International Journal of Computers, Communications & Control
Vol. I (2006), No. 4, pp. 21-34

On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems

Hajer Bouzaouache, Naceur Benhadj Braiek

Abstract: The problem of global exponential stability for a class of nonlinear
singularly perturbed systems is examined in this paper. The stability analysis
is based on the use of basic results of integral manifold of nonlinear singularly
perturbed systems, the composite Lyapunov method and the notations and properties
of Tensoriel algebra. Some of the derived results are presented as linear matrix
inequalities (LMIs) feasibility tests. Moreover, we pointed out that if the global
exponential stability of the reduced order subsystem is established this is equivalent
to guarantee the global exponential stability of the original high order closed loop
system. An upper bound ε1 of the small parameter ε , can also be determined up
to which established stability conditions via LMI’s are maintained verified. A
numerical example is given to illustrate the proposed approach.

Keywords: Nonlinear singularly perturbed system, Integral manifold, Lyapunov sta-
bility, Kronecker product, Linear matrix inequalities (LMIs).

1 Introduction

Stability analysis and control of nonlinear singularly perturbed systems have been widely studied in
the literature [2], [6], [7], [12], [13]. In a two time scale framework, the stability study of the controlled
systems using the Lyapunov stability method [15] and the integral manifold approach as a means for the
control of nonlinear systems based on the singular perturbation method have been developed in recent
years [10], [11], [14], [16], [17]. The approaches proposed in this direction. differ by imposing different
conditions on the smoothness properties of the used functions, different assumptions and different classes
of Lyapunov functions.

In this paper, we are concerned with the global exponential stability of polynomial singularly per-
turbed systems when the chosen design manifold is an exact integral one. Further extension of some
previous results [11], [17] are suggested and leads to effective global exponential stability conditions via
LMIs [8] which can be easily verified when using LMI toolbox of Matlab.

The contribution of the present paper is based, on one hand, on the use of the Lyapunov method which
is a powerful tool for combined controller design and stability analysis, the definition of appropriate
Lyapunov functions for the reduced systems and the corrected system via the integral manifold approach
and on the other hand, on the notations and properties of the tensoriel product [9].

Our paper is organized as follows: in section 2 we present the considered description of the studied
systems which allows important algebraic manipulations and some results from the literature on integral
manifolds for nonlinear singularly perturbed systems. Some useful notations and needed assumptions are
introduced in section 3. Exploiting the stability statements about singularly perturbed systems possessing
integral manifolds and using the composite Lyapunov technique, we propose in section 4 an appropriate
control law that insures the existence of an attractive integral manifold and furthermore insures stability
of the studied systems when the dynamics are restrictive to the integral manifold. The stability results
proving the global exponential stability of polynomial singularly perturbed systems are also given and
presented as linear matrix inequalities feasibility tests. Finally an illustrative example is treated and some
conclusions are drawn.

Copyright c© 2006 by CCC Publications



22 Hajer Bouzaouache, Naceur Benhadj Braiek

2 Studied systems and integral manifolds

The class of systems to be considered in this paper are described by the following state equations:
{

ẋ = f (x, z) (a)
ε ż = g(x, z) + l(x, z)u (b)

(1)

where x ∈ Rn1 is the state of the slow subsystem (1-a), z ∈ Rn2 is the state of the fast subsystem, u ∈
Rpis the input control. ε is a small positive parameter. f , g and l are analytic vector fields which are
sufficiently many times continuously differentiable functions of their arguments. Using the Kronecker
power of vectors, these functions can be written in the polynomial form as [7]:





f (x, z, ε) =
r
∑

i=1

i+1
∑
j=1

Fi jx
[i+1− j] ⊗z[ j−1]

g(x, z) =
r
∑

i=1

i+1
∑
j=1

Gi jx
[i+1− j] ⊗z[ j−1]

l(x, z) =
r
∑

i=1

i+1
∑
j=1

Li j(Im ⊗(x[i+1− j] ⊗z[ j−1]))

(2)

In general the stability of the reduced order subsystems for a class of nonlinear singularly perturbed
systems cannot guarantee the stability of the original full order system even with the additional stability
of the boundary layer subsystem but when an attractive manifold is designed, the stability problem of
the original system reduces to a stability problem of a low dimensional system on the manifold. Subse-
quently, in the context of control system design, our goal is to find an appropriate control law that insures
the existence of an attractive integral manifold and furthermore insures stability of the studied systems
(1) when the dynamics are restrictive to the integral manifold.

