STATE AND UNKNOWN INPUTS ESTIMATION FOR A CLASS OF NONLINEAR SYSTEMS

Copyright (c) 2005 IFAC. All rights reserved 16th Triennial World Congress, Prague, Czech Republic

STATE AND UNKNOWN INPUTS ESTIMATION FOR A CLASS OF NONLINEAR SYSTEMS M. Farza ∗ M. M’Saad ∗ F.L. Liu ∗ B. Targui ∗ ∗

GREYC, UMR 6072 CNRS, Universit´e de Caen, ENSICAEN, 6 Bd Mar´echal Juin, 14050 Caen Cedex, FRANCE. [email protected]

Abstract: A high gain observer is proposed for a class of multi-output nonlinear systems with unknown inputs in order to simultaneously estimate the whole state as well as the unknown inputs. The gain of this observer does not require the resolution of any dynamical system and is analytically given. Moreover, its tuning is reduced to the choice of two real numbers. The performances of the proposed observer are demonstrated in c 2005 IFAC simulation through an illustrative example.Copyright ° Keywords: Nonlinear system, High gain observer, Unknown inputs.

1. INTRODUCTION

server for the simultaneous estimation of the non measured states and the unknown inputs. The proposed approach does not necessitate the output differentiation and it only assumes that the dynamics of these inputs are bounded without making any hypothesis on how these inputs vary. This paper is organized as follows. In the next section, the class of nonlinear systems which is the basis of the observer design is introduced. Section 3 is devoted to the observer synthesis. For sake of clarity, only relevant results are given and corresponding proofs are reported in appendices. Section 4 is devoted to a simulation example in order to highlight the performance of the proposed observer.

Over the last twenty years, many researches have focused on the observer design for linear systems with unknown inputs (Johnson, 1975; Kudva et al., 1980; Hou and M¨uller, 1992; Guan and Saif, 1991; Darouach et al., 1994). In most cases, the objective was to estimate the non measured state variables and the proposed observers do not provide any information on the unknown inputs. In a relatively recent paper (Corless and Tu, 1998), the authors proposed a LMI based observer in order to jointly estimate the missing states and the unknown inputs. However, strong conditions are assumed to ensure the convergence of the inputs estimates. In a more recently paper (Xiong and Saif, 2003), the authors proposed reduced order observers to simultaneously estimate state and the unknown inputs when the latter vary slowly. Other results on unknown observers synthesis for some particular classes of nonlinear systems can be found in (Xiong and Saif, 2001; Farza et al., 2004; Ha and Trinh, 2004). In this paper, one proposes a full order high gain ob-

2. PROBLEM FORMULATION Consider MIMO systems of the form: ½

97

x˙ = f (w, x) ¯ = x1 y = Cx

(1)



1



x  x2    with x =  .  ;  ..  xq

Ã



 f 1 (w, x1 , x2 )  f 2 (w, x1 , x2 , x3 )      .. f (w, x) =  ; .   q−1  f (w, x)  f q (w, x)

C¯ = [In1 , 0n1 ×n2 , 0n1 ×n3 , . . . , 0n1 ×nq ]

fe11 (u, x1 , x2 ) fe21 (u, x1 , x2 )

!

and lowing ones fe1 (u, x1 , x2 ) = µ 1 ¶ G1 (u, s) G1 (u, s) = . The following two rank G12 (u, s) conditions¡are assumed ¢ to be satisfied: (i) Rank G11 (u, s) = m, for all u ∈ U and for all t ≥ 0  ∂ fe11 1 2 1  2 (u, x , x ) G1 (u, s)  ∂x  = n2 + m (ii)Rank   ∂ fe1  2 1 2 1 (u, x , x ) G (u, s) 2 ∂x2 for all x ∈ IRn , u ∈ U and t ≥ 0 (H5) The time derivative of the unknown input v(t) is a completely unknown function, ε(t), which is uniformly bounded that is sup kε(t)k ≤ βε where βε > 0

