Realization theory for rational systems:
Minimal rational realizations
Jana Němcová
CWI, Amsterdam, The Netherlands
Jan H. van Schuppen
CWI, Amsterdam, The Netherlands
Abstract
The study of realizations of response maps is a topic of control and system theory. Realization theory is used in system identification and control synthesis.
A minimal rational realization of a given response map p is a rational realization of p
such that the dimension of its state space equals the transcendence degree of the observation
field of p. We relate minimality of rational realizations with their rational observability,
algebraic controllability and canonicity. We show that the existence of a minimal rational
realization is implied by the existence of a rational realization. We also specify the relation
between birational equivalence of rational realizations and the properties of being canonical
and minimal. Furthermore, we briefly discuss the procedures for checking various properties
of rational realizations.
Keywords: Rational systems, realization theory, minimality, algebraic controllability, rational
observability.
Mathematics Subject Classification (2000): 93B15, 93B27, 93C10
Published in Acta Applicandae Mathematicae, 2009 (online).
DOI: 10.1007/s10440-009-9464-y
1
Introduction
By a rational system we mean an initialized dynamical system with the dynamics and the output
function determined by rational functions. Rational systems are widely used as models of various
biological phenomena. The dynamics of a rational system then describes the behaviour of a biological phenomenon like for example the changes of the concentrations of certain enzymes in a cell
within some time-period. The outputs of the system correspond for example to the measurements
of the concentrations of the observed enzymes in an experiment. Measurable stimuli influencing
the phenomenon, like different concentrations of glucose, can be considered as inputs. Rational
systems model metabolic networks and enzyme-catalyzed reactions, see [24]. Their applications
can be found also in economy, physics and engineering.
The study of control and system theoretic properties (as controllability, observability, and
identifiability) of rational systems is motivated by the real life problems. The existence of a control
law which achieves a particular control objective is often phrased in terms of controllability. The
existence of an observer which allows estimation of the state of the system from observations is
characterized by observability. Identifiability of a system with unknown parameters implies that
these parameters can be determined uniquelly from the measurements. Many biological models
contain uncertainties in the form of unknown parameters which have to be determined to fully
characterize the model. Since minimality and canonicity (both controllability and observability)
of rational realizations play a crucial role in the characterization of identifiable rational systems
with parameters, the results of this paper are applicable in modeling of metabolic, signaling, and
genetic networks.
The algebraic geometric framework in which we study minimality of rational realizations is
adopted from [4]. Algebraic geometry is one of the mathematical fields which are useful in control
and system theory. In [8, 12, 13], the algebraic geometric approach to system theory of linear
dynamical systems is presented. Within the class of nonlinear systems, the systems with an
1
REALIZATION THEORY FOR RATIONAL SYSTEMS
2
obvious algebraic structure, like polynomial and rational systems, are studied by the means of
algebraic geometry in [1, 2, 4]. The application of these algebraic techniques can lead to the
development of new or more efficient procedures and algorithms for checking control and system
theoretic properties of rational systems or for constructing the systems having these properties.
Minimal rational realizations are one of the topics of realization theory which is a subfield of
control and system theory. Realization theory studies the problem of finding a dynamical system,
within a certain class of systems, and an initial state of this system such that it corresponds to
an apriori given input-output or response map. The correspondence is given by getting the same
output after applying the same input to the system and to the map. For polynomial systems,
realization theory was developed by using algebraic (algebraic geometric) techniques in [3, 5]. The
algebraic geometric framework for rational systems and the solution of the immersion problem of
a smooth system into a rational system can be found in [4]. The problem of the existence of a
rational system realizing a given response map is treated in [29].
Minimal realizations within the class of linear systems realizing a given response map are
defined as linear realizations of this map for which the state space dimension is minimal overall
such linear systems. For linear systems, the property of being minimal in this sense is proven to be
equivalent to the property of being observable and controllable. Many papers deal with extension
of the same concept of minimality to the world of nonlinear systems (see for example [21, 22, 34,
35, 37]).
Minimal realizations within the class of polynomial discrete-time systems were firstly defined
in [32] by Sontag as minimal-dimensional realizations, i.e. as realizations having the state spaces
of minimal dimension within all realizations. The dimension of the state space X was understood
as the transcendence degree of the polynomial functions on X. In [5], Bartosiewicz generalizes
discrete-time polynomial case to continuous-time case. He defines the so-called algebraically minimal polynomial realizations by using the concept of transcendence degree introduced by Sontag for
minimal dimensionality. Bartosiewicz proves that algebraically minimal polynomial realizations
are algebraically observable and algebraically controllable. We will proceed in the same way for
rational realizations.
We define minimality for rational realizations according to the papers [4, 5]. Analogous to
realization theory of linear systems, we want the property of being a minimal rational realization
to be equivalent to being a canonical (rationally observable and algebraically controllable) rational realization and a minimal-dimensional rational realization. These requirements are fulfilled
under additional assumptions by defining a minimal realization as a realization whose state space
dimension equals the transcendence degree of the observation field of the map realized by the
system. We prove that minimal realizations are unique up to a birational equivalence and that
canonical realizations of the same response map are birationally equivalent.
Apart from our paper [29] on the existence of a rational realization for a given response map, we
are aware of two papers, [4] and [36], concerning rational systems. In [36] the problem of minimality
of rational realizations is not considered. In [4], even the problem of rational realizations is not
considered and thus neither the problem of minimal rational realizations. In spite of this, one
can observe analogies between our definition of minimal rational realizations and the minimal
dimension of rational systems to which a smooth system can be immersed, studied in [4]. These
analogies follow from the fact that in both problems a studied object is related to a rational
system. A minimal rational system to which a smooth system can be immersed is defined as a
rational system to which the smooth system can be immersed and such that its dimension equals
the transcendence degree of the observation field of the smooth system. Note that in this definition
the observation field of a smooth system is considered instead of the observation field of a response
map which is considered in the definition of minimal rational realizations.
The short history of the development of knowledge of minimality is sketched in [33].
The organization of the paper is as follows. Terminology, notation and mathematical preliminaries adopted from [29] are provided in Section 2. We also state there some results developed
in [29] since they are needed in the subsequent sections. In Section 3 we recall algebraic independence of polynomial and rational functions on varieties. It is also a preliminary section. Minimal
rational realizations are studied in Section 4. We prove that canonical rational realizations are
minimal and that minimal rational realizations are algebraically controllable and rationally observable if specific algebraic conditions are fulfilled. We also analyze the cases when a rational
realization, which is not rationally observable, is or is not minimal. Minimality and minimaldimensionality are shown to be equivalent properties of rational realizations. Moreover we prove
that the existence of a minimal rational realization for a response map is given by the existence
REALIZATION THEORY FOR RATIONAL SYSTEMS
3
of its rational realization. In Section 5 we study the relation of birational equivalence within the
rational systems realizing the same response map. Every rational realization which is birationally
equivalent to a minimal rational realization of the same map is shown to be minimal. On the other
hand, we show that two canonical rational realization of the same response map are birationally
equivalent. Section 6 contains an example of a canonical rational realization for a given response
map and an example of a rational realization which is birationally equivalent to it. In Section 7
we propose, due to the theory developed in [29] and in the preceding sections, the procedures
for checking rational observability of rational systems, minimality of rational realizations, and
for constructing a rational realization for a given map. The way how to develop a procedure for
checking algebraic controllability of a rational system is shortly discussed as well.
2
Preliminaries
This paper is a continuation of the paper [29]. In [29] we have introduced and developed further
the framework which is originally adopted from [4, 5]. For the present paper to be self-contained,
we recall the content of [29], namely the setting for rational systems and an overview of the results
concerning the existence of rational realizations.
2.1
Polynomial and rational functions on varieties
Let R[X1 , . . . , Xn ] be the ring of all polynomials in n variables with real coefficients. Because the
field R is an integral domain, so is the algebra R[X1 , . . . , Xn ]. We denote by R(X1 , . . . , Xn ) the
field of quotients of R[X1 , . . . , Xn ]. Later we use also the notation Q(S) for the field of quotients
of an integral domain S. Hence, for example, Q(R[X1 , . . . , Xn ]) = R(X1 , . . . , Xn ).
By a real affine variety X we mean a subset of Rn of zero points of finitely many polynomials
from R[X1 , . . . , Xn ]. We say that a variety is irreducible if we cannot write it as an union of two
non-empty varieties which are its strict subvarieties.
