M Triki

M Farza

M M'saad

T Maatoug

B Dahhou

Observer synthesis for a class of MIMO non triangular nonlinear systems

Keywords: Nonlinear systems, High gain observers, Sliding mode observers, MIMO systems

A high gain observer is proposed for a class of nonlinear systems involving some uncertainties. In the absence of the uncertainties, the considered class of systems becomes a canonical form that characterizes a class of uniformly observable systems and the observation error exponentially converges to zero. In the presence of uncertainties, it is shown that the observation error can be made as small as desired by appropriately specifying the design parameter of the observer gain. Two main contributions are worth to be emphasized: the first one is related to the structure of the considered class of systems which does not assume a complete triangular structure. That is, each block may contain nonlinearities which depend on the whole state. The second one lies in the simplicity of the observer gain design since its calibration is reduced to the choice of a single design parameter. More specifically, it involves a design function that has to satisfy a mild condition which is given. Different expressions of such a function are proposed. Of particular interest, it is shown that high gain observers and sliding mode like observers can be derived by considering particular expressions of the design function. An example with simulation results is given for illustration purposes.

I. INTRODUCTION

The need to study the observer design problem for nonlinear dynamical systems is, from a control point of view, well understood by now. The list of references herein covers part of the recent works done in that area ( [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF], [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF], [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF], [START_REF] Souleiman | Direct transformation of nonlinear systems into state affine MISO form and nonlinear observers design[END_REF], [START_REF] Rajamani | Observers for Lipschitz Nonlinear Systems[END_REF], [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF], [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF], [START_REF] Farza | Observer design based on triangular form generated by injective map[END_REF], [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF]). In this paper, one aims at designing a high gain observer for a class of nonlinear uncertain systems satisfying some regularity assumptions. The general framework of this observer design is based on the contributions given in [START_REF] Bornard | A graph approach to uniform observability of nonlinear multi output systems[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF]. The gain of the proposed observer is issued from the resolution of a constant Lyapunov algebaric equation and it is explicitly given. Its tuning is achieved through the choice of a single parameter whatever the dimension of the considered system is. More specifically, one exhibits a state observer for nonlinear systems which are diffeomorphic to: ẋ = Ax + ϕ(u, x) + ε(t) y = Cx (1)

1 Groupe de Recherches en Informatique, Image, Automatique et Instrumentation de Caen (GREYC) Ensicaen UMR 6072 CNRS, Bd Maréechal Juin, 14050 Caen, France. mfarza@greyc.ensicaen.fr where the state x =     

x 1 x2 . . .

x q      ∈ IR n , with x k =      x k 1 x k 2 . . . x k λ k      ∈ IR n k , x k i ∈ IR p k , i = 1, . . . , λ k , k = 1, . . . , q, q k=1 n k = q k=1
p k λ k = n with p k ≥ 1 and λ k ≥ 2; the

output y =      y 1 y 2 . . . y q      ∈ IR p with y k ∈ IR p k , k = 1, . . . , q and q k=1 p k = p; A =    A 1 . . . A q   , A k =      0 I p k 0 . . . . . . 0 . . . 0 I p k 0 . . . 0 0      , C =    C 1 . . . C q   , C k = I p k 0 . . . 0 and the nonlinear function field ϕ(u, x) =      ϕ 1 (u, x) ϕ 2 (u, x) . . . ϕ q (u, x)      ∈ IR n ; ϕ k (u, x) =      ϕ k 1 (u, x) ϕ k 2 (u, x) . . . ϕ k λ k (u, x)      ∈ IR n k
where for k = 1, . . . , q, the element

ϕ k i (u, x) ∈ IR p k
has the structural dependence on the states:

• for 1 ≤ i ≤ λ k -1: ϕ k i (u, x) = ϕ k i (u, x 1 , x 2 , . . . , x k-1 , x k 1 , x k 2 , . . . , x k i , x k+1 1 , x k+2 1 , . . . , x q 1 ) (2)
• for i = λ k :

ϕ k λ k (u, x) = ϕ k λ k (u, x 1 , x 2 , . . . , x q ) (3)
The function ε(t) is a completely unknown function. Many components of this function are zero since its structure can be described as follows:

ε(u, x) =      ε1 (t) ε2 (t) . . . εq (t)      ∈ IR n with • εk ≡ 0 for k = 2, . . . , q. • ε1 i ≡ 0 for i = 1, . . . , λ 1 -1. • ε1 λ1 ≡ ε(t) where ε(t) is an unknown bounded function i.e. ∃η > 0; ∀t ≥ 0; ε(t) ≤ η (4)
Notice that in the absence of uncertainties, the class of considered systems generalizes that considered in [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF] in two directions. Firstly, the output x k 1 of each sub-block k is not a scalar as in [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF] but belongs to IR p k . Secondly, unlike in [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF]where the nonlinearity intervening in the last equation of each sub-block, namely ϕ k λ k (u, x), assumes the same triangular state dependence as the previous variables of the same sub-block, i.e. ϕ k λ k (u, x) satisfies (2) with i = λ k , a more general state dependence (3)is assumed in this paper. Moreover, it has been also shown in [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF] that in the case where no uncertainty is considered, the class of systems the authors considered is diffeomorphic to system (1) with q = 1 which means that in [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF], all the output belong to the same block. In the case where the uncertainties are considered, system (1) includes the class of systems considered in ( [START_REF] Farza | Observer design based on triangular form generated by injective map[END_REF]). In fact, many systems involving unknown inputs can be put under form [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF] and this allows to design nonlinear observers that simultaneously estimate the state variables as well as the unknown inputs as shown in [START_REF] Farza | Unknown inputs observers for a class of nonlinear systems[END_REF]. This paper is organized as follows. In section 2, the observer design is given and a full convergence analysis is detailed. It is shown that the observation error exponentially converges to zero in the absence of uncertainties, and can be made as small as desired in the presence of uncertainties. The latter result is achieved by appropriately specifying the design parameter involved in the observer gain. In section 3, different expressions of the observer design function are proposed giving rise to different observers. Simulation results are given in section 4 to illustrate the effectiveness of the proposed observer design.

II. OBSERVER DESIGN

As generally used in the high gain observer methodology, one assumes that system (1) satisfies the following Lipschitz assumption:

Assumption 1: ϕ(u, x) is a globally Lipschitz nonlinear function with respect to x uniformly to u. Before giving a candidate observer, one introduces the following notations:

• Let ∆ k (θ) be the diagonal matrix defined by:

∆ k (θ) = diag I p k , 1 θ δ k I p k , . . . , 1 θ δ k (λ k -1) I p k (5)
where θ > 0 is a real number and one defines δ k which indicates the power of θ as follows:

         δ k = 2 q-k q i=k+1 (λ i -1) for k = 1, . . . , q -1 δ q = 1 (6)
Notice that for any k = 1, . . . , q -1, one has

δ k 2 = (λ k+1 -1)δ k+1 (7)
Since λ k ≥ 2, the δ q 's constitute a decreasing sequence of positive real numbers, i.e.

δ 1 ≥ δ 2 ≥ . . . ≥ δ q > 0 (8)
It is easy to show that the following identities hold:

• ∆ k (θ)A k ∆ -1 k (θ) = θ δ k A k • ∆ -1 k (θ)C k = C k (9)
• Let S k be the unique solution of the algebraic Lyapunov equation

S k + A T k S k + S k A k = C T k C k ( 10
)
where A k and C k are defined in system (1). It can be shown that the explicit solution of ( 10) is symmetric positive definite for every θ > 0 and in particular, one has

S -1 k C T k = (C 1 n k I p k , . . . , C n k n k I p k ) T • ∀ξ k ∈ IR p k , let K k (ξ k ) ∈ IR p k
be a vector of smooth functions satisfying:

ξ k T K k (ξ k ) ≥ 1 2 ξ k T ξ k (11)
A candidate observer for system (1) is described by the following dynamical system: [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF].

ẋk = A k xk +ϕ k (u, x)-θ δ k ∆ -1 k (θ)S -1 k C T k K k (C k e k ) (12) for k = 1, . . . , q with • x =      x1 x2 . . . xq      ∈ IR n , xk =      xk 1 xk 2 . . . xk λ k      ∈ IR n k , xk i ∈ IR p k , i = 1, . . . , λ k , k = 1, . . . , q, q k=1 n k = n. • xk 1 = x k 1 (output injection) for k = 1, . . . , q. • xk i = xk i if i = 1. • e k = xk -x k . • The function K k 's satisfy condition
• u and y are known inputs and outputs of system (1). Indeed, one can establish the following fundamental result.

