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Asian Journal of Control, Vol. 5, No. 4, pp. 513-527, December 2003

513

IMPLICIT TRIANGULAR OBSERVER FORM DEDICATED TO A SLIDING MODE OBSERVER FOR SYSTEMS WITH UNKNOWN INPUTS T. Boukhobza, M. Djemai, and J.P. Barbot ABSTRACT It has been presented in previous works that every uniformly observable single-output system can be put on a triangular observation form. For this structure a special kind of sliding mode observer has been designed by authors, which ensures a finite-time state reconstruction using a step by step observation algorithm. In this paper, we show that the multi-output case is more delicate to study especially when the system has some unknown inputs. Thus, in order to generalizes the triangular observer form, from single to multi-output case, we define an Implicit Triangular Observer (ITO) form. For such a form, two results are given. Firstly, we design a finite time converging observer for all values of the unknown inputs. Secondly, we give the necessary and sufficient condition, including a matching condition, for the existence of a coordinate change to put the system into this form. It is also shown that this class of systems is a subset of the uniform observable class of systems.

KeyWords: Sliding mode observer, multi-output systems, unknown inputs, Implicit Triangular observer form.

I. INTRODUCTION The techniques employed for the observation of the systems with unknown inputs are applicable to many problems ([33]) such as the fault detection and disturbance rejection. We can distinguish two approaches: An approach which consists in observing the unknown inputs or at least in determining the thresholds such as in the diagnosis as described in [8,9,13,15,17,18,24,29] for the linear and time variant case and [16,20,35] for the bilinear case. The other approach consists in observing the states of the system in the presence of unknown inputs which can be regarded as disturbances. The present work deals with the problem of the observation of systems with unknown inputs in the multi-output case. The objective is to obtain an observer converging for all possible values of bounded unknown Manuscript received January 15, 2003; accepted April 1, 2003. T. Boukhobza is at IUT de Kourou, Universit´e des Antilles Guyane, Avenue Bois Chaudat, B.P. 725, 97 387 Kourou CEDEX, France. M. Djemai and J.P. Barbot are with Equipe Commande des Syst`emes (ECS), ENSEA, 6 Av. du Ponceau, 95014 Cergy Cedex, France.

inputs. We treat on the case where the system can be put on the “ITO(Implicit Triangular Observer) form”. For such a form we construct an observer which carries out the objective of convergence defined above and we give the matching condition. To solve this problem, we use a step by step sliding mode observation algorithm which generalize the idea of the geometrical linearization of the observation error dynamics. In fact, the idea to linearize the observation error dynamics presented in [21,22], is developed by the authors for the output and output derivative injection form ([7]), and a step by step sliding mode observer with linear error dynamics was introduced. The works [31,32], gave the conditions under which a nonlinear system can be put with a coordinate change on the so-called ”output injection form”. Always for a single output, and using sliding mode method to obtain a finite time convergence instead of an exponential one as in [28], in [2] the authors gave a triangular observation form which is a generalization of the output injection form [21]. This work will be recalled in section 2 of the present paper. Thus, in this paper a more general form is introduced : the ITO form. The paper is organised as follows : In the 2nd section we present the single output triangular form, the matching condition is given for this case, and the diffi-

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culty to generalize it to the multi-output case on the basis of an example. Section 3 is devoted to the triangular implicit observer form. In the next Section we construct a sliding mode observer for such a form. This observer converges in finite time independently from the input. In Section 5, we set the necessary and sufficient conditions under which we obtain the ITO form. In Section 6, we give the matching condition in the multi-output case, i.e. we give the geometrical conditions which must verify a nonlinear system with unknown inputs for the existence of a local coordinate change which put it on the triangular implicit form with a total rejection of the unknown inputs.

1)-steps we obtain a convergence of all the observer state components. More precisely in [2], we have presented a new type of sliding mode observer under the BIBS1 assumption in finite time2. This observer allows to obtain a uniform stable observable error dynamics for all the systems of the form (1). To illustrate our purpose, we use a step by step sliding mode observer for the system (1) which authorizes a finite time convergence of the observation error at each step. Applying the results of [2] of those of Drakunov et al. in [10], we have the following observer for the system (1):

II. SOME RECALLS ON TRIANGULAR FORM

ξˆ = ξˆ + g (ξ , u ) + λ sign(ξ − ξˆ ) 2 1 1 1 1 1  1 ξˆ = E ξˆ + g (ξ , ξ , u ) + λ sign(ξ − ξˆ )  1 3 2 1 2 2 2 2   2   ξˆ = E ξˆ + g (ξ , ξ … , ξ , u ) n −1 n−2  n n −1 1 n −1 2 (3)   ˆ + λn −1 sign(ξ n −1 − ξ n −1    ξˆn = En −1  f n (ξ1 , ξ 2 … , ξ n ) + g n (ξ1 , ξ 2 … , ξ n , u )    + λn sign(ξ n − ξˆn )  

2.1 Single output case In [3,6], the authors show that all uniformly observable system can be put on a special form called the single-output triangular form: ξ1 = ξ 2 + g1 (ξ1 , u )  ξ 2 = ξ3 + g 2 (ξ1 , ξ 2 , u )    ξ n −1 = ξ n + g n −1 (ξ1 ,… , ξ n −1 , u ) ξ = f (ξ ) + g (ξ , u ) n n  n

(1)

where y = ξ1 and with gi (., u = 0) = 0 for any i ∈ {1, …, n}. We remark that this form is close to the parametric strict-feedback form introduced in the adaptive control context ([19,23]) and it is especially similar to the triangular form used in [4]. Moreover, it is proved in [14] that a single output system x = f ( x) + g ( x)u y = h( x )