(2)

where the state x ∈ IRn with xk ∈ IRnk , k = 1, . . . , q q X and p = n1 ≥ n2 ≥ . . . ≥ nq , nk = n; the input k=1

w(t) ∈ U the set of bounded absolutely continuous functions with bounded derivatives from IR+ into W a compact subset of IRs ; the output y ∈ IRp and f (u, x) ∈ IRn with f k (u, x) ∈ IRnk . The functions f k are assumed to satisfy the following hypothesis: (H1) Each function f k (w, x), k = 1, . . . , q − 1 satisfies the following rank condition: ∂f k Rank( k+1 (w, x)) = nk+1 ∀x ∈ IRn ; ∀w ∈ W ∂x Moreover, one assumes that ∃αf , βf > 0 such that for all k ∈ {1, . . . , q − 1}, ∀x ∈ IRn , ∀w ∈ µ ¶T ∂f k ∂f k 2 W , αf Ink+1 ≤ (w, x) (w, x) ≤ ∂xk+1 ∂xk+1 2 βf Ink+1 (H2) For 1 ≤ k ≤ q − 1; the map xk+1 7→ f k (w, x1 , . . . , xk , xk+1 ) from IRnk+1 into IRnk is one to one. System (1) has been considered in (Hammouri and Farza, 2003) and it characterizes a subclass of U uniformly observable systems. In this paper, one shall suppose that a subset of the inputs is unknown. More precisely, one shall suppose that the µ vector ¶ w(t) can u(t) be partitioned as follows w(t) = where v(t) v(t) ∈ IRm is completely unknown and u(t) ∈ IRs−m is fully known. The objective then consists in synthesizing an observer to simultaneously estimate the vector of unknown inputs v(t) and the non measured states without assuming any model for the unknown inputs. The synthesis of such observer necessitates the adoption of some hypothesis which will be stated in due courses. At this step, one assumes the following: (H3) For k = 1, . . . , q, each function f k has the following structure: • f k (u, v, x) = fek (u, x1 , . . . , xk+1 )+Gk (u(t), s(t))v where s(t) is a known signal with a bounded time derivative; Gk (u(t), s(t)) ∈ IRnk and G1 (u(t), s(t)) satisfies the following rank condition: Rank(G) = Rank(G1 ) = m, ∀ u ∈ U and ∀ t ≥ 0 Moreover, one supposes that ∃αG , βG > 0 such that 2 2 ∀u ∈ U , ∀t ≥ 0, 0 < αG Im ≤ (G1 )T G1 ≤ βG Im . Notice that hypothesis (H1) and (H2), satisfied by f k , still be satisfied by fek . (H4) The x1 can be partitioned as follows: µ 1output ¶ x1 x1 = with x11 ∈ IRm1 , x12 ∈ IRp−m1 and x12 m ≤ m1 < p. Such a partition induces the fol-

t≥0

is a real number. To summarize, the nonlinear system which will be considered with view to observer synthesis can be written under the following condensed form:  1 1 X˙ = feX (u, X11 , X12 , X22 ) + G1X (u, s)X21 + ε¯(t)    ˙2 2 1 2 e X = + G2X (u, s)X21 µ fX (u, X11 , X 1) ¶  y1 = X1 = x1  y = y2 = X12 = x12 µ 1¶ µ 1 ¶(3) 1 X X = x 1 1 where X = ∈ IRn+m ; X 1 = ∈ X21 = v X2  2  X1 = x12  X22 = x2    IRm1 +m ; X 2 =   ∈ IRn−m1 ; ε¯(t) = ..   .