Theorem 2.1: Assume that system (1) satisfies assumption (1), then:

∀M > 0; ∃θ 0 > 0; ∀θ ≥ θ 0 ; ∃λ θ > 0; ∃µ θ > 0; ∃β θ > 0 such that for k ∈ {1, . . . , q}: xk (t) -x k (t) ≤ λ θ e -µ θ t x(0) -x(0) + β θ η for every admissible control u s.t. Essup u(t) ≤ M where η is the upper bound of ε given in (4) . Moreover, λ θ is polynomial in θ, lim θ→∞ µ θ = +∞ and lim θ→∞ β θ = 0.
Proof of Theorem (2.1): set the estimation error e(t) = x(t) -x(t) and let e k (t) be the k ′ th subcomponent of e(t) . For writing convenience and as long as there is no ambiguity, one shall omit the time t for each variable. One has:

ėk = A k e k + ϕ k (u, x) -ϕ k (u, x) -θ δ k ∆ -1 k (θ)S -1 k C T k K k (C k e k ) -εk ( 13
)
where

u is an admissible control such that u ∞ ≤ M , M > 0 is a given constant. For k = 1, • • • , q, set ēk = ∆ k (θ)e k (14)
From equation ( 13) and using identities (9), one gets:

ėk = θ δ k A k ēk -θ δ k S -1 k C T k K k (C k ēk ) + ∆ k (θ)(ϕ k (u, x) -ϕ k (u, x)) -∆ k (θ)ε k (15) Set V k (ē k ) = θ -2σ k 1 ēk T S k ēk ( 16
)
where the σ k 1 's are some reals that shall be specified later

and let V (ē) = q k=1 V k (ē k ) where S = diag(S 1 , • • • , S q
), be the candidate Lyapunov function. One has:

Vk = 2θ -2σ k 1 ēk T S k ėk = 2θ δ k -2σ k 1 ēk T S k A k ēk -2θ δ k -2σ k 1 ēk T C T k K k (C k ēk ) + 2θ -2σ k 1 ēk T S k Λ k (θ)(ϕ k (u, x) -ϕ k (u, x)) -2θ -2σ k 1 ēk T S k Λ k (θ)ε k
Using the algebraic Lyapunov equation [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], one gets:

Vk = -θ δ k -2σ k 1 ēk T S k ēk + θ δ k -2σ k 1 ēk T C T k C k ēk -2θ δ k -2σ k 1 C k ēk T K k (C k ēk ) + 2θ -2σ k 1 ēk T S k ∆ k (θ)(ϕ k (u, x) -ϕ k (u, x)) -2θ -2σ k 1 ēk T S k ∆ k (θ)ε k
Using property [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF] satisfied by the function K k , one obtains:

Vk ≤ -θ δ k -2σ k 1 ēk T S k ēk + θ δ k -2σ k 1 ēk T C T k C k ēk -2θ δ k -2σ k 1 C k ēk T C k ēk + 2θ -2σ k 1 ēk T S k ∆ k (θ)(ϕ k (u, x) -ϕ k (u, x)) -2θ -2σ k 1 ēk T S k ∆ k (θ)ε k = -θ δ k -2σ k 1 ēk T S k ēk -θ δ k -2σ k 1 ēk T C T k C k ēk + 2θ -2σ k 1 ēk T S k ∆ k (θ)(ϕ k (u, x) -ϕ k (u, x)) -2θ -2σ k 1 ēk T S k ∆ k (θ)ε k ≤ -θ δ k V k +2θ -2σ k 1 S k ēk ∆ k (θ)(ϕ k (u, x) -ϕ k (u, x)) +2θ -2σ k 1 S k ēk ∆ k (θ)ε k ≤ -θ δ k V k + +2 λ k max V k λ k i=1 1 θ σ k i (ϕ k i (u, x) -ϕ k i (u, x)) +2 λ k max V k α k,1 η θ σ k 1 +(λ1-1)δ1
where λ k max is the maximum eigenvalue of S k ,