(2)

can be put on the triangular form (1) if and only if it is uniformly observable. The particularity of such a form is that it is possible to define a step by step observation algorithm allowing the reconstruction of all the state with finite-time convergence. Therefore, using the sliding mode properties of finite time convergence, in [3,6], on the basis of Drakunov and Utkin previous work [10], we introduce a sliding mode observer dedicated to this triangular single-output system. This sliding mode observer consists in a step by step convergence algorithm. Thus, if the system is uniformly observable, in (n −

where ξ i = ξˆi + Ei −1λi −1 sign(ξ i −1 − ξˆi −1 ) for i = 2, …, n – 1 (by convention ξ1 = ξ1 ), with signav is an averaging value of the discontinuous sign function which can be obtained using a low pass filter whose high cutoff frequency preserves the slow component of the motion but is small enough to eliminate the high frequency well known under the name “chattering phenomenon” on the sliding surface. The need of a low pass filter is due to the fact that in the Fillipov’s resolution method, the discontinuous sign function, which is not defined at zero, generates an infinite frequency chattering around its equivalent value signeq. Thus, to eliminate such a phenomenon, which becomes in practice unacceptable, a low pass filter is introduced (see also [10]). The Ei are used for an anti-peaking strategy ([27]). This anti-peaking strategy comes from the idea that we do not inject the observation error information before reaching the sliding manifold linked with this information. More precisely Ei = 0 if there exists j ∈{1, i} such that ξ j − ξˆ j ≠ 0 , else Ei = 1. In the observer structure, these functions allow ξ i − ξˆi to converge to zero if all the ξ j − ξˆ j with j < i have converged to zero before. 1 2

Bounded Input Bounded State We assume that the BIBS condition is verifed by the nonlinear system, the aim of the paper is not to characterize the systems which are BIBS.

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

Moreover, at each step, only a sub-dynamic of dimension one is assigned. Let us explain briefly this step by step convergence. In fact, at step 1, the sliding term is only injected in the dynamics of ξˆ1 , thus if λ1 > | ξ 2 − ξˆ2 |max , we have convergence in finite time of ξˆ1 to ξ1. Therefore, using the equivalent vector [30], we obtain ξ 2 = ξˆ2 + λ1 sign(ξ1 − ξˆ1 ) ξ 2 . After this first step, we obtain the real value of ξ2. Iterating this argumentation, we obtain at the step 2, ξ 3 = ξ 3 ,… , and finally, at the step n − 1, ξ n = ξ n . The following theorem summarizes this result: Theorem 1. ([2]) Considering the BIBS system (1) and the observer (3), for any bounded initial state ξ(0), ξˆ(0) and any bounded input u, there exists λi such that the observer state ξˆ converges in finite time to ξ.

515

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

3

2.5

3

2.5

3

Fig. 1. x1(-) and xˆ1 (-). 0.8

Proof. See ([2]).

0.6

Example 1. Let us consider the following system Σ which is in the triangular observer form x1 = x2 − x13u x2 = x3 + x2 x1u x3 = −3 x3 − 3 x2 − x1 − x33 − u y = x1

0.2

(4)

−0.4 −0.6 0

xˆ1 = xˆ2 − x13u + λ1 sign( x1 − xˆ1 ) xˆ3 = −3 x3 − 3 x2 − x1 − x33 − u + λ3 sign2 ( x3 − xˆ3 ) y = x1

0 −0.2

For this system the observer (3) takes the form

xˆ2 = xˆ3 + x2 x1u + λ2 sign1 ( x2 − xˆ2 )

0.4

0.5

1.5

2

Fig. 2. x2(-) and xˆ 2 (-).

(5)

with

1

0.6 0.4 0.2

x2 = xˆ2 + λ1 sign1 ( x1 − xˆ1 ) and

0 −0.2 −0.4

x3 = xˆ3 + λ2 sign2 ( x2 − xˆ2 ) , where signi functions are designed as noted in section 3.

−0.6 −0.8 −1 −1.2

This approach has been tested by simulation with the following initial conditions x = (1, 0.5, 0.5)T and xˆ = (0, 0, 0)T. Moreover, we have chosen a first-order low pass filter with a cut frequency equal to 100hz and observation gain λ1, λ2 and λ3 respectively equal to 4, 2 and 2. Moreover the function “sign” is approximated by a saturation function with a slow rate equal to 104.

But xˆ2 will only reach x2 when xˆ1 will be equal to x1. • In Fig. 3, xˆ3 reaches x3 in finite time 1s.

• In Fig. 1, xˆ1 reaches x1 in finite time 0.25s. • In Fig. 2, xˆ2 also reaches x2 in finite time 0.75s.

Now, starting from the same initial conditions, we add an output noise in order to show the behavior of the

−1.4 0

0.5

1

1.5

2

Fig. 3. x3(-) and xˆ3 (-).

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1.6

0.8

1.4

0.6

1.2

0.4

1

0.2

0.8 0

0.6 −0.2

0.4

−0.4

0.2 0

0

0.5

1

1.5

2

2.5

3

−0.6

0

0.5

1

1.5

2

2.5

3

2.5

3

Fig. 6. x2(-) and xˆ 2 (-).

Fig. 4. x1(-) and xˆ1 (-). 0.5

1.8 1.6 1.4

0

1.2 1 0.8

−0.5

0.6 0.4 0.2 0

−1

0

0.5

1

1.5

2

2.5

3

Fig. 5. y = x1(-) and xˆ1 (-).

observer. Following the work of Yaz and Azemi ([34]), in [3] we have proposed to use a saturation function with a dead zone for the observer in the case of the extended injection form. This reduces the observer sensitivity to the noise, but we were obliged to change the observer gain as follows λ1 = λ2 = λ3 = 4 in order to recover a time response quite similar to the previous simulation.

0

0.5

1

1.5

2

Fig. 7. x3(-) and xˆ3 (-).

step observation algorithm is the convergence of the proposed sliding mode observer independently from the ξn dynamics. If in this dynamics, some unknown terms are present, it does not change any thing in the observation results if the states ξi (i = 1, ..n) stay bounded. So we have ([1,33]): Proposition 1. Let us consider the system

• In Figs. 4, 5, 6 and 7, we see that the observer state xˆ reaches the neighborhood of the system state x in finite time. But we also see that the noise is not totally suppressed in the observer. We can reduce this noise with some minor modifications by introducing an asymptotic gain or a sign function modified with respect to the noise output knowledge [34] for example. 2.2. Observer matching condition (OMC) An immediate consequence of the previous step by

ξ 2 + g1 (ξ1 , u )  ξ1        g ( , , u ) ξ ξ ξ + 3 2 1 2  ξ2     =       ξ n −1   ξ n + g n −1 (ξ1 ,… , ξ n −1 , u )   ξ   f (ξ ) + g (ξ , u ) + l (ξ , w)  n   n   n

(6)

where y = ξ1, gi (., 0) = 0 for any i ∈ {1, …, n} and w is an unknown bounded input. Then, for any bounded initial state ξ(0), ξˆ(0) and any bounded inputs u and w, there exists λi, so that the observer state ξ (3) converges in finite time to ξ.

