COPRIME FACTORIZATIONS OF MISO FRACTIONAL
TIME-DELAY SYSTEMS
LE HA VY NGUYEN
AND CATHERINE BONNET∗
Abstract. In this paper we consider MISO fractional systems with delays that are given by
their transfer function. For a large class of such systems, we determine left coprime factorizations as
well as associated Bézout factors, and guarantee the existence of right coprime factorizations.
Key words. MISO systems, fractional systems, delay systems, H∞ -stability, coprime factorizations
AMS subject classifications. 15A23, 26A33, 93D15
1. Introduction. Fractional models have been of increased interest for about
twenty years in many domains, for example, mechanics, electronics, electromagnetic,
viscoelasticity, biology,. . . . Being able to model long memory systems, they provide
more compact models in comparison with classical models of integer order.
The transfer function of many fractional system models is a rational function of
sα , where s is the Laplace variable and 0 < α < 1. Some examples are electrochemical processes, flexible structures [11], viscoelastic materials [6], and non-laminated
suspension systems [4]. If we consider delays which often occur in real systems, we
obtain fractional time-delay systems.
There are few works on controller synthesis for this class of systems. In the framework of fractional representation approach to synthesis problems [10], the works in
[1, 2] derive coprime factorizations of the transfer function of the system and the
corresponding Bézout factors for SISO fractional time-delay systems. The work [3]
derives these expressions in terms of operators used in a pseudo state-space representation for a large class of MIMO infinite-dimensional systems. This representation is
not trivial to obtain for fractional time-delay systems. Moreover, fractional systems
are often described by transfer functions by means of frequency identification, for example, [11], [9]. Recently, PID controllers have been designed for a class of fractional
time delay systems with only one unstable pole [5].
The aim of this work is to determine explicit expressions of left coprime factorizations and the corresponding Bézout factors for a large class of MISO fractional
time-delay systems considering H∞ stability. The MISO structure, which is a particular and simple MIMO case, might be encountered in communication networks, for
example [8].
The determination of coprime factorizations and associated Bézout factors is the
first step for determining the set of all stabilizing controllers.
The paper is organized as follows. Section 2 provides some definitions. In section
3, the class of systems of interest is presented. In section 4, we show the main result
that gives explicit expressions of left coprime factorizations and associated Bézout
factors over H∞ of the transfer function of the systems under study. Some examples
are provided to illustrate this result. The existence of right coprime factorizations is
also proved. Section 5 gives conclusions and perspectives.
2. Preliminaries.
∗ INRIA Saclay - Île-de-France, Equipe DISCO, LSS-SUPELEC, 3 rue Joliot Curie, 91192 Gifsur-Yvette, France (le-ha-vy.nguyen@inria.fr, catherine.bonnet@inria.fr)
1
2
L. H. V. NGUYEN AND C. BONNET
• C+ = {s ∈ C | Re(s) > 0}.
• H∞ denotes the space of functions that are analytic and bounded in C+ with
kGkH∞ = sups∈C+ |G(s)| and M(H∞ ) denotes the set of matrices whose
components are in H∞ .
• A system G is said to have a right coprime factorization (N, M ) over H∞ if
G = N M −1 , det M 6= 0, N , M ∈ M(H∞ ) and there exist X, Y ∈ M(H∞ )
such that XM + Y N = I. Then X, Y are called right Bézout factors.
f, N
e ) over H∞ if
• A system G is said to have a left coprime factorization (M
−1
f N
e , det M
f 6= 0, M
f, N
e ∈ M(H∞ ) and there exist X,
e Ye ∈ M(H∞ )
G=M
f
e
e
e
e
e
such that M X + N Y = I. Then X, Y are called left Bézout factors.
In order to guarantee a unique value of the transfer function involving sα , we will
consider the complex variable s in the principle branch C\R− , that is arg(s) ∈ (−π, π).
We refer to poles (resp. roots) in the closed right half-plane C+ as unstable poles
(resp. roots).
