Adv Syst Sci Appl 2023; 04:119–147
Published online at https://ijassa.ipu.ru.

Identifiability of a Family of Dynamical Systems:
Application to Crops Identification

Abdes Bernoussi1*, Edyta Woźniak2, Abdelaâziz Belfekih1

1Faculty of Sciences and Techniques Tangier, University Abdelmalek Essaadi, Tangier, Morocco
2Centrum Badań Kosmicznych PAN, Warsaw, Poland

Abstract: In this paper we consider the problem of identifying a system among a family of
given systems. Thus, from measurements collected on an unidentified system but that is part of
a family of known model systems, we seek to determine this unidentified system. This differs
from identifying the parameters of a given system through experimental observations [15].
The determination (identification) in a given family not always being possible, we refer to the
identifiable family as any family for which this identification is possible. We thus introduce
the concept of identifiability of a family of systems through a given measurement function.
For localized linear systems we give algebraic characterizations that use the notion of system
observability. We then propose algorithms which, in case of identifiability of the family and by
a process of elimination, identify the system to which the collected measurements correspond.
We have given some examples to illustrate these algorithms. We have also added an exemplified
extension to discrete localized systems.

Keywords: dynamical systems, identifiability, observability, remote sensing

1. INTRODUCTION

The modeling of a given phenomenon consists in describing it mathematically by the
equations that describe its dynamics. Moreover, in the model we have to take into account
various input-output parameters that allow interaction with the system. For the output, the
information are obtained via a measurement function. This function actually models the
sensors installed on this system and provides the collected measurements.

In general, the purpose of collecting measurements through these means of observation
is to estimate, when possible, the state of the system which evolves over time, which
subsequently allows corrections or adjustments of actions that one might have to exert on the
system. So knowing the model of the dynamics of the system and the measurement function,
the observability concept consists in the possibility to reconstruct the state of the system
and even predict its future evolution and this in order to control it to achieve predetermined
goals. The observability concept was introduced and characterized for localized systems
by Kalman [11, 12]. Due to the interest of this concept, many works were devoted to it
in many aspects for numerous applications [4, 19]. It was later generalized to distributed
systems [2, 8, 18]. Different degrees of observability were then introduced depending on the
considered problem: ”total” observability ( [10]), partial observability ( [6]) and regional
observability ( [1, 7]). The global state of the system can be determined by measurements in
the case of total observability; the ”visible” part of this state can be determined in the case

∗Corresponding author: abernoussi@uae.ac.ma



120 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

of partial observability and the trace of the state of the distributed system over an observable
region can be determined in the case of regional observability.

The problem that we are considering in this paper differs both in the data of the
problem and in the goal to be achieved. We assume that we have a family of models and
a measurement function, then the problem is the following: can we determine, from the
collected measurements, the model of our system knowing that it is in the given family?
If we manage to determine it, we will be able to do everything that could be done on a system
of which we have both the model and the measurements.

As it is not always possible to identify the system (or, more precisely, the model) that
can generate the measurements among any family of systems (see an example below), we
introduce the mathematical concept of identifiability of a family of systems: this is the
ability to differentiate the systems of this family from one another by a given measurement
function, which then makes it possible to identify exactly which of them corresponds to the
phenomenon under study. We define this concept for dynamic systems governed by equations
of the ”Cauchy problem” type in a fairly general way.

We will then distinguish two levels of identifiability: the first level is the identifiability
relative to a given function of measurement g, which we will refer to as the g identifiability (or
the C identifiability for linear systems); while the second level is absolute identifiability which
corresponds to the existence of a measurement function g which ensures g identifiability. It
is immediate to see that the first level, which relates to a measurement function, can only
be achieved if the second level is achieved, which is an intrinsic property of the family of
systems independently of the measurement function. We noted that the observability was
generally characterized by a rank condition of a given matrix. The computation of such rank
depends on the particularity of the matrices and for large matrices many numerical methods
are used [9, 14].

The study of the identifiability of a family of localized linear systems, both continuous (in
time) and discrete, reveals an abstract system and uses its “observability”.

The concept of identifiability, introduced in this paper, and observability one differ in
their philosophies. While observability is the possibility for a measurement function that is
associated with a system to reconstruct the state of that system from collected measurements,
identifiability is the possibility to differentiate a group of systems from one another using
a set of measurements, and to identify the system that generated those measurements. This
study is motivated by several real issues. Let’s quote two:

• In agriculture, can we identify the crop growing on a given plot of land from
measurements collected with satellite images? This is the problem of identifying a crop
by comparing its signature with the signatures of a number of crops.

• Among a certain number of physical systems, only one of these systems emits (or has
emitted) a signal; can we know which of these systems is sending this signal?
This paper is organized as follow: In a first part we define the identifiability of a

family of systems and we give a characterization for the linear localized systems by a rank
condition, similar to the one given for the observability or the controllability of the linear
systems. We then propose algorithms that test the identifiability of a family of systems, and
other algorithms which determine, for an identifiable family, the evolving system that has
generated the collected measurements. These algorithms are based on an elimination process.
Numerical simulations and an application are provided to illustrate our approach by applying
the developed algorithms. We then propose algorithms that test the identifiability of a family
of systems, and other algorithms that determine, for an identifiable family, the evolving
system that has generated the collected measurements. These algorithms are based on a
process of elimination. Numerical simulations and an application are provided to illustrate
our approach by applying the developed algorithms.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 121

2. IDENTIFIABILITY OF A FAMILY OF SYSTEMS:
INTRODUCTION OF THE CONCEPT

In this section we introduce the concept of identifiability of a family of two, and more
then many systems and this for systems that are not necessarily linear. Next, we give some
characteristics of the general case before dedicating ourselves to finite dimensional linear
systems in the following sections.

Consider two systems (S1) and (S2) governed by the following state equations (t > 0) :

(S1)

{
ż (t) = f1 (z (t))

z (0) = z10 ∈ Z
(S2)

{
ż (t) = f2 (z (t))

z (0) = z20 ∈ Z

where fi : Z → Z and Z an Hilbert space (states space). We assume that for each z10 ∈ Z,
the first equation admits a unique solution z1 (t) , t ⩾ 0, and that for each z20 ∈ Z, the second
equation admits a unique solution z2 (t) , t ⩾ 0.

Consider on the other hand an output function

y (t) = g (z (t)) , t ∈ [0, T ] (2.1)

where g : Z → Y and Y an Hilbert space (observations space). z (t) in (2.1) is one of the
states of these systems (S1) or (S2) , but we do not know which one. Actually, we have one
of two statements:

y (t) = g
(
z1 (t)

)
, t ∈ [0, T ] or y (t) = g

(
z2 (t)

)
, t ∈ [0, T ]

To be able to identify the system that has generated the collected measurements, it is
necessary that through the output y(t), the systems (S1) and (S2) give different signals.
Moreover, it is necessary that this difference appears for any state of (S1) and any state of
(S2) because when they have the same state then their outputs coincide and in this case the
origin of the measurements cannot be detected.
These conditions allow us to define the identifiable families concept to which we give the
following definition:
Definition 2.1:
(i) we say that the family of the two systems F = {(S1) , (S2)} is identifiable on [0, T ] by
the observation (2.1), or g−identifiable on [0, T ], if the obtained measurements of (S1) are
different from the obtained measurements of (S2) and this for all possible states of (S1) and
all possible states of (S2) , provided that at least one of the states is non-zero:

∀z1 (.) state of (S1) and ∀z2 (.) state of (S2) checking
z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, T ] , we have g (z1 (.)) ̸= g (z2 (.)) on [0, T ]

(2.2)

(ii) We say that the family of the two systems F = {(S1) , (S2)} is identifiable on [0, T ] , if
there exist a space Y and an application g : Z → Y such as the family F i.e. g−identifiable
on [0, T ] .

Remark 2.1:
According to (2.2), the family F = {(S1) , (S2)} is not g−identifiable if there exist a state
z1 (t) of the system (S1) and a state z2 (t) of the system (S2), non-zero, such that g (z1(t)) =
g (z2(t)) , ∀t ∈ [0, T ]. In this case we can have the same measurement for one or more states
of (S1) and one or more states of (S2) .

We have an important particular case which consists in taking Y = Z and g = Id; the
observation in this case is y (t) = z (t) , t ∈ [0, T ] , which corresponds to direct and complete
access to the state of the system. In this case we speak of Id−identifiable family whose

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



122 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

definition is as follows

∀z1 (.) state of (S1) and ∀z2 (.) state of (S2) satisfying
z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, T ] we have z1 (.) ̸= z2 (.) on [0, T ]

(2.3)

We then have the following proposition:
Proposition 2.1:
The family F = {(S1) , (S2)} is identifiable on [0, T ] if, and only if it is Id−identifiable on
[0, T ] .

