Abstract This paper proposes a method to discuss the stability analysis of multiple models. The proposed stability analysis is based on the use of scalar Lyapunov functions and the properties of M-matrices. An example is included to illustrate the proposed method

Didier Maquin

José Ragot

Mohammed Chadli

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On the stability analysis of a class of multiple models
Mohammed CHADLI, Didier MAQUIN, José RAGOT
Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039
2, Avenue de la forêt de Haye, 54516 Vandœuvre les Nancy – France.
Phone: (33) 3 83 59 57 02, Fax: (33) 3 83 59 56 44
Email: {Mohamed.Chadli, Didier.Maquin, José.Ragot}@ensem.inpl-nancy.fr
Abstract
This paper proposes a method to discuss the stability
analysis of multiple models. The proposed stability
analysis is based on the use of scalar Lyapunov functions
and the properties of M-matrices. An example is included
to illustrate the proposed method.
Keywords: multiple models, stability analysis, Lyapunov
methods, Linear Matrix Inequalities (LMIs). 
1 Introduction
The multiple model representation, which includes the
Polytopic Linear Differential Inclusion [7] and the Takagi-
Sugeno model [4], consists to construct nonlinear
dynamics by means of interpolating the behaviour of
several LTI subsystems
&( ) ( ) ( )x t x t A x ti i
i
n
=
=
∑µ a f
1
(1)
where x t p( )∈r  is the state vector, n is the number of
LTI subsystems and the matrices Ai
p p∈ ×r  define the ith
LTI model. The nonnegative functions µ i .a f called
activation functions, satisfy the following properties
µ
µ
i
i
n
i
x t
x t i n
( )
( ) ,...,
a f
a f k p
=
≥ ∀ ∈
R
S|
T|
=
∑ 1
0 1
1 (2)
Many researches have addressed the issue of stability of
such multiple models. The stability depends on the
existence of a common positive definite matrix satisfying a
set of LMIs [3][7]. To reduce the conservativness of some
previous constraints, a nonquadratic Lyapunov functions
have also been used [1][6].
Some results use the properties of M-matrices to study
subclass of systems which admit a Common Quadratic
Lyapunov Function (CQLF), see [5], [8] and references
therein. In this paper, with the aid of a Scalar Lyapunov
Function (SLF) and the properties of M-matrices,
sufficient conditions for global exponential stability are
established for the class of multiple models accepting a
CQLF. 
Next, the following useful notation is introduced.
λmin ( )X  and λmax ( )X  denote respectively the minimum
and the maximum eigenvalues of the matrix X,
x x xF
T2 =  is the Frobenius norm, xi  is the absolute
value of the real variable xi  and y > 0 is a positive vector,
i.e. all of its entries are positive.
2 Main result
In general, there is not any systematic method allowing to
obtain a quadratic Lyapunov function. However, in the
particular case where matrices Ai  are Hurwitz and
commute pairwise, it is possible to build, in a systematic
way, a CQLF [2]. Thereafter, an approach stressing
another class of multiple models admitting CQLF will be
presented. This study is based on the properties of M-
matrices and the SLF candidate of the form
V x t x t P x tT( )a f a f a f a f= α (3)
where: P Pi i
i
n
α αa f =
=
∑ ,
1
, P Pi i
T= > 0  and α i ∈
+r .
In the following, we will suggest a new sufficient global
exponential stability condition of multiple model (1). 
Theorem 1 : Suppose that there exists symmetric positive
definite matrices Pi > 0  and symmetric matrices Sii > 0
and Sij  such that
i) A P P A Si
T
i i i ii+ ≤ − ,      i n∈ 1,...,k p (4a)
ii) A P P A Sj
T
i i j ji+ ≤   i j i j n≠ ∈, , , ...,a f k p1 2 (4b)
iii) Φ  is a M-matrix. (4c)
where 
Φ=
− −
−
−
− −
F
H
GGGGGG
I
K
JJJJJJ
−
−
λ λ λ
λ λ
λ
λ λ λ
min max max
max min
max
max max min
.
. .
. . .
.
S S S
S S
S
S S S
n
n n
n n n nn
11 12 1
21 22
1
1 1
b g b g b g
b g b g
e j
b g e j b g
a f
a f
(5)
Then the multiple model (1) is globally exponentially
stable.
Proof : Consider the SLF candidate (3). This is a radialy
unbounded Lyapunov function since that : 
λ α λ αmin max ,P x V x P x xF F
pa fb g a f a fb g2 2≤ ≤ ∈r (6)
The derivative of the SLF (3) along the trajectory of the
continuous multiple model (1) gives with conditions (4a)
