ξ22 )2

Theorem 3.2. The polynomial system ΣD is locally observable at an initial state σ ∈ Rn if and only if

− (2η32 − η33 + (η1 + η22 )2 ), ξ14 − η14 ⟩ J3 = J2 + · · · .

V(JID ) = V(I)

Here, J2 = J3 is obtained. Thus, we define J ⊂ R[ξ , η] as J2 . By substituting η = 0 ∈ R3 , we obtain J D = ϕ0 (J ) ⊂ R[ξ ] as

(36)

holds. As a result, the global and local observability conditions are obtained for discrete-time systems, and the obtained conditions have similar forms to those for continuous-time systems. Example 3.1. We give an example in which the global and local observability at an initial state of a discrete-time polynomial system are tested. The state equation is described by 2x23 [t ] − x33 [t ] x[t + 1] = x1 [t ] + x22 [t ] + x23 [t ]u[t ] , x21 [t ]





y[t ] = x2 [t ]. It is easily checked that observability rank condition for discretetime systems (27) does not hold at the origin. Thus, the local observability at the origin of the system cannot be determined from the rank condition. The observability of the system is thus tested using the main results of this work. Here, we test for local observability for two cases of the system: (i) No input is considered (u[t ] = 0, ∀t ∈ N). (ii) An input is considered. First, we test for global observability at the origin in case (i). We generate an ideal I ⊂ R[ξ ](:= R[ξ1 , ξ2 , ξ3 ]) as

J D := ϕ0 (J3 ) = ⟨ξ1 , ξ2 , 2ξ32 ⟩, and V(J D ) = V(I) is obtained. From Theorem 3.1, the system is globally observable at the origin. Note that, the system in case (i) is not globally observable at the origin, and the observability rank condition (27) is not satisfied at the origin. 4. Conclusion In this paper, we have derived necessary and sufficient conditions for global and local observability at an initial state for continuous-time and discrete-time polynomial systems. To derive the conditions, the results of commutative algebra play an important role. In particular, the local observability condition is obtained owing to the localization of polynomial rings. All obtained conditions are checked by simply computing the common zeros of a finite set of polynomials. By our approach, we can check the local observability at an initial state for the polynomial systems even if the observability rank condition is not satisfied at the initial state. In the examples, we test the global and local observability of polynomial systems using the proposed criteria. Appendix A. Ascending chains of ideals

I = ⟨ξ1 , ξ2 , ξ3 ⟩, and the following ideals, which are subsets of R[ξ , η] (:= R[ξ1 , ξ2 , ξ3 , η1 , η2 , η3 ]): J0 = ⟨ξ2 − η2 ⟩, J1 = J0 + ⟨ξ1 + ξ22 − (η1 + η22 )⟩,

We give some lemmas for ascending chains of ideals Lemma A.1. The ascending chain (9) satisfies the strong inclusion property (10).

J2 = J1 + ⟨2ξ32 − ξ33 + (ξ1 + ξ22 )2 − (2η32 − η33 + (η1 + η22 )2 )⟩,

Proof. For simplicity, a single-output autonomous system is considered as follows:

J3 = J2 + · · · .



Here, J2 = J3 is obtained. Thus, we define J ⊂ R[ξ , η] as J2 . By substituting η = 0 ∈ R3 , we obtain J D = ϕ0 (J ) ⊂ R[ξ ] as

J D = ⟨ξ1 , ξ2 , 2ξ32 − ξ33 ⟩, and V(J D ) ̸= V(I) is obtained. From Theorem 3.1, the system is not globally observable at the origin. Then, we test for local observability at the origin in case (i). The minimal primary decomposition of J D is computed as

J = q1 ∩ q2 , D

q1 = ⟨ξ1 , ξ2 , ξ32 ⟩, q2 = ⟨ξ1 , ξ2 , 2 − ξ3 ⟩.

x˙ (t ) = f (x(t )), x(0) = x0 , y(t ) = h(x(t )),

and the ideals J0 ⊂ R[ξ , η] and Jt ⊂ R[ξ , η] (t = 1, 2, . . .) are computed as J0 = ⟨h(ξ ) − h(η)⟩, Jt = Jt −1 + ⟨Ltf h(ξ ) − Ltf h(η)⟩. Assume that JN = JN +1 for a certain N ∈ N. Then, there exists at ∈ R[ξ , η](t = 0, 1, . . . N ) such that LfN +1 h(ξ ) − LNf +1 h(η) =

N  t =0

at (ξ , η)(Ltf h(ξ ) − Ltf h(η)).