Xq2 = xq ¶ µ 1 ¶ 0 fe1 (u, x1 , x2 ) 1 , ; feX (u, X11 , X12 , X22 ) = ε(t) 0   e1 f2 (u, x1 , x2 )  fe2 (u, x1 , x2 , x3 )    1 2 feX (u, X11 , X 2 ) =  ; GX (u, s) = ..   . q e f (u, x)  1  G2 (u, s) µ 1 ¶  G2 (u, s)  G1 (u, s)   2 ; GX (u, s) =  . .. 0   . q G (u, s)

µ

3. OBSERVER SYNTHESIS Before giving the equations of the proposed observer, one shall introduce some notations and preliminary results. • Let θ1 , θ2 > 0 be two real numbers and let ∆1 (θ1 ) and ∆2 (θ2 ) be the following two block diagonal matrices: 1 Im ) (4) θ1 1 1 1 ∆2 (θ2 ) = diag(Ip−m1 , Ip−m1 , . . . , q−1 Ip−m1 ) θ2 θ2 ∆1 (θ1 ) = diag(Im1 ,

98

µ ∂ f¯2 diag Ip−m1 , 12 (u, s, ξ11 , ξ12 , ξ22 ), (13) ∂ξ2 ∂ f¯12 ∂ fe2 (u, s, ξ11 , ξ12 , ξ22 ) 2 (u, ξ11 , ξ12 , ξ22 , ξ32 ), . . . , 2 ∂ξ2 ∂ξ3 ! q−1 Y ∂ fek ∂ f¯12 2 (u, s, ξ11 , ξ12 , ξ22 ) (u, ξ11 , ξ12 , . . . , ξk+1 ) 2 ∂ξ22 ∂ξk+1

• For i = 1, 2, let Si be the unique solution of the algebraic Lyapunov equation : (5)

Si + ATi Si + Si Ai − CiT Ci = 0 · ¸ 0 Im 1 where A1 = and 0 0 

0 Ip−m1 0 0  .. . Ip−m1 A2 =   .. 0 . Ip−m1 0 ... 0 0

     

k=2

Notice that according to Hypothesis (H1),(H4) and lemma 1, the matrices Λ1 and Λ2 are of full rank.

(6)

Now, consider the following dynamical system:  1 ˆ˙ 1 = feX ˆ 11 , X ˆ 12 , X ˆ 22 ) + G1X (u, s)X ˆ 21  X (u, X    + −1 −1 T 1 ˆ    − θ1 Λ1 (u, s)∆1 (θ1 )S1 C1 (X1 − y1 ) ˙2 2 1 ˆ2 2 e ˆ ˆ ˆ1 X = fX (u, ³ X1 , X ) +´GX (u, s)X2   −1 T ˆ 2 1 ˜2  − θ2 Λ+ ∆−1  2 u, s, X1 , X 2 (θ2 )S2 C2 (X1 − y2 )    ˆ 11 − y1 ) − 2θ1 D(u, s)(X (14) µ 1¶ µ 1¶ ˆ X x ˆ n+m 1 1 ˆ = ˆ = where X with X ∈ ˆ 2 ∈ IR vˆ X  1 x ˆ2 2 x ˆ  m1 +m m1 m ˆ2 1 IR ,x ˆ1 ∈ IR , vˆ ∈ IR , X =  .  ∈  ..  x ˆq n−m1 p−m1 nk 2 k IR , x ˆ1 ∈ IR , x ˆ ∈ IR , k = 2, . . . , q; ˜2 = X ˆ 2 − D(u, s)(X ˆ 1 − y1 ), Λ1 and Λ2 are respecX 1 tively given by (12) and (13); ∆k (θk ) and Sk−1 CkT , k = 1, 2 are respectively given by (4) and (8); the matrix D is given by (9); θ1 , θ2 > 0 are real numbers.

are respectively 2m1 ×2m1 and q(p−m1 )×q(p−m1 ) square matrices and C1 = [Im1 0m1 ] ; C2 = [Ip−m1 0p−m1 . . . 0p−m1 ] (7) are respectively m1 × 2m1 and (p − m1 ) × q(p − m1 ) rectangular matrices. It can be shown that S1 and S2 are symmetric positive definite and that one has: 