σ k i = σ k 1 + (i -1)δ k and α k,1 is the Kroneker symbol, i.e α k,1 = 1 if k = 1 and 0 otherwise. Therefore, Vk ≤ -θ δ k V k + 2ρ k λ k max V k λ k i=1 q l=1 λ l j=1 χ k,i l,j θ σ l j -σ k i θ -σ l 1 ēl j + 2 λ k max V k α k,1 η θ σ k 1 +(λ1-1)δ1 where ρ k =sup{ ∂ϕ k i ∂x l j (u, x) ; x ∈ IR n and u ∞ ≤ M } and χ k,i l,j = 0 if ∂ϕ k i ∂x l j (u, x) ≡ 0, χ k,i l,j = 1 otherwise. Now, one has Vk ≤ -θ δ k V k + 2ρ k λ k max V k λ k i=1 q l=1 λ l j=1 χ k,i l,j θ σ l j -σ k i θ -σ l 1 ēl + 2 λ k max V k α k,1 η θ σ k 1 +(λ1-1)δ1 ≤ -θ δ k V k + 2ρ k λ k max V k λ k i=1 q l=1 λ l j=1 χ k,i l,j θ σ l j -σ k i √ V l λ l min + 2 λ k max V k α k,1 η θ σ k
where λ l min is the minimum eigenvalue of S l . Thus,

Vk ≤ -( θ δ k V k ) 2 + 2ρ k µ S θ δ k V k λ k i=1 q l=1 λ l j=1 χ k,i l,j θ σ l j -σ k i - δ k 2 - δ l 2 θ δ l V l +2 λ k max V k α k,1 η θ σ 1 1 +(λ1-1)δ1
where µ S = λ max (S) λ min (S) .

Please notice the last term appearing in the right hand side of the last inequality is to be dropped for k = 1. As a result, the real σ k 1 appearing in the denominator of this term is replaced by σ 1 1 . The remaining of the proof is as follows. One firstly assumes that the reals σ k 1 , k = 1, . . . , q are chosen such that the following conditions are satisfied:

0 = σ 1 1 > σ 2 1 > . . . > σ q 1 ( 17
) if χ k,i l,j = 1 then σ l j -σ k i - δ k 2 - δ l 2 ≤ - 1 2 (18)
σ k 1 + (λ k -1)δ k is constant for k = 1, . . . , q (19)
Then, one shall shows that such a choice is possible and provides a set of σ k 1 's satisfying such conditions. Now, suppose that condition (18) holds and assume that θ ≥ 1. Then, one gets:

Vk ≤ -( θ δ k V k ) 2 + 2ρ k µ S θ δ k V k λ k i=1 q l=1 λ l j=1 θ -1 2 θ δ l V l +2 λ k max V k α k,1 η θ σ 1 1 +(λ1-1)δ1 ≤ -( θ δ k V k ) 2 + 2λ k ρ k µ S θ -1 2 θ δ k V k q l=1 λ l j=1 θ δ l V l +2 λ k max V k α k,1 η θ σ 1 1 +(λ1-1)δ1 Now, set V * k = θ δ k V k for k = 1, • • • , q and V * = q k=1 V * k .
Notice that

θV ≤ V * ≤ θ δ1 V (20)
Then,

Vk ≤ -V * k + 2λ k ρ k µ S θ -1 2 V * k q l=1 λ l j=1 V * l +2 λ k max V k η θ σ 1 1 +(λ1-1)δ1 ≤ -V * k + 2λ k nρ k µ S θ -1 2 V * k √ V * +2 λ k max V k η θ σ 1 1 +(λ1-1)δ1 ≤ -V * k + 2λ k nρ k µ S θ -1 2 V * +2 λ k max V k η θ σ 1 1 +(λ1-1)δ1
Hence,

V ≤ -(1 -2n 2 ρµ S θ -1 2 )V * + 2 λ max √ V η θ σ 1 1 +(λ1-1)δ1 where ρ = max{ρ k , 1 ≤ k ≤ q} and λ max = max{λ k max , 1 ≤ k ≤ q}. Using (20), one gets V ≤ -θ(1 -2n 2 ρµ S θ -1 2 )V + 2 λ max √ V η θ σ 1 1 +(λ1-1)δ1
This leads to:

V (ē(t)) ≤ exp -θ(1 -2n 2 ρµ S θ -1 2 ) 2 t V (ē(0)) + 2 √ λ max η θ(1 -2n 2 ρµ S θ -1 2 )θ σ 1 1 +(λ1-1)δ1 (21)
and therefore one obtains for k = 1, . . . , q:

V k (ē k ) ≤ exp -θ(1 -2n 2 ρµ S θ -1 2 ) 2 t V (ē(0)) + 2 √ λ max η θ(1 -2n 2 ρµ S θ -1 2 )θ σ 1 1 +(λ1-1)δ1 (22)
On one hand, from the definition of V and V k (equation ( 16)), one has

θ σ k 1 V k (ē k ) λ k max ≤ ēk ≤ θ σ k 1 V k (ē k ) λ k min ( 23
)
On another hand, from the definition of ēk (equation ( 14)), one has:

ēk ≤ e k ≤ θ (λ k -1)δ k ēk (24)
Finally, from the definition of V k and according to ( 17) and ( 5), one has

( V (ē) ≤ λ max ē ≤ λ max e (25)
Combining ( 22), ( 23) and ( 24), one gets:

e k ≤ θ σ k 1 +(λ k -1)δ k √ λ max √ λ min exp -θ(1 -2n 2 ρµ S θ -1 2 ) 2 t e(0) + 2 √ λ max η √ λ min θ σ k 1 +(λ k -1)δ k θ(1 -2n 2 ρµ S θ -1 2 )θ σ 1 1 +(λ1-1)δ1 = θ σ k 1 +(λ k -1)δ k √ λ max √ λ min exp -θ(1 -2n 2 ρµ S θ -ε ) 2 t e(0) + 2 √ λ max η √ λ min θ -σ 1 λ 1 +σ k λ k θ(1 -2n 2 ρµ S θ -1 2 ) = θ (λ1-1)δ1 √ λ max √ λ min exp -θ(1 -2n 2 ρµ S θ -ε ) 2 t e(0) + 2 √ λ max η √ λ min 1 θ(1 -2n 2 ρµ S θ -1 2 ) (26)
The last equality comes from property (19) satisfied by the σ k 1 's. The parameters λ θ , µ θ and β θ of the theorem are hence:

λ θ = θ (λ1-1)δ1 √ λ max √ λ min , µ θ = -θ(1 -2n 2 ρµ S θ -1 2 ) 2 β θ = 2 √ λ max √ λ min 1 θ(1 -2n 2 ρµ S θ -1 2 )
To end the proof of the theorem, one shall exhibit a set of reals σ k 1 satisfying conditions ( 17), ( 18) and (19). Before providing the set of σ k 1 's, one firstly notes that according to the state dependence given by ( 2) and (3), the case where χ k,i l,j = 1, k, l ∈ {1, . . . , q} and j ∈ {2, . . . , λ l }, occurs if and only if one of the following three situations is met:

• k > l and i takes any value in {1, . . . , λ k }.

• k = l and i ∈ {1, . . . , λ k } with i ≥ j. • k < l and i = λ k . Now, for k = 1, . . . , q and i = 1, . . . , λ k , let us define the σ k i 's as follows:

σ k i = σ k 1 + (i -1)δ k where σ k 1 = -(λ k -1)δ k + (λ 1 -1)δ 1 (27)
and the δ k 's are given by [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF]. According to (27), one has:

σ l j -σ k i - δ k 2 - δ l 2 = σ l 1 + (j -1)δ l -σ k 1 -(i -1)δ k - δ k 2 - δ l 2 = -(λ l -1)δ l + (j -1)δ l +(λ k -1)δ k -(i -1)δ k - δ k 2 - δ l 2 = (λ k -i - 1 2 )δ k + (j -λ l - 1 2 )δ l (28)
Let us now check condition (18) by considering the three cases listed above.

• k > l and i takes any value in {1, . . . , λ k }: Please notice that since k > l, one has k ≥ 2. Equality (28) becomes:

σ l j -σ k i - δ k 2 - δ l 2 ≤ (λ k - 3 2 )δ k - δ l 2 = (λ k -1)δ k - δ k 2 - δ l 2 ≤ (λ k -1)δ k - δ k 2 - δ k-1 2 (29)
The last inequality results from [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF]. Using [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF], one obtains

σ l j -σ k i - δ k 2 - δ l 2 ≤ - δ k 2 ≤ - 1 2 (30)
• k = l and i ∈ {1, . . . , λ k } with i ≥ j: Equation (28) specializes as follows:

σ l j -σ k i - δ k 2 - δ l 2 = (j -i -1)δ k ≤ -δ k ≤ -δ q ≤ - 1 2 • k < l and i = λ k : Equation (28) becomes: σ l j -σ k i - δ k 2 - δ l 2 = - δ k 2 + (j -λ l - 1 2 )δ l ≤ - δ k 2 - δ l 2 ≤ - 1 2
To end the proof of the theorem , one has to check property [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF]. It is easy to check from (27) that σ 1 1 = 0. Thus, it remains to show that

σ k+1 1 -σ k 1 > 0 for k = 1, . . . , q -1 (31)
On one hand, one has:

σ k 1 = -(λ k -1)δ k + (λ 1 -1)δ 1
On another hand, one has:

σ k+1 1 = -(λ k+1 -1)δ k+1 + (λ 1 -1)δ 1 = - δ k 2 + (λ 1 -1)δ 1
This leads to

σ k+1 1 -σ k 1 = - δ k 2 + (λ k -1)δ k = (λ k - 3 2
)δ k > 0 III. SOME PARTICULAR DESIGN FUNCTIONS Some particular expressions of the design function K k that satisfy condition [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF] shall be given. It will be shown that each of these expressions yields a well known observer. Indeed, one has:

• The usual high gain design function given by

K k (e k 1 ) = ρ e k 1 ( 32
)
where ρ is a positive scalar satisfying ρ ≥ 1 2 . Notice that the function K k is bounded as soon as e k 1 is.

• The design function involved in the actual sliding mode framework

K k (e k 1 ) = ρ sign(e k 1 ) (33)
where ρ is a positive scalar and 'sign' is the usual signum function. It is worth mentioning that the required property [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF]

IV. EXAMPLE

Consider the following dynamical system

               ẋ1 = x 2 -tanh(x 1 ) ẋ2 = x 3 2 + ε(t) -x 4 -sin(x 5 ) ẋ3 = x 4 -x 3 2 -tanh(x 3 ) ẋ4 = x 5 + x 2 /(1 + x 2 2 ) + cos(x 4 ) ẋ5 = -x 2 * sin(x 2 ) -x 3 5 y = (x 1 x 3 ) T (35)
where the uncertainty term ε(t) is given by ε(t) = 2cos(t).

It is clear that system (35) is under form (1) with λ 1 = 2 and λ 2 = 3. An observer of the form (12) can be hence designed for the estimation of the state without any prior knowledge on the expression of the uncertainties. Please notice that one has δ 1 = 2(λ 2 -1) = 4 according to [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF]. Simulation results corresponding to the non measured state variables are given in figure 1. This clearly shows the ability of the observer in providing accurate state estimates as the curves of x 2 and x2 become quickly superimposed in spite of different corresponding initial conditions. Indeed, these simulations have been carried out using the following initial conditions x i (0) = 1 for the model and x1 = x3 = 1, xi = -1 for observer. The value of the design parameter θ was set to 8.

V. CONCLUSION

A high gain observer for a large class of nonlinear MIMO systems has been proposed in this paper. Unlike previous works related to high gain observer design, a complete triangular structure is not assumed. Moreover, the gain of the proposed observer involves a well defined design function. The latter provides a unified framework for the high gain observer design, namely classical high gain observers and several versions of sliding mode like observers are obtained by considering particular expressions of the design function.

Fig. 1 .

 1 Fig. 1. Simulation results for x i , i = 2, 4, 5

  where tanh denotes the hyperbolic tangent function and ρ and k o are positive scalars. One can easily show that the design function (34) satisfies the property (11) for relatively high values of k o . More particularly, recall that one has lim

	holds in the case of bounded input
	bounded state systems for relatively high values of
	ρ. However, this design function induces a chattering
	phenomena which is by no means suitable in practical
	situations.
	• The design functions that are commonly used in the sliding mode practice, namely
	K k (e k 1 ) = ρ tanh(k o e k 1 ) = ρ tanh(k o e k 1 )(34)
	ko-→+∞	tanh(k o e k 1 ) = sign(e k 1 ).

ENIS, Département de Génie électrique, BP W,

Sfax, Tunisia (e-mail:moncef trik1@yahoo.fr ) 3 CNRS ; LAAS ; 7 avenue du colonel Roche, F-31077 Toulouse, France (dahhou@laas.fr) 4 Université de Toulouse ; UPS ; INSA ; INP ; ISAE ; LAAS ; F-31077 Toulouse, France

+(λ1-1)δ1