Proof

• It is immediate to see that if F is Id− identifiable on [0, T ] , then it is identifiable on [0, T ]
since it is g−identifiable with Y = Z and g = Id.

• Conversely if F is identifiable on [0, T ], then there exists Y and g : Z → Y such that
F be g−identifiable on [0, T ] . For a state z1 (.) of (S1) and a state z2 (.) of (S2) , with
z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, T ] , we have g (z1 (.)) ̸= g (z2 (.)) on [0, T ] . We deduce that
z1 (.) ̸= z2 (.) on [0, T ] because z1 (.) = z2 (.) on [0, T ] would give g (z1 (.)) = g (z2 (.)) on
[0, T ] . This proves that F is Id−identifiable on [0, T ].

Remark 2.2:
It is therefore sufficient, to know if F = {(S1) , (S2)} is identifiable on [0, T ] to check (2.3) (if
it is Id−identifiable on [0, T ]). However, since in practice the measurement function is given,
it is necessary to test if the family is C−identifiable.

This concept can be generalised to several systems. Consider a family F =
{(S1) , . . . , (SN)} of N systems and suppose that each system (Si) is governed by the state
equation

(Si)

{
ż (t) = fi (z (t)) , t > 0

z (0) = zi0 ∈ Z

with fi : Z → Z. We also assume that for each zi0 ∈ Z, this equation admits a unique solution
denoted by zi (.) . The output equation is always given by (2.1), we consider the following
definitions which generalize the definition 2.1.
Definition 2.2:
(i) We say that the family of systems F = {(S1) , . . . , (SN)} is g−identifiable on [0, T ] if
every subfamily {(Si) , (Sj)}, i ̸= j, is g−identifiable on [0, T ] .
(ii) We say that the family of systems F = {(S1) , . . . , (SN)} is identifiable on [0, T ] if every
subfamily {(Si) , (Sj)} is identifiable on [0, T ] .

As in the case of two systems, we will say that the family {(S1) , . . . , (SN)} is
Id−identifiable on [0, T ] if each subfamily {(Si) , (Sj)} is Id−identifiable on [0, T ] .

Proposition 2.2:
The family of systems F = {(S1) , . . . , (SN)} is identifiable on [0, T ] if and only if it is
Id−identifiable on [0, T ] .

Proof
The family F is identifiable on [0, T ] if and only if each subfamily {(Si) , (Sj)}, i ̸= j, is
identifiable on [0, T ] , or if and only if, thanks to the proposition 2.1, each {(Si) , (Sj)} , i ̸= j,
is Id−identifiable on [0, T ] , which is equivalent to what the family F i.e. Id−identifiable
on [0, T ] .

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 123

3. IDENTIFIABILITY OF FAMILIES OF FINITE
DIMENSIONAL LINEAR SYSTEMS

Consider a family of two systems whose dynamics are governed by the two following linear
state equations:

(S1)

{
ż (t) = A1z (t)

z (0) = z10 ∈ Rn
, (S2)

{
ż (t) = A2z (t)

z (0) = z20 ∈ Rn

with A1 and A2 are two square matrices of order n. The state of (S1) , noted by z1 (.), is
given by z1 (t) = exp (tA1) z

1
0 and the state of (S2) , noted by z2 (.), is given by z2 (t) =

exp (tA2) z
2
0 . The system is augmented by the following linear measurement function

(E) y (t) = Cz (t) , 0 ⩽ t ⩽ T

with C a matrix with q rows and n columns. z (t) is either z1 (t) or z2 (t); So we have
y (t) = Cz1 (t) , t ∈ [0, T ] , or y (t) = Cz2 (t), t ∈ [0, T ] .

Remark 3.1:
The matrices A1 and A2 are necessarily of the same order n since we use the same output
(and therefore the same matrix C) to measure each of the two systems.

Note that the output (E) is of the form (2.1) with the choice g(u) = Cu, u ∈ Rn; hence,
and in all that follows, the families g−identifiable will be called C−identifiable and families
Id−identifiable will be called In−identifiable.

Let us go back to the definition 2.1. The characterisation (2.2) takes the form

∀z1 (.) state of (S1) and ∀z2 (.) state of (S2) satisfying
z1 (.) ̸= 0 or z2 (.) ̸= 0 on [0, T ] we have Cz1 (.) ̸= Cz2 (.) on [0, T ]

By contraposition, and denoting zk0 = zk(0) the initial state of the system (Sk), we obtain the
characterisation of the C−identifiability of a family of two systems as given in the following
proposition:
Proposition 3.1:
(i) The family F = {(S1) , (S2)} is C−identifiable on [0, T ] if and only if the only states of
the two systems which give the same measurements are the null states (generated by zero
initial conditions):

z1 (.) state of (S1) , z
2 (.) state of (S2)

and Cz1 (.) = Cz2 (.) on [0, T ]

}
⇒ z10 = z20 = 0 (3.4)

(ii) The family F = {(S1) , (S2)} is identifiable on [0, T ] if, and only if, there exists an integer
q ⩾ 1 and a matrix C ∈Mq,n (R) such that F is C−identifiable on [0, T ] .
(iii) F = {(S1) , (S2)} is identifiable on [0, T ] if, and only if, it is In−identifiable:

z1 (.) state of (S1) z2 (.) state of (S2)
and z1 (.) = z2 (.) on [0, T ]

}
⇒ z10 = z20 = 0

This proposition which characterises C−identifiable (resp. identifiable) families of two
linear systems can be easily generalised to families of several linear systems.
Example 3.1:
(i) Consider the two systems {(S1) , (S2)} given by the following state equations:

(S1)

(
ż1 (t)
ż2 (t)

)
=

(
1 0
0 2

)(
z1 (t)
z2 (t)

)
, (S2)

(
ż1 (t)
ż2 (t)

)
=

(
3 0
0 1

)(
z1 (t)
z2 (t)

)
Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



124 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

with the measurement function y (t) = z1 (t) + z2 (t) , 0 ⩽ t ⩽ T. This corresponds to C =

( 1 1 ) . For z10 =

(
1
0

)
and z20 =

(
0
1

)
the respective states are z1 (t) =

(
et

0

)
and

z2 (t) =

(
0
et

)
which give Cz1 (t)− Cz2 (t) = ( 1 1 )

(
et

0

)
− ( 1 1 )

(
0
et

)
= 0,

∀t ⩾ 0; the family {(S1) , (S2)} is therefore not C−identifiable.
(ii) Consider now the two systems {(S1) , (S

′
2)} given by the following state equations:

(S1)

(
ż1 (t)
ż2 (t)

)
=

(
1 0
0 2

)(
z1 (t)
z2 (t)

)
and (S ′

2)

(
ż1 (t)
ż2 (t)

)
=

(
3 0
0 4

)(
z1 (t)
z2 (t)

)

with the same measurement function y (t) = z1 (t) + z2 (t) , 0 ⩽ t ⩽ T. For z10 =

(
a
b

)
and

z20 =

(
c
d

)
the respective states are z1 (t) =

(
aet

be2t

)
and z2 (t) =

(
ce3t

de4t

)
which give

Cz1 (t)− Cz2 (t) = aet + be2t − ce3t − de4t. The family of functions {et, e2t, e3t, e4t} is free,
equality aet + be2t − ce3t − de4t = 0, t ∈ ]0, T [ , gives a = b = c = d = 0 or z10 = z20 = 0,
which proves that the family {(S1) , (S2)} is C−identifiable.

Before characterising the C-identifiable families, we will recall in the following section
the definition of the observable system as well as its algebraic characterisation. Then we will,
in this case, give the expression of the state from the collected measurements. This will be
useful for us to characterize algebraically the C−identifiable families.

3.1. Observable System and State Reconstruction
The observability concept of a localised system reflects the ability of an output to reconstruct
the state of the system in a “faithful”manner. Let us now the definition of observable system.

Consider a linear system whose dynamics is described by the following state equation

(S)

{
ż (t) = Az (t) , t > 0

z (0) = z0 ∈ Z = Rn

where A is a square matrix of order n representing the dynamics of the system, and consider
the output function given by the following equation:

y (t) = Cz (t) , t ∈ [0, T ] (3.5)

where C is a given matrix with q rows and n columns.