· S1−1 C1T =

2Im1 Im 1

 Cq1 Ip−m1 ¸  Cq2 Ip−m1    ; S2−1 C2T =   (8) ..   . Cqq Ip−m1

j! i!(j − i)! • Let D(u, s) be the following (n − m1 ) × m1 rectangular matrix:  ¡ ¢+  G12 (u, s) G11 (u, s) ¡ ¢+   2  G (u, s) G11 (u, s)    D(u, s) =  (9) ..    . ¡ ¢+ Gq (u, s) G11 (u, s) where Cji =

One now states the main result (a sketch of the proof is given in the appendix): Theorem 1:Suppose that system (3) satisfies hypothesis (H1) to (H5). Then,

where the notation (·)+ means the left inverse of (·). • ∀ξ11 ∈ IRm1 , ∀ξ12 ∈ IRp−m1 , ∀ξ22 ∈ IRn2 , set:

∃θ1,0 > 0; ∃θ2,0 > 0; ∀θ1 > θ1,0 ; ∀θ2 > θ2,0 ; ∃λ > 0; ∃µ(θ1 , θ2 ) > 0; ∃M (θ1 , θ2 ) > 0;

f¯12 (u, s, ξ11 , ξ12 , ξ22 ) = fe21 (u, ξ11 , ξ12 , ξ22 ) ¡ ¢+ −G12 (u, s) G11 (u, s) fe11 (u, ξ11 , ξ12 , ξ22 ) (10)

ˆ ∀u ∈ U ; ∀X(0) ∈ Rn+m ; one has : ke(t)k ≤ λθ1 exp (−µ(θ1 , θ2 )t) ke(0)k+M (θ1 , θ2 )βε

One states the following (see the appendix for the proof): Lemma 1 Under hypothesis (H1) and (H4), one has: µ ¯2 ¶ ∂ f1 1 2 2 Rank (u, s, ξ1 , ξ1 , ξ2 ) = n2 ∂ξ22 ¡ ¢ • Set Λ1 (u, s) = diag Im1 , G11 (u, s)

ˆ where e(t) = X(t) − X(t) with X(t) is the unknown trajectory of system (3) associated to the inˆ put u, X(t) is any trajectory of system (14) associated to the input u and the outputs y1 and y2 ; βε is the upper bound of kε(t)k given in hypothesis (H5). Moreover, one has: lim µ(θ1 , θ2 ) = +∞

(11)

and

θ1 ,θ2 →∞

lim

θ1 ,θ2 →∞

M (θ1 , θ2 ) = 0.

Remark: Observe that for ε(t) = 0, i.e. when the unknown inputs are constant, the convergence of the estimation error is exponential. In the case where kε(t)k 6= 0 but bounded by βε , the asymptotic estimation error can be made as small as desired by choosing values of θ1 and θ2 high enough.

(12)

• ∀ξ11 ∈ IRm1 , ∀ξ12 ∈ IRp−m1 , ∀ξk2 ∈ IRnk , k = 2, . . . , q, let Λ2 be the following block diagonal matrix: Λ2 (u, s, ξ11 , ξ12 , ξ22 , . . . , ξq2 ) =

99

4. EXAMPLE

3

X4

2.5

Consider the following dynamical system:  x˙ 1 = (a − x3 )x4 − x3 v(t) − x1     x˙ 2 = x3 x4 + (a − x3 )v(t) − x2    x˙ 3 = x4 (1 + x24 ) − x33 − 10sin(t)v(t) x˙ 4 = x5 − x34 − 2cos(t)v(t)     x˙ 5 = 5sin(2t)v(t)    y = [x1 x2 x3 ]T

2

1.5

ESTIMATED

1

0.5

(15)

0

SIMULATED 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4.5

5

4.5

5

20

X5

18 16 14

where x = [x1 x2 x3 x4 x5 ]T ∈ IR5 with xi ∈ IR, v(t) is the unknown input and a 6= 0 is a real number. To simplify, no known input has been considered. For simulation purposes, the following expression (unknown by the observer) has been used for the unknown input: v(t) = 5sin(5t) (16)