By a polynomial (polynomial function) on a variety X we mean a map p : X → R for which
there exists a polynomial q ∈ R[X1 , . . . , Xn ] such that p = q on X. We denote by A the algebra of all polynomials on X. It is a finitely generated algebra of polynomials, i.e. there exist
ϕ1 , . . . , ϕk ∈ A such that A = R[ϕ1 , . . . , ϕk ]. Moreover, the irreducibility of X implies that A
is an integral domain, and we can define Q which denotes the field of quotients of A, especially
Q = R(ϕ1 , . . . , ϕk ). The elements of Q are called rational functions on X. Note that the elements
of Q do not have to be defined on all of X.
On Rn we consider the Zariski topology so that the closed sets are defined as real affine varieties.
We refer to an open/closed/dense set in Zariski topology as to a Z-open/Z-closed/Z-dense set.
See [5] for the above introduced terminology.
2.2
Rational vector fields
Let X be an irreducible real affine variety, let A be the algebra of polynomials on X, and let Q
denote the field of rational functions on X. Then a rational vector field f on an irreducible real
affine variety X is a R-linear map f : Q −→ Q such that f (ϕ · ψ) = f (ϕ) · ψ + ϕ · f (ψ) for all
ϕ, ψ ∈ Q. We say that the rational vector field f is well-defined at the point x ∈ X if it maps the
ring Ox = { cb |b, c ∈ A, c(x) 6= 0} ⊆ Q to itself, i.e. f (Ox ) ⊆ Ox . A trajectory of a rational vector
field f from a point x0 at which f is well-defined is the map x : [0, T ) → X with T ∈ (0, ∞] for
d
(ϕ ◦ x)(t) = (f ϕ)(x(t)) and x(0) = x0 for t ∈ [0, T ) and for ϕ ∈ A. It is sufficient to
which dt
consider ϕ ∈ A in this definition as it is explained in [4]. Note that for any rational vector field f
and any point x0 at which f is well-defined there is an unique trajectory of f from x0 defined on
the maximal interval [0, Tmax ) (Tmax may be infinite), [4, Theorem 1].
2.3
Rational systems
As an input space U we consider an arbitrary set of input values in Rm , i.e. U ⊆ Rm . As an output
space we take the space Rr . We define rational systems by a slight modification of [4, Definition
2].
Definition 2.1 [29, Definition 3.1] A rational system Σ is a quadruple (X, f, h, x0 ) where
REALIZATION THEORY FOR RATIONAL SYSTEMS
4
(i) X ⊆ Rn is an irreducible real affine variety,
(ii) f = {fα |α ∈ U } is a family of rational vector fields on X,
(iii) h : X → Rr is an output map with rational components, i.e. h = (h1 , . . . , hr ) and hi ∈ Q
for i = 1, . . . , r,
(iv) x0 ∈ X and h, fα are defined at x0 for all α ∈ U .
The set of points at which the output function h is defined and at which at least one of the
rational vector fields fα , α ∈ U is defined is a Z-dense open subset of X, [4]. Note that if the set
U is finite, then also the set of points at which all components of h are defined and at which all
rational vector fields fα , α ∈ U are defined is a Z-dense open subset of X.
As the space of input functions we consider the set U pc of piecewise constant functions u :
[0, Tu ) → U where Tu ∈ [0, ∞) depends on u. Let u ∈ U pc , we represent u as u = (α1 , t1 )(α2 , t2 ) . . .
Pi
Pi+1
. . . (αn , tn ), n ∈ N, which means that for t ∈ [ j=0 tj , j=0 tj ) the input u(t) = αi+1 ∈ U for
Pn
i = 0, 1, . . . , n − 1, t0 = 0. Then Tu = j=1 tj and we call the interval [0, Tu ) a time domain of u.
The empty input e is such input that Te = 0. Note that any input of U pc can be represented in different ways which we consider being equivalent. For example, u = (α1 , t1 )(α1 , t2 )(α2 , 0)(α3 , t3 ) =
(α1 , t1 + t2 )(α3 , t3 ). If u = (α1 , t1 ) . . . (αn , tn ) and v = (β1 , s1 ) . . . (βk , sk ) are both inputs, then
(u)(v) is an input represented as (u)(v) = (α1 , t1 ) . . . (αn , tn )(β1 , s1 ) . . . (βk , sk ). To express that
an input u was applied only on the time interval [0, t) ⊆ [0, Tu ) we use the notation u[0,t) .
Below we define the sets of admissible inputs and admissible inputs of a given rational system.
The necessity of these assumptions on the inputs is explained in [29].
g
Definition 2.2 [29, Definition 4.1] A subset U
pc of the set U pc of input functions is called a set
of admissible inputs if:
g
g
(i) ∀u ∈ U
pc ∀t ∈ [0, Tu ) : u[0,t) ∈ U pc ,
g
g
(ii) ∀u ∈ U
pc ∀α ∈ U ∃t > 0 : (u)(α, t) ∈ U pc ,
g
(iii) ∀u = (α1 , t1 ) . . . (αk , tk ) ∈ U
pc ∃δ > 0 ∀ti ∈ [0, ti + δ), i = 1, . . . , k :
g
u = (α1 , t1 ) . . . (αk , tk ) ∈ U
pc .
The trajectory of a rational system Σ with an initial state x0 ∈ X corresponding to the constant
input u = (α, Tu ) ∈ U pc is the trajectory of the rational vector field fα from x0 at which fα is
d
defined, i.e. it is the map x(·; x0 , u) : [0, Tu ) → X for which dt
(ϕ ◦ x)(t; x0 , u) = (fα ϕ)(x(t; x0 , u))
and x(0; x0 , u) = x0 for t ∈ [0, Tu ) and for ϕ ∈ A. The trajectory of a rational system Σ with an
initial state x0 ∈ X is for the empty input e equal to x0 , i.e. x(0; x0 , e) = x0 . The trajectory of
a system Σ from P
an initial state x0 corresponding to an input u = (α1 , t1 )(α2 , t2 ) . . . (αnu , tnu ) ∈
nu
U pc with Tu =
j=1 tj is the map x(·; x0 , u) : [0, Tu ) → X such that x(0; x0 , u) = x0 , and
Pi−1
Pi−1
Pi−1
Pi
x(t; x0 , u) = xαi (t − j=0 tj ; x( j=0 tj ; x0 , u), u) for t ∈ [ j=0 tj , j=0 tj ) and for i = 1, . . . , nu
Pi−1
where xαi (·; x( j=0 tj ; x0 , u), u) : [0, ti ) → X is a trajectory of a vector field fαi from the initial
Pi−1
Pi−1
state x( j=0 tj ; x0 , u) = xαi−1 ( j=0 tj ; x0 , u) for i = 2, . . . , nu , and from the initial state x0 for
i = 1. Since a trajectory of a system Σ with an initial state x0 ∈ X does not need to exist for
every input u ∈ U pc , we define the set of admissible inputs for a given Σ = (X, f, h, x0 ).
g
Definition 2.3 [29, Definition 3.2] We define the set of admissible inputs U
pc (Σ) for a system
Σ = (X, f, h, x0 ) as a subset of the set of input functions U pc such that:
g
∀u ∈ U
pc (Σ) ∃ x(·; x0 , u) : [0, Tu ) → X.
2.4
Response maps
g
g
Let U
pc be a set of admissible inputs. We say that a function ϕ : U pc → R is analytic at the
g
g
switching time-points of the inputs from U pc if for every input u = (u1 , t1 ) . . . (uk , tk ) ∈ U
pc the
function
ϕu1 ,...,uk (t1 , . . . , tk ) = ϕ((u1 , t1 ) . . . (uk , tk ))
is analytic, i.e. we can write ϕu1 ,...,uk in the form of convergent formal power series in k indeterg
minates. We assume that if (u)(α, 0)(v) ∈ U
pc , then ϕ((u)(α, 0)(v)) = ϕ((u)(v)).
REALIZATION THEORY FOR RATIONAL SYSTEMS
5
g
Definition 2.4 [29, Definition 4.3, Corollary 4.6] We denote the set of real functions ϕ : U
pc → R
g
g
which are analytic at the switching time-points of the inputs from U pc by A(U pc → R). Since
g
g
A(U
pc → R) is an integral domain, we also define Q(U pc → R) as the field of fractions of
g
A(U pc → R).
g
g
Note that A(U
pc → R) is an integral domain due to the definition of admissible inputs U pc and
g
due to the fact that ϕ((u)(α, 0)(v)) = ϕ((u)(v)) for an admissible input (u)(v) ∈ U pc . Then we
apply the knowledge that the ring of convergent formal power series in finitely many indeterminates
is an integral domain. See [29, Theorem 4.5] for the details of the proof.