The basic ideas of exploiting the integral manifold method are:

– If an integral manifold Σ of systems described by (1) is established, so that if the initial states start
on Σ, the trajectory of the system remains on Σ thereafter.

– When restricted to the integral manifold Σ, the dynamics of the system should insure stability of
the equilibrium.

– The integral manifold should be attractive so that if the initial conditions are off Σ, the solution
trajectory asymptotically converges to Σ.

According to these important issues of the integral manifold method, let’s present the definition, and the
properties of integral manifold of nonlinear systems.
Definition 1. [16] The set Σ ⊂ R×Rn is said to be an integral manifold (invariant manifold) for the
differential equation: Ẋ = N(t, X ), X , N ∈Rn if for (t0, X0) ∈ Σ, the solution (t, X (t)), X (t0) = X0, is in Σ
for t ∈R. If (t, X (t)) ∈ Σ for only a finite interval of time, then Σ is said to be a local integral manifold.
Lemma 1. [11] Consider the following system:

{
ẋ = f (t, x, y, ε)
ε ẏ = g(t, x, y, ε)

(3)

x, f ∈Rn, y, g ∈Rm,t ∈R, ε a small parameter. And suppose the following hypotheses hold:
– The algebraic equation g(t, x, y, 0) = 0 has an isolated solution y = h0(t, x), ∀t ∈R, ∀x ∈ Bx
– The functions f , g, and h0 are twice continuously differentiable (∈C2) ∀t ∈R,∀x ∈ Bx. ∀ε ∈ [0, ε0)

and for ‖y−h0(t, x)‖ ≤ ϕ̄y where ε0 and ϕ̄y are positive real constants.



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 23

– The eigenvalues λi = λi(t, x), i = 1, 2, . . . , m of the matrix Z(t, x) :=
(

∂ g
∂ y

)
(t, x, h0(t, x), 0) satisfy

the inequality

Re [λi] ≤ −2β < 0 ∀t ∈R,∀x ∈ Bx (4)

Then there exists ε ≤ ε1 such that ∀ε ∈ [0, ε1), the singularly perturbed system (3) has an m-dimensional
local integral manifold

Σε : y = h0(t, x) + H(t, x, ε) = h(t, x, ε) (5)

where h(t, x, ε) is defined for all x ∈ Bx and ε ≤ ε1 and is continuously differentiable (∈ C1)
The function h(t, x, ε) ∈ C1 satisfies the so-called manifold equation :

ε
∂ h
∂ t

+ ε
∂ h
∂ x

f (t, x, h, ε)g(t, x, h, ε) (6)

which is obtained by substituting y by h in equation (3).
On this manifold, the flow of systems (3) is governed by the n-dimensional reduced system

ẋ = f (t, x, h(t, x, ε), ε) (7)

Furthermore, if for x ∈ Bx and p integer we have f (t, x, y, ε) ∈ C p+1, g(t, x, y, ε) ∈ C p+2 and h0(t, x) ∈
C p+2 then h ∈ C p

3 Useful notations and assumptions

In our study we make use of the following lemma 2 and Assumptions 1-2. The lemma 2 is concerned
with a Kronecker transformation of vectors. More properties of the Kronecker product are given in the
Appendix.