12 10

6 4

T

2

ESIMATED

2

SIMULATED

0 0

0.5

1

1.5

2

2.5

3

3.5

4

8

v

6

4

It is easy to see that system (15) is under form (3) with: 1

ESTIMATED

8

2

0

3

x = [x1 x2 x3 ] ; x = x4 ; x = x5 ;   (a − x3 )x4 − x1  ; fe2 (x) = x5 −x34 ; fe1 (x1 , x2 ) =  x3 x4 − x2 3 2 x4 (1 + x4 ) − x3   −x3 (t) fe3 (x) = 0; G1 (s(t)) =  a − x3 (t)  ; −10sin(t)

−2

−4

ESTIMATED

−6

−8

0

0.5

1

1.5

2

SIMULATED 2.5

3

3.5

4

Fig. 1. Estimation of x4 , x5 and v

Concerning the partition of x1 needed in hypothesis (H4), one can consider the following one (the only possible partition in this example): x11 = [x1 x2 ]T and x12 = x3 . Now, one can easily check hypothesis (H1) to (H5) and an observer under form (14) can be used in order to achieve the required estimations.

Conclusion: A high gain observer has been designed for a class of nonlinear systems. The appealing features of the proposed observer are its implementation and calibration simplicity. The performances of the proposed observer have been demonstrated in simulation through an example. The use of the proposed observers in real experiments related to bioreactors and induction motors will be treated in upcoming works. Appendix: Proofs Proof of Lemma 1. Let

4.1 simulation Results

P (t) =

An observer of the form (14) has been used in order to estimate x4 , x5 and v. This observer has been simulated using data issued from simulation. In order to simulate practical situations, each of the measurements of x1 , x2 and x3 has been corrupted by a uniformly distributed random signal produced by SIMULINK with zero mean value and a standard deviation respectively equal to 10−3 , 10−3 and 3.2 10−4 . In figure 1, the true time evolutions of x4 , x5 and v (issued from model simulation) are compared with their respective estimates provided by the observer. Notice that corresponding curves are almost superimposed. The employed values of θ1 and θ2 are respectively equal to 60 and 15. The initial conditions for the model and the observer are: x1 (0) = x ˆ1 (0) = 1; x2 (0) = x ˆ2 (0) = 1; x3 (0) = x ˆ3 (0) = 1; x4 (0) = 2; x5 (0) = 10; x ˆ4 (0) = 0; x ˆ5 (0) = 0; vˆ(0) = −1. The obtained results clearly show the good agreement between the estimated and simulated variables. Recall that the expression of the unknown input (equation (16)) introduced for simulation purposes is ignored by the observer.

Since rank(P (t)) = p for all t ≥ 0 and p ≥ n2 + m (according to (ii) of (H4)), one has:

G2 (s(t)) = −2cos(t); G3 (s(t)) = 5sin(2t)

µ

Im 1 0 ¡ 1 ¢+ 1 −G2 (u(t), s(t)) G1 (u(t), s(t)) Ip−m1

¶

n2 + m = Rank   e1  ∂ f1 1 2 2 1 2 (u, ξ1 , ξ1 , ξ2 ) G1 (u, s)    P (t) ·  ∂ξ2    ∂ fe1  2 1 2 2 1 ((u, ξ1 , ξ1 , ξ2 ) G2 (u, s) 2 ∂ξ2   ∂ fe11 1 2 2 1  ∂ξ 2 (u, ξ1 , ξ1 , ξ2 ) G1 (u, s)   2 = Rank   ∂ f¯2  1 1 2 2 (u, s, ξ1 , ξ1 , ξ2 ) 0 2 ∂ξ µ ¯2 2 ¶ ∂ f1 1 2 2 = Rank (u, s, ξ , ξ , ξ ) +m 1 1 2 ∂ξ22 This leads to (11). Sketch of the proof of Theorem 1. One shall introduce two changes of coordinates. Consider the follown+m ing first one:µT (u, → IRn+m ¶s) : IR µ 1¶ 1 x ¯ x ¯1 1 X 7→ x ¯= = T (u, s)X with x ¯ = ∈ x ¯2 x ¯12