Y. Kawano, T. Ohtsuka / Automatica (

(η)) f (ξ ) ∂ξ ∂(Ltf h(ξ ) − Ltf h(η)) −Ltf +1 h(η) = f (η) ∂η

Ltf +1 h(ξ ) =

∂(

(ξ ) −

Ltf h

holds for all t ∈

Ltf h

N, we can compute LNf +1 h

LNf +1 h(ξ ) − LNf +1 h(η) =

9

Here we derive some lemmas to prove the theorems in this paper. To derive these lemmas, we use the following results (Atiyah & Macdonald, 1969; Cox et al., 1992, 2005; Kunz, 1984).

(ξ )−

∂(LNf h(ξ ) − LNf h(η)) ∂ξ

LNf +1 h

(η) as follows:

∂ξ

t =0

+

  ∂ at (ξ , η)(Ltf h(ξ ) − Ltf h(η)) ∂η

N −1   ∂ at (ξ , η) t =0

∂ξ

f (ξ ) +

 Lemma B.3. Suppose that I ⊂ R[x] is an ideal. Then, the following hold.

f (η)

∂ at (ξ , η) f (η) ∂η

(i) I ⊂ I(V(I ))√ . (ii) V (I ) = V ( I ).

 

∈ JN −1 + JN = JN , which implies JN +2 = JN , and consequently we obtain JN = JN +1 = JN +2 = · · ·. The proof is readily extended to the generic system ΣC .  Lemma A.2. The ascending chain (33) satisfies the strong inclusion property (34). Proof. For simplicity, a single-output autonomous system is considered as follows: x[t + 1] = f (x[t ]), x[0] = x0 , y[t ] = h(x[t ]),



and the ideals J0 and Jt (t = 1, 2, . . .) are computed as J0 = ⟨h(ξ ) − h(η)⟩, Jt = Jt −1 + ⟨h(f t (ξ )) − h(f t (η))⟩. Assume that JN = JN +1 for a certain number N ∈ N, then there exists at ∈ R[ξ , η](t = 0, 1, . . . , N ) such that at (ξ , η)(ht (ξ ) − ht (η)).

t =0

By computing hN +2 (ξ ) − hN +2 (η), we have hN +2 (ξ ) − hN +2 (η) = hN +1 (f (ξ )) − hN +1 (f (η))

=

N 

at (f (ξ ), f (η))(ht (f (ξ )) − ht (f (η)))

t =0

=

N 

I ⊂ J then V(I ) ⊃ V(J ). V ⊂ W then I(V ) ⊃ I(W ). V ̸⊂ W then I(V ) ̸⊃ I(W ). I ⊂ J then I(V(I )) ⊂ I(V(J )).

Lemma B.2. Suppose that V ⊂ Rn is an algebraic variety. Then, V(I(V )) = V holds.

f (ξ )

   · Ltf h(ξ ) − Ltf h(η) + at (ξ , η) Ltf +1 h(ξ ) − Ltf +1 h(η)

hN +1 (ξ ) − hN +1 (η) =

If If If If

Note that (iv) follows from (i) and (ii).



N 

Lemma B.1. Suppose that I , J ⊂ R[x] are ideals and V , W ⊂ Rn are algebraic varieties. Then the following inclusion properties hold. (i) (ii) (iii) (iv)

f (ξ )

(∂ LNf h(ξ ) − LNf h(η))

f (η) ∂η    N −1  ∂ at (ξ , η)(Ltf h(ξ ) − Ltf h(η))

+

=

–

Appendix B. Some technical lemmas

Since

=

)

at (f (ξ ), f (η))(ht +1 (ξ ) − ht +1 (η))