Lemma 2. [6] Given X =
(

x
z

)
∈ Rn; x ∈ Rn1 , z ∈ Rn2 and n = n1 + n2 there exists a matrix

(i)
M ∈

Rn
i×ni making possible a transformation which introduces the change of coordinates that forms the new

following Kronecker power of vector:

(̂
x
z

)[i]
=




x[i]

:
x[i− j] ⊗z[ j]

:
z[i]



∈Rni

such that

X [i] =
(

x
z

)[i]
=

(i)
M

(̂
x
z

)[i]
(8)

with 



(i)
M = (

(i−1)
M ⊗In)

(i)
U

(i)
V

(1)
M = In

∣∣∣∣∣∣

1 ≤ j ≤ (i−1)
ni =

i
∑
j=0

n
i− j
1 n

j
2

(9)



24 Hajer Bouzaouache, Naceur Benhadj Braiek

and

(i)
V =




Ini1
a
b 0

. . .
0 ab

. . .
Ini2




︸ ︷︷ ︸∣∣∣∣ (i+1) blocs columns2i blocs rows

(i)
U =




U
n

(i−1)
1 ×n

. . . 0
U

n
(i−k)
1 n

(k−1)
2 ×n

0
. . .

U
n
(i−1)
2 ×n




f or j ∈ {2, ...i}
{

a = U
n2×n(i− j+1)1

⊗I
n
( j−2)
2

b = I
n
(i− j+1)
1 ×n

( j−1)
2

(10)

The permutation matrix denoted Un×m is defined in [9]

Un×m =
n

∑
i=1

m

∑
k=1

( ei
(n)

eTk
(m)

)⊗(eTk
(m)

ei
(n)

) (11)

This matrix is square (nm×nm) and has precisely a single 1 in each row and in each column.
To clarify the meaning of

(i)
M, consider the following example:

For n = 3 (n1 = 2, n2 = 1), i = 2; X [2] =
(

x
z

)[2]
=

(2)
M

(̂
x
z

)[2]

X [2] =
(

x
z

)[2]
=




1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 0 0 1 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 1 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1







x[2]

x⊗z
z[2]




Assumption 1. There exists a continuously differentiable function V1(t, x) : R×Rn1 →R+ such that the
following inequalities hold:

∀t ∈R, x ∈Rn1

α1 ‖x‖2 ≤ V1(t, x) ≤ α2 ‖x‖2 (12)
‖∇xV1(t, x)‖ ≤ α3 ‖x‖2 (13)



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 25

∇tV1(t, x) + (∇xV1(t, x))T f (t, x, 0, 0) ≤ −2γ1V1 (14)
where α1, α2, α3, and γ1 are positive constants.
Assumption 2. There exists a continuously differentiable function V2(t, z) : R×Rn2 →R+ such that the
following inequalities hold:

∀t ∈R, z ∈Rn2
β1 ‖z‖2 ≤ V2(t, z) ≤ β2 ‖z‖2 (15)

dV2(t, z)
dt

≤ −2
ε

γ2V2(t, z) (16)

(17)

where β1, β2 and γ2 are positive constants.

4 Main results

Given the system (1), (2), we have to determine an adequate feedback control u that, starting from
any initial states, will attract exponentially the trajectories of the closed loop system along the chosen
design manifold to the equilibrium point at the origin.

In what follows, we assume that hypothesis of lemma 1 are satisfied, and hence the singularly per-
turbed system (1) has an n2 dimensional integral manifold:

z = h0(x) (18)

satisfying the equation (1-b). The flow of system (1), on this manifold, is governed by the n1 dimensional
reduced system:

ẋ = f (t, x, h0(t, x, ε), ε) (19)

This result can be reached by the design of a desired control u satisfying:

l(x, z)u = −g(x, z) + A(z−h0) + ε
dh0(x)

dx
f (x, z) (20)

where A is a Hurwitz matrix. Specifically, we choose the design manifold in this paper to be equal to
z = h0(x) = 0. Then, the control in equation (20) becomes:

l(x, z)u = −g(x, z) + Az (21)
and the nonlinear singularly perturbed system (1) can be written as:





ẋ =
r
∑

i=1

i+1
∑
j=1

Fi jx
[i+1− j] ⊗z[ j−1] (a)

ε ż = Az (b)
(22)

It is then clear, that the fast subsystem and the fast states of the system (22) are attracted toward the
manifold as quickly as desired by the choice of the Hurwitz matrix A.