A response map is a map from a function space of inputs to a set of values of outputs. By
a response map we evaluate the output of a system at the time t after applying the input u to
the system only on the time domain [0, t) ⊆ [0, Tu ). Therefore, according to Definition 2.2(i), the
g
maps with the components in A(U
pc → R) can be considered being response maps.
r
g
g
Definition 2.5 Let U
pc be a set of admissible inputs. A map p : U pc → R is called a response
g
g
map if its components pi : U
pc → R, i = 1, . . . , r are such that pi ∈ A(U pc → R).
The reason for posting an extra assumption, namely analyticity at switching time-points of
the inputs, on response maps is that this property is necessary for well-definedness of observation
algebras and observation fields of response maps. These objects are used to solve the realization
problem for rational systems.
Before stating the definition of observation algebra and observation field of a response map,
g
we define Dα derivations of the maps from U
pc to R which are the derivations of the real functions
g
g
defined on U pc at the switching time-points of the inputs of U
pc .
g
Definition 2.6 [29, Definition 4.2] Let U
pc be a set of admissible inputs. We define Dα derivation
g
of a real function ϕ : U pc → R for α ∈ U as
(Dα ϕ)(u) =
d
g
ϕ((u)(α, t))|t=0+ for all u ∈ U
pc .
dt
g
Note that, by Definition 2.2(i),(ii), if u ∈ U
pc then for every α ∈ U there is a sufficiently small
g
g
t > 0 such that (u)(α, t) ∈ U pc , and (u)(α, t)[0,s) ∈ U
pc for every s ∈ [0, Tu + t). Therefore Dα
g
derivations are well-defined for the functions from U pc to R. To simplify the notation, we use
Dα ϕ instead of Dα1 . . . Dαi ϕ where α is the multiindex α = (α1 , . . . , αi ). Remark that the maps
g
g
p:U
pc → R which are analytic at the switching time-points of the inputs of U pc are also smooth
with respect to Dα derivations.
r
g
g
Definition 2.7 [29, Definition 5.9] Let U
pc be a set of admissible inputs, and let p : U pc → R
be a response map. The observation algebra Aobs (p) of p is the smallest subalgebra of the algebra
g
A(U
pc → R) which contains the components pi , i = 1, . . . , r of p, and which is closed with respect
to the derivations Dα , α ∈ U . The observation field Qobs (p) of p is the field of quotients of Aobs (p).
2.5
Existence of rational realizations
r
g
g
Consider a response map p : U
pc → R where U pc is a set of admissible inputs. The problem of
rational realization of p consists of determining a rational system Σ = (X, f, h, x0 ) such that
g
g g
p(u) = h(x(Tu ; x0 , u)) for all u ∈ U
pc , and U pc ⊆ U pc (Σ).
We call such system Σ a rational realization of a response map p.
Next we specify when we speak about algebraically controllable and rationally observable
rational systems realizing a given response map. These concepts are due to [29] and [4].
r
g
Definition 2.8 A rational system Σ = (X, f, h, x0 ) realizing a response map p : U
pc → R is said
to be algebraically controllable (from the initial state x0 ∈ X) if the reachable set from x0 ,
g
g
R(x0 ) = {x(Tu ; x0 , u) ∈ X|u ∈ U
pc ⊆ U pc (Σ)},
is Z-dense in X.
REALIZATION THEORY FOR RATIONAL SYSTEMS
6
Definition 2.9 Let Σ = (X, f = {fα |α ∈ U }, h, x0 ) be a rational system and let Q denote the field
of rational functions on X. The observation algebra Aobs (Σ) of Σ is the smallest subalgebra of the
field Q containing all components hi , i = 1, . . . , r of h, and closed with respect to the derivations
given by rational vector fields fα , α ∈ U (fα ϕ ∈ Aobs (Σ) for every ϕ ∈ Aobs (Σ) and for every
α = (α1 , . . . , αk ), αi ∈ U ). The observation field Qobs (Σ) of the system Σ is the field of quotients
of Aobs (Σ). The rational system Σ is called rationally observable if Qobs (Σ) = Q.
Definition 2.10 We call a rational realization of a response map canonical if it is both rationally
observable and algebraically controllable.
Let us recall the notion of input-to-state map and a map related to it which are derived from
the trajectories of a system for the admissible inputs.
g
Definition 2.11 [29, Definition 5.2] Let U
pc be a set of admissible inputs, let Σ = (X, f, h, x0 )
g
g
be a rational system such that U
⊆
U
(Σ),
and let A be the algebra of polynomial functions on
pc
pc
g
g
X. We define the input-to-state map τ : U pc → X as the map τ (u) = x(Tu ; x0 , u) for u ∈ U
pc .
∗
∗
∗
g
The map τ determined by τ is defined as τ : A → A(U
→
R)
such
that
τ
(ϕ)
=
ϕ
◦
τ
for
all
pc
ϕ ∈ A.
In the following proposition we state necessary conditions for a response map to be realizable
by a rational system.
g
Proposition 2.12 [29, Proposition 5.12 and 5.13] Let U
pc be a set of admissible inputs and let
r
g
p : U pc → R be a response map which is realizable by a rational system Σ = (X, f, h, x0 ). Let
g
τ :U
pc → X be as in Definition 2.11. Then
∗
∗
(i) the map τext
: Aobs (Σ) → Aobs (p) defined as τext
ϕ = ϕ ◦ τ for every ϕ ∈ Aobs (Σ) is a
well-defined surjective homomorphism,
∗
∗
→ Aobs (p) is an isomor)), where τc∗ : Aobs (Σ)/Ker τext
(ii) Qobs (p) = τc∗ (Q(Aobs (Σ)/Ker τext
∗
phism derived from the map τext : Aobs (Σ) → Aobs (p),
(iii) Qobs (p) is finitely generated.
∗
Note that because τc∗ is an isomorphism, and Aobs (Σ), Aobs (p) are integral domains, Ker τext
is a prime ideal of Aobs (Σ).
The next proposition specifies a sufficient condition for rational realizability of a response map.
g
g
Proposition 2.13 [29, Proposition 5.15] Let U
pc be a set of admissible inputs and let p : U pc →
r
g
R be a response map. If there exists a field F ⊆ Q(U
pc → R) such that
g
(i) F is finitely generated by elements of A(U
pc → R),
(ii) F is closed with respect to Dα derivations, i.e.
∀i ∈ N, ∀αj ∈ U, j = 1, . . . , i : Dα1 . . . Dαi F ⊆ F,
(iii) Qobs (p) ⊆ F ,
then p has a rational realization.
The main theorem concerning the existence of rational realizations and characterization of all
maps realizable by rational systems has the following form based on two propositions above.
g
g
Theorem 2.14 [29, Theorem 5.17 and 6.1] Let U
pc be a set of admissible inputs and let p : U pc →
Rr be a response map. The following statements are equivalent:
(i) the observation field Qobs (p) of p is finitely generated
(ii) p is realizable by a rational system,
(iii) p has a rational realization which is rationally observable,
(iv) p has a rational realization which is canonical.
REALIZATION THEORY FOR RATIONAL SYSTEMS
3
7
Transcendence degree
Let X be an irreducible real affine variety, and let A and Q denote the algebra of polynomials on
X and the field of rational functions on X, respectively. We call the elements ϕ1 , . . . , ϕs ∈ A (or
Q) algebraically independent over R if there does not exist a non-zero polynomial p of s variables
with real coefficients such that p(ϕ1 , . . . , ϕs ) = 0 in A (Q). In the same way we define algebraic
independence of the elements of any field extension of R. Let F be A, Q, or any field extension
of R. We denote by trdeg F the transcendence degree of F over R which is defined as the largest
number of elements of F which are algebraically independent over R. An arbitrary subset of F
which has cardinality trdeg F and which consists only of algebraically independent elements, is
called a transcendence basis of F . For more details on transcendence degree, basis and extensions
see [7, 9, 26, 30, 38].
Definition 3.1 Let F be a subfield of a field G. An element g ∈ G is said to be algebraic over F
if there exist elements f0 , . . . , fj ∈ F , j ≥ 1, not all equal to zero, such that
f0 + f1 g + · · · + fj g j = 0.
In the following three propositions we state some properties of transcendence degree used
within this paper. The statements can be found or derived from statements in [9, 26, 39].
Proposition 3.2 Let F be a subfield of a field G. Then trdeg F ≤ trdeg G.
Proposition 3.3 Let F be a subfield of a field G such that trdeg F = trdeg G. If the elements
of G \ F are not algebraic over F then F = G.