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

In [3], conditions were given to obtain the form (6) starting from nonlinear single input single output affine system using a local diffeomorphism. Proposition 2. Let us consider the BIBS system x = f ( x) + g ( x)u + l ( x) w y = h( x )

(7)

where x is the state, y the output, w is the unknown input and u the known input, can be put on the form (6) using a local coordinate change if and only if:

x Xi

The generalization of the form (6) to the multioutput case is one of the main purpose of this paper. For this, it is necessary to define a new step by step sliding mode observation algorithm. The main problem in the multi-output case is the definition of the triangular observer form. This is due to the fact that, firstly, states information are not all the time in the same derivatives outputs. Secondly, we do not want to use the input derivatives in our observer, thus, because it may not be C1, as in the sliding modes control case for example.

yi = xi ,1 , i = 1,… , p

j = 1,… , p

Example 2. Let us consider the uniformly observable system x1 = x2 + φ1 ( x1 , x4 )u x2 = x3 + φ2 ( x1 , x2 , x4 )u x3 = φ3 ( x)u + l3 ( x) w x4 = x5 + φ4 ( x1 , x2 , x4 )u

(10)

x5 = φ4 ( x)u + l5 ( x) w y1 = x1

y2 = x4

with x = (x1, x2, x3, x4, x5)T . Using a step by step algorithm, we obtain: xˆ1 = xˆ2 + φ1 ( x1 , x4 )u + λ1 sign( x1 − xˆ1 )

A trivial idea is to define an explicit triangular observer form as the simplest generalization of the single-output triangular form defined in (1). This gives

where

(9)

where w is the unknown input and l is analytic function vectors of appropriate dimensions. Unfortunately this form is not sufficiently general again as it is shown in the example hereafter.

2.3. Multi-output case introduction example

 xi ,1 = xi ,2 + g1 ( x j ,1 )u  x = x + g ( x , x )u i ,3 2 j ,1 j ,2  i ,2    xi , µ −1 = xi , µ + g µ −1 ( x j ,1 ,… , x j , µ −1 )u i i i  i  xi , µi = f n ( x) + g µi ( x)u yi = xi ,1 , i = 1,… , p j = 1,… , p

( xi ,1 , xi ,2 , xi ,3 ,… , xi , µi )T ∈ ℜµi

 xi ,1 = xi ,2 + g1 ( x j ,1 )u  x = x + g ( x , x )u i ,3 2 j ,1 j ,2  i ,2    xi , µ −1 = xi , µ + g µ −1 ( x j ,1 ,… , x j , µ −1 )u i i i  i  xi , µi = f n ( x) + g µi ( x)u + l ( x) w

Proof. The proof of this proposition is obvious using the results of [14] on the relationship between the uniform observability and the triangular form and the Theorem 1. Remark 1. The second item is called the observer matching condition. This condition is completely different from the well known matching condition for the control case given in [11].

( X 1T , X 2T ,… , X Tp )T ∈ ℜn ,

is the state, u ∈ ℜm is the input, y ∈ ℜp is the output, f, g are analytic function vectors of appropriate dimensions and the numbers µi are the observability indices and are defined for example in [21]. Unfortunately this form is not sufficiently general. Moreover, the system (8) affected by the perturbation w verify the OMC3 can be written as:

• The unperturbed system ((2) with w = 0) is uniformly observable with respect to the known input u, • The relative degree of the unknown input w is equal to n − 1,

„

517

xˆ2 = E1[ xˆ3 + φ2 ( x1 , x2 , x4 )u + λ2 sign( x2 − xˆ2 )] xˆ3 = E2 [φ3 ( x1 , x2 , x3 , x4 , x5 )u + λ3 sign( x3 − xˆ3 )] (11) xˆ4 = E2 [ xˆ5 + φ4 ( x1 , x2 , x3 , x4 )u + λ4 sign( x4 − xˆ4 )] xˆ5 = E4 [φ5 ( x1 , x2 , x3 , x4 , x5 )u + λ5 sign( x5 − xˆ5 )]

(8)

The observer parameters are assumed to be well defined as in [3], where the auxiliary states are defined by: x2 = xˆ2 + E1λ1 sign( x1 − xˆ1 ), x3 = xˆ3 + E2 λ2 sign( x2 − xˆ2 ) 3

Observer Matching Condition.

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and

α1i (ζ 0 ,… , ζ i )

= ζ 1i +1 + φ1i +1 (ζ 0 ,… , ζ i )u

x5 = xˆ5 + E2 λ4 sign( x4 − xˆ4 ), while the commutation parameters Ei are defined such as: E1 = 1 if x1 = xˆ1 otherwise E1 = 0; and E2 = 1 if ( x2 = xˆ2 , x5 = xˆ5 and E1 = 1), otherwise E1 = 0. At step 1, from x1 we can obtain x2 as x2 = x1 − φ1(x1, x4)u: the nonlinear terms in x1 are function of x1, x4 and u which are known. Nevertheless, we obtain nothing from the equation of x4 as the nonlinear term φ4 is function of x2, x3 and x5, which are not obtained yet. At step 2, from x2 we obtain x3 as φ2 depends only on x1, x2 and x4. At step 3, from x4 we also obtain the state x5. After three steps we obtain all the state variables. Moreover, we can see that the direction of y1 is used twice before y2’s one. From the steps previously described, whatever the values of the input w, we can deduce, using trivial generalization of [2], a sliding mode observer for such systems. It is important to note that this observer converges whatever the values of the input w. It is easy to verify that we can not transform the system (10) into the triangular explicit form (8) but the step by step observation algorithm works. Nevertheless as we will see later in a other example : it won’t be the more general form.