The reduced order system of (22) is obtained by setting ε = 0 as:

ẋ = f (x, 0) =
r

∑
i=1

Fiix
[i] (23)

The boundary layer system is given, in the fast time scale τ tε , by:

dz∗(τ)
dτ

= Az∗(τ) (24)



26 Hajer Bouzaouache, Naceur Benhadj Braiek

Now to study the stability of the system (22), let’s consider that the reduced order system (23) and the
boundary layer system (24) have respectively V1(x) and V2(z) as quadratic Lyapunov candidate functions
verifying Assumptions 1-2 and defined as follows :

V1(x) = x
T P1x (25)

V2(z) = z
T P2z (26)

where P1, P2 are symmetric positive definite matrices solutions of the following Lyapunov equations:

V̇1(x) ≤ −xT Q1x (27)

V̇2(z) ≤ −zT Q2z (28)
Q1, Q2 are also positive definite matrices.

Based on results of the stability theory [15] and others derived in previous work [1], [5], (27) and
(28) are formulated as follows:

τ T1 (M
T
1 P1 + P1M1)τ1 ≤ −Q1 (29)

(
A
ε

)T
P2 + P2

(
A
ε

)
≤ −Q2 (30)

where

P1 =




P1 0
P1 ⊗In1

. . .
0 P1 ⊗Ins−11


 (31)

and

M1 =




λ11Π11 λ12Π12 ··· λ1sΠ1s
...

. . .
...

...
. . .

...
λs1Πs1 ··· ··· λssΠss




(32)

with

Πk− j+1, j =




mat
(nk− j ,n j )

(F 1
T

kk )

mat
(nk− j ,n j )

(F 2
T

kk )

...
mat

(nk− j ,n j )
(F n

T

kk )




(33)

For the corrected system (22), we define the following Lyapunov functions:

V (x, z, ε) = X T Eε PX (34)

where

X =
[

x
z

]
∈Rn; P =

(
P1 0
0 P2

)
and Eε =

(
In1 0
0 ε In2

)
(35)

The derivative of V (x, z, ε) along the trajectories of (22) is given by:

V̇ (x, z, ε) = X T Eε PẊ + Ẋ T Eε PX (36)



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 27

In the above equation, we need to explicit the derivative of the state vector X . So, we begin by writing
the state equations (22) in the following form:

Ẋ =

.(
x
z

)
=

r

∑
i=1

Λi
(̂

x
z

)[i]
(37)

with

Λ1 =
(

F11 F12
0 Aε

)

and for i > 1:

Λi =
(

Fi1 ···Fi j ···Fi(i+1)
On2×αi

)
(38)

Using the results given by the lemma 2, it follows from equation (37) that

Ẋ =
r

∑
i=1

Λi
(i)
M +X [i] (39)

where
(i)+

M is the Moore-Penrose pseudo inverse of
(i)
M defined in (A.4).

The derivative of the composite Lyapunov function (34) is then written:

V̇ (x, z, ε) = 2
r

∑
k=1

X T (Eε PΛk
(k)
M +)X k (40)

Using the property of the vec-function (1), we have:

V̇ (x, z, ε) = 2
r

∑
k=1

V Tk X
[k+1] (41)

where

Vk = vec(Eε PΛk
(k)
M +) (42)

Knowing that all polynomials with even degree (2s) can be represented as a symmetric quadratic
form. Thus, we assume in the following development that r is odd: r = 2s−1, and it comes out:

V Tk X
[k+1] =

hk

∑
j=gk

λk− j+1, jX [k+1− j]Nk− j+1, jX [ j] (43)

where λk− j+1, j are arbitrary reals verifying:

hk

∑
j=gk

λk− j+1, j = 1 (44)

and for k = 1, . . . , 2s−1:
gk = sup(0, k + 1−s) and hk = inf(s, k) (45)

for j = gk, . . . , hk:
Nk− j+1, j = mat

(nk− j+1,n j )
(Vk) (46)

Applying the properties [1], one obtains:

Nk− j+1, j = mat
(nk− j+1,n j )

(
vec(Eε PΛk

(k)
M +)

)
= Unk− j×n(Eε P⊗Ink− j ).Mk− j+1, j (47)



28 Hajer Bouzaouache, Naceur Benhadj Braiek

with

Mk− j+1, j =




mat
(nk− j ,n j )

(B1
T

k )

mat
(nk− j ,n j )

(B2
T

k )

...
mat

(nk− j ,n j )
(Bn

T

k )




andBk=Λk
(k)
M + (48)

where Bik is the i-th row of the matrix Bk:

Bk=




B1k
B2k
...

Bnk


 (49)

By (47) and from the relation (48), we obtain:

X [k− j+1]
T
Nk− j+1, jX [ j]

= X [k− j+1]
T
Unk− j×n(Eε P⊗Ink− j )Mk− j+1, jX [ j]

= X [k− j+1]
T
(Eε P⊗Ink− j )Mk− j+1, jX [ j]

(50)

Consequently, we have:

V Tk X
[k+1] =

hk

∑
j=gk

λk− j+1, jX [k− j+1]
T
Nk− j+1, jX

[ j] = X T (Pε Mk)X (51)

with

X =




X
X [2]

...
X [s]


 (52)

and

Pε =




Eε P 0
Eε P⊗In

. . .
0 Eε P⊗Ins−1


 (53)

Let’s note that Pε is a symmetric positive matrix, and V̇ (X , ε) (41) can be written as:

V̇ (X , ε) = 2
2s−1
∑
k=1

V Tk X
[k+1] = X T (Pε Mε + M

T
ε Pε )X (54)

with

Mε =
2s−1
∑
k=1

Mk =




λ11M11 λ12M12 ··· λ1sM1s
...

. . .
...

...
. . .

...
λs1Ms1 ··· ··· λssMss




(55)

When considering the nun-redundant form, the vector X can be written as:

X = τ X̃ (56)



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 29

where

τ =




T1 0
. . .

0 Ts


 and X̃ =




X̃
...

X̃ [s]


 (57)

>From (54) and (56) we easily obtain:

V̇ (X , ε) = X̃ T τ T (Pε Mε +MTε Pε )τ X̃ (58)

Let us denote the largest eigenvalue of the matrix Pε by λmax(Pε ), the smallest eigenvalue of Q by
λmin(Q). Where the matrix Q verifies: V̇ (X , ε) ≤−X T QX . The positive definiteness of Pε and Q implies
that these scalars are all strictly positive. Since matrix theory shows that:

Pε ≤ λmax(Pε )I; λmin(Q)I ≤ Q (59)
We have

X T QX ≥ λmin(Q)
λmax(Pε )

X T [λmax(Pε )I] X ≥
λmin(Q)
λmax(Pε )

X T [λmax(Eε P)I] X (60)

Otherwise
X T X ≥ ‖X‖2 and Eε P ≤ λmax(Eε P)I (61)

Hence, (54) will satisfies the following condition:

V̇ (X , ε) ≤ −2γV (X , ε) where γ = 1
2
· λmin(Q)

λmax(Pε )
(62)

It comes out
V (X , ε) ≤ V (X0)e−2γ(t−t0) (63)

Considering the previous developments, we state now our main result:
Theorem 1. Assume the following assumptions hold:

(i) Lemma 1 satisfied
(ii) Assumptions 1 - 2 are satisfied

The system (1) is globally exponentially stable (GES), if there is, for all ε < ε1, ε1 > 0 a feasible solution
to the LMI : 