Proposition 3.4 Let F and G be field extensions of R such that there exists a field isomorphism
i : F → G. Then trdeg F = trdeg G.
4
Minimal rational realizations
The state spaces of rational systems which we consider are irreducible real affine varieties. Therefore, to determine the dimension of a rational system, and thus the dimension of its state space,
we need to determine the dimension of a variety.
The dimension of an irreducible affine variety is defined as the degree of the affine Hilbert
polynomial of the corresponding ideal of polynomials which vanish on the studied variety. There
are other ways to describe this dimension and we follow one of them relating the Hilbert polynomial
approach with the number of algebraically independent polynomials on a variety. According to [10,
Section 9.5, Theorem 2], the dimension of an irreducible real affine variety X equals the maximal
number of polynomials on X which are algebraically independent over R. Moreover, it equals the
maximal number of rational functions on X which are algebraically independent over R. This is
due to the definitions of rational functions on X and their algebraic independence over R. The
complete proof can be found in [10, Section 9.5, Theorem 6]. We use this characterization of the
dimension of an irreducible real affine variety by the transcendence degree of the field of rational
functions on X as its definition.
Definition 4.1 Let X be an irreducible real affine variety and let Q denote the field of rational
functions on X. We define the dimension of X as dim X = trdeg Q.
Note that trdeg Q also corresponds to the dimension of the rational vector fields on X considered as a vector space over Q, [20, Corollary to Theorem 6.1].
r
g
g
Definition 4.2 Let U
pc be a set of admissible inputs and let p : U pc → R be a response map. We
call a rational realization Σ = (X, f, h, x0 ) of p a minimal realization of p if dim X = trdeg Qobs (p).
Note that all minimal rational realizations of the same response map have the same dimension.
g
Lemma 4.3 Let Σ be a rational realization of a response map p defined on a set U
pc of admissible
inputs. Then
trdeg Qobs (p) ≤ trdeg Qobs (Σ).
REALIZATION THEORY FOR RATIONAL SYSTEMS
8
Proof:
From Proposition 2.12(ii) and from Proposition 3.4, it follows that trdeg Qobs (p) =
∗
trdeg Q(Aobs (Σ)/Ker τext
). Moreover
∗
trdeg Q(Aobs (Σ)/Ker τext
) ≤ trdeg Q(Aobs (Σ)) = trdeg Qobs (Σ).
Therefore trdeg Qobs (p) ≤ trdeg Qobs (Σ) which was to be proved.
2
Hence, all rational realizations of a considered response map have the dimension higher (not
necessarilly strictly higher) than the dimension of a minimal rational realization of the same map.
4.1
Minimal versus canonical realizations
We relate the properties of rational realizations of being minimal, rationally observable, and
algebraically controllable. Firstly, we prove that canonicity of a rational realization implies its
minimality and that a rational realization which is algebraically controllable but not rationally
observable is minimal under a certain condition.
Proposition 4.4 Let Σ = (X, f, h, x0 ) be a canonical rational realization of a response map p
g
defined on a set U
pc of admissible inputs. Then Σ is also a minimal rational realization of p.
Proof:
Let Σ be a canonical rational realization of p. Then Σ is algebraically controllable,
∈ Aobs (Σ)
which means, according to Definition 2.8, that X = Z-cl R(x0 ). Therefore, if f = ffnum
den
where fnum , fden ∈ A, fden 6= 0 is such that f = 0 on R(x0 ), it implies that fnum = 0 on
R(x0 ) and moreover that fnum = 0 on X. Otherwise Z-cl R(x0 ) ( X. Consider the map
∗
∗
τext
: Aobs (Σ) → Aobs (p), which is defined in Proposition 2.12 as τext
(ϕ) = ϕ◦τ for all ϕ ∈ Aobs (Σ).
We get that
∗
Ker τext
= {f ∈ Aobs (Σ)|f = 0 on R(x0 )} = {f ∈ Aobs (Σ)|f = 0 on X},
∗
∗
∼
and thus Ker τext
is a zero ideal in Aobs (Σ). Consequently, Aobs (Σ)/Ker τext
= Aobs (Σ), and
∗
∼
furthermore Q(Aobs (Σ)/Ker τext ) = Qobs (Σ). Thus, by Proposition 3.4,
∗
trdeg Q(Aobs (Σ)/Ker τext
) = trdeg Qobs (Σ).
∗
) → Qobs (p) defined in Proposition 2.12(ii) is an isomorphism,
As the map τc∗ : Q(Aobs (Σ)/Ker τext
it follows from Proposition 3.4 that
∗
trdeg Qobs (p) = trdeg Q(Aobs (Σ)/Ker τext
).
By the last two equalities, trdeg Qobs (p) = trdeg Qobs (Σ). Because Σ is also a rationally observable
rational realization of p, by Definition 2.9, Qobs (Σ) = Q and trdeg Qobs (Σ) = trdeg Q. Hence,
trdeg Qobs (p) = trdeg Q and finally, by the definition of the dimension of a variety,
dim X = trdeg Q = trdeg Qobs (p).
This proves that the system Σ is a minimal rational realization of p.
2
As a direct consequence of Theorem 2.14 and Proposition 4.4 we get a sufficient condition for
the existence of a minimal rational realization formulated in the following theorem.
Theorem 4.5 If a response map p defined on a set of admissible inputs has a rational realization,
then p has also a minimal rational realization.
Proposition 4.6 If Σ is an algebraically controllable rational realization of a response map p
defined on a set of admissible inputs such that it is not rationally observable, and such that the
elements of Q \ Qobs (Σ) are algebraic over Qobs (Σ), then Σ is minimal.
Proof: Because Σ is not rationally observable, Qobs (Σ) $ Q. In spite of this, since the elements
of Q \ Qobs (Σ) are algebraic over Qobs (Σ),
trdeg Qobs (Σ) = trdeg Q.
REALIZATION THEORY FOR RATIONAL SYSTEMS
9
From the algebraic controllability, in the same way as in the proof of Proposition 4.4 (by applying
Definitions 2.11 and 2.8, and Propositions 2.12 and 2.12), it follows that
∗
trdeg Qobs (Σ) = trdeg Q(Aobs (Σ)/Ker τext
) = trdeg Qobs (p).
Therefore trdeg Qobs (p) = trdeg Q = dim X and thus the rational realization Σ of p is minimal.
2
For the reversed implication, i.e. rational observability and algebraic controllability being
determined by minimality, we study two problems. One is whether minimality of a rational
realization means that the realization is already rationally observable, and the other one is whether
a minimal rational realization is algebraically controllable.
The next proposition answers the first question, whether a minimal rational realization is
rationally observable.
Proposition 4.7 If Σ = (X, f, h, x0 ) is a minimal rational realization of a response map p defined
on a set of admissible inputs such that the elements of Q \ Qobs (Σ) are not algebraic over Qobs (Σ),
then Σ is rationally observable.
Proof:
Let Σ = (X, f, h, x0 ) be as in the proposition. From minimality of Σ it follows that
trdeg Q = trdeg Qobs (p). Further, by Proposition 2.12(ii) and by Proposition 3.4, trdeg Q =
∗
trdeg Q(Aobs (Σ)/Ker τext
). Because Qobs (Σ) is a subfield of Q, it follows from Proposition 3.2
that trdeg Qobs (Σ) ≤ trdeg Q. Consequently we obtain that
∗
trdeg Q = trdeg Q(Aobs (Σ)/Ker τext
) ≤ trdeg Qobs (Σ) ≤ trdeg Q.
Hence, trdeg Q = trdeg Qobs (Σ). By the assumption that the elements of Q \ Qobs (Σ) are not
algebraic over Qobs (Σ) and by Proposition 3.3, Q = Qobs (Σ) which proves rational observability
of Σ.
2
From Proposition 4.7 it follows that if a rational realization Σ of a response map p is not
rationally observable then Σ is not minimal or the elements of Q \ Qobs (Σ) are algebraic over
Qobs (Σ). In the following proposition we prove that if a rational realization Σ is not rationally
observable and if the elements of Q\Qobs (Σ) are not algebraic over Qobs (Σ), then Σ is not minimal.
Thus, if we know that a rational realization Σ is minimal and that the elements of Q \ Qobs (Σ) are
algebraic over Qobs (Σ) then we are still not able to say, without additional information, whether
Σ is rationally observable or not.
Proposition 4.8 If Σ is a rational realization of a response map p defined on a set of admissible
inputs such that it is not rationally observable and the elements of Q \ Qobs (Σ) are not algebraic
over Qobs (Σ), then Σ is not minimal.