and (4b)
& ( ) ( ) ( ) ( )V x t x t x t S x tT i i
i
n
iia f a f≤ − +
=
∑α µ
1
x t x t S x tT i
j i
n
j ji
i
n
( ) ( ) ( )
:
α µ
≠=
∑∑
11
a f (7)
Then taking into account the following properties :
x S x S xT ji ji F≤ λmax ( )
2 (8a)
λ λmin max( ) ( )S x x S x S xii F
T
ii ii F
2 2≤ ≤ (8b)
the inequality (7) becomes 
& ( ) ( )V x t x t x t Fa f a fb g≤ −ν αΦ 2 (9)
where ν µ µx t x t x tna fb g a f a fb g= 1 ( ) ,..., ( )  and
α α α= 1, ,K n
Tb g  are positive vectors and the matrix Φ  is
defined in (5). Since Φ  is an M-matrix there exists a
positive vector α  such that Φα > 0 . The exponential part
of the proof is obtained by considering definitions (6) and
(9).
It is important to note that, in some cases, the obtaining of
an M-matrix is not direct. Such a matrix may be obtained
by replacing the matrices Sii  by S Iii + τ  where τ  is a real
parameter and the matrix I p p∈r .  is the identity matrix.
An M-matrix that depends on λ τmin Siib gc h+  is then
computed by an iterative method modifying the τ
parameter. This case is illustrate by the following
example.
3 Numerical example
Consider the multiple model (1) with 
n = 2 , A1 =
F
HG
I
KJ
0 1
-0.06 -1
, A2 =
F
HG
I
KJ
0 2.04
-0.5 -1.5
(10)
Conditions of theorem 1 fail to derive directly an M-
matrix and then to prove the stability of this example. By
modifying these conditions as stated above, the theorem 1
leads to solve the following LMIs constraints:
P P S I S I1 2 11 220 0 0 0> > + > + >, , ,τ τ (11a)
A P P A S I A P P A S IT T1 1 1 1 11 2 2 2 2 22+ ≤ − + + ≤ − +τ τb g b g, (11b)
A P P A S A P P A ST T1 2 2 1 12 2 1 1 2 21+ ≤ + ≤, (11c)
Solving the LMI system (11), we obtain
P1 =
F
HG
I
KJ
1.094 1.011
1.011 1.649
 , P2 =
F
HG
I
KJ 
0.8462 0.6593
0.6593 1.3542
τ = =
F
HG
I
KJ03 11. , S
60.537 7.998
7.998 637.672
, S22 =
F
HG
I
KJ
329.503 -30.139
-30.139 686.174
(12a)
S12 =
F
HG
I
KJ
666.874 105.686
105.686 -643.808
, S21 =
F
HG
I
KJ
-265.461 -109.216
-109.216 -75.108
 (12b)
The expression (5) with the result in (12) allow the matrix
Φ  to be deduced:
Φ =
F
HG
I
KJ
60.726 -675.341
-25.417 327.274
(13)
As Φ  is clearly an M-matrix, the considered multiple
model (10) is exponentially stable 
4 Conclusion
Using the properties of M-matrices, sufficient conditions
for the global exponential stability for multiple models are
obtained. The method studies classes of multiple models
admitting CQLF. An heuristic method, based on LMIs
formulation, to compute such an M-matrix is proposed
when this latter is not directly obtained. The result can be
extended to discrete case.
5 References
[1] M. Chadli, D. Maquin, J. Ragot, “Nonquadratic
stability analysis of Takagi-Sugeno models”, In Proc.
of the 41th IEEE CDC, Las Vegas, Nevada, pp. 2143-
2148, 2002.
[2] K. S. Narendra, J. Balakrishnan, “ A common
Lyapunov function for stable LTI systems with
common A-matrices”, IEEE Trans. on Automatic
Control, 39(12), pp. 2469-2471, 1994.
[3] K. Tanaka, T. Ikeda and H.O Wang, “Fuzzy
regulators and fuzzy observers: relaxed stability
conditions and LMI-based design”, IEEE Trans. on
Fuzzy Systems, 6(2), pp. 250-256, 1998.
[4] T. Takagi and M. Sugeno, “Fuzzy identification of
systems and its application to modelling and control”,
IEEE Trans. on Systems, Man, Cybernetics, 15, (1),
116-132, 1985.
[5] M. Akar and U. Ozguner, “Decentralized techniques
for the analysis and control of T-S fuzzy systems”,
IEEE. Trans. on Fuzzy Systems, 8, (6), pp. 691-704,
2000.
[6] M. Johansson, A. Rantzer, K. Arzén, “Piecewise
quadratic stability of fuzzy systems”, IEEE Trans. on
Fuzzy Systems, 7(6), pp. 713-722, 1999. 
[7] S. Boyd, L. Elghaoui, E. Feron and V. Balakrishnan.
Linear matrix inequalities in systems and control
theory, Philadelphia, PA: SIAM, 1994.
[8] Y. Mori, T. Mori and Y. Kuroe, “Somme new
subclasses of systems having a common quadratic
Lyapunov function and comparison of known
subclasses”. In Proc. of the 40th IEEE CDC, Orlando,
Florida, pp. 2179-2180, 2001.


On the stability analysis of a class of multiple models

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On the stability analysis of a class of multiple models

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