III. IMPLICIT TRIANGULAR OBSERVER (ITO) FORM For the sake of simplicity, we will first study the case without unknown inputs. Let us consider the following multi-output system: x = f ( x) + g ( x)u

(12)

y = h( x ) n

ανi i +1 (ζ 0 ,… , ζ i )

= ζ νii++11 + φνii++11 (ζ 0 ,… , ζ i )u

α1k0 −1 (ζ 0 ,

, ζ k0 −1 )

= ζ 1k0 + φ1k0 (ζ 0 ,… , ζ k0 −1 )u

ανkk0 −1 (ζ 0 ,

, ζ k0 −1 )

= ζ νkk0 + φνk20 (ζ 0 ,… , ζ k0 −1 )u

0

0

β1 (ζ )

= ψ 1 (ζ , u )

β p (ζ )

= ψ p (ζ , u )

(13) where span{ dα ij , dβ k i = 0, …, k0 – 1, j = 1, …, vi+1, k = 1, …, p} = n and ζ 0 (y1, …, yp)T. The function α ij is the jth implicit function used at the step i of the observation algorithm. At the first step i = 0, the functions α 0j for j = 1..ν0 represent combination of the outputs. The goal of such a combination is to eliminate the input and the unknown state variables from the nonlinear terms of each equation. For the ensuing steps i = 1 … k0, the function α i j , for j = 1 … νi is a combination of all the known state variables until the step i − 1. At each step i, νi new directions of the state vector are observed. The βj functions are used to complete the observer dynamics dimension to n, but they are not useful for the observer convergence. The νi are not the observability index for the output yi. As we will show in the simple example hereafter, these characteristic numbers are linked to the step by step observation algorithm. Example 3. Let us consider the following system:

p

m

where x ∈ ℜ is the state, u ∈ ℜ is the input and y ∈ ℜ is the output. the functions f, g and h are assumed sufficiently smooth. Then we defined a more general class of system in (8) for which the step by step observation algorithm works. Definition 1. The ITO form for the multi-output system (12) is given by

α10 (ζ 0 ) = ζ 11 + φ10 (ζ 0 )u αν01 (ζ 0 ) = ζ ν11 + φν01 (ζ 0 )u

x1 = − x1 + x3 ( x4 + 1) + ux4 x2 = − x2 + x3 ( x4 − 1) + ux4 x3 = − x3 + x1 x4 u

(14)

x4 = φ4 ( x1 , x2 , x3 , x4 )u y1 = x1

y2 = x2

The observability index of x3 are r31 = 1 if x4 ≠ −1 and r32 = 1 if x4 ≠ 1. For the state x4, we obtain r41 = r42 = 1 if u ≠ 0. As we see, the observability index are not constant and depends on the input and the state vector. Let us now rewrite the system into the ITO form:

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

then we obtain

We take the following local coordinate change:

ζ 10 = x1 , ζ 20 = x2 , ζ 11 = x3 and ζ 12 = x3 x1 ( x4 + 1). The functions α are :

α10 =

(ζ 10 − ζ 20 ) (ζ 0 )2 and α11 = 1 − ζ 11 . 2 2

We obtain the following system :

α10 = − 1 1

(ζ 10 − ζ 20 ) + ζ 11 2 0 2 1

1 1

α = −(ζ ) + ζ − ζ

(15)

2 1

We deduce from the example that, contrary to the observability index, ν1 = ν − 2 = 1. We do not specify the values of β1 and β2 for sake of simplicity. We can also see through this example, that the functions α are a combination of the output only for the first step α0 but are not for next steps. This fact justifies the implicit triangular form denomination. The ITO form, as we see is a triangular form which does not use the input derivatives as in the triangular forms of [5,25,26]. Now, we prove the observability for the ITO form introduced in the definition 1. For this, from (13), we define the following pseudo observability matrix: dy1         dy p     dα10 (ζ 0 )         0 0   dαν1 (ζ )      Oo =   dα1i (ζ 0 ,… , ζ i )        i 0 i  dαν i +1 (ζ ,… , ζ )         dα1k0 −1 (ζ 0 ,… , ζ k0 −1 )        k0 −1  dαν k (ζ 0 ,… , ζ k0 −1 )  0  

519

  dζ 10       1 dζ p     1 0 0 d [ζ 1 + φ1 (ζ )u ]       1 0 0   d [ζ ν1 + φν1 (ζ )u ]     Oo =   i +1 i +1 0 i  d [ζ 1 + φ1 (ζ ,… , ζ )u       d [ζ νi +1 + φνi +1 (ζ 0 ,… , ζ i )u ]  i +1 i +1       k k k −1 0  d [ζ 1 0 + φ1 0 (ζ ,… , ζ 0 )u ]       d [ζ k0 + φ k0 (ζ 0 ,… , ζ k0 −1 )u ]  ν2  ν k0  which leads to a triangular matrix with rank (Oo ) = n Thus, we can conclude to the observability. Practically, and to illustrate the efficiency of the ITO form, we construct in the next section a sliding mode observer converging in finite time.

IV. SLIDING MODE OBSERVER FOR ITO FORM Using the ITO form (13), a sliding mode observer can be written so that at each step we obtain the states ζ i and consequently at the step k0 the whole state is obtained. Let us assume that at step i we know all the states ζ j, j = 1, …, i and then from the equations

α1i (ζ 0 , ζ 1 ,… , ζ i ) = ζ 1i +1 + φ1i +1 (ζ 0 , ζ 1 ,… , ζ i )u α 2i (ζ 0 , ζ 1 ,… , ζ i ) = ζ 2i +1 + φ2i +1 (ζ 0 , ζ 1 ,… , ζ i )u

ανi i +1 (ζ 0 , ζ 1 ,… , ζ i ) = ζ νii++11 + φνii++11 (ζ 0 , ζ 1 ,… , ζ i )u we construct the observer

αˆ1i (ζ 0 , ζ 1 ,… , ζˆ i ) = Ei −1[ζˆ1i +1 + φ1i +1 (ζ 0 , ζ 1 ,… , ζ i )u +λ1i sign(α1i (ζ 0 , ζ 1 ,… , ζ i ) −αˆ1i (ζ 0 , ζ 1 ,… , ζˆ i ))]

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αˆ 2i (ζ 0 , ζ 1 ,… , ζˆ i ) = Ei −1[ζˆ2i +1 + φ2i +1 (ζ 0 , ζ 1 ,… , ζ i )u

inclusions [12]

+λ2i sign(α 2i (ζ 0 , ζ 1 ,… , ζ i ) −αˆ 2i (ζ 0 , ζ 1 ,… , ζˆ i ))]