3. A class of MISO fractional time-delay systems. We consider the following class of systems
G(s) = e−sh1 R1 (sα ), . . . , e−shn Rn (sα ) ,
(3.1)
where 0 ≤ hk ∈ R for k = 1, . . . , n are the delays; α ∈ R, 0 < α < 1; Rk (sα ) =
qek (sα )/e
pk (sα ), where pek (sα ) and qek (sα ) are polynomials of integer degree respectively
dk and nk in sα , pek (sα ) and qek (sα ) have no common roots, and deg pek (sα ) ≥ deg qek (sα )
for k = 1, . . . , n; and s ∈ C\R− .
The following notations will be of intense use later.
The polynomial p(sα ) denotes the lowest common denominator of Rk (sα ) for
k = 1, . . . , n, and d denotes the degree in sα of p(sα ). Then Rk (sα ) are rewritten as
α
Rk (sα ) = qk (sα )/p(sα ), where qk (sα ) are
.
Qpolynomials insQ
N
N0
m0j
α
α m0
α
mi
α
We can decompose p(s ) = (s )
, where
i=1 (s − bi )
j=1 (s − cj )
bi ∈ D := {σ ∈ C\{0} | − πα/2 < Arg(σ) < πα/2}, cj ∈ C\D and m0 , mi , m0j ∈ Z+
1/α
for i = 1, . . . , N and j = 1, . . . , N 0 . Hence si = bi are the non-zero unstable roots
in s of p(sα ).
Q
Q 0
N
N
m0jk
α
mik
α
(s
−
b
)
(s
−
c
)
,
Similarly, we write pek (sα ) = (sα )m0k
i
j
i=1
j=1
where m0k , mjk , m0jk ∈ Z+ for i = 1, . . . , N , j = 1, . . . , N 0 and k = 1, . . . , n. It is
obvious that m0k ≤ m0 , mjk ≤ mj , and m0jk ≤ m0j .
4. Left coprime factorizations and Bézout factors.
4.1. Left coprime factorizations.
Proposition 4.1. Let G(s) be given by (3.1). Then
f(s) =
(4.1) M
−sh1
1
p(sα )
e (s) =
and N
e
q1 (sα ), . . . , e−shn qn (sα )
(sα + 1)d
(sα + 1)d
is a left coprime factorization over H∞ of G.
f(s)−1 N
e (s) = G(s).
Proof. It is obvious that M
f(s) ∈ H∞ . Also, each component of N
e (s) is in H∞ , and then
We see that M
e
N (s) ∈ M(H∞ ).
For all roots σ of p, there exists at least one 1 ≤ k ≤ n such that qk (σ) 6= 0.
Pn
ek | + |M
f|) > 0 which ensures that (M
f, N
e ) is a left coprime
Thus inf s∈C+ ( k=1 |N
factorization over H∞ of G.
COPRIME FACTORIZATIONS OF MISO FRACTIONAL DELAY SYSTEMS
3
4.2. Bézout factors. Here we propose Bézout factors for two particular cases
of systems (3.1). The first proposition gives simple result for systems involving at
most one unstable pole for each element of the transfer matrix.
Proposition 4.2. Let G(s) be given by (3.1) with
(4.2)
Rk (sα ) =
ak
with ak , σk ∈ R for k = 1, . . . , n.
sα − σk
Then Bézout factors corresponding to the left coprime factorization (4.1) are given by
(4.3)
(sα + 1)d −
e
X(s)
=
Pn
k=1 e
p(sα )
−shk
qk (sα )Yek
,
h
iT
Ye (s) = Ye1 , . . . , Yen ,
(4.4)
where Yek for k = 1, . . . , n are constant and satisfy
(4.5)
n
X
e−shk qk (sα )Yek = (sα + 1)d
k=1
at s = b1/α for all b ∈ D = {σk |1 ≤ k ≤ n, σk ≥ 0}.