Lemma B.4. Suppose that I , J ⊂ R[x] are ideals and V , W ⊂ Rn are algebraic varieties. Then the following equalities hold. (i) V(I + J ) = V(I ) ∩ V(J ). (ii) V(I ∩ J ) = V(I ) ∪ V(J ). (iii) I(V ∪ W ) = I(V ) ∩ I(W ). Lemma B.5. Suppose that I , J ⊂ R[x] are ideals and p ⊂ R[x] is a prime ideal. Then, (I ∩ J )p = Ip ∩ Jp holds. The following lemmas are used to prove the theorems in this paper. Lemma B.6. Suppose that I , J ⊂ R are ideals and p is a prime ideal. If I ̸⊂ p and J ⊂ p, then (I ∩ J )p = Jp . Proof. Since it is trivial that (I ∩ J )p ⊂ Jp , we show that (I ∩ J )p ⊃ Jp . From the assumption I ̸⊂ p, there exists some a ∈ I such that a ̸∈ p. For this a and for all b ∈ J, ab ∈ IJ ⊂ I ∩ J holds. Thus, for all b/r ∈ Jp (b ∈ J , r ̸∈ p), b/r = ab/ar ∈ (J ∩ I )p holds. Note that since p is prime, r ̸∈ p and a ̸∈ p imply ar ̸∈ p.  Lemma B.7. Suppose that q ⊂ R is a primary ideal and p ⊂ R is a prime ideal. If q ⊂ p, then qp ∩ R = q.

√

Proof. Since p is a prime ideal, q ⊂ p yields q ⊂ p. Then, we √ prove that q ⊂ p implies qp ∩ R = q. The inclusion property qp ∩ R ⊃ q is trivial, and thus we show that qp ∩ R ⊂ q by contradiction. Assume that (qp ∩ R) ̸⊂ q. Then for a ∈ (qp ∩ R) \ q, the numerator of a belongs to q and the denominator of a does not belong to p, which implies that there exists b ̸∈ p such that ab ∈ q. √ Therefore, ab ∈ q is satisfied, irrespective of a ̸∈ q and b ̸∈ q. This contradicts the fact that q is primary.  Lemma B.8. Suppose that I ⊂ R is an ideal and p ⊂ R is a prime ideal. Then Ip ∩ R is the intersection of the ideals that comprise the minimal primary decomposition of I and is also a subset of p.

t =0

∈ J N + J N +1 = J N , which implies JN +2 = JN , and consequently we obtain JN = JN +1 = JN +2 = · · ·. The proof is readily extended to the generic system ΣD . 

Proof. Suppose that q1 ∩ q2 ∩ · · · ∩ qt is the minimal primary decomposition of I, where qi satisfies qi ⊂ p (i = 1, . . . , s) and qi ̸⊂ p (i = s + 1, . . . , t ) for some s ≤ t. Then, Ip = q1 p ∩ · · · ∩ qs p holds (Atiyah & Macdonald, 1969), and Lemma B.7 implies Ip ∩ R = q1 ∩ · · · ∩ qs . 

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Y. Kawano, T. Ohtsuka / Automatica (

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Lemma B.9. Suppose that I , J ⊂ R are ideals and p ⊂ R is a prime ideal. Then, we have

(B.2). Computing I(V(I(V(I ))p )) using (B.2) and (iii) of Lemma B.4, we obtain

(Ip : Jp ) ∩ R = (Ip ∩ R) : (Jp ∩ R).

I(V(I(V(I ))p )) = I (X1 ∪ · · · ∪ Xs )

(B.1)