αˆνi i +1 (ζ 0 , ζ 1 ,… , ζˆ i ) = Ei −1[ζˆνii++11 + φνii++11 (ζ 0 , ζ 1 ,… , ζ i )u +λνii +1 sign(ανi i +1 (ζ 0 , ζ 1 ,… , ζ i ) −αˆνi i +1 (ζ 0 , ζ 1 ,… , ζˆ i ))] The observer (16) is a step by step observer and we use an iterative argument to explain it. Step 1. We have at any time the information on ζ 0 = y. Let us consider the first sub-dynamics,

α10 (ζ 0 ) = ζ 11 + φ11 (ζ 0 )u 0 2

0

1 2

1 2

0 = ei1 − λ10 sign(eα0 ,i ) and this gives [30]

ζ i1 = ζˆi1 + λi0 signav (eα0 ,i ) where the sign function is low pass filtered with an arbitrary large frequency band width in order to preserve the right and side function convexity at the next step and obtain signav equal to signeq which implies ζ i1 = ζ i1 . For step i (for i ∈ {2, …, k0 − 1}), we assume that we know all the states ζ j (i.e. ζ j = ζ j ) for j = 1, …, i. Thus we have

α1i (ζ 0 , ζ 1 ,… , ζ i )

= ζ 1i +1 + φ 1i +1 (ζ 0 , ζ 1 ,… , ζ i )u

α 2i (ζ 0 , ζ 1 ,… , ζ i )

= ζ 2i +1 + φ i2+1 (ζ 0 , ζ 1 ,… , ζ i )u

(17)

0

α (ζ ) = ζ + φ (ζ )u αν01 (ζ 0 ) = ζ ν11 + φν11 (ζ 0 )u and we consider the related observer sub-dynamics

α10 (ζˆ 0 ) = ζˆ11 + φ11 (ζ 0 )u + λ10 sign(α10 (ζ 0 ) − α10 (ζˆ 0 ))

ανi i +1 (ζ 0 , ζ 1 ,… , ζ i ) = ζ νii++11 + φνii++11 (ζ 0 , ζ 1 ,… , ζ i )u where only ζ i+1 is unknown and appears linearly in the subdynamics (17). Therefore, as Ei = 1 we have the following observer sub-dynamics

α1i (ζ i ) =

ζˆ1i +1 + φ1i +1 (ζ 0 , ζ 1 ,… , ζ i )u

(

α 20 (ζˆ 0 ) = ζˆ21 + φ21 (ζ 0 )u + λ20 sign(α 20 (ζ 0 ) − α 20 (ζˆ 0 ))

+λ1i sign α1i (ζ 0 , ζ 1 ,… , ζ i ) − α1i (ζ i )

α 2i (ζ i ) =

)

ζˆ2i +1 + φ2i +1 (ζ 0 , ζ 1 ,… , ζ i )u

(

αν01 (ζˆ 0 ) = ζˆν11 + φν11 (ζ 0 )u + λν01 sign(αν01 (ζ 0 ) − αν01 (ζˆ 0 ))

+λ2i sign α 2i (ζ 0 , ζ 1 ,… , ζ i ) − α 2i (ζ i )

)

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then the observer error eα0 ,i = α i0 (ζ 0 ) − α i0 (ζˆ 0 )

ανi i +1 (ζ i ) = ζˆνii++11 + φνii++11 (ζ 0 , ζ 1 ,… , ζ i )u

(

+λνii +1 sign ανi i +1 (ζ 0 , ζ 1 ,… , ζ i ) − ανi i +1 (ζ i )

dynamics is for i ∈ {1, …, ν1} eα0 ,i = (ζ i1 − ζˆi1 ) − λi0 sign(eα0 ,i ) Thus, the condition in order to guarantee the stability of eα0 is that for any i ∈ {1, …, ν1} and ∀t > 0

λ > ζ (t ) − ζˆi1 (t ) 0 i

1 i

1 i

e

The existence of such a λi0 is given by the fact that firstly ζ i1 is bounded and secondly , before the convergence of ζˆ 0 to ζ 0 the observer is in an open loop with respect to the observer gain, so ζˆi1 is consequently bounded from the BIBS in finite time assumption. Then, after a finite time t1 all ζˆi0 goes to ζ i0 . Thus, we have after t1 in the meaning of the differential

)

thus the observation error dynamics in eαi , j is eαi , j = eζi +1 − λ ij sign(eαi , j )

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Consequently, from (19) we see that eαi , j converges to zero in a finite time. Thus, ∃ ti,j > 0 such that ∀t > ti , j

eαi = 0 ⇒ eij+1 − λ ij sign(eαi , j ) = 0

∀t > ti , j

ζ ij +1 = ζˆ ij +1 + λ ij sign(eαi , j ) = ζ ij+1

then

and the step i is finished after the time

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

ti

max j∈{1,…,ν i } (ti , j ) .

Finally, at step k0 as all ζ ij have converged to ζ , the last sub-dynamics observation error i j

eβk0, j = ( β kj 0 (ζ 0 , ζ 1 ,… , ζ k0 ) − β kj 0 (ζ 0 , ζ 1 ,… , ζ k0 −1 , ζˆ k0 ))

521

whole state of a multi-output system in the ITO form. Now, we want to characterize the necessary and sufficient conditions in order to obtain this form starting from (20) using a local diffeomorphism. Roughly speaking, we will “algebraise” the structured implicit triangular observer form. For this, let us begin by giving some definitions

is equal to Definition 2. Considering the system eβk0, j = −λ kj 0 sign(eβk0, j ) k0 +1 j

> 0 and without any other conwhich is stable if λ dition, this last sub-dynamics is just given in order to ensure that ekj0 is bounded but we know the whole state at the step k0 − 1 thank to ζ . Consequently, let us state Theorem 2. For any system in ITO form (13), there exists a step by step sliding mode observer of the form (16) such that the observation error converges to zero, independently of the input, step by step in finite time.

x = f ( x) + g ( x)u y = h( x )

where f(x) and g(x) are nonlinear analytic functions, x ∈ ℜn is the state, u ∈ ℜm is the input and y ∈ ℜp is the output. Let us define the following sets E0

F0

span{dhi , i = 1,… , p}

Moreover, for k = 1, …, n − p, we define: Sk is the greater set of analytic function so that

Proof. The proof is explicitely detailed in the paragraph before the theorem statement. „

Lg ( S k ) ∈ F k −1

Remark 2. The ζˆ ij +1 dynamics is implicitly obtained from the αˆ k , k = i + 1, …, k0 and βˆ j = 1, …, p dynamics.