Definition 3.1:
The system (S) is said to be observable for the measurement function (3.5) on [0, T ] if two
different states of the system give two different measurements:

z (t) ̸= z̃ (t) on [0, T ] ⇒ Cz (t) ̸= Cz̃ (t) on [0, T ]

We then say that (A,C) is observable on [0, T ] .

Thanks to the linear character of the above equations, we can see that this definition is
equivalent to the following

(Cz (t) = 0 , t ∈ [0, T ]) ⇒ z0 = 0 (3.6)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 125

where z0 = z (0) is the initial state of the system. Let us introduce the matrix O with nq rows
and n columns

O =


C
CA
CA2

...
CAn−1


called observability matrix of the system (S), and the symmetric square matrix M of order
n:

M =

∫ T

0

exp
(
tAT

)
CTC exp (tA) dt (3.7)

Then we have the following result ( [1]):
Proposition 3.2:
(i) The system (S) is observable on [0, T ] for the measurement function (3.5) if and only if
rank (O) = n, or again if and only if the matrix M is invertible.
(ii) If the system (S) is observable on [0, T ] for the observation (3.5) and if we have the
measurements ymes (t) , t ∈ [0, T ], then the state of the system can be determined by

z (t) = exp (tA) z0 , t ⩾ 0, (3.8)

where

z0 = M−1

∫ T

0

exp
(
sAT

)
CTymes (s) ds (3.9)

The observability of a localised system is often characterised by the rank condition:
rank (O) = n.

3.2. Characterisation of Identifiable Families
Let us consider the family of two systems F = {(S1) , (S2)} as well as the observation (E)
and let us look for a condition equivalent to condition (3.4) of the C−identifiability of F ,
where z1(t) is the state of (S1) and z2(t) is the state of (S2). We have

Cz1 (t)− Cz2 (t) = C exp (tA1) z
1
0 − C exp (tA2) z

2
0

= [ C exp (tA1) −C exp (tA2) ]

[
z10
z20

]
= [ C −C ] exp

(
t

[
A1 O
O A2

])[
z10
z20

]
= C exp (tA ) ξ0

where we have noted that

A =

[
A1 O
O A2

]
, C = [ C −C ] , ξ0 =

[
z10
z20

]
The condition given by (3.4) then becomes

(C exp (tA ) ξ0 = 0 , t ∈ [0, T ]) ⇒ ξ0 = 0

which is exactly the characterisation of the observability of the couple (A ,C ) . The matrices
A and C can be associated to an abstract linear system of order 2n and an observation of
order q; then we have the following:

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



126 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

Proposition 3.3:
The family F = {(S1) , (S2)} is C−identifiable on [0, T ] if, and only if, the following 2n
order system:

(S0)

{
ξ̇ (t) = A ξ (t) , t > 0

ξ (0) = ξ0 ∈ Rn ×Rn
(3.10)

is observable on [0, T ] for the output

η (t) = C ξ (t) , t ∈ [0, T ] (3.11)

We deduce a characterisation of the C−identifiability of {(S1) , (S2)} via the
observability of (S0) . This observability is verifiable by calculating the rank of the
observability matrix of the latter system. Indeed the system (S0) is observable if and only
if its observability matrix

O =


C

C A
C A 2

...
C A 2n−1


has rank 2n. And since for any integer k we have C A k =

[
C (A1)

k −C (A2)
k
]
, then O

is expressed as a function of A1, A2 and C. Noting it DC , this matrix becomes

DC =


C −C

CA1 −CA2

C (A1)
2 −C (A2)

2

...
...

C (A1)
2n−1 −C (A2)

2n−1

 (3.12)

Thus, we obtain an algebraic characterisation of the C−identifiability of a family of two
systems and thus of its identifiability. This gives us an algebraic characterisation of the
C−identifiability of a family of two systems and hence of its identifiability:

Theorem 3.1:
(i) The family F = {(S1) , (S2)} is C−identifiable on [0, T ] if and only if the matrix DC ,
called the C−identifiability matrix of F , has rank 2n:

rank (DC) = 2n

(ii) The family F = {(S1) , (S2)} is identifiable if and only if the matrix DIn , called matrix
of identifiability matrix of the family F , is of rank 2n:

rank (DIn) = 2n

Remark 3.2:
1. The (ii) in theorem 1 shows that the C−identifiability and identifiability do not depend on
the observation time interval; it suffices that it is of non-zero length.
2. If A1 = A2, the problem of identifying does not arise because we have the same system.
Note that in this case, and as one might expect, the family F is not C−identifiable since the
matrix of C−identifiability is in this case of rank ⩽ n (by adding in DC each column k to
column n+ k we get n null columns that can be eliminated).

The theorem 3.1 proves that:

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 127

• If rank (DIn) < 2n then the family {(S1) , (S2)} is non identifiable and consequently non
C−identifiable for none matrix C (i.e. none given output).

• If rank (DIn) = 2n then the family {(S1) , (S2)} is identifiable and therefore there are C
matrices for which this family is C−identifiable i.e. rank (DC) = 2n.

Example 3.2:

(i) Let us take the example 3.1 where A1 =

(
1 0
0 2

)
and A2 =

(
3 0
0 4

)
. The

identifiability matrix is then

DIn =

 I2 −I2
A1 −A2

(A1)
2 − (A2)

2

(A1)
3 − (A2)

3

 =



1 0 −1 0
0 1 0 −1
1 0 −3 0
0 2 0 −4
1 0 −9 0
0 4 0 −16
1 0 −27 0
0 8 0 −64


DIn is of rank 4 so the family {(S1) , (S2)} is identifiable.
(ii) Now take C = ( 1 0 ); the C−identifiability matrix is given by

DC =

 C −C
CA1 −CA2

C (A1)
2 −C (A2)

2

C (A1)
3 −C (A2)

3

 =

 1 0 −1 0
1 0 −3 0
1 0 −9 0
1 0 −27 0


which is of rank 2. The family {(S1) , (S2)} is therefore not C−identifiable (although it is
identifiable).
Remark 3.3:
Consider the matrix C̃ defined by blocks of type (q, n) , having 2n row blocks, 2n column
blocks, the diagonal blocks are equal to C and non-diagonal blocks are zero (C̃ is therefore
of type (2nq, 2n2)). So we have DC = C̃ DIn which gives us

rank (DC) ⩽ rank (DIn) ⩽ 2n , ∀C

Inequality allowing to find that any C-identifiable family is identifiable.
In the case where the matrices A1 and A2 commute, we have a characterisation of

identifiability with a reduced dimension matrix. The proof uses the following lemma:
Lemma 3.1:
Let A be a square matrix of order n with real coefficients. So

(exp (tA) z0 = Ct on ]0, T [) ⇒ z0 = 0 (3.13)

if and only if the following matrix has rank n
A
A2

...
An

 (3.14)

where Ct denotes a constant vector.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



128 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

Proof
Since d

dt
exp (tA) z0 = A exp (tA) z0, ∀t ⩾ 0, then we have equivalence

(exp (tA) z0 = Ct , ∀t ∈ [0, T ]) ⇔ (A exp (tA) z0 = 0 , ∀t ∈ [0, T ])

and the implication (3.13) becomes

(A exp (tA) z0 = 0 , ∀t ∈ [0, T ]) ⇒ z0 = 0

This implication is none other than the characterisation (3.6) of the observability for the
following observed system{

η′ (t) = Aη (t) , t ⩾ 0

η0 ∈ Rn
;

y (t) = Aη (t)

t ∈ [0, T ]

whose observability matrix is exactly the matrix (3.14). The proposition 3.2 then makes it
possible to obtain the equivalence.

Using this lemma, we have the following proposition:
Proposition 3.4:
Suppose that the matrices A1 and A2 commute. Then family {(S1) , (S2)} is identifiable if and
only if the following matrix has rank n

M =


A2 − A1

(A2 − A1)
2

...
(A2 − A1)

n


Proof
The characterisation of identifiability given by (3.4) becomes, when C = In ,(

exp (tA1) z
1
0 = exp (tA2) z

2
0 , ∀t ∈ [0, T ]

)
⇒ z10 = z20 = 0

Since A1 and A2 commute the equality exp (tA1) z
1
0 = exp (tA2) z

2
0 is equivalent to

exp (t (A2 − A1)) z
2
0 = z10 , ∀t ∈ [0, T ] . Taking A = A2 − A1 in the lemma 3.1, the family

{(S1) , (S2)} is identifiable by the observation (E) on [0, T ] if and only the matrix M has
rank n.