Proof: To prove that the rational realization Σ of p is not minimal it is sufficient to prove, by
Lemma 4.3, that trdeg Qobs (p) < trdeg Q.
Since Σ is not rationally observable, Qobs (Σ) $ Q. Moreover, as the elements of Q \ Qobs (Σ)
are not algebraic over Qobs (Σ), trdeg Qobs (Σ) < trdeg Q.
∗
Because trdeg Q(Aobs (Σ)/Ker τext
) ≤ trdeg Qobs (Σ) and because by Proposition 2.12(ii) there
∗
∗
exists an isomorphism τc : Q(Aobs (Σ)/Ker τext
) → Qobs (p), it follows by Proposition 3.4 that
∗
trdeg Qobs (p) = trdeg Q(Aobs (Σ)/Ker τext
) ≤ trdeg Qobs (Σ).
Therefore, trdeg Qobs (p) ≤ trdeg Qobs (Σ) < trdeg Q which proves that Σ is not minimal.
2
In the following proposition we answer the second question, whether a minimal realization is
also algebraically controllable.
Proposition 4.9 Let Σ = (X, f, h, x0 ) be a minimal rational realization of a response map p
g
defined on a set U
pc of admissible inputs. If the elements of Q \ Qobs (Σ) are not algebraic over
Qobs (Σ), then the rational realization Σ is algebraically controllable.
REALIZATION THEORY FOR RATIONAL SYSTEMS
10
Proof: Let Σ = (X, f, h, x0 ) and the corresponding Q be as in the proposition. Let us assume
that the elements of Q \ Qobs (Σ) are not algebraic over Qobs (Σ).
Generally, Z-cl R(x0 ) ⊆ X. To prove that X = Z-cl R(x0 ), and thus to prove that the rational
realization Σ is algebraically controllable, we show that Z-cl R(x0 ) ⊇ X.
As Σ is a minimal rational realization, trdeg Q = trdeg Qobs (p). This implies, by Proposition 2.12(ii) and by Proposition 3.4, that
∗
trdeg Q = trdeg Q(Aobs (Σ)/Ker τext
).
(1)
Further, from Proposition 4.7, Σ is rationally observable. Therefore Q = Qobs (Σ) and from (1) it
follows that
∗
trdeg Qobs (Σ) = trdeg Q(Aobs (Σ)) = trdeg Q(Aobs (Σ)/Ker τext
).
Because the field Q(Aobs (Σ)) of fractions of the integral domain Aobs (Σ) has the same transcen∗
dence degree over R as Aobs (Σ), and because the field Q(Aobs (Σ)/Ker τext
) of fractions of the
∗
∗
integral domain Aobs (Σ)/Ker τext has the same transcendence degree over R as Aobs (Σ)/Ker τext
,
see [30, Proposition 2.2.22], it follows that
∗
trdeg Aobs (Σ) = trdeg (Aobs (Σ)/Ker τext
) < ∞.
(2)
∗
Because Ker τext
is a prime ideal in Aobs (Σ), (2) implies, from [30, Proposition 2.2.27], that
∗
Ker τext = (0) in Aobs (Σ).
Next we prove that
g
∀0 6= ϕ ∈ A ∃u ∈ U
(3)
pc : ϕ(x(Tu ; x0 , u)) 6= 0
which is equivalent to
∀ϕ ∈ A : ϕ 6= 0 on X ⇒ ϕ 6= 0 on R(x0 ).
(4)
g
Assume that (3) is not true. Let 0 6= ϕ ∈ A be such that ϕ(x(Tu ; x0 , u)) = 0 for all u ∈ U
pc .
where
ϕ
,
ϕ
∈
A
(Σ),
and
ϕ
,
ϕ
=
6
0.
As Q = Qobs (Σ), we can write 0 6= ϕ = ϕϕnum
num
den
obs
num
den
den
nnum
dnum
Furthermore, 0 6= ϕnum = nden and 0 6= ϕden = dden for 0 6= nnum , nden , dnum , dden ∈ A.
Therefore ϕnden dnum = nnum dden ∈ Q which implies that
g
∀u ∈ U
pc :
ϕ(x(Tu ; x0 , u))nden (x(Tu ; x0 , u))dnum (x(Tu ; x0 , u))
(5)
= nnum (x(Tu ; x0 , u))dden (x(Tu ; x0 , u)).
∗
∗
(ϕnum ) 6= 0. Hence, there exists
= (0), τext
Because 0 6= ϕnum ∈ Aobs (Σ) and because Ker τext
g
u∈U
such
that
ϕ
(x(T
;
x
,
u))
=
6
0,
and
consequently
pc
num
u
0
g
∃u ∈ U
pc : nnum (x(Tu ; x0 , u)) 6= 0.
(6)
∗
Because τext
(ϕden ) is well-defined, it implies that
g
∀u ∈ U
pc : dden (x(Tu ; x0 , u)) 6= 0.
(7)
g
From (6) and (7), there is u ∈ U
pc such that nnum (x(Tu ; x0 , u))dden (x(Tu ; x0 , u)) 6= 0. Then, by
g
(5), there exists u ∈ U pc such that
ϕ(x(Tu ; x0 , u))nden (x(Tu ; x0 , u))dnum (x(Tu ; x0 , u)) 6= 0.
g
This contradicts the assumption that ϕ(x(Tu ; x0 , u)) = 0 for all u ∈ U
pc . Therefore (3), and hence
(4) is valid. Thus,
∀ϕ ∈ A : ϕ = 0 on R(x0 ) ⇒ ϕ = 0 on X.
(8)
From (8) it follows that
I1 = {ϕ ∈ A|ϕ = 0 on R(x0 )} ⊆ {ϕ ∈ A|ϕ = 0 on X} = I2 ,
and therefore {x ∈ X|ϕ(x) = 0 for all ϕ ∈ I1 } ⊇ {x ∈ X|ϕ(x) = 0 for all ϕ ∈ I2 } which means
that Z-cl R(x0 ) ⊇ X.
2
REALIZATION THEORY FOR RATIONAL SYSTEMS
11
Remark 4.10 According to the proof of Proposition 4.9, we can state the following: If Σ =
(X, f, h, x0 ) is a minimal rational realization of a response map p defined on a set of admissible
inputs, and if Σ is rationally observable, then Σ is also algebraically controllable.
Theorem 4.11 Let Σ = (X, f, h, x0 ) be a rational realization of a response map p defined on a
set of admissible inputs such that the elements of Q \ Qobs (Σ) are not algebraic over Qobs (Σ).
Then Σ is canonical if and only if Σ is minimal.
Proof:
4.2
This follows directly from the Propositions 4.7, 4.9, and 4.4.
2
Minimal versus minimal-dimensional realizations
Definition 4.12 A rational realization of a response map defined on a set of admissible inputs is
called minimal-dimensional if its state space dimension is minimal, i.e. there does not exist another
rational realization of the same map such that its state space has a strictly lower dimension.
Theorem 4.13 A rational realization Σ of a response map p defined on a set of admissible inputs
is minimal if and only if Σ is minimal-dimensional.
Proof: (⇐) Let Σ be a minimal-dimensional rational realization of p. If dim X = trdeg Qobs (p),
then Σ is also minimal. Let us assume that dim X 6= trdeg Qobs (p). Then by Lemma 4.3,
dim X > trdeg Qobs (p). From Theorem 4.5 follows that there exists a minimal rational realization
Σ0 of p. Because Σ0 is minimal, dim X 0 = trdeg Qobs (p). Therefore, dim X > dim X 0 which
contradicts the minimal-dimensionality of Σ. Hence, dim X = trdeg Qobs (p), and Σ is a minimal
realization of p.
(⇒) It follows directly from Lemma 4.3.
2
5
Birational equivalence of rational realizations
The topic of this section is the relation between minimality and birational equivalence of rational
realizations. We prove that every rational realization of a response map defined on a set of
admissible inputs which is birationally equivalent to a minimal rational realization of the same
map, is itself minimal. On the other hand, we show that canonical rational realizations are
birationally equivalent. Therefore minimal rational realizations are birationally equivalent if they
are canonical. This is for example the case when the assumptions of Theorem 4.11 are satisfied.
Note that birational equivalence of irreducible varieties is a weaker equivalence relation than
isomorphism. That means that the set of varieties birationally equivalent to a given variety
contains many different nonisomorphic varieties.
The terminology of rational mappings between varieties and birational equivalence of varieties
is adopted from [10]. For the completeness of the paper we recall some definitions and statements
below.