Ek

L f S k ∩ ( F k −1 ) ⊥

Fk

F k −1 ∪ E k

Remark 3. We will note that in the multi-input case, the characterization of BIBS systems is not usual. However, we make this assumption work on systems not being unstable in finite time for reasons of stability of the observer, the aim of this paper being not the geometrical characterization of BIBS systems, but the observability forms associated with a certain type of sliding mode observer.

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Roughly speaking, Sk represents the invariant part of F under the transformation Lg, Ek the new state variables obtained by step k of the observation algorithm and Fk is the state variable part obtained since step k. The constraint imposed here is not to use the input derivatives. Let us take the following notations: ζ i is a basis of i E and ν i dim( E i ). k−1

Lemma 1. For the system (20), if there exists k0 so that S k0 = ∅, then ∀k ≥ k0, Sk = ∅ and dim(Fk) = dim( F k0 ) .

As it can be seen in the last step, the knowledge of the functions βj is not necessary to the convergence of ~ the zeta to ζ in finite time. As a consequence of this fact, we can state that some unknown terms as unknown inputs can occur in all ζ functions. An outline of the proof of stability and conditions of convergence of the observer presented above can be exposed briefly as being similar to those stated in paper [2]. For this, it is necessary to use a particular function sign: signi(.) is equal to zero if there exists j ∈{1, i} such as ζ lj − ζˆ lj ≠ 0 , else signi(.) is equal to the usual sign(.) function.

Lemma 2. For the system (20), ∀i, j ∈ {0, n − p} such as i < j we have Ei ∩ Ej = ∅ and Si ⊆ Sj.

V. CHARACTERIZATION OF THE ITO FORM

Lemma 3. For the system (20), if there exists k0 so that E k0 = ∅, then ∀k ≥ k0, Ek = ∅ and Fk = F k0 .

Previously, we have highlighted the efficiency of a sliding mode observer to reconstruct in finite time the

Proof. If E k0 = ∅, we have F k0 +1 = F k0 , then S k0 +1 = S k0 and for any k > k0 we have Fk = F k0 and Sk = S k0 . „

Proof. From the definition of Ek, we can state that S k0 = ∅ ⇒ E k0 +1 = ∅ and then it is clear that S k0 +1 = ∅. Thus for any k ≥ k0, Ek = ∅ and Fk = F k0 . „

Proof. Starting from the definition of Ek, it is clear that Ek ∩ Fk−1 = ∅ and we deduce from the definition of Fk−1, that Ek ∩ Ej = ∅, and this for any j < k. Moreover, if Lg(Sk) ∈ Fk−1, as Fk−1 ⊆ Fk, then Lg(Sk) ∈ Fk. „

Asian Journal of Control, Vol. 5, No. 4, December 2003

522

Remark 4. In this case the observation algorithm does not give a new variable. In the case where dim( F k0 ) ≠ n, this can possibly lead to an unobservability of the system or at least a system which is not directly observable, i.e without the use of u knowledge. Let us take the following notations: ζ0 (yj, j = 1, …, p) and ζ i ( ζ ij , j = 1, …, νi)T such as dζ i is a basis for Ei for i = 1, …, k0 where k0 is the least integer so that dim( F k0 ) = n. Now, we can give the necessary and sufficient conditions to put the system in the ITO form. Remark 5. We see that if at step i any new state variable is obtained, then no state variable will be obtained for any step j > i. Consequently, at each step we must obtain at least one new state, so the maximal step number is n − p.

Example 4. Let us consider the following locally observable system x1 = x2 x1 − x3u x2 = x3 + g ( x1 , x2 , x4 )u x3 = φ ( x, u )

(21)

x4 = − x2 − x3 x1u y1 = x1 and

y2 = x4

For (21), the transformation to explicit triangular form is impossible. In fact, if we take y1 = x1 and x2 y2′ = 1 − x4 , we have 2 y1 = x2 x1 − x3u y2′ = x2 ( x12 + 1) ξ 2,1

Theorem 3. The system (20) can be put in an ITO form (13) using a local diffeomorphism if and only if

(22)

But

ξ 2,1 = ( x3 + g ( x1 , x2 , x4 )u )( x12 + 1)

dim( F n − p ) = n

Proof. Sufficiency If dim(Fn−p) = n then, after n − p steps, we obtain n independent variables ζ ij , i = 0, …, n – p, j = 1, …, νi. As there exists a non singular coordinate change Ψ so that ζ = ( ζ ij , i = 0, …, n – p, j = 1, …, νi)T = Ψ(x) because Ψ(x)’s Jacobian is equal to dim(F n−p). In this case, we can define the functions αi of the form (13). From the definition of the sets Ei, Fi and Si, let the subset Ti of Si for i = 1, …, k0 so that LfTi = Ei. We also know that T i ⊂ F i−1. Let us consider the states ζ i such that dζ i is a basis of Ei, which means4 that ζ i can be written as Lf χ i where χ i ∈ Ti. Thus, χ i can be written in the basis of F i−1 i.e χi = α(ζ 0, …, ζ i−1). This gives

χ i = α (ζ 0 ,… , ζ i −1 ) = L f χ i + Lg χ i u

+2 x2 x1 ( x2 x1 − ux3 ) has a nonlinear term depending on x3 which is not yet known. Thus, using only the derivation operation on the two directions we can not obtain a new variable ξ2,2 or ξ1,2. Therefore, this system must be transformed into the ITO form. In order to design a sliding mode observer, we compute the sets Ei and Fi, we will obtain E0 = F0 = span{dx1, dx4}, thus, S1 = span{d(x1 − x4)}, i.e. at step 1 d x12 we can only obtain the variable ( − x4 ) = x2 ( x12 +1) dt 2 because of the g1(x) = −x3, g4(x) = −x3x1 and dx3 ∉ F0. Then, E1 = span{d(x2( x12 + 1))} and a basis of F1 is {dx1, dx2, dx4}. At the second and last step, we deduce x3 from the x2 dynamics. In two steps we obtain the whole state, but using in the first step outputs combination and we can write the system in the ITO form with ζ 10 = x1, ζ 11 = x12 x2 + x2 , ζ 12 = x3, ζ 20 = x4

= ζ i + φ i (ζ 0 ,… , ζ i −1 )u

α10 (ζ 10 , ζ 20 ) = ζ 11

with

α11 (ζ 10 , ζ 11 ) = ζ 12 + g ′(ζ 10 , ζ 11 )u φ i (ζ 0 ,… , ζ i −1 )

Lg χ i .