Example 3.3:
Consider the systems (S1) and (S2) whose dynamics are defined by the matrices

A1 =

(
1 0
0 1

)
and A2 =

(
2 1
2 −1

)
In this case the identifiability matrix DI2 admits 8 lines. Note that the matrices A1 and A2

commute, and let us apply the previous proposition; the matrix M admits 4 rows and it is
equal to

M =

[
A2 − A1

(A2 − A1)
2

]
=

 1 1
2 −2
3 −1
−2 6


M is of rank 2, so the family {(S1) , (S2)} is identifiable.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 129

On the other hand, the observability of both systems (S1) and (S2) cannot guarantee
the C−identifiability of the family {(S1) , (S2)}; however this observability is necessary as
shown the following proposition:

Proposition 3.5:
If the family {(S1) , (S2)} is C−identifiable then each system of this family is observable by
the observation defined by C. The converse is not true.

Proof
• Let z1 (.) be a state of (S1) such that Cz1 (t) = 0, t ∈ [0, T ] , and let z2 (.) be a state of
(S2) such that Cz2 (t) = 0, t ∈ [0, T ]; then Cz1 (t)− Cz2 (t) = 0, t ∈ [0, T ] . The family
{(S1) , (S2)} being C−identifiable, we get z10 = 0 and z20 = 0. This proves that both (S1) are
observable and (S2) is observable.
• The converse is not true. Simply consider both systems in R2 and the observation defined
by

A1 =

(
1 0
0 2

)
, A2 =

(
2 0
1 1

)
, C = ( 1 −1 )

We verify that {(S1) , (S2)} is not C−identifiable while every system is observable.

As a consequence of proposition 3.5 we have:

Corollary 3.1:
If one of the two systems is not observable for the observation given by C then the family
{(S1) , (S2)} is not C−identifiable.

Remark 3.4:
If the family {(S1) , (S2)} is identifiable then we can find at least one matrix C which makes
it possible to differentiate these systems (and sometimes several matrices). By introducing a
comparison criterion between these C matrices, we can take the one that verifies a certain
optimality. We take this into account in our outlook

Using the above we can give a first algorithm which detects if a family of two systems is
C−identifiable or not:

Algorithm 1 C−identifiability test of a family of 2 systems.

1. Form the matrix DC ;

2. If rank (DC) = 2n: Display “Family C−identifiable.”; STOP

3. Else, Display “Family non C−identifiable.”

3.3. Algorithm for Identifying the Searched System in a Family
Once the possibility of differentiating between two specific systems is acquired, the problem
arises of the effective determination of the system which has generated the collected
measurements {ymes (t) , t ∈ [0, T ]} . The algorithm we present allows to identify this system.
It relies on the C−identifiability of the family {(S1) , (S2)} and its unfolding is done in two
steps. First, we estimate the states of the two systems from measurements collected over an
arbitrary interval [0, T1]; As a second step, and with measurements collected over a second
interval [T1, T2] that we will determine, comparisons are made which eliminate the system
which cannot generate these measurements.
Principle. Suppose the family {(S1) , (S2)} is C−identifiable over an interval of time [0, T ].

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



130 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

a) We assume that measurements are taken ymes (t) on a time interval [0, T1] (T1 < T )
and supposing that these measurements are non-zero. If the system (S1) generated these
measurements then its state, which is necessarily non-zero, is given by

z1 (t) = exp (tA1) z
1
0 , t ⩾ 0

with

z10 = (M1)
−1

∫ T1

0

exp
(
sAT

1

)
CTymes (s) ds

where we have noted (k = 1, 2)

Mk =

∫ T1

0

exp
(
sAT

k

)
CTC exp (sAk) ds (3.15)

If on the other hand it is the system (S2) which generated these measurements then its state,
which is necessarily non-zero, is given by

z2 (t) = exp (tA2) z
2
0 , t ⩾ 0

with

z20 = (M2)
−1

∫ T1

0

exp
(
sAT

2

)
CTymes (s) ds

b) If we have the information that the system sought is one of the two systems then we have
only two cases to study. In the general case, we have four cases (the first three of which lead
to outputs):

Case 1. Cz1 (T1) = ymes (T1) and Cz2 (T1) ̸= ymes (T1), then the measurements come from
(S1) and not from (S2); (S1) is therefore the sought system;

Case 2. Cz1 (T1) ̸= ymes (T1) et Cz2 (T1) = ymes (T1), then the measurements come from (S2)
and not from (S1); (S2) is therefore the sought system;

Case 3. Cz1 (T1) ̸= ymes (T1) and Cz2 (T1) ̸= ymes (T1), then the measurements come neither
from (S1) nor from (S2);

Case 4. Cz1 (T1) = ymes (T1)Cz2 (T1) = ymes (T1), so we cannot differentiate these systems on
[0, T1]; we then apply the following:
c) The family {(S1) , (S2)} is C−identifiable on [T1, T ] and one of the states is non-zero,
we can find an instant T2 such that

T2 ∈ ]T1, T [ and Cz1 (T2) ̸= Cz2 (T2) (3.16)

Indeed in the opposite case we will have Cz1 (t) = Cz2 (t) , ∀t ∈ ]T1, T [ , and
C−identifiability from {(S1) , (S2)} gives z10 = z20 = 0 in contradiction with non-zero
measurements.
d) We resume step b) with T2 instead of T1; the output is then made by one of the first
three cases only, the last case cannot be realised thanks to (3.16).

In this algorithm, tolerance plays a key role. We set a tolerance ε > 0 and for two vectors
u, v ∈ Rn, we will try to fulfill the condition u = v by the test ∥u− v∥ < ε and to fulfill the
condition u ̸= v by the test ∥u− v∥ ⩾ ε, where ∥u∥ denotes the norm of the vector u. This
gives the following algorithm:

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 131

Algorithm 2 Identification of the system sought in a family C−identifiable from 2 systems.

1. Take measurements ymes (t) on [0, T1];

2. Compute initial states z10 and z20 , and the states z1 (t) and z2 (t);

3. If
∥∥Cz1 (T1)− ymes (T1)

∥∥ ⩾ ε : Reject the system (S1);

4. If
∥∥Cz2 (T1)− ymes (T1)

∥∥ ⩾ ε : Reject the system (S2);

5. If only one system is not rejected: It is the sought system; STOP

6. If both systems are rejected: Go to 13

7. Determine T2 > T1 such that
∥∥Cz1 (T2)− Cz2 (T2)

∥∥ ⩾ ε;

8. Take the measurements ymes (t) on [T1, T2];

9. If
∥∥Cz1 (T2)− ymes (T2)

∥∥ ⩾ ε : Reject the system (S1);

10. If
∥∥Cz2 (T2)− ymes (T2)

∥∥ ⩾ ε : Reject the system (S2);

11. If only one system is not rejected: It is the sought system; STOP

12. If no system is rejected: Display “Tolerance too large.”; STOP

13. Display “No system matches the measurements or tolerance too low.”

Determination of the moment T2

The existence of T2 is ensured by the C−identifiability of the family but this moment is
unknown. To determine this we present two methods, both at random:
Method 1 (Monte-Carlo method) we choose T2 ∈ ]T1, T [ randomly (according to the normal
law) and, as long as we have Cz1 (T2) = Cz2 (T2), we do a another choice of T2.
Method 2

• We choose a time step δ > 0; we start with T2 = T1 + δ and while we have Cz1 (T2) =
Cz2 (T2) we increase T2 by δ: T2 ← T2 + δ. If we do not fulfill this condition, we have

Cz1 (kδ) = Cz2 (kδ) , k = 0, 1, 2, . . . ,
T

δ

We can then avoid this situation by slightly changing δ.
• We can also take δ large enough and T2 = T1 + δ, and as long as we have Cz1 (T2) =
Cz2 (T2), we change δ by δ/2 and resume T2 = T1 + δ; the condition Cz1 (T2) =
Cz2 (T2) will be then checked at a certain step, except in very special cases where the
changing δ fixes the problem.

3.4. Generalisation to Families of Several Systems
In applications, it is rare to have to identify one system among two systems. The most usual
way is to seek to identify it among several given systems (in a database). Therefore, we
assume that a given family F = {(S1) , . . . , (SN)} of N linear systems is governed by

(Si)

{
ż (t) = Aiz (t) , t > 0

z (0) = zi0 ∈ Rn
i = 1, . . . , N

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



132 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

where the matrices Ai which model their dynamics are all of dimension n. We also assume
that we have an observation

(E) y (t) = Cz (t) , t ∈ [0, T ] (3.17)

where z (t) is the state of the system and C a matrix of type (q, n). We can generalise the
theorem 3.1 which characterises C−identifiability and identifiability to families of several
systems.