Definition 5.1 [10, Chapter 5.5, Definition 4] Let X ⊆ Rm and X 0 ⊆ Rn be irreducible real
affine varieties. A rational mapping from X to X 0 is a function φ represented by
φ(x1 , . . . , xm ) = (f1 (x1 , . . . , xm ), . . . , fn (x1 , . . . , xm )) ,
where fi ∈ R(x1 , . . . , xm ) are such that φ is defined at some point of X, and if φ is defined at a
point (a1 , . . . , am ) ∈ X, then φ(a1 , . . . , am ) ∈ X 0 .
Similarly as the rational functions on a variety, rational mappings between varieties do not
have to be defined everywhere. They are defined on Z-dense subsets of respective varieties, [10,
Chapter 5.5, Proposition 8].
Definition 5.2 [10, Chapter 5.5, Definition 9(i)] Two irreducible varieties X and X 0 are birationally equivalent if there exist rational mappings φ : X → X 0 , ψ : X 0 → X such that
(i) there exists a point p ∈ X such that φ is defined at p and ψ is defined at φ(p),
REALIZATION THEORY FOR RATIONAL SYSTEMS
12
(ii) there exists a point p0 ∈ X 0 such that ψ is defined at p0 and φ is defined at ψ(p0 ),
(iii) φ ◦ ψ = 1X 0 and ψ ◦ φ = 1X (recall that these equalities hold on Z-dense subsets of X 0 and
X, respectively).
Theorem 5.3 [10, Chapter 9.5, Corollary 7] Let the irreducible real affine varieties X and X 0 be
birationally equivalent. Then dim X = dim X 0 .
Theorem 5.4 [10, Chapter 5.5, Theorem 10] Let X and X 0 be irreducible real affine varieties and
let Q and Q0 denote the field of rational functions on X and X 0 , respectively. Then the varieties
X and X 0 are birationally equivalent if and only if there exists an isomorphism of the fields Q and
Q0 which is the identity on the constant functions R ⊂ Q.
In [4, Definition 8] Bartosiewicz introduces the concept of isomorphic rational systems. Because
rational realizations are rational systems with fixed initial states, and because in [4, Definition 8]
only rational systems without fixed initial state are considered, we slightly modify Bartosiewicz’s
definition to define isomorphic rational realizations.
Definition 5.5 We say that rational realizations Σ = (X, f, h, x0 ), Σ0 = (X 0 , f 0 , h0 , x00 ) of the
same response map p defined on a set of admissible inputs with the same input space U and the
same output space Rr are isomorphic if
(i) the state spaces X and X 0 are birationally equivalent (with the corresponding rational mappings φ : X → X 0 , ψ : X 0 → X),
(ii) h0 φ = h,
(iii) fα (ϕ ◦ φ) = (fα0 ϕ) ◦ φ for ϕ ∈ Q0 , α ∈ U ,
(iv) φ is defined at x0 , and φ(x0 ) = x00 .
Definition 5.6 Let Σ = (X, f, h, x0 ) and Σ0 = (X 0 , f 0 , h0 , x00 ) be rational realizations of the same
response map p defined on a set of admissible inputs with the same input space U and the same output space Rr . We say that Σ and Σ0 are birationally equivalent if there exists a field isomorphism
i : Q0 → Q such that
(i) i is the identity on the constant functions R ⊂ Q0 ,
(ii) i(h0 ) = h,
(iii) fα (i(ϕ)) = i(fα0 ϕ) for ϕ ∈ Q0 , α ∈ U ,
(iv) (i(ϕ))(x0 ) = ϕ(x00 ) for all ϕ ∈ Q0 such that ϕ is defined at x00 and i(ϕ) is defined at x0 .
Note that Definitions 5.5 and 5.6 are equivalent, thus the rational realizations of the same
response map and with the same input and output spaces are isomorphic if and only if they are
birationally equivalent. In the proof of Theorem 5.4 it is shown that a field isomorphism i : Q0 → Q
from Definition 5.6 can be chosen as i = φ∗ defined as i(ϕ) = ϕ ◦ φ for all ϕ ∈ Q0 where φ is
a rational mapping from Definition 5.5. From the same theorem we also get that if i : Q0 → Q
is a field isomorphism then there is a rational mapping φ : X → X 0 such that i = φ∗ . Then it
follows that the conditions (ii) and (iii) of Definition 5.6 are only rewritten conditions (ii) and (iii)
of Definition 5.5, and that the condition (i) of Definition 5.6 corresponds to the condition (i) of
Definition 5.5 according to Theorem 5.4. The rational function i(ϕ) = φ∗ (ϕ) = ϕ ◦ φ is defined at
x0 if and only if ϕ ◦ φ is defined at x0 . Therefore, i(ϕ) is defined at x0 if and only if ϕ is defined
at φ(x0 ) and φ is defined at x0 . Because ϕ(x00 ) = (i(ϕ))(x0 ) = ϕ(φ(x0 )) for all ϕ ∈ Q0 defined
at x00 , φ(x0 ) ∈ X 0 , and because the rational functions Q0 on X 0 distinguish the points of X 0 , the
equivalence of the conditions (iv) follows.
Theorem 5.7 Let Σ be a minimal rational realization of a response map p defined on a set of
admissible inputs. Then every rational realization of p with the same input and output space as Σ
which is birationally equivalent to Σ is minimal.
REALIZATION THEORY FOR RATIONAL SYSTEMS
13
Proof:
Let Σ be a minimal rational realization of a response map p and let Σ0 be a rational
realization of p which is birationally equivalent to Σ. As Σ and Σ0 are birationally equivalent, the
state spaces X and X 0 are birationally equivalent according to Definition 5.6(i) and Theorem 5.4.
Then, by Theorem 5.3, and because Σ is minimal, dim X 0 = dim X = trdeg Qobs (p). Therefore
the system Σ0 is minimal.
2
Theorem 5.8 Let Σ = (X, f, h, x0 ) and Σ0 = (X 0 , f 0 , h0 , x00 ) be canonical rational realizations of
the same response map p defined on a set of admissible inputs with the same input space U and
the same output space Rr . Then Σ and Σ0 are birationally equivalent.
Proof:
Let Σ = (X, f, h, x0 ) and Σ0 = (X 0 , f 0 , h0 , x00 ) be canonical rational realizations of
∗
0∗ :
the same map p. By Proposition 2.12, the maps τc∗ : Q(Aobs (Σ)/Ker τext
) → Qobs (p) and τc
0
0∗
Q(Aobs (Σ )/Ker τext ) → Qobs (p) are field isomorphisms.
∗
Consider a map Ψ : Aobs (Σ) → Aobs (Σ)/Ker τext
defined as Ψ(ϕ) = [ϕ] for all ϕ ∈ Aobs (Σ).
It is a surjective homomorphism. Because Σ is algebraically controllable, it follows from the
∗
is a zero ideal in Aobs (Σ). Therefore the map Ψ is
proof of Proposition 4.4 that Ker τext
also injective, and thus Ψ is an isomorphism. We extend the isomorphism Ψ : Aobs (Σ) →
∗
Aobs (Σ)/Ker τext
to the field isomorphism which we denote by the abuse of notation as Ψ,
∗
∗
Ψ : Qobs (Σ) → Q(Aobs (Σ)/Ker τext
). Note the because Aobs (Σ) and Aobs (Σ)/Ker τext
are in∗
tegral domains, the fields Qobs (Σ) = Q(Aobs (Σ)) and Q(Aobs (Σ)/Ker τext ) are defined. Because
Σ is rationally observable, Q = Qobs (Σ), and the field isomorphism Ψ is the field isomorphism
∗
Ψ : Q → Q(Aobs (Σ)/Ker τext
).
Analogously, for the rational system Σ0 we derive the field isomorphism Ψ0 : Q0 → Q(Aobs (Σ0 )/
0∗
).
Ker τext
−1
0∗ ◦ Ψ0 . We show that i
Consider a field isomorphism i : Q0 → Q defined as i = Ψ−1 ◦ τc∗ ◦ τc
satisfies the conditions of Definition 5.6 which proves that the rational realizations Σ and Σ0 are
birationally equivalent.
−1
0∗ , τ
c∗ , Ψ−1 are the identities on constant functions R, the field
Since all isomorphisms, Ψ0 , τc
isomorphism i is the identity on R ⊂ Q0 , and thus i satisfies Definition 5.6(i).