Necessity Suppose that dim(F n−p) < n, then there exists k0 < n − p such that dim( E k0 ) = 0 . Thus, from Lemma 2 we find that dim( F n − p ) = dim( F k0 −1 ) . This means that ∀d Ξ ∈ F k0 −1 , Lg(Ξ) depends on the unobserved state part at the step k0 − 1. Thus, φik0 −1 depends on the unobservable states and consequently, and we do not have the form (13). „

β1 (ζ 12 ) = φ ′(ζ , u ) β 2 (ζ ) = −ζ 12 − ζ 10ζ 12 u y1 = ζ 10 y2 = ζ 20 4

k −1

Any one-form of F 0 is generated by a function dependent only on y , y ,… , y[ p ] .

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

verify the system

with

α10 (ζ 10 , ζ 20 )

(ζ 10 ) 2 − ζ 20 , α11 (ζ 10 , ζ 11 ) 2

x = f ( x) + g ( x)u + l ( x) w

ζ 11 , 1 + (ζ 10 ) 2

β1 (ζ 12 ) ζ 12 0 2

and β 2 (ζ ) = ζ and φ ′, g ′ denote respectively the functions φ and g with respect to ζ instead of x. Due to the observability singularity given by −1 − ζ 10 u = 0, we decide to use only information on y1. This example shows that the ITO form is more general than the triangular form (8).

VI. GENERALIZED OBSERVER MATCHING CONDITION In this section, we generalize the result of the proposition 2. Consider the following ITO form with q unknown input w ∈ < ℜ :

y = h( x )

Definition 3. Considering the system (24), we define the following sets Ew0 = Fw0

Lg ( S wk ) ∈ Fwk −1

α

(ζ ,

(23)

,ζ

) = ζ

+ φ (ζ ,… , ζ

Fwk

Fwk −1 ∪ Ewk

Example 5. Let us consider the system: x = f ( x) + ug(x) with

, ζ k0 −1 ) = ζ 1k0 + φ1k0 (ζ 0 ,… , ζ k0 −1 )u 0

L f S wk ∩ ( Fwk −1 ) ⊥

Proof. The proof is similar to the theorem 3, where the condition Ll ( S wk ) = 0 ensures that there is no term depending on w in the α i dynamics. „

ανi i +1 (ζ 0 ,… , ζ i ) = ζ νii++11 + φνii++11 (ζ 0 ,… , ζ i )u

2 2

Ewk

dim( Fwn − p ) = n

α 2i (ζ 0 ,… , ζ i ) = ζ 2i +1 + φ2i +1 (ζ 0 ,… , ζ i )u

k0 2

Ll ( S wk ) = 0

and

α1i (ζ 0 ,… , ζ i ) = ζ 1i +1 + φ1i +1 (ζ 0 ,… , ζ i )u

k0 −1

and

Theorem 4. The system (24) can be put in an ITO form (23), using a local diffeomorphism, if and only if

αν01 (ζ 0 ) = ζ ν11 + φν01 (ζ 0 )u

0

span{dhi , i = 1,… , p}

Moreover, for k = 1, …, n − p, we define: Slk is the greater set of analytic function such that

α 20 (ζ 0 ) = ζ 21 + φ20 (ζ 0 )u

α1k0 −1 (ζ 0 ,

k0 −1

x1 = x3 x1 + u1 x1 − u2 x5

)u

x2 = x3 + u2 x5 x1 x3 = x4 + u1 ( x2 + x12 + x3 ) + u2 x5

k0 −1 νk 0

α

0

(ζ ,

,ζ

k0 −1

) = ζ

k0 νk 0

k0 ν2

(24)

to obtain the form (23) after a change of coordinates. As in the previous section, let us use the following definition

α10 (ζ 0 ) = ζ 11 + φ10 (ζ 0 )u

k0 −1 2

523

0

+ φ (ζ ,… , ζ

k0 −1

)u

x4 = − x4 + x5 ( x42 + 1) + u1 x3 x1 + u2 x4

β1 (ζ ) = ψ 1 (ζ , u ) + ϕ1 (ζ , w) β 2 (ζ ) = ψ 2 (ζ , u ) + ϕ 2 (ζ , w) β p (ζ ) = ψ p (ζ , u ) + ϕ p (ζ , w) As the unknown term depending on w occurs on the p last lines (in the β dynamics), the step by step observation algorithm works well and ζ converges in finite time to ζ in the observer (18). Thus, we have rejected the unknown input w. The objective now is to find what conditions must

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x5 = φ ( x, u1 , u2 ) with y1 = x1 ,

y2 = x2

This example cannot be put in the injection output form or injection output and derivatives output form. Already, the derivative of the output depends not linearly on a part of the non measurable state Nevertheless, it checks the conditions of setting in the ITO form. Indeed, the calculation of subspaces E, F gives:

Asian Journal of Control, Vol. 5, No. 4, December 2003

524

E0

F0

0.08

{dx1 , dx2 }

x2 x2 As dLg ( 1 + x2 ) ∈ F 0 , one deduce that E1 = L f ( 1 + 2 2 x2 ) = span{d ( x3 (1 + x12 ))}. In this case F1 = span{dx1, dx2, dx3}. Moreover, as dLg(x1 + x3) ∈ F1, then E2 = Lf (x1 + x3) = span{d(x4 + x3x1)} and F2 = span{dx1, dx2, dx3, dx4}. Finally, dLg(x4) ∈ F2, E3 = Lf(x4) = span{d(x5( x42 + 1))} and F 3 = span{dx1, dx2, dx3, dx4, dx5}. dim(F 3) = 5, as the system is of dimension 5, according to the Theorem, the system (25) is put under ITO form Let us consider the following diffeomorphism: ζ = Ψ(x)

0.06 0.04 0.02 0 −0.02 −0.04 −0.06 0

0.5

1

1.5

2 0

Fig. 8. ζ 1

ζ 0 = (ζ 10 , ζ 21 )T = ( x1 , x2 )T ζ 1 = x3 (1 + x12 )

2.5

3

3.5

4

3

3.5

4

0 and ζˆ1 .