Let us introduce the matrices

D ij
C =


C −C

CAi −CAj

C (Ai)
2 −C (Aj)

2

...
...

C (Ai)
2n−1 −C (Aj)

2n−1


i, j = 1, . . . , N

i ̸= j

Then we have the following characterisations:
Theorem 3.2:
(i) The family of systems F = {(S1) , . . . , (SN)} is C−identifiable if and only if

rank
(
D ij

C

)
= 2n , i, j = 1, . . . , N , i ̸= j

(ii) The family of systems F = {(S1) , . . . , (SN)} is identifiable if and only if

rank
(
D ij

In

)
= 2n , i, j = 1, . . . , N , i ̸= j

From this theorem we deduce the following algorithm which allows to test the
C−identifiability of family F :

Algorithm 3 C−identifiability test for a family of systems.

1. For i = 1, . . . , N and for j = 1, . . . , N, j ̸= i

2. Determine the matrix D ij
C ;

3. If rank
(
D ij

C

)
< 2n : Display “Family {(Si) , (Sj)} non C−identifiable.”; STOP

4. next j ; next i

5. Display “Family C−identifiable.”

For a family {(S1) , . . . , (SN)} C−identifiable we generalise the previous algorithm to
the case of N systems with a slight adaptation imposed by the case of several systems using
the algorithm below.

To identify the system sought we take in the first step the measurements on [0, T1] and we
calculate the state of each system; then, using the measurements collected on a second interval
[T1, T2] that is determined by one of the methods we make comparisons that eliminate systems
that cannot generate these measurements. This last operation is repeated for more than one
system, as long as the observation generated by its state coincides in T2 with the measurement
obtained. In more details:

a) We take measurements on [0, T1] (arbitrary), and we determine the state zi (t) of each
system (Si) by

zi0 (t) = exp (tA1) z
i
0 , t ⩾ 0, i = 1, . . . , N

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 133

with

zi0 = (Mi)
−1

∫ T1

0

exp
(
sAT

i

)
CTymes (s) ds

b) At each step τ we reject each system whose output associated with its state at the time
Tτ does not coincide with the measurement at this instant. If a single system is not rejected,
it is the system sought.

c) If all the systems are rejected it is that, or none of the systems does not correspond to
the measurements, or the tolerance is too low.

d) Otherwise, we take two non-rejected systems, we determine an instant Tτ+1 > Tτ

where their observations at this instant are distinct, we take measurements on [Tτ , Tτ+1] and
we start again at step b) with this time Tτ+1 instead of Tτ .
Step by step the number of non-rejected systems is reduced, until one is reached and then and
only then the sought system is found.
If at some stage the number of non-rejected systems is ⩾ 2 and is not reduced, it is because
the tolerance is too large: we change tolerance and resume at step b).
Remark 3.5:
In some cases the only tests in T1 are enough to determine the sought system (see the example
in section 3.5).

This gives the algorithm 4.

Algorithm 4 Identification of the sought system in a family C−identifiable.

1. Take the measurements ymes (t) on [0, T1];

2. Compute the initial states zi0 and the states zi (t) , 1 ⩽ i ⩽ N ;

3. Reject each system (Sk) that checks
∥∥Czk (T1)− ymes (T1)

∥∥ ⩾ ε;

4. If all systems are rejected: Go to 14

5. If only one system is not rejected: this is the sought system; STOP

6. Else, choose 2 non-rejected systems (Si) and (Sj) , i ̸= j;

7. Determine T2 > T1 such that
∥∥Czi (T2)− Czj (T2)

∥∥ ⩾ ε;

8. Take measurements ymes (t) on [T1, T2];

9. Reject every system (Sk) that matches
∥∥Czk (T2)− ymes (T2)

∥∥ ⩾ ε;

10. If all systems are rejected: Go to 14

11. If only one system is not rejected: this is the sought system; STOP

12. If the number of unrejected systems has not decreased: Display “Tolerance too large.”; STOP

13. Do T1 ← T2 and resume at 6

14. Display “None of the systems in the family matches the measurements or tolerance too low.”

Remark 3.6:
The algorithm 3is applied once to find out if the systems registered in the database form a
C−identifiable family. It is in the affirmative that we apply algorithm 4 and we reapply it
each time we have an identification to make both with this family of systems and for any
measurements collected.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



134 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

3.5. Numerical Example
Consider a family of three systems whose dynamics are modeled by the matrices

A1 =

(
1 0
0 2

)
, A2 =

(
3 0
0 4

)
, A3 =

(
2 −1
1 2

)
, C = ( 1 1 ) (3.18)

where C intervenes in the measurement function. To verify the C−identifiability of the family
{(S1) , (S2) , (S3)} we calculate the matrices D ij

C

D12
C =

 1 1 −1 −1
1 2 −3 −4
1 4 −9 −16
1 8 −27 −64

 , D13
C =

 1 1 −1 −1
1 2 −3 −1
1 4 −7 1
1 8 −13 9


D23

C =

 1 1 −1 −1
3 4 −3 −1
9 16 −7 1
27 64 −13 9


(3.19)

Since they are all of rank 4 then the family {(S1) , (S2) , (S3)} is C−identifiable. We consider
the tolerance ε = 0.01.

• First example of measurements

1) We take measurement on [0, T1] with T1 = 1. We obtain ymes (t) = 2e4t − e3t , t ∈
[0, 1] .

2) We compute the matrices Mi by (3.15), we obtain

M1 =

(
1
2
e2 − 1

2
1
3
e3 − 1

3
1
3
e3 − 1

3
1
4
e4 − 1

4

)
, M2 =

(
1
6
e6 − 1

6
1
7
e7 − 1

7
1
7
e7 − 1

7
1
8
e8 − 1

8

)
M3 ≃

(
25.70 0.22
0.22 1.09

) (3.20)

which make it possible to calculate the initial states

z10 ≃
(
−23.70
19.06

)
, z20 =

(
−1.0
2.0

)
, z30 ≃

(
5.76
−9.08

)
and from this the states of the three systems

z1 (t) ≃
(
−23.7et
19.06e2t

)
, z2 (t) =

(
−e3t
2e4t

)
, z3 (t) ≃

(
5.76 cos t+ 9.08 sin t
5.76 sin t− 9.08 cos t

)
3) (i) For (S1) : since ∣∣Cz1 (T1)−

(
2e4T1 − e3T1

)∣∣ ≃ 12.69 ⩾ ε

we reject (S1).
(ii) For (S2): since

∣∣Cz2 (T1)−
(
2e4T1 − e3T1

)∣∣ = 0.0 < ε, we keep (S2).
(iii) For (S3) : since

∣∣Cz3 (T1)−
(
2e4T1 − e3T1

)∣∣ ≃ 298.95 ⩾ ε, we reject (S3) .
4) Only (S2) is not rejected: therefore (S2) is the sought system.

• Second example of measurements

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 135

1) We take the measurement on [0, T1] with T1 = 1. We get ymes (t) = 4e2t cos (t) ,
t ∈ [0, 1] .

2) The matrices Mi are already calculated by (3.20); we calculate the initial states:

z10 ≃
(

12.52
−5.88

)
, z20 ≃

(
4.27
−1.31

)
, z30 =

(
2.0
2.0

)
and the states

z1 (t) ≃
(

12.52et

−5.88e2t
)

, z2 (t) ≃
(

4.27e3t

−1.31e4t
)

, z3 (t) = e2t
(

2 cos t− 2 sin t
2 cos t+ 2 sin t

)
3) (i) For (S1) : since

∣∣Cz1 (T1)− 4e2T1 cos (T1)
∣∣ ≃ 25.38 ⩾ ε, we reject (S1) .

(ii) For (S2) : since
∣∣Cz2 (T1)− 4e2T1 cos (T1)

∣∣ ≃ 32.47 ⩾ ε, we reject (S2) .

(iii) For (S3) : since
∣∣Cz3 (T1)− 4e2T1 cos (T1)

∣∣ = 0.0 < ε, we keep (S3) .
4) (S3) is the only system not rejected; so (S3) is the sought system.

Finally we obtain the following assignments:

plot 1 2
System (S2) (S3)

4. CASE OF DISCRETE FINITE DIMENSIONAL LINEAR SYSTEMS

We note that the majority of environmental phenomena has a very slow evolution; this allows
us to assume for one of these systems that are modelling these phenomena that, after an
instant tk, its state only shows a significant change at an instant tk+1. The measurements can
then be considered as taken at instants tk. All these considerations lead us to consider what
becomes of everything discussed earlier when the systems have discrete models.