100

µ



 x ¯21   IRm1 +m ; x ¯11 ∈ IRm1 ; x ¯12 ∈ IRm ; x ¯2 =  ...  ∈

C¯1 = µ

x ¯2q IR ,x ¯21 ∈ IR ,x ¯2k ∈ IR , k = 2, . . . , q and where T (u, s) is the following (n + m) × (n + m) non singular square matrix :   Im 1 0 0 ... ... 0  0 Im 0 ... ... 0    .    −γ12 (u, s) 0 Ip−m1 0 . . . ..    T (u, s) =  . . ..   −γ22 (u, s) 0  . 0 I . n2     .. . . . . . .  . . . 0  . −γq2 (u, s) 0 . . . 0 Inq (17) ¡ 1 ¢+ ∆ ∆ 2 1 2 with γ1 (u, s) = G2 (u, s) G1 (u, s) and γk (u, s) = ¡ ¢ + Gk (u, s) G11 (u, s) , k = 2, . . . , q. The objective of this transformation is to generate a subsystem that does not depend on the unknown input (v = x ¯12 ). Indeed, one can show that this transformation puts system (3) under the following form: n−m1

p−m1

C¯2 =

nk

 1 z˙    y1  z˙ 2   y2



= A1 z 1 + ψ 1 (u, s, z 1 , z 2 ) + Λ1 (u, s)¯ ε(t) 1 1 1 = h1 (z ) = C1 z = z1 (21) = A2 z 2 + ψ 2 (u, s, y1 , z11 , z 2 ) = h2 (z 1 , z 2 ) = C2 z 2 + γ12 C1 z 1

−1 T −θ1 ∆−1 z 1 ) − y1 ) 1 (θ1 )S1 C1 (h(ˆ zˆ˙ 2 = A2 zˆ2 + ψ 2 (u, s, y1 , zˆ11 , zˆ2 ) ∂Φ2 ∂ f¯2 + 2 (u, s, y1 , Φc2 (ˆ z 2 )) 1 (u, s, ξ11 , Φc2 (ˆ z 2 ))(ˆ z11 − z11 ) ∂x ¯ ∂x ¯1 −1 T −θ2 ∆−1 z 1 , zˆ2 ) − y2 ) 2 (θ2 )S2 C2 (h(ˆ ∂Φ2 − 2 (u, s, y1 , Φc2 (ˆ z 2 )) ¯ Ã ∂x µ ¶+ ! ∂Φ 2 c 2 Λ+ z )) − (u, s, y1 , Φc2 (ˆ z 2 )) 2 (u, s, y1 , Φ2 (ˆ ∂x ¯2

dγ12 (u, s)¯ x11 dt ∆ g¯2 (u, s, x ¯1 , x ¯2 ) = −γ 2 (u, s)fe1 (u, x1 , x2 ) ∆

dt

x21 2 ¯ f1 (u, s, y1 , x ¯2 )

zˆ˙ 1 = A1 zˆ1 + ψ 1 (u, s, zˆ1 , zˆ2 )

g¯12 (u, s, x ¯11 ) = −

−

zq2

where C1 and C2 are given by (7) and each of the functions ψ i (u, s, z 1 , z 2 ) is triangular with respect to z i , i = 1, 2. Now, consider the following dynamical system:

fek (u, x1 , . . . , xk+1 )

k

(20)

k=1

1

1

Ip−m1 0(p−m1 )×(n−p) 0(n−p)×(p−m1 ) 0(n−p)×(n−p)