ε > 0
∃PT1 = P1 > 0
∃PT2 = P2 > 0
∃PT = P > 0(

A
ε
)T

P2 + P2
(

A
ε
)
≤ −Q2

τ T1 (M
T
1 P1 + P1M1)τ1 ≤ −Q1

τ T (MTε Pε + Pε Mε )τ ≤ −Q

(64)

M1, τ1 and P1 are given by (32), (57) and (31). Mε , τ and Pε are given by (55), (57) and (53). Moreover,
the Lyapunov function that demonstrates the G.E.S is given by: V (x) = X T Eε Pε X

Now, let’s evaluate the convergence rate of the full order system (22). In view of (12), (15), and (27),
we have for all t ∈R, x ∈Rn1 and z ∈Rn2 :

V1(x) ≤ (α2 ‖x0‖2 + ε β2 ‖z0‖2)e−2γ(t−t0) (65)
>From (12) and (15) we have:

‖x‖ ≤




(√
α2
α1
‖x0‖

)2
+




√
ε β2
α1

‖z0‖



2



1/2

e−γ(t−t0) ≤



√
α2
α1
‖x0‖+

√
ε β2
α1

‖z0‖

 e−γ(t−t0)

(66)



30 Hajer Bouzaouache, Naceur Benhadj Braiek

Identically, using (15) and (16), we obtain:

β1
∥∥z2

∥∥ ≤ V2(z) ≤ β2
∥∥z20

∥∥ e−2(γ2/ε)(t−t0) (67)

then

‖z‖ ≤
√

β2
β1
‖z0‖e−(γ2/ε)(t−t0) (68)

>From (66) and (68), we can write in the case γ ≤ (γ2/ε) that

‖X‖2 ≤ 2η e−2γ(t−t0)

where

η = max







√
α2
α1
‖x0‖+

√
ε β2
α1

‖z0‖



2

,

(√
β2
β1
‖z0‖

)2
 (69)

which implies that the norm ‖X‖ of the state vector converges to zero exponentially, with a rate γ . The
convergence rates of the reduced systems can be calculated:

γ1 =
1
2
· λmin(Q1)

α2
, γ2 =

λmin(Q2)
2β2

>From above, we state the following second result:
Theorem 2. Assume the following assumptions hold:
(i) Lemma 1 satisfied
(ii) Assumptions 1 - 2 are satisfied
(iii) There exists a Lyapunov function V (t, X , ε) that satisfies equation ((34))
Then the original nonlinear singularly perturbed system ((1)) is globally exponentially stable under the
proposed control ((21)) and with the convergence rate γ ((63)).

Moreover, note that when we proves that the limit of γ as ε → 0 , tends to the convergence rate of the
reduced order system. This implies that under the proposed control (21), the global exponential stability
of the initial studied system is equivalent to that of the reduced order system.

5 Illustrative Example

To illustrate the previous derived results, we consider a third order nonlinear singularly perturbed
system defined by the following equations:





ẋ1 = −x1 + x2 + 0.1x1z
ẋ2 = −x1 −0.09x2 + 2z + 0.05x1z
ε ż = 4x1 −4x2 + z + 0.5x21 −x22 + 10u

This system can be described by the following model using the Kronecker product and the power of
vectors which allowed important algebraic manipulations.

{
ẋ = F11x + F12z + F21x

[2]
+ F22x⊗z + F23z

[2]

ε ż = G11x + G12z + G21x
[2]

+ G22x⊗z + G23z
[2]

+ Bu



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 31

where

F11 =
[
−1 1
−1 −0.09

]
, F12 =

[
0
2

]
, G11 =

[
4 −4

]
, G12 = 1,

F21 = 02×4, F22 =
[

0.1 0
0 0.5

]
, F23 = 02×1

G21 =
[

0.5 1 0
]
, G22 =

[
0 0

]
, G23 = 0, B = 10.