4.1. Identifiability of a Family of Finite Dimensional Discrete Linear Systems
Suppose the two dynamical systems are governed by the following two recurring equations

(S1)

{
zk+1 = A1zk k = 0, 1, . . .

z0 ∈ Z
(S2)

{
zk+1 = A2zk k = 0, 1, . . .

z0 ∈ Z

The state of (S1) is denoted z1k and the state of (S2) is denoted z2k. Suppose also that the
measurements are discrete

(E) yk = Czk , k = 0, 1, . . . ,m− 1

where zk is either z1k or z2k. If ymes
k , k = 0, 1, . . . ,m− 1, are collected measurements; so we

have either ymes
k = Cz1k , k = 0, 1, . . . ,m− 1 or else ymy

k = Cz2k , k = 0, 1, . . . ,m− 1. We
take up the same problem of the identification posed in the previous sections.
Definition 4.1:
(i) We say that the family {(S1) , (S2)} is C−identifiable for the first m measurements if[

z1k state of (S1) , z
2
k state of (S2) and

∃k ⩽ m− 1 such as z1k ̸= 0 or z2k ̸= 0

]
⇒

(
∃j ⩽ m− 1 such as Cz1j ̸= Cz2j

)
(ii) We say that the family {(S1) , (S2)} is identifiable for the first m measurements if there
exist an integer q ⩾ 1 and a matrix C of type (q, n) such that this family is C−identifiable for
the first m measurements.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



136 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

By taking the contrapositive, and making the initial states z10 and z20 appear, the definition
of C−identifiability becomes[

z1k state of (S1) , z
2
k state of (S2)

and Cz1k = Cz2k, k = 0, . . . ,m− 1

]
⇒ z10 = z20 = 0 (4.21)

We will adopt this implication to characterize the families C−identifiable formed by two
discrete systems.

In the general case, the sought system is among N systems of a given family. We assume
that these systems have the equations

(Si)

{
zk+1 = Aizk , k = 0, 1, 2, . . .

z0 = zi0 ∈ Rn i = 1, . . . , N

where the matrices Ai, which model their dynamics, are all of dimension n. We can generalize
the definition of identifiability to families of several systems as follows:
Definition 4.2:
(i) The family of systems {(S1) , . . . , (SN)} is said to be C−identifiable for the first m
measurements if each subfamily {(Si) , (Sj)} , i ̸= j, is C−identifiable for the observation
via the first m measurements.
(ii) The family of systems {(S1) , . . . , (SN)} is said to be identifiable for the first m
measurements if each subfamily of systems {(Si) , (Sj)} , i ̸= j, is identifiable for the first
m measurements.

4.2. Observable Discrete System and State Reconstruction
As for continuous systems in time, we use the observability of an abstract discrete system for
the characterisation of C−identifiability.
Consider a dynamical system governed by the following recurring equation of state:

(S)

{
zk+1 = Azk , k = 0, 1, 2, . . .

z0 ∈ Z = Rn

where A is a given square matrix of order n with real coefficients. The state of the system
has the expression zk = Akz0 , k = 0, 1, 2, . . .. This equation of state is augmented by a
measurement function (output equation ):

yk = Czk , k = 0, 1 . . . ,m− 1 (4.22)

where C is a given matrix with q rows and n columns.
Definition 4.3:
The system (S) is said to be observable for observation (4.22) on the discrete interval
0, 1, . . . ,m− 1 if for two states any distinct zk and z̃k of this system correspond to two distinct
measurements:

(∃k ⩽ m− 1 such as zk ̸= z̃k) ⇒ (∃j ⩽ m− 1 such as Czj ̸= Cz̃j)

We then say that the couple (A,C) is observable for the first m measurements (or on
0, 1, . . . ,m− 1).

One can easily see that this definition is equivalent to the following characterisation:

(Czk = 0 , k = 0, 1, . . . ,m− 1) ⇒ z0 = 0

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 137

Consider the linear map K : z0 ∈ Rn → {yk}0⩽k⩽m−1 ∈ (Rq)m. Since yk = CAkz0,
0 ⩽ k ⩽ m− 1, the matrix expression O of the application K is given by

O =


C
CA
CA2

...
CAm−1


The system (S) is then observable with m measurements if, and only if, K is injective,
which is equivalent to rank (O) = n. The matrix M = OTO is symmetric; it is expressed as
functions of A and C by

M =
m∑
k=1

(
Ak−1

)T
CTCAk−1

Then we have:

Proposition 4.1:
(i) The system (S) is observable for the observation (4.22) with the first m measurements if
and only if rank (O) = n, or even if and only if the matrix M is invertible.
(ii) If the system (S) is observable for the observation (4.22) with the first m measurements
and if we have the measurements ymes

k , k = 0, 1, . . . ,m− 1, then the state of the system is
given by

zk = Akz0 ; k = 0, 1, 2, . . .

where

z0 = (M)−1
m−1∑
j=0

(
Aj

)T
CTymes

j

Corollary 4.1:
The system (S) is not observable for the observation (4.22) on the discrete interval
0, 1, . . . ,m− 1 if

mq < n

This shows that for the system to be observable, a necessary but not sufficient condition
must exist and that the number of measurements should be greater than or equal to the value
n/q.

4.3. Characterisation of the C−identifiability of a Family
Similar to the 3.2 section, we have, in the case of a family with two systems,

Cz1k − Cz2k = CAk
1z

1
0 − CAk

2z
2
0 =

[
C (A1)

k −C (A2)
k
] [ z10

z20

]
= [ C −C ]

[
A1 O
O A2

]k [
z10
z20

]
= C A kξ0

where

A =

[
A1 O
O A2

]
, C = [ C −C ] , ξ0 =

[
z10
z20

]
Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



138 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

The characterisation (4.21) of the C−identifiability of the family {(S1) , (S2)} then becomes(
C A kξ0 = 0, k = 0, 1, . . . ,m− 1

)
⇒ ξ0 = 0

which characterises the observability of the couple (A ,C ) over the time interval
0, 1, . . . ,m− 1. This couple models the dynamics of a certain abstract discrete system and a
certain observation, which allows us to have the following characterisation:

Proposition 4.2:
The family {(S1) , (S2)} is C−identifiable for the first m measurements if and only if the
following system of order 2n

(S0)

{
ξk+1 = A ξk k = 0, 1, . . .

ξ0 ∈ Z × Z = Rn ×Rn

is observable during the time interval 0, 1, . . . ,m− 1 by the observation

ηk = C ξk , k = 0, 1, . . . ,m− 1

We know (proposition 4.1) that the observability of the system (S0) over 0, 1, . . . ,m− 1
is equivalent to its matrix observability O of rank 2n.

O =


C

C A
...

C A m−1


With the remark that C A k =

[
C (A1)

k −C (A2)
k
]

the matrix O can be expressed in
terms of A1, A2 and C. This matrix becomes

DC =


C −C

CA1 −CA2

C (A1)
2 −C (A2)

2

...
...

C (A1)
m−1 −C (A2)

m−1


We then get the following theorem:

Theorem 4.1:
(i) The family {(S1) , (S2)} is C−identifiable for the first m measurements if, and only if, the
so-called C−identifiability matrix has rank 2n :

rank (DC) = 2n

(ii) The family {(S1) , (S2)} is identifiable for the first m measurements if and only if, the
so-called identifiability matrix has rank 2n:

rank (DIn) = 2n

We deduce from this theorem that for a given number m of measurements, if rank (DIn) <
2n then the family {(S1) , (S2)} is not identifiable and therefore it is not C−identifiable
for any matrix C. If on the other hand rank (DIn) = 2n then this family is identifiable and
therefore there are matrices C for which this family {(S1) , (S2)} is C−identifiable.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 139

Corollary 4.2:
The family {(S1) , (S2)} cannot be C−identifiable with the first m measurements if

m <
2n

q
(4.23)

Proof
DC admits mq rows so rank (DC) ⩽ mq. If mq < 2n then rank (DC) < 2n and in this case
the family {(S1) , (S2)} is not C−identifiable.

Remark 4.1:
1. In the particular case of a scalar measurement (q = 1) the condition (4.23) becomes
m < 2n; a minimum of 2n measurements is therefore necessary. If besides the system states
are scalars (n = 1) a minimum of 2 measurements is required.