1.2

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ζ 2 = x4

1 0.8

ζ 3 = x5 ( x42 + 1)

0.6

In these coordinates, the system (25) will be written as:

0.4 0.2

α10 = ζ 1 + u1 (ζ 10 ) 2 1 1

2

1

0 0 2 1

α = ζ + u1 (ζ /((ζ ) + 1) + ζ 0 1

0 1

0 1

1

0 2

−0.2

0 2 1

+ζ (ζ + 1)) + ζ ζ /((ζ ) + 1)

(27)

α12 = ζ 3 − ζ 2 + u1ζ 1ζ 10 /((ζ 10 )2 + 1) + u2ζ 2

−0.4 0

0.5

1

1.5

2 0

Fig. 9. ζ 2

2.5

0 and ζˆ2 .

β1 = φ1 (ζ , u1 , u2 ) αˆ11 (ζ 0 , ζˆ1 ) = ζˆ 2 + u1 (ζ 1 /((ζ 10 ) 2 + 1) + ζ 20 + ζ 10 (ζ 10 + 1))

β 2 = φ2 (ζ , u1 , u2 )

+ζ 10ζ 1 /((ζ 10 )2 + 1)λ 1 sign1 (α 1 (ζ 0 , ζ 1 )

with

−αˆ 1 (ζˆ 0 , ζˆ1 ))

0 2 1

α10

(ζ ) + ζ 20 2

α11

ζ 1 /((ζ 10 )2 + 1) + ζ 10α12

(28)

ζ2

and to complete the coordinates change β1 ζ 3, which gives

ζ

0 2

and β 2

2 5 4

+λ 2 sign2 (α 2 (ζ 0 , ζ 1 , ζ 2 ) −αˆ 2 (ζˆ 0 , ζ 1 , ζˆ 2 ))

βˆ1 (ζ 0 , ζ 1 , ζ 2 , ζˆ 3 ) = φ1 (ζ 0 , ζ 1 , ζ 2 , ζ 3 , u1 , u2 ) +λ13 sign3 ( β1 (ζ 0 , ζ 1 , ζ 2 , ζ 3 )

φ1 ( x, u1 , u2 ) = x3 + u2 x5 x1 2 4

αˆ12 (ζ 0 , ζ 1 , ζˆ 2 ) = ζˆ 3 − ζ 2 + u1ζ 10ζ 1 /(ζ 10 ) 2 + 1) + u22ζ

2 4

φ2 ( x, u1 , u2 ) = φ ( x, u1 , u2 )( x + 1) + 2 x x ( x + 1) +u1 x3 x1 + u2 x4

φ1(ζ, u1, u2) and φ2(ζ, u1, u2) are obtained by substituting x = Ψ−1(ζ). Starting from (27), we construct the sliding modes observer: αˆ10 (ζˆ 0 ) = ζˆ1 + u1 (ζ 10 ) 2 + λ10 sign0 (α10 (ζ 0 ) − αˆ10 (ζˆ 0 ))

− βˆ1 (ζˆ 0 , ζ 1 , ζ 2 , ζˆ 3 ))

βˆ2 (ζ 0 , ζ 1 , ζ 2 , ζˆ 3 ) = φ2 (ζ 0 , ζ 1 , ζ 2 , ζ 3 , u1 , u2 ) +λ23 sign3 ( β 2 (ζ 0 , ζ 1 , ζ 2 , ζ 3 )

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− βˆ2 (ζˆ 0 , ζ 1 , ζ 2 , ζˆ 3 )) Figures (8,…,12) show simulations in ζ coordinates. We can see the well step by step convergence. At step 3, all the states ζˆ converge to ζ. Gains are: λ10 = 15, λ1 = 4, λ2 = 2, and λ3 = 1.

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

can obtain in the multi-output case with unknown inputs. Furthermore, we give a geometric characterization of the class of systems which can be put into an ITO form using a diffeomorphism including a matching condition. These characterization is constructive i.e. it allows to obtain the coordinate change. We show also that this class is a subset of the uniformly observable class systems. A step by step sliding mode observer was presented for a such class of ITO form and some illustrative examples were presented.

1

0.5

0

−0.5 0

REFERENCES 0.5

1

1.5

2

2.5

3

3.5

4

1 1 Fig. 10. ζ 1 and ζˆ1 .

40 35 30 25 20 15 10 5 0 −5

0

0.5

1

1.5

2

2.5

3

3.5

4

3

3.5

4

Fig. 11. ζ 2 and ζˆ2 . 1

1

2 0 −2 −4 −6 −8 −10 −12 −14 −16 0

525

0.5

1

1.5

2 2

Fig. 12. ζ 1

2.5

and ζˆ1 . 2

VII. CONCLUSION Summarizing the results of this paper, we have introduced the ITO form which extends as well as possible the single-output triangular observer form. This form is to our knowledge the most general triangular form we

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Taha Boukhobza is currently Assistant Professor at University of Antilles-Guyane (West IndiesGuiana), France. He obtained his Ph.D. in Automatics at the University of Orsay in November 1997. He has specialized in the observability forms for sliding mode observers and the high order control techniques. His research interests always include sliding mode observers and controllers.

Mohamed Djemai is actually Assistant professor at Ecole Nationale Suprieure d’Electronique et de ses Applications (ENSEA), Cergy Pontoise, France, with ECS team of control systems. He received the Electrical engineering diploma in 1991 from the ”Ecole Polytechnique d’Alger” of Algiers, and the Ph.D. degree in automatics from Paris XI University (France) in 1996. His main reasearch activities deal with singular perturbation, sliding mode mode control and observrs, fault detection and residual generation and synchronization of chaotics systems with application to cryptography. The domains of applications are robotics and electrical machine. ([email protected])

T. Boukhobza et al.: Implicit Triangular Observer Form Dedicated to A Sliding Mode Observer

Jean Pierre Barbot is actually professor at Ecole Nationale Suprieure d’Electronique et de Ses Applications (ENSEA), Cergy Pontoise, France. He is the head of ECS team of control systems. He received his Ph.D. degree in automatics from

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Paris XI University (France) in 1991 and HDR in 1999. His main reasearch activities deal with singular perturbation in continuous time and under sampling, sliding mode mode control and observrs, fault detection and residual generation and synchronization of chaotics systems with application to cryptography. The domains of applications are robotics and electrical machine.