In the case where all σk are positive (σk ≥ 0) and distinct, the unique solution of
e
Yk is given by
1/α
eσk
Yek =
(4.6)
hk
(σk + 1)n
.
qk (σk )
In the case where σ1 = . . . = σn = σ > 0, then Yek satisfy the single equation
(4.7)
n
X
e−σ
1/α
hk
ak Yek = σ + 1.
k=1
Proof. From the left Bézout identity, we obtain
(4.8)
α
d
e
f−1 (1 − N
e Ye ) = (s + 1) −
X(s)
=M
Pn
k=1
e−shk qk (sα )Yek (s)
p(sα )
.
h
iT
e
If we choose Ye (s) = Ye1 (s), . . . , Yen (s) ∈ M(H∞ ) such that the numerator of X(s)
e
vanishes at s = b1/α for all b ∈ D, then X(s)
is analytic on C+ since s = b1/α are
e
also the roots of the denominator of X(s).
In this particular case, it suffices to choose
e
e
constants Y1 , . . . , Yn such that
(4.9)
n
X
e−shk qk (sα )Yek = (sα + 1)d
k=1
e
at s = b1/α for all b ∈ D to have X(s)
∈ H∞ .
4
L. H. V. NGUYEN AND C. BONNET
For all b ∈ D, we see that qk (b) = 0 if b 6= σk . Then it remains to solve
(4.10)
X
e−b
1/α
hk
qk (b)Yek = (b + 1)d ,
k:1≤k≤n,σk =b
which in general gives infinitely many solutions.
When all σk for k = 1, . . . , n are positive and distinct, the number of equations is
equal to the number of unknowns d = n. We deduce then the unique solution (4.6).
The case where σ1 = . . . = σn = σ > 0 corresponds to d = 1. Trivially we have
p(sα ) = sα − σ and qk (sα ) = ak . Thus we have to solve
n
X
(4.11)
e−σ
1/α
hk
ak Yek = σ + 1,
k=1
which gives a unique solution if n = 1 and infinitely many solutions if n > 1.
We illustrate the proposition by the next simple example.
Example 4.3.
−s
e−s
e
G(s) = √ , √
s
s−1
√
√ √
√
√
√
√
It is easy to see that p( s) = s( s − 1), q1 ( s) = s − 1, and q2 ( s) = s.
Then we obtain
−s √
√ √
√ e ( s − 1) e−s s
s( s − 1)
f
e
√
M (s) = √
, N (s) =
, √
.
( s + 1)2
( s + 1)2 ( s + 1)2
This system corresponds to the case of unstable and distinct poles. So we obtain
T
Ye (s) = [−1, 4e] ,
√
√
√
( s + 1)2 + e−s ( s − 1) − 4e1−s s
e
√ √
X(s) =
.
s( s − 1)
Remark 4.4. The result of Proposition 4.2 can be extended to systems with
(4.12)
Rk (sα ) =
(sα
ak
− σk )e
p0k (sα )
where the polynomials pe0k (sα ) have no unstable zero. This allows us to treat models of
non-laminated suspension systems obtained in [4], whose fractional transfer function
is proved to have only one unstable pole.
The next proposition considers a more general class of fractional systems.
Proposition 4.5. Let G(s) be given by (3.1) and suppose that the multiplicity
of the root at zero of p(sα ) verifies the condition m0 α ≤ 1. Then Bézout factors
corresponding to the left coprime factorization (4.1) are given by
(4.13)
(4.14)
Pn
(sα + 1)d u(sα ) − k=1 e−shk qk (sα )µk (sα )
e
X(s)
=
,
p(sα )u(sα )
T
µ1 (sα )
µn (sα )
e
Y (s) =
,...,
,
u(sα )
u(sα )
COPRIME FACTORIZATIONS OF MISO FRACTIONAL DELAY SYSTEMS
5
where u(sα ) is a polynomial in sα of degree greater or equal to d whose zeros are
PN
stable, and µk (sα ) for k = 1, . . . , n are polynomials in sα of degree m0k + i=1 mik
satisfying
(sα + 1)d u(sα ) −
(4.15)
n
X
e−shk qk (sα )µk (sα ) ∼s→0 sm0 α ,
k=1
and
"
α
d
α
(s + 1) u(s ) −
(4.16)
n
X
#(l)
e
−shk
α
α
qk (s )µk (s )
=0
k=1
1/α
at s = bi for 0 ≤ l ≤ mi − 1.
Proof. It is obvious that Ye (s) ∈ M(H∞ ).
The condition (4.15) is satisfied if, in the development of the denominator of
e
X(s)
around zero, the powers of s whose order is smaller than m0 α are cancelled.