When implemented using the LMI toolbox of Matlab, the proposed LMI’s conditions proves that the
numerical studied system which is initially instable can be globally exponentially stabilised by the given
controller with the considered A = −1 for all ε < ε1 = 0.5 in view of theorem 1:

u = −0.4x1 + 0.4x2 −0.1z−0.05x21 + 0.1x22

Figure 1: State trajectories of the controlled studied system

Hence for all ε < ε1 = 0.5 and from any initial states, the trajectories of the system are steered to the
origin along the integral manifold with the convergence rate 0.1 in view of (62). Indeed, it is shown in
Fig. 1 that the state trajectories of the controlled system ( – – –) are bounded by the function (—).

6 Conclusion

In this paper, the global exponential stabilisation for nonlinear singularly perturbed control systems
is investigated. In the stability study, the composite Lyapunov method was applied and the global expo-
nential stability of the equilibrium of the full control system was established for all ε < ε1. The upper
bound ε1 for which the stability properties are guaranteed can be reached after a number of iterations on
ε when resolving the proposed LMI’s conditions via the LMI Toolbox of Matlab. A numerical example
has been provided to illustrate the proposed results.



32 Hajer Bouzaouache, Naceur Benhadj Braiek

7 Appendix

Notations: The dimensions of the matrices used here are the following:
A(p×q), B(r ×s), C(s×h), D(s×h), E(n× p),
P(n×n), X (n×1) ∈Rm, Y (m×1) ∈Rm, Z(q×1) ∈Rq
Let’s consider the following notations:
In: n×n identity matrix;
0n×m: n×n zero matrix;
0zero matrix of convenient dimensions ;
AT : transpose of matrix A;
A > 0 (A ≥ 0): symetric positive definite (semi definite matrix A);
ek
(q)

: q dimensional unit vector which has 1 in the kth element and zero elsewhere.

The kth row of a matrix such as A is denoted Ak. and the kth column is denoted A.k. The ik element of A
will be denoted aik.
The Kronecker product of A and B is denoted A⊗B a (p.r×q.s) matrix, and the i−th Kronecker’s power
of A denoted i A[i] = A⊗A⊗···⊗A s a ( pi ×qi) matrix.

The nun-redundant j-power X̃ [ j] of the state vector X introduced in [9] is defined as:
X̃ [1] = X [1] = X



∀ j ≥ 2 X̃ [ j] =
[
x

j
1, x

j−1
1 x2,··· , x

j−1
1 xn, x

j−2
1 x

2
2

x
j−2
1 x2x3,··· , x

j−2
1 x2xn,

···x j−21 xn2 ,··· , x
j−3
1 x

3
2
,··· , x jn

]
(A.1)

where the repeated components of the redundant j-power are omitted. Then we have the following
relation: 



∀ j ∈N ∃!Tj ∈Rn

j×α j ; α j =
(

n + j −1
j

)

X [ j] = TjX̃ [ j]
(A.2)

thus, one possible solution for the inversion can be written as:
X̃ [ j] = T +j X

[ j] (A.3)
where T +j is the Moore-Penrose pseudo inverse of Tj given by:

T +j =
(

T Tj Tj
)−1

T Tj (A.4)

and α j stands for the binomial coefficients.
An important vector valued function of matrix denoted vec(.) was defined as [9]:

vecpq×1(A) =




A.1
A.2
...
A.q


 (A.5)

A matrix valued function is a vector denoted mat(n,m)(.) was defined in [1] as follows: If V is a
vector of dimension p = n×n then Mmat(n,m)(V ) is the n×m matrix verifying:

V = vec(M) (A.6)
Among the main properties of this product presented in [9], [1], we recall the following useful ones:

(A⊗B) (C ⊗D) = (AC)⊗(BD) (A.7)
(A⊗B)T = AT ⊗BT (A.8)

B⊗A = Ur×p (A⊗B)Uq×s (A.9)
X ⊗Y = Un×m (Y ⊗X ) (A.10)

vec (EAC) =
(
CT ⊗E

)
vec (A) (A.11)

vec
(
AT

)
= Up×qvec (A) (A.12)

∀i ≤ k X [k] = Uni×nk−i X [k] (A.13)



On Guaranteed Global Exponential Stability Of Polynomial Singularly
Perturbed Control Systems 33

References

[1] E. Benhadj Braiek , F. Rotella, M. Benrejeb, Algebraic criteria for global stability analysis of non-
linear systems, SAMS Journal, Vol.17, pp. 211-227, 1995.