= I(X1 ) ∩ · · · ∩ I(Xs ). Proof. From the definition of the ideal quotient, for all b ∈ Jp , a ∈ Ip : Jp satisfies ab ∈ Ip , and consequently for all b ∈ Jp ∩ R, a ∈ (Ip : Jp ) ∩ R satisfies ab ∈ Ip , and a, b ∈ R implies ab ∈ R. Thus, ab ∈ Ip ∩ R holds, which implies a ∈ (Ip ∩ R) : (Jp ∩ R). Therefore, (Ip : Jp ) ∩ R ⊂ (Ip ∩ R) : (Jp ∩ R) holds. From the definition of the ideal quotient, for all b ∈ Jp ∩ R, a ∈ (Ip ∩ R) : (Jp ∩ R) satisfies ab ∈ Ip ∩ R. For all c ̸∈ p, (ab)/c ∈ Ip holds. That is, for all b/c ∈ Jp (b ∈ J , c ̸∈ p), (ab)/c ∈ Ip holds, which implies a ∈ Ip : Jp . Since a ∈ (Ip ∩ R) : (Jp ∩ R) holds, a ∈ R holds. Thus, a ∈ (Ip : Jp ) ∩ R holds. This implies (Ip ∩ R) : (Jp ∩ R) ⊂ (Ip : Jp ) ∩ R holds.  Restricting the ring to the polynomial ring, we clarify some relations between a local ring and the minimal decomposition of an algebraic variety corresponding to an ideal of the polynomial ring. Lemma B.10. Suppose that p ⊂ R[x] is a prime ideal satisfying p = I(V(p)), and suppose the minimal decomposition of an algebraic variety (Cox et al., 1992) V(I ) ⊂ Rn corresponding to an ideal I ⊂ R[x] is represented as V(I ) = X1 ∪ X2 ∪ · · · ∪ Xt , where each Xi satisfies V(p) ⊂ Xi (i = 0, 1, . . . , s ≤ t ) and V(p) ̸⊂ Xi (i = s + 1, s + 2, . . . , t ). Then, we have V(I(V(I ))p ) = X1 ∪ · · · ∪ Xs .

(B.2)

Proof. By computing V(I ) using the decomposition and (iii) of Lemma B.4, we obtain I(V(I )) = I (X1 ∪ · · · ∪ Xt )

= I(X1 ) ∩ · · · ∩ I(Xt ).

(B.3)

From (ii) of Lemma B.1, V(p) ⊂ Xi (i = 0, . . . , s) leads to I(V(p)) ⊃ I(Xi ) (i = 0, . . . , s), and the assumption p = I(V(p)) leads to p ⊃ I(Xi ) (i = 0, . . . , s). Since Xi (i = 0, . . . , t ) is an irreducible variety (Cox et al., 1992, 2005; Kunz, 1984) I(Xi ) (i = 0, . . . , t ) is a prime ideal, and consequently I(Xi ) (i = 0, . . . , t ) is primary. From Lemma B.8, (B.3) implies I(V(I ))p ∩ R[x] = I(X1 ) ∩ · · · ∩ I(Xs ).

(B.4)

Computing V(I(V(I ))p ) using (B.4) and (ii) of Lemma B.4, we obtain V(I(V(I ))p ) = V (I(X1 ) ∩ · · · ∩ I(Xs ))

= V(I(X1 )) ∪ · · · ∪ V(I(Xs )).

(B.5)

From Lemma B.2, we have V(I(X1 )) ∪ · · · ∪ V(I(Xs )) = X1 ∪ · · · ∪ Xs .

(B.6)

Eqs. (B.4) and (B.9) imply (B.8). By using (B.8), we show I(V(I ))p ∩ R[x] = I(V(Ip ))p ∩ R[x].

I(V(Ip )) ⊂ I(V(I(V(I ))p )) = I(V(I ))p ∩ R[x].

(B.11)

Since (I(V(I ))p ∩ R[x])p ∩ R[x] = I(V(I ))p ∩ R[x] generally holds for any ideal I ⊂ R[x], (B.11) leads to the following inclusion property. I(V(Ip ))p ∩ R[x] ⊂ (I(V(I ))p ∩ R[x])p ∩ R[x]

= I(V(I ))p ∩ R[x].

(B.12)

On the other hand, from (iv) of Lemma B.1, the general inclusion property Ip ∩ R[x] ⊃ I leads to I(V(Ip )) ⊃ I(V(I )), I(V(Ip ))p ∩ R[x] ⊃ I(V(I ))p ∩ R[x].