Then this fact gives m0 equations of unknown coefficients of µk (sα ). The second
PN
PN
condition (4.16) introduces i=1 mi equations. Hence in total, there are m0 + i=1 mi
Pn
Pn PN
PN
equations of
m0k + k=1 i=1 mik + nN unknowns. As m0 + i=1 mi ≤
Pn
Pnk=1 PN
m
+
0k
k=1
k=1
i=1 mik + nN , the system of equations admits solutions.
e
The boundedness of X(s)
at zero is assured by (4.15). Then following the same
e
arguments as in the proof of the previous proposition, we can conclude that X(s)
∈
H∞ .
The left Bézout identity is satisfied.
Remark 4.6. When m0 α > 1, for the condition (4.15), the development of e−shk
around zero will contain powers of s, which might not be multiples of sα . This imposes
that µk might no longer be polynomials in sα .
The following example illustrates how to apply the proposition.
Example 4.7.
e−s
e−s
G(s) = 1/3
,
s −1 s
It is obvious that p(s1/3 ) = s(s1/3 − 1), then d = 4, m0 = 3, and b1 = 1 with
m1 = 1. We have q1 (s1/3 ) = s, and q2 (s1/3 ) = s1/3 − 1.
From Proposition 4.1, we obtain
1/3
f(s) = s(s − 1) ,
M
(s1/3 + 1)4
e (s) =
N
h
i
1
−s
−s 1/3
e
s,
e
(s
−
1)
(s1/3 + 1)4
From Proposition 4.5, we can choose u(s1/3 ) = (s1/3 + 1)4 , which has no unstable
roots.
Also, µ1 (s1/3 ) and µ2 (s1/3 ) have the form
µ1 (s1/3 ) = β01 + β11 s1/3 ,
µ2 (s1/3 ) = β02 + β12 s1/3 + β22 s2/3 + β32 s.
e
As s → 0, the numerator of X(s)
is developed as
(s1/3 + 1)4 u(s1/3 ) − e−s q1 (s1/3 )µ1 (s1/3 ) − e−s q2 (s1/3 )µ2 (s1/3 )
= 1 + β02 + (8 − β02 + β12 )s1/3 + (28 − β12 + β22 )s2/3 + O(s).
6
L. H. V. NGUYEN AND C. BONNET
The first condition of Proposition 4.5 is satisfied if and only the powers of s whose
exponent is smaller than m0 α vanish, thus giving β02 = −1, β12 = −9, and β22 = −37.
e
From the second condition, at non-zero unstable pole s = 1 of X(s),
we must
have
(s1/3 + 1)4 u(s1/3 ) − e−s q1 (s1/3 )µ1 (s1/3 ) − e−s q2 (s1/3 )µ2 (s1/3 ) = 0
This gives β01 + β11 = 256e.
One possible solution is β32 = 0, β01 = 256e, and β11 = 0.
In conclusion, we obtain the Bézout factors as follows
−1 − 9s1/3 − 37s2/3
256e
e
,
,
Y (s) =
(s1/3 + 1)4
(s1/3 + 1)4
(s1/3 + 1)8 − e1−s 256s − e−s (s1/3 − 1)(−1 − 9s1/3 − 37s2/3 )
e
.
X(s)
=
s(s1/3 − 1)(s1/3 + 1)4
4.3. Existence of right coprime factorizations.
Proposition 4.8. Let G(s) be given by (3.1) and suppose that the multiplicity
of the root at zero of p(sα ) verifies the condition m0 α ≤ 1. Then G(s) possesses right
coprime factorizations.
Proof. The system G(s) admits a left coprime factorization given by (4.1). Since
H∞ is a Hermite ring, then by Corollary 4.14 of [7], we conclude that there exist right
coprime factorizations for G(s).
5. Conclusion. Explicit expressions of a left coprime factorization over H∞ of
a large class of MISO fractional time-delays systems as well as corresponding Bézout
factors are given. The existence of right coprime factorizations of the systems is
also derived. In order to obtain the parametrization of all stabilizing controllers,
determining the right coprime factorizations is the objective of a forthcoming work.
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