[2] P. Borne, J.C Gentina, On the stability of large nonlinear systems. Structured and simultaneous
Lyapunov for system stability problem, Joint. Aut. Cont. Conf., Austin, Texas, 1974.

[3] P. Borne, M. Benrejeb, On the stability of a class of interconnected systems. Application to the forced
Working condition, Actes 4th IFAC Symposium MTS, Fredericton, Canada, 1977.

[4] P. Borne and G. Dauphin-Tanguy, Singular perturbations, boundary layer problem, Systems and
Control Encyclopedia, M.G.Singh Eds, Pergamon press, III I-2, 1987.

[5] H. Bouzaouache, E. Benhadj Braiek, On the stability analysis of nonlinear systems using polynomial
Lyapunov functions, IMACS World Congress, Paris, 2005.

[6] H. Bouzaouache, Sur la représentation d’état des systèmes non linéaires singulièrement perturbés
polynomiaux, Conférence Tunisienne de Génie Electrique CTGE ,pp. 420-424, 2004.

[7] H. Bouzaouache, E. Benhadj Braiek, M. Benrejeb, A reduced optimal control for nonlinear singu-
larly perturbed systems", Systems Analysis Modelling Simulation Journal.Vol 43, pp. 75-87, 2003.

[8] S. Boyd, L. El Ghaoui and Feron E. Balakrishnan, Linear Matrix Inequalities in System and Control
Theory, SIAM, Philadelphia,1994.

[9] J. W. Brewer, Kronecker product and matrix calculus in system theory, IEEE Trans.on Circ.Syst, 25
, Nř9, pp. 772-781, 1978.

[10] C.C. Chen, Research on the feedback control of nonlinear singularly perturbed systems, PhD The-
sis, Inst. Electrical Eng., NatSun Yat Sen Univ, Taiwan, Roc, 1994.

[11] F. Ghorbel and M.W.. Spong, Integral manifolds of singularly perturbed systems with application
to rigid-link flexible joint multibody systems, International Journal of Nonlinear Mechanics 35, pp.
133-155, 2000.

[12] M. Innocenti, L. Greco, L. Pollini, Sliding mode control for two time scale systems : stability
issues, Automatica, Vol. 39,pp. 273-280,2003.

[13] P.V. Kokotovic, H.K. Khalil, O’Reilly, Singular perturbation methods in control: analysis and de-
sign, academic Press , New york, 1986.

[14] P.M. Sharkey and J. O’Reilly, Exact design manifold control of a class of NLSP systems, IEEE
Trans. Automat. Contr., Vol.32, pp.933-937, 1987.

[15] J.J.E. Slotine, Applied nonlinear control, Englenwood Cliffs, NJ, Prentice Hall, 1991.

[16] V.A. Sobolev, Integral manifolds and decomposition of singularly perturbed systems, Systems Con-
trol Letters, Vol.5, pp.169-179, 1984.

[17] M.W.Spong, K. Khorasani, P.V. Kokotovic, An Integral Manifold approach to feedback control of
flexible joint robots, IEEE J. Robot. Automat.3 (4), pp. 291-300, 1987.



34 Hajer Bouzaouache, Naceur Benhadj Braiek

Hajer Bouzaouache, Naceur Benhadj Braiek
Laboratory of Study and Automatic Control of Processes

Address: Polytechnic School of Tunisia (EPT) BP.743, 2078 La Marsa, Tunisia.
E-mail: hajer.bouzaouache,naceur.benhadj@ept.rnu.tn

Received: November 6, 2006