(B.13)

Thus, (B.12) and (B.13) lead to (B.10). Next, we show I(V(Ip ))p ∩ R[x] = I(V(Ip )). We represent the minimal primary decomposition of an ideal Ip ∩ R[x] as Ip ∩ R[x] = q1 ∩ · · · ∩ qs , where, from Lemma B.8, qi ⊂ p (i = 1, . . . , s). By computing I(V(Ip )) using the decomposition of Ip ∩ R[x] and (ii) and (iii) of Lemma B.4, we obtain I(V(Ip )) = I(V(q1 ∩ · · · ∩ qs ))

= I(V(q1 ) ∪ · · · ∪ V(qs )) = I(V(q1 )) ∩ · · · ∩ I(V(qs )).

(B.14)

By computing I(V(Ip ))p ∩ R[x] using (B.14), from Lemma B.5, we have I(V(Ip ))p ∩ R[x] = (I(V(q1 )) ∩ · · · ∩ I(V(qs )))p ∩ R[x]

  = I(V(q1 ))p ∩ · · · ∩ I(V(qs ))p ∩ R[x].

(B.15)

From (iv) of Lemma B.1, qi ⊂ p (i = 1, . . . , s) implies I(V(qi )) ⊂ I(V(p)) (i = 1, . . . , s), and the assumption p = I(V(p)) implies I(V(qi )) ⊂ p (i = 1, . . . , s). If qi is primary, then I(V(qi )) is prime, and consequently I(V(qi )) is primary. Lemma B.7 implies I(V(qi ))p ∩ R[x] = I(V(qi )). Thus, (B.15) and (B.14) lead to the following equalities: I(V(Ip ))p ∩ R[x] = I(V(q1 ))p ∩ · · · ∩ I(V(qs ))p ∩ R[x]



Lemma B.11. Suppose that I ⊂ R[x] is an ideal and p ⊂ R[x] is a prime ideal satisfying p = I(V(p)). Then, we have I(V(I ))p ∩ R[x] = I(V(Ip ))p ∩ R[x] = I(V(Ip )).

(B.10)

From (i) of Lemma B.3, we have the inclusion property I ⊂ I(V(I )). Since the inclusion property of ideals holds irrespective of localizing the ring, I ⊂ I(V(I )) leads to Ip ⊂ I(V(I ))p , and consequently, Ip ∩ R[x] ⊂ I(V(I ))p ∩ R[x]. From (iv) of Lemma B.1 and (B.8), we have



Thus, (B.5) and (B.6) lead to (B.2).

(B.9)

(B.7)



= I(V(q1 )) ∩ · · · ∩ I(V(qs )) = I(V(Ip )). This completes the proof.



Eq. (B.7) implies V(I(V(I ))p ) = V(I(V(Ip ))p ) = V(I(V(Ip ))).

Proof. First we show I(V(I ))p ∩ R[x] = I(V(I(V(I ))p )).

(B.8)

The minimal decomposition of V(I ) ⊂ R is represented as X1 ∪ · · · ∪ Xt , where each Xi satisfies V(p) ⊂ Xi (i = 0, 1, . . . , s ≤ t ) and V(p) ̸⊂ Xi (i = s + 1, s + 2, . . . , t ). Lemma B.10 implies n

Since V(I(V(Ip ))) = V(Ip ), we have V(I(V(I ))p ) = V(Ip ). Eq. (B.16) and Lemma B.10 lead to the following lemma.

(B.16)

Y. Kawano, T. Ohtsuka / Automatica (

Lemma B.12. Suppose that I ⊂ R[x] is an ideal, p ⊂ R[x] is a prime ideal satisfying p = I(V(p)) and X0 ∪ · · · ∪ Xt is the minimal decomposition of an algebraic variety V(I ) ⊂ Rn , where each Xi satisfies V(p) ⊂ Xi (i = 0, 1, . . . , s) and V(p) ̸⊂ Xi (i = s + 1, s + 2, . . . , t ) for some s ≤ t. Then, we have V(Ip ) = X0 ∪ · · · ∪ Xs .