(19) ¶

zk2 ∈ IRp−m1 , k = 1, . . . , q. One can show that this transformation puts system (18) under the following form (see e.g. (Hammouri and Farza, 2003)):

k = 2, . . . , q − 1 ∆ f¯2 (u, s, x ¯1 , x ¯2 ) = feq (u, x)

dγk2

¶

    ¯2   ∂ f 1 2 2 2 ¯   ¯ )f2 (u, s, y1 , x ¯ ) 2 (u, s, y1 , x   ∂ x ¯ 2   with   ..   . ! Ã   q−2   Y ∂ f¯k2 2 2 2  ¯ (u, s, y , x ¯ ) f (u, s, y , x ¯ ) 1 1 q−1 ∂x ¯2k+1

∆ f¯12 (u, s, x ¯11 , x ¯2 ) = fe21 (u, x1 , x2 ) −γ 2 (u, s)fe1 (u, x1 , x2 )

k

x ¯2q



∆ f¯11 (u, s, x ¯11 , x ¯21 , x ¯22 ) = fe11 (u, x1 , x2 )

q

0m×m1

0m1 ×m 0m×m

where Im1 is the m1 × m1 identity matrix and the notation 0i×j means the i × j null matrix. Now, consider the second change of coordinates: µ ¶ µ 1 ¶Φ = Φ1 x ¯ : IRn+m −→ IR2m1 +q(p−m1 ) , x ¯= 7→ Φ2 x ¯2 µ 1¶ z z= . More precisely, one has: z2 2m1 • Φ1 : IRµm1 +m ¶ −→ IR µ 1¶ 1 x ¯1 z1 ∆ 1 x ¯1 = → 7 z = = Φ1 (u, s, x ¯1 ) = x ¯12 z21 µ ¶ x ¯11 1 Λ1 (u, s)¯ x = with zk1 ∈ IRm1 , k = G11 (u, s)¯ x12 1, 2 and where the matrix Λ1 is given by equation (12). n−m1 q(p−m1 ) • Φ2 : IR  2  −→ IR  2 x ¯1 z1 2 x  z22  ¯ 2 ∆     x ¯2 =  .  −→ z 2 =  .  = Φ2 (u, s, y1 , x ¯2 ) =  ..   .. 

 1 x ¯˙ = f¯1 (u, s, x ¯11 , x ¯21 , x ¯22 ) + G(u, s)¯ x1 + ε¯(t)    ∆  ¯ 1 (¯  x1 ) = C¯1 x ¯1  y1 = h 2 2 1 ¯ (18) x ¯˙ = f (u, s, x ¯1 , x ¯2 ) + g¯2 (u, s, x ¯11 , x ¯2 )  ∆  1 2 1 2 2 ¯  x1 y = h2 (¯ x ,x ¯ )=x ¯1 + γ1 (u, s)¯    2 ¯1 = C¯2 x ¯2 + γ12 (u, s)C¯1 x ¶ µ 1 f¯1 (u, s, x ¯11 , x ¯21 , x ¯22 ) ¯22 ) = ¯21 , x where f¯1 (u, s, x ¯11 , x 0   ¯2 2 1 ¯22 ) ¯1 , x ¯1 , x f1 (u, s, x  f¯22 (u, s, x ¯11 , x ¯21 , x ¯22 , x ¯23 )    f¯2 (u, s, x ¯11 , x ¯2 ) =  ; ..   . 2 1 2 ¯ fq (u, s, x ¯1 , x ¯ ) µ ¶ µ ¶ 1 0 0 G1 (u, s) ; ε¯(t) = ; G(u, s) = ε(t) 0 0   2 1 g¯1 (u, s, x ¯1 ) 1 2  g¯22 (u, s, x ¯ , x ¯22 )  1 ¯1 , x   2 2 1 ¯ )= g¯ (u, s, x ¯1 , x  with ..   . g¯q2 (u, s, x ¯11 , x ¯21 , x ¯22 )

1 1 ∆ 2 1 f¯k (u, s, x ¯1 , x ¯2 ) =

Im1

1

(u, s)¯ x11 , k = 2, . . . , q

−1 T θ2 ∆−1 z 1 , zˆ2 ) − y2 ) 2 (θ2 )S2 C2 (h(ˆ

101

(22)