∗
and τ ∗ (see Proposition 2.12),
Because Ψ0 (h0 ) = [h0 ], and because, by the definitions of τext
0
0
0∗
0∗ ([h ]) = p, it follows that
τext (h ) = p and thus τc
i(h0 ) = (Ψ−1 ◦ τc∗
−1
0∗ ◦ Ψ0 )(h0 ) = Ψ−1 (τ
c∗
◦ τc
−1
0∗ ([h0 ]))) = Ψ−1 (τ
c∗
(τc
−1
(p)).
Since Ψ and τc∗ are field isomorphisms such that Ψ(h) = [h] and τc∗ ([h]) = p, for their inverses
−1
−1
Ψ−1 and τc∗ it holds that Ψ−1 ([h]) = h and τc∗ (p) = [h]. Then,
i(h0 ) = Ψ−1 (τc∗
and the field
Since Q0
combination
Consider an
−1
(p)) = Ψ−1 ([h]) = h,
(9)
isomorphism i satisfies the condition (ii) of Definition 5.6.
= Qobs (Σ0 ) = Q(Aobs (Σ0 )), every element ϕ ∈ Q0 can be written as a rational
of finitely many elements of the set {h0 , fα0 h0 |α = (α1 , . . . , αi ), i ∈ N, αj ∈ U }.
arbitrary ϕ ∈ Q0 . Without loss of generality we assume that there exist i ∈ N,
and ϕnum , ϕden ∈ R[X1 , . . . , Xi ] such that ϕ =
0
0
ϕnum (fα
h0 ,...,fα
h0 )
k
1
i
where αj = (αj1 , . . . , αj j ),
0 h0 ,...,f 0 h0 )
ϕden (fα
α
1
i
h0 ), and ϕden (fα0 1 h0 , . . . , fα0 i h0 ) 6= 0. Then, for
kj ∈ N ∪ {0}, and αjl ∈ U (if kj = 0, then fα0 j h0 =
any α ∈ U ,
ϕnum (fα0 1 h0 , . . . , fα0 i h0 )
−1
0∗ ◦ Ψ0 )
fα (i(ϕ)) = fα (Ψ−1 ◦ τc∗ ◦ τc
.
ϕden (fα0 1 h0 , . . . , fα0 i h0 )
0∗ ◦ Ψ0 : Q0 → Q
The field isomorphism τc
obs (p) is, according to the proof of Proposition 2.12 stated
0∗ ◦ Ψ0 )(f 0 h0 ) = D p for any α = (α , . . . , α ), i ∈ N, α ∈ U . Further, because
in [29], such that (τc
α
1
i
j
α
Σ0 is rationally observable, thus Qobs (Σ0 ) = Q0 , for ϕ =
0
0
ϕnum (fα
h0 ,...,fα
h0 )
1
i
0 h0 ,...,f 0 h0 )
ϕden (fα
α
1
0∗ ◦ Ψ0 )(ϕ) =
(τc
ϕnum (Dα1 p, . . . , Dαi p)
.
ϕden (Dα1 p, . . . , Dαi p)
i
∈ Q0 it holds that
REALIZATION THEORY FOR RATIONAL SYSTEMS
14
−1
: Qobs (p) → Q is such that (Ψ−1 ◦
By the same token, the field isomorphism Ψ−1 ◦ τc∗
−1
τc∗ )(Dα p) = fα h for any α = (α1 , . . . , αi ), i ∈ N, αj ∈ U , and
−1
ϕnum (Dα1 p, . . . , Dαi p)
ϕnum (fα1 h, . . . , fαi h)
−1 c
∗
)
(Ψ ◦ τ
=
.
ϕden (Dα1 p, . . . , Dαi p)
ϕden (fα1 h, . . . , fαi h)
Therefore,
fα (i(ϕ))
ϕnum (fα0 1 h0 , . . . , fα0 i h0 )
−1
0∗ ◦ Ψ0 )
= fα (Ψ−1 ◦ τc∗ ) (τc
ϕden (fα0 1 h0 , . . . , fα0 i h0 )
−1
ϕnum (Dα1 p, . . . , Dαi p)
−1 c
∗
)
= fα (Ψ ◦ τ
ϕden (Dα1 p, . . . , Dαi p)
ϕnum (fα1 h, . . . , fαi h)
= fα
ϕden (fα1 h, . . . , fαi h)
−1
ϕnum (Dα1 p, . . . , Dαi p)
−1 c
∗
) Dα
= (Ψ ◦ τ
ϕden (Dα1 p, . . . , Dαi p)
ϕnum (fα0 1 h0 , . . . , fα0 i h0 )
−1
0
−1 c
0∗
∗
c
= i(fα ϕ),
◦ τ ◦ Ψ ) fα
= (Ψ ◦ τ
ϕden (fα0 1 h0 , . . . , fα0 i h0 )
(10)
and the field isomorphism i satisfies the condition (iii) of Definition 5.6.
From the definition of rational systems, h, fα are defined at x0 , and h0 , fα0 are defined at x00 ,
for α ∈ U . Because Σ and Σ0 are rationally observable, to prove that (i(ϕ))(x0 ) = ϕ(x00 ) for
all ϕ ∈ Q0 such that ϕ is defined at x00 and i(ϕ) is defined at x0 it is sufficient to prove that
(i(ϕ))(x0 ) = ϕ(x00 ) for all ϕ ∈ {h0 , fα0 h0 |α = (α1 , . . . , αi ), i ∈ N, αj ∈ U }. From (9), i(h0 ) = h,
and then (i(h0 ))(x0 ) = h(x0 ). Further, since Σ and Σ0 realize the same map, h(x0 ) = h0 (x00 ).
Therefore, (i(h0 ))(x0 ) = h0 (x00 ). Consider any α = (α1 , . . . , αi ) such that i ∈ N and αj ∈ U . From
(9) and (10), i(fα0 h0 ) = fα (i(h0 )) = fα h. Since Σ and Σ0 realize the same map it follows also that
(fα0 h0 )(x00 ) = (fα h)(x0 ). Therefore, (i(fα0 h0 ))(x0 ) = (fα h)(x0 ) = (fα0 h0 )(x00 ) which completes the
proof of the claim that i satisfies the condition (iv) of Definition 5.6.
Finally, the systems Σ and Σ0 are birationally equivalent.
2
6
Example
In [29] we have constructed a rational realization Σ = (X, f, h, x0 ) for the response map p :
R Tu u(s)
g
g
U
pc → R given as p(u) = exp( 0
(1+s)2 ds), where U pc is the set of all piecewise constant inputs
u : R → R. The rational system Σ = (X, f = {fα |α ∈ R}, h, x0 ) realizing p is of the form:
= R2 ,
x1 ∂
∂
fα (x1 , x2 ) = α 2
+
,α ∈ R
x2 ∂x1
∂x2
h(x1 , x2 ) = x1 ,
X
x0
=
(1, 1).
We show that this rational realization of p is canonical (both rationally observable and algebraically
controllable), and consequently, by Proposition 4.4 and Theorem 4.13, also minimal and minimaldimensional.
To check rational observability of the system Σ it is sufficient to remark that since x1 , x2 ∈
Qobs (Σ) ⊆ R(x1 , x2 ), the observation field Qobs (Σ) equals R(x1 , x2 ) = Q.
Since x1 (t) = eαt/(t+1) , x2 (t) = 1 + t are describing the trajectories of Σ for the constant input
u with the value α ∈ R, the reachable set of Σ from x0 is a set R(x0 ) = {[a, b] ∈ R2 |a > 0, b >
1} ∪ {[1, 1]}. We can steer the system Σ to the state [a, b] ∈ R(x0 ) by applying the input with the
b
log a till the time b − 1. Because the varieties in R2 are: R2 , finite set of points,
value α = b−1
union of an algebraic plane curve and a finite set of points, then the smallest irreducible variety
containing R(x0 ) is R2 . Thus Σ is algebraically controllable from x0 = (1, 1).