(B.17)

Lemma B.13. Suppose that I ⊂ R[x] is an ideal and p ⊂ R[x] is a prime ideal satisfying p = I(V(p)). Then, V(Ip ) = V(p) holds if and only if V(p) ∪ X1 ∪ · · · ∪ Xt is the minimal decomposition of V(I ) ⊂ Rn . In particular, there exists a variety Xj such that Xj ) V(p) if V(Ip ) ̸= V(p) holds. Proof. We can represent the minimal decomposition Xi of V(I ) as V(p) ⊂ Xi (i = 0, 1, . . . , s) and V(p) ̸⊂ Xi (i = s + 1, s + 2, . . . , t ) for some s ≤ t. Lemma B.12 implies V(Ip ) = X0 ∪ · · · ∪ Xs . Since, from (ii) of Lemma B.1, I ⊂ p implies V(I ) ⊃ V(p), there exists a variety Xj such that Xj ⊃ V(p). We assume j = 0, that is, Xj = X0 , without loss of generality. Thus, the fact that V(p) is an irreducible variety and the property of the minimal decomposition X0 ̸⊂ X1 ∪ · · · ∪ Xt imply that V(Ip ) = V(p) if and only if X0 = V(p). Since V(I ) ⊃ V(p) holds, X0 ) V(p) holds if V(Ip ) ̸= V(p) holds.  Lemma B.12 leads to the following lemma.

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Cox, D., Little, J., & O’Shea, D. (2005). Using algebraic geometry. Springer. Forsman, K. (1993). Some generic results on algebraic observability and connections with realization theory. In Proceedings of the 2nd European control conference (pp. 1185–1190). Hermann, R., & Krener, A. J. (1977). Nonlinear controllability and observability. IEEE Transactions on Automatic Control, 22(5), 728–740. Isidori, A. (1995). Nonlinear control systems. Springer-Verlag. Kunz, E. (1984). Introduction to commutative algebra and algebraic geometry. Birkhäuser. Ne˘sić, D. (1998). A note on observability tests for general polynomial and simple Wiener–Hammerstein systems. Systems and Control Letters, 35, 219–227. Nijmeijer, H. (1982). Observability of autonomous discrete-time non-linear systems: a geometric approach. International Journal of Control, 36(5), 867–874. Nijmeijer, H., & van der Schaft, A. J. (1991). Nonlinear dynamical control systems. Springer-Verlag. Ohtsuka, T. (2005). Model structure simplification of nonlinear systems via immersion. IEEE Transactions on Automatic Control, 50(5), 607–618. Ohtsuka, T. (2008). Algebraic structures in nonlinear systems over rings obtained by immersion. SIAM Journal on Control and Optimization, 47(4), 1961–1976. Sontag, E. D. (1979). On the observability of polynomial systems, I; finite-time problems. SIAM Journal on Control and Optimization, 17, 139–151. Sontag, E. D. (1990). Mathematical control theory. Springer-Verlag. Tibken, B. (2004). Observability of nonlinear systems—an algebraic approach. In Proceedings of the 43rd IEEE conference on decision and control (pp. 4824–4825).

Yu Kawano received his M.S. degree from Osaka University, Japan, in 2011. Since 2011, he has been a Ph.D. student of the Department of Systems Innovation at the Graduate School of Engineering Science at Osaka University. In 2012, he was a research student at the Institute of Cybernetics at Tallinn University of Technology, Estonia. His research interest includes nonlinear control theory.

Lemma B.14. Suppose that I is an ideal and p ⊂ R is a prime ideal such that p = I(V(p)). Then, we have

√

V( I p ) = V(Ip ).

(B.18)

√

Proof. From (ii) of Lemma B.3, we have V( I ) = √V(I ), which implies that the minimal decompositions of both V( I ) and V(I ) are the same. Thus, Lemma B.12 implies (B.18).  References Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Westview Press. Cox, D., Little, J., & O’Shea, D. (1992). Ideals, varieties and algorithms. Springer.

Toshiyuki Ohtsuka received his B.S., M.S., and Ph.D. degrees from Tokyo Metropolitan Institute of Technology, Japan, in 1990, 1992, and 1995, respectively. From 1995 to 1999, he was an Assistant Professor at the University of Tsukuba. In 1999, he joined Osaka University, where he is currently a Professor of the Department of Systems Innovation at the Graduate School of Engineering Science. His research interests include nonlinear control theory and real-time optimization with applications to aerospace engineering and mechanical engineering.