2. There exist non-identifiable families for a given number m of measurements. An
example of such families is given by the family of the two systems modeled by the matrices

A1 =

(
1 0
0 2

)
and A2 =

(
3 0
0 2

)
with m = 4 since its identifiability matrix DIn is of rank < 4. However, and contrary to the
case of continuous models in time, it is possible that an unidentifiable family of discrete
systems for a number of measurements m becomes identifiable for a greater number m′ ⩾ m.

3. As for continuous linear systems in time (remark 3.3) we can show that

rank (DC) ⩽ rank (DIn) ⩽ 2n , ∀C

inequality allowing to find that any family C−identifiable is identifiable.

Example 4.1:
Consider the two systems

(S1) zk+1 = A1zk , (S2) zk+1 = A2zk

where

A1 =

(
1 0
0 2

)
, A2 =

(
3 0
0 4

)
and m = 4

• The identifiability matrix of the family {(S1) , (S2)} is given by

DIn =



1 0 −1 0
0 1 0 −1
1 0 −3 0
0 2 0 −4
1 0 −9 0
0 4 0 −16
1 0 −27 0
0 8 0 −64


and its rank is 4. The family {(S1) , (S2)} is therefore identifiable for 4 measurements.
• For observation

(E) yk = Czk , k = 0, 1, 2, 3 with C = ( 1 0 )

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



140 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

The C−identifiability matrix of the family {(S1) , (S2)} is given by

DC =

 C −C
CA1 −CA2

C (A1)
2 −C (A2)

2

C (A1)
3 −C (A2)

3

 =

 1 0 −1 0
1 0 −3 0
1 0 −9 0
1 0 −27 0


which is of rank 2. The family {(S1) , (S2)} is therefore not C−identifiable.
• Since the family {(S1) , (S2)} is identifiable we can find a matrix C ′ ̸= C such that this
family is C ′−identifiable. It is the case of the following observation:

(E ′) yk = C ′zk , k = 0, 1, 2, 3 with C ′ = ( 1 1 )

The C ′−identifiability matrix of this family is given by

DC′ =

 C ′ −C ′

C ′A1 −C ′A2

C ′ (A1)
2 −C ′ (A2)

2

C ′ (A1)
3 −C ′ (A2)

3

 =

 1 1 −1 −1
1 2 −3 −4
1 4 −9 −16
1 8 −27 −64


which is of rank 4. The family {(S1) , (S2)} is therefore C ′−identifiable.

We have similar results for the case of continuous systems, which we give in the following
proposition:

Proposition 4.3:
(i) When A1 and A2 commute the family {(S1) , (S2)} is identifiable by the first m
measurements if, and only if, the matrix with mn rows

M =


A2 − A1

(A2 − A1)
2

...
(A2 − A1)

m


is of rank n.
(ii) For the first m measurements, if the family {(S1) , (S2)} is C−identifiable then each
system of this family is observable. The opposite is false.

We therefore deduce from this proposition, as well as from Corollary 4.2, that if one of
the systems is not observable by the first m measurements or if m < n = q then the family
formed by these two systems cannot be C−identifiable for these measurements.

For families with several systems the algebraic characterisation of the C−identifiability
and identifiability of a family of systems {(S1) , . . . , (SN)} is done through the following
matrices

D ij
C =


C −C

CAi −CAj

C (Ai)
2 −C (Aj)

2

...
...

C (Ai)
m−1 −C (Aj)

m−1

 i, j = 1, . . . , N
i ̸= j

Theorem 4.2:
(i) The family of systems {(S1) , . . . , (SN)} is C−identifiable for first m measurements if and

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 141

only if
rank

(
D ij

C

)
= 2n , i, j = 1, . . . , N , i ̸= j

(ii) The family of systems {(S1) , . . . , (SN)} is identifiable for the first m measurements if and
only if

rank
(
D ij

In

)
= 2n , i, j = 1, . . . , N , i ̸= j

4.4. Algorithm for the Identification of the Sought System
For a given family F = {(S1) , . . . , (SN)} of N discrete systems, we give below the first
algorithm that determines, when it exists, the first integer for which F is C−identifiable.
A second algorithm identifies in the F family the system that generated the measurements,
assuming that this family is C−identifiable.

Consider an integer m0 ⩾ 2 (to be set by the user). We search for the first integer m ⩽ m0,
if it exists, that ensures the C−identifiability of the family F . E [x] is always the integer part
of x we have the following algorithm:

Algorithm 5 C−identifiability test of a family and determination of m.

1. If 2n
q is an integer, take m← 2n

q Else take m← E
[
2n
q

]
+ 1

2. For i = 1, . . . , N and for j = 1, . . . , N, j ̸= i

3. Determine D ij
C ; If rank

(
D ij

C

)
< 2n go to 6

4. next j ; next i

5. Display “Family C−identifiable for m =”m; STOP

6. m← m+ 1; Si m ⩽ m0 resume in 2

7. Else, Display “Family non C−identifiable for m ⩽”m0

Suppose in the following that the family F is C−identifiable for an integer m, determined
by the algorithm 5, and that we have collected m measurements ymes

0 , . . . , ymes
m−1 assumed to

be non-zero. For each system (Si) of the family, we can estimate the state zik by

zik = Ak
i z

i
0 (4.24)

where

zi0 = (Mi)
−1

m−1∑
k=0

(
Ak

i

)T
CTymes

k and Mi =
m−1∑
k=0

(
Ak

i

)T
CTCAk

i (4.25)

Let us then denote by (Si0) the system (still unknown) which has given these measurements.
So, other than system (Si0), the state of each system (Si), i ̸= i0, check Czik ̸= ymes

k for
at least one integer k ⩽ m− 1; otherwise we would have(

Czi0k = ymes
k , k = 0, 1, . . . ,m− 1

)
and

(
Czik = ymes

k , k = 0, 1, . . . ,m− 1
)

which cannot be achieved because of the C−identifiability of the subfamily {(Si0) , (Si)} .
So just test (Czik = ymes

k , k = 0, 1, . . . ,m− 1) for each system (Si) and to eliminate it
if it does not fulfill this test. At the end of this step we should, theoretically, obtain a single
not eliminated system. Practically three cases can arise:

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



142 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

Case 1. Several systems check for equalities Czik = ymes
k , i.e. ∥Czik − ymes

k ∥ < ε , k =
0, . . . ,m− 1. In this case the tolerance ε is too large to be able to identify the sought system;
Case 2. No system fulfills these equalities: ∥Czik − ymes

k ∥ ⩾ ε , k = 0, 1, . . . ,m− 1 , 1 ⩽
i ⩽ N. In this case, either the tolerance is too low, or no system in the family has generated
these measurements;
Case 3. Only one system checks for ties; this is then the sought system.We deduce from all
the above the following algorithm:

Algorithm 6 Identification of the sought system in a family C−identifiable by m
measurements.

1. Take the measurements ymes
k ; k = 0, 1, . . . ,m− 1;

2. Compute the states zik , i = 1, . . . , N ;

3. For each system (Si)

4. For k = 0, 1, . . . ,m− 1

5. If
∥∥Czik − ymes

k

∥∥ ⩾ ε: Reject (Si) and go to 7

6. next k

7. next system (Si)

8. If all systems are rejected: Display “None of the systems generated these measurements, or tolerance was
too low.”; STOP

9. If the number of non-rejected systems is ⩾ 2: Display “Tolerance too large.”; STOP

10. The only non-rejected system is the sought system.

Remark 4.2:
The algorithms of the discrete case are slightly different from those of the continuous case (in
time). This results from the very nature of these systems.

4.5. Numerical Example
Consider a family of three systems whose dynamics are modeled by the matrices (3.18).

We determine the first integer m for which the family of these 3 systems is C−identifiable.
For m < 4 the matrices D ij

C have less of 4 rows and therefore their ranks are < 4. For m = 4

the D ij
C are given by (3.19) which all have rank 4 = 2n; hence the C−identifiability of this

family for measurements in k = 0, 1, 2, 3. In what follows we take ε = 0.01 as tolerance.
• First example of measurements

The observation at moments k = 0, 1, 2, 3 gave the measurements
{ymes

0 , ymes
1 , ymes

2 , ymes
3 } = {1, 5, 23, 101} .

1) Compute of the states: We first calculate the matrices Mi by (4.25); we obtain

M1 =

(
4 15
15 85

)
, M2 =

(
820 1885
1885 4369

)
, M3 =

(
228 −120
−120 84

)
(4.26)

so the initial states, computed by (4.24), are

z10 =

(
−523

23
1694
115

)
, z20 =

(
−1
2

)
, z30 =

(
65
22

−449
66

)
Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 143

2) For (S1) : we test
(i) |Cz10 − ymes

0 | = 1036
115

⩾ ε : we reject (S1) .
3) For (S2) : we test
(i) |Cz20 − ymes

0 | = 0.0 < ε;
(ii) |Cz21 − ymes

1 | = |CA2z
2
0 − 5| = 0.0 < ε;

(iii) |Cz22 − ymes
2 | =

∣∣C (A2)
2 z20 − 23

∣∣ = 0.0 < ε;
(iv) |Cz23 − ymes

3 | = C (A2)
3 z20 = 0.0 < ε.