An example of a rational realization of p which is birationally equivalent to Σ is the rational
system Σ0 which is specified as follows. The state space is the unit sphere in R3 , i.e. X 0 =
REALIZATION THEORY FOR RATIONAL SYSTEMS
15
{(x, y, z) ∈ R3 |x2 + y 2 + z 2 − 1 = 0}. The irreducible varieties X and X 0 are birationally
y
x
, 1−z
) and its
equivalent since the stereographic projection Ψ : X 0 → X given as Ψ(x, y, z) = ( 1−z
x21 +x22 −1
2x1
2x2
0
inverse Φ : X → X given as Φ(x1 , x2 ) = x2 +x2 +1 , x2 +x2 +1 , x2 +x2 +1 are the rational mappings
1
2
1
2
1
2
such that Ψ ◦ Φ = 1X and Φ ◦ Ψ = 1X 0 on a Z-dense subsets of X, X 0 , respectively. Then we
derive the rational vector fields fα0 , α ∈ R on X 0 from the rational vector fields fα , α ∈ R on X
0
according to Definition 5.6. Therefore, the rational vector fields
α ∈ R as
on X are defined for
αx2 (1−z)2
αx2 (1−z 2 )
(1−z)(2αx(1−z−x2 )−y 3 ) ∂
∂
∂
2
0
fα (x, y, z) =
2y 2
∂x + 1 − z − y −
y2
∂y + y(1 − z) +
y2
∂z ,
the output function is h0 (x, y, z) =
7
x
1−z ,
and the initial state is x00 = (2/3, 2/3, 1/3).
Computational algebra for realization
To apply the results of realization theory for rational systems, which we developed in this paper and
in [29], to systems biology and to engineering we need procedures to check algebraic controllability
and rational observability of rational systems. For example, in metabolic networks one is provided
a rational system from first principles. It is not obvious whether this system is algebraically
controllable and/or rationally observable because of the modeling assumptions and because of
the modular way these networks are formulated. It is important to be able to decide whether a
system has these properties since they imply minimality of the system and thus also relatively
easier computations.
In this section we discuss the procedures for checking algebraic controllability, rational observability and minimality of rational systems and realizations. We shortly mention the possibility of
constructing rational realizations for response maps defined on a set of admissible inputs. However,
there are not yet ready-made algorithms available.
r
g
g
Construction of a rational realization Consider a response map p : U
pc → R where U pc is
a set of admissible inputs.
1. Check whether the observation field Qobs (p) is finitely generated (then p is realizable by
g
a rational system, see Theorem 2.14), and then find ϕ1 , . . . , ϕk ∈ A(U
pc → R) such that
Qobs (p) = R(ϕ1 , . . . , ϕk ).
2. Determine the functions viα and wj such that
Dα ϕi = viα (ϕ1 , . . . , ϕk ) and pj = wj (ϕ1 , . . . , ϕk )
for i = 1, . . . , k, j = 1, . . . , r, and α ∈ U .
3. A rational realization of p is then
X = Rk ,
k
X
∂
, α ∈ U,
fα =
viα
∂x
i
i=1
hj (x1 , . . . , xk ) = wj (x1 , . . . , xk ), j = 1 . . . r,
x0 = (ϕ1 (e), . . . , ϕk (e)) where e is an empty input, i.e. Te = 0.
This procedure follows the steps of the constructive proof of Proposition 2.13, see [29]. The
algorithms or procedures for the first two steps of the construction procedure are still an open
problem.
The other three procedures for checking rational observability and algebraic controllability of
rational systems, and minimality of rational realizations are based only on the verification of the
definitions (Definition 2.9, Definition 2.8, Definition 4.2) of the corresponding properties.
Procedure for checking rational observability of a rational system
system Σ = (X, f, h, x0 ).
Consider a rational
1. Calculate the observation algebra Aobs (Σ) of a rational system Σ.
2. Calculate the observation field Qobs (Σ) as a field of fractions of Aobs (Σ).
REALIZATION THEORY FOR RATIONAL SYSTEMS
16
3. Check whether B ⊆ Qobs (Σ), where B is a set of generators of the algebra of polynomials
on X, i.e. a set of generators of the quotient ring R[X1 , . . . , Xn ]/I where I is the ideal of
polynomials vanishing on X.
4. If B ⊆ Qobs (Σ), then Qobs (Σ) = Q, where Q is the field of rational functions on X, and
the rational system is algebraically observable. Otherwise the system is not algebraically
observable.
The fact that the observation field Qobs (Σ) is finitely generated and that a set B of generators
of the algebra of polynomials on X is a finite set could simplify the computations for the second
and the third step of the procedure. The third step of the procedure above could be executed
element-wise. The algorithms for checking whether an element of B (and therefore an element of
the field Q of rational functions on X) is also an element of the field Qobs (Σ) are described in [27]
and [28].
Procedure for checking algebraic controllability of a rational system To check algebraic
r
g
controllability of a rational realization Σ = (X, f, h, x0 ) of a response map p : U
pc → R we have
to be able to construct the set of points in X which can be reached from a given initial state by
g
g
applying only admissible inputs U
pc ⊆ Upc (Σ). Afterwards we have to check whether this set is
Z-dense in the state space.
We say that a rational system Σ = (X, f, h, x0 ) is algebraically controllable from an initial state
g
x0 if Z-cl({x(Tu ; x0 , u)|u ∈ U
pc (Σ)}) = X. It is algebraically controllable if it is algebraically
controllable from all admissible initial states. Therefore to check algebraic controllability of a
rational system Σ by definition means to construct all reachable sets from admissible initial states
and check whether they are Z-dense in X.
Another way how to check algebraic controllability of a rational system is to look for an
equivalent characterization of algebraic controllability. For rational realizations, Proposition 4.9
is such characterization. However, this proposition works only in the cases when the field Q of
rational functions on X is such that the elements of Q \ Qobs (Σ) are not algebraic over Qobs (Σ).
Procedure for checking minimality of a rational realization Consider a rational realization Σ = (X, f, h, x0 ) of a given response map p defined on a set of admissible inputs.
1. Calculate dim X of an irreducible real affine variety X as the degree of the affine Hilbert
polynomial of the corresponding ideal (ideal generated by the polynomials defining the variety X).
2. Compute the observation field Qobs (p) of the map p.
3. Calculate the transcendence degree of a field Qobs (p).
4. If dim X = trdeg Qobs (p), then the rational realization Σ of p is minimal. Otherwise Σ is
not minimal.
The first step of the procedure above is already implemented in Maple (see the command “HilbertDimension”). To calculate the observation field Qobs (p) of a response map p we proceed similarly
as if we calculate the observation field of the system. Firstly we calculate the observation algebra
of p as the smallest algebra containing pi , i = 1, . . . , r which is closed with respect to certain
derivations (see Definition 2.7) and then we build a field of fractions of its elements. The algorithms for computing the transcendence degree of field extensions of a field are presented in [27]
and (of not necessarily purely transcendental field extensions) in [28]. There are other algorithms
for the same problem which can be found in the references therein. These algorithms can be used
for computing the transcendence degree of an observation field since an observation field as we
defined it is a field extension of R.
Further research is needed for specifying more details of these procedures or developing new
procedures. For that reason a deeper study of differential and computational algebra would be
useful. An introduction to the differential algebra, developed by Ritt [31] and by Kolchin [25], is
provided by [23]. Especially for problems of control and system theory, several researchers developed a framework using differential algebra, see for example [14, 15, 16, 17], and the references
therein. Concerning the references on computational algebra, see for example the handbook [18]
and the text books [6, 10, 11, 19].
REALIZATION THEORY FOR RATIONAL SYSTEMS
8
17
Conclusions
This paper deals with minimal rational realizations of response maps defined on a set of admissible
inputs. The main results concern the existence of a minimal rational realization for a given
response map, and the properties of minimal realizations. We proved that if a response map
is realizable by a rational system, then there exists also a minimal rational realization of this
map. Minimal realizations are not unique. Every rational realization of a response map which
is birationally equivalent to a minimal rational realization of the same map is itself minimal. In
addition to the equivalence of minimality and minimal-dimensionality of rational realizations, we
derived the relations between the properties of rational realizations such as algebraic controllability,
rational observability, and minimality. We proved that canonical (both algebraically controllable
and rationally observable) realizations are minimal, and that two canonical rational realizations of
the same response map are birationally equivalent. Further, under a certain algebraic condition,
minimal realizations are canonical, and realizations which are not rationally observable are not
minimal. By assuming the negation of this algebraic condition, we showed that algebraically
controllable rational realizations are minimal. Finally, we provided an example and the first
sketches of the procedures for the construction of a rational realization of a given response map,
and for checking algebraic controllability, rational observability, and minimality of rational systems
and realizations.
We have chosen to work with the field R of real numbers due to the applications in real life
problems of biology and engineering. Because algebraic geometry is not restricted only to the field
of real numbers, our results could be generalized to arbitrary field k. For the computations and
developing procedures, computable fields like the field of rational numbers could be of interest. We
restricted our attention only to the state spaces defined as irreducible varieties which simplified
our approach. The generalization to reducible varieties is also possible.
Acknowledgements
The authors are very grateful to Mihály Petreczky for his careful reading of the paper and for his
very useful comments and advice.
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