On conserve (S2).
4) For (S3) : we test
(i) |Cz30 − ymes

0 | = 160
33

⩾ ε : We reject (S3) .
(S2) is the only system not rejected; so (S2) is the sought system.

• Second example of measurements

The observation at moments k = 0, 1, 2, 3 gave the measurements
{ymes

0 , ymes
1 , ymes

2 , ymes
3 } = {1, 5, 15, 35} .

1) The matrices Mi are always given by (4.26); we calculate the initial states by (4.24):

z10 =

(
−101

23
564
115

)
, z20 =

(
74 039
29 355
−3028

5871

)
, z30 =

(
2
−1

)
2) For (S1) we test :
(i) |Cz10 − ymes

0 | = 56
115
≃ 0.48 ⩾ ε : we reject (S1) .

3) For (S2) we test:
(i) |Cz20 − ymes

0 | = 0.0 < ε;
(ii) |Cz21 − ymes

1 | = |CA2z
2
0 − 5| = 778

1545
≃ 0.5 > ε : we reject (S2) .

4) For (S3) , we keep it because:
(i) |Cz30 − ymes

0 | = 0.0 < ε;
(ii) |Cz31 − ymes

1 | = |CA3z
3
0 − 5| = 0.0 < ε;

(iii) |Cz32 − ymes
2 | =

∣∣C (A3)
2 z30 − 15

∣∣ = 0.0 < ε;
(iv) |Cz33 − ymes

3 | =
∣∣C (A3)

3 z30 − 35
∣∣ = 0.0 < ε.

(S3) is the only system not rejected; so (S3) is the sought system.
Finally we obtain the following assignments:

plot 1 2
System (S2) (S3)

5. APPLICATION TO THE CROP IDENTIFICATION USING RADAR

In this section we present the use of Radar for crop identification as an application of the
general approach given in the previous sections. We recall the Radar principle.

The Radar, considered as a sensor, is a satellite-based Synthetic Aperture Radar (SAR).
It transmits microwave pulses and receives the signals backscattered from the objects on
the Earth, which are recorded as an image data and data products. Radar signal backscatter
depends mainly on the physical properties of the observed objects: their roughness, geometry
and dielectric constant.

In case of vegetation it is mostly linked to the plant external structure, biomass and water
content ( [13]). The possibility of observation of crop characteristics is also dependent on
sensor properties, e.g.:

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144 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

• the wavelength used for the observation and the associated depth of penetration, for instant:
when using C-band (5,6 cm) we detect mainly plant’s leaves, but when using L-band (23
cm) mostly only branches are visible.

• the polarization of the transmitted and received signal. Using different polarization we can
observe better different kind of objects. Cross-polarization is better to use for observation
of non-structured crops (cereals, maize), co-polarization is more suitable for detection of
structured crops (vineyards, hop).
To use the Radar for crop identification, it is necessary to develop a classification

algorithm/approach. First, it should be noted that even if crops evolve continuously in time
we study them using discrete measurements as shown in figure 5.1.

Fig. 5.1. Evolution of a crop (Potato) (BCCH - the phenological development stages of plants)

For identification of crops using Radar we proceed in two steps as in the mathematical
approach:

1. First step: dataset and crops dynamics. To assume that the dynamics of the crops, and
consequently the discrete states, are known, a reference dataset is needed. This requires ”a
first” measurement to construct the database.

In the case of crops, ground truth data are needed to know the different types of crops,
their dynamics and states in specific fields, as well as the measurements of radar backscatter
must be acquired for these fields using satellite sensors (e.g.: Sentinel-1). It must be noticed
that the SAR measurement can be realized using different polarization at the same time so
several parameters can be derived to extract more information about crops, e.g.: polarimetric
coherence matrices and H/α ( [5]), which show the degree of the depolarization of the signal
in the contact with the object, its scatter type or entropy. Such multidimensional observation
increases the accuracy of the systems identification ( [16]). Also the speckle noise must
be removed. Using these measurements we estimate the standard dynamic of each crop
(reference models of crops) which we want to identify (figure 1). So the dataset consists
to have the equivalent of figure 1 for each crop as in figure 2.

2. Second step: identification. To identify a system on a given plot, where the crop type is
unknown, we consider two steps:

2.1 We must ensure that our sensor can distinguish crops. This is done through the analysis
of our reference database. In figures 5.3 we show accordingly a case where one crop can be
identified whereas the other cannot. It must be noticed that different parameters can or cannot
identify the systems.

2.2 If the sensor can identify the crops, a ”second measurement” is made. We collect
measurements of all parameters for all the fields in the area where we want to identify

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



IDENTIFIABILITY OF A FAMILY OF DYNAMICAL SYSTEMS 145

Fig. 5.2. Reference models of different crops (dataset)

Fig. 5.3. Examples of identifiable crops (a) and non-identifiable crops (b).

unknown crop types (systems) and use the equivalent of the algorithm given in the
mathematical approach. In the case of crop classification using radar we will look for all fields
which can give the similar measurements given by the standard crops (reference models of
crops). This step is carried out using one specific classifier (e.g. Random Forest (RF) ( [3])
which compares directly measurements to the reference models of the systems (crops) and
identifies crops.

In the mathematical approach, we have pointed out that measurements must be made over
a time interval I = ]0, T ] in the continuous case and in the discrete cases, the measurements
must be greater than a given threshold iterations. It is the case for crop identification using
Radar. As an example, in figure 5.4, we show that in order to identify the systems we have to
consider a relatively greater time.

In fact, we could include this (the time of the measurements) in our classification
algorithm and choose this time according to this data. We can mention that different times of
measurement have better or worse potential for the identification.

The case study carried out in Poland using Sentinel-1 SAR images ( [17]) showed that,
e.g. sugar beet and winter rapeseed systems can be very well discriminated from other 23
crops (systems) – F1 score which evaluate an accuracy of the classification of a specific class
is 0.98 and 0.99, respectively. So Sentinel-1 can be considered as an identifying sensor for
these systems. On the contrary, the cases of spring triticale and spring wheat systems, which
were very poorly discriminated 0.53 and 0.6, respectively.

In our mathematical approach, the first measurement consists in searching for standard
states of systems (reference models of crops) given through the measurement of radar
parameters. The second measurement consists in the identification among systems of both:
mathematical and radar algorithms.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



146 A. BERNOUSSI, E. WOŹNIAK & A. BELFEKIH

Fig. 5.4. Example of two systems identifiable or non-identifiable depending on the time of measurement.

6. CONCLUSION

In this paper, we considered the problem of identifying a family of dynamical systems.
For the case of finite dimensional linear systems, we have given characterizations and
proposed algorithms to determine the sought system, among N systems, from the collected
measurements.

We have developed numerical examples and an application to illustrate our approach.
It would be interesting to extend this work, and in particular the algorithms, to nonlinear
systems and to distributed parameter systems. It would also be very useful to automate
the identification of the system sought for a possible application to the automation of the
identification of cultures using Radar. These issues are currently under investigation.

ACKNOWLEDGEMENTS
This work have been supported by MESRSFC, CNRST and ADD, Morocco under the
research project Al Khawarizmi/2020/11 ”Intelligent management tool for irrigation water
and forest heritage”.

REFERENCES

1. Belfekih, A., El Jai, A. & Bernoussi A. (2023). Observabilité et Opacité – Systèmes Spatio-temporels
[Observability and Opacity of distributed parameter systems]. Perpignan, France: Presses Universitaires
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Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	Identifiability of a family of systems:  Introduction of the concept
	Identifiability of families of finite  dimensional linear systems
	Observable System and State Reconstruction
	Characterisation of Identifiable Families 
	Algorithm for Identifying the Searched System in a Family
	Generalisation to Families of Several Systems
	Numerical Example 

	Case of discrete finite dimensional linear systems
	Identifiability of a Family of Finite Dimensional Discrete Linear Systems
	Observable Discrete System and State Reconstruction
	Characterisation of the C-identifiability of a Family
	Algorithm for the Identification of the Sought System
	Numerical Example

	Application to the crop identification using Radar
	Conclusion