µ where zˆ =

zˆ1 zˆ2

¶

· ∈ IR

2m1 +q(p−m1 )



; zˆ = 1

zˆ11 zˆ21

¸

Now, for θ1 > c1 , θ2 > c4 , choose θ1 such that 2(q−1) c2 θ2 θ1 > max(θ2q−1 , θ2 − c4 + c1 , c1 + ). One (θ2 − c4 ) √ c3 can show that: V˙ ≤ −η(θ2 − c4 )V + βε V where θ1 cθ2q−1 η = 1− p > 0. Using the fact (θ1 − c1 )(θ2 − c4 ) that k¯ e(t)k ≤ ke(t)k ≤ θ1 k¯ e(t)k, one can show r h η i λ2 that: ke(t)k ≤ θ1 exp − (θ2 − c4 )t ke(0)k + λ1 2 c3 √ βε . This ends the proof of Theorem 1. η λ1 (θ2 − c4 )

∈

 zˆ12  zˆ22    IR2m1 , zˆi1 ∈ IRm1 ; zˆ2 =  .  ∈ IRq(p−m1 ) with  .. 

zˆq2 ∈ IR , k = 1, . . . , q; Φci denotes the converse function of Φi ; θ1 , θ2 > 0 are two real numbers and the variable ξ11 ∈ IRm1 is given by the Mean value Theorem: ˆ¯11 , x ˆ¯2 ) − f¯2 (u, s, x ˆ¯2 ) = f¯2 (u, s, x ¯11 , x 2 ¯ ∂f ˆ¯2 )(x ˆ¯11 − x (u, s, ξ11 , x ¯11 ) = ∂x ¯11 ∂ f¯2 (u, s, ξ11 , Φc2 (ˆ z 2 ))(ˆ z11 − z11 ). ∂x ¯11 One shall show that system (22) can be written under form (14) in the original coordinates x. Indeed, proceeding as in (Farza et al., 2004), one can show that system (22) can be written in the coordinates x ¯ as follows: zˆk2

p−m1

REFERENCES Corless, M. and J. Tu (1998). State and Input Estimation for a Class of Uncertain Systems. Automatica 34(6), 757–764. Darouach, M., M. Zasadzinski and S.J. Xu (1994). Full-Order Observer for Linear Systems with Unknown Inputs. IEEE Trans. on Aut. Control 39(3), 606–609. Farza, M., M. M’Saad and L. Rossignol (2004). Observer design for a class of MIMO nonlinear systems. Automatica 40, 135–143. Gauthier, J.P., H. Hammouri and S. Othman (1992). A simple observer for nonlinear systems - application to bioreactors. IEEE Trans. on Aut. Control 37, 875–880. Guan, Y. and M. Saif (1991). A Novel Approach to the Design of Unknown Inputs Observers. IEEE Trans. on Aut. Control 36(5), 632–635. Ha, Q.P. and H. Trinh (2004). State and input simultaneous estimation for a class of nonlinear systems. Automatica 40, 1779–1785. Hammouri, H. and M. Farza (2003). Nonlinear observers for locally uniformly observable systems. ESAIM J. on Control, Optimisation and Calculus of Variations 9, 353–370. Hou, M. and P. C. M¨uller (1992). Design of observers for linear systems with unknown inputs. IEEE Trans. on Aut. Control AC-37, 871–875. Johnson, C. D. (1975). On observers for linear systems with with unknown and inaccessible inputs. International Journal of Control 14, 825–831. Kudva, J., N. Viswanadham and A. Ramakrishna (1980). Observers for linear systems with unknown inputs. IEEE Trans. on Aut. Control AC25, 113–115. Xiong, Y. and M. Saif (2001). Sliding Mode Observer for Nonlinear Uncertain Systems. IEEE Trans. on Aut. Control 46(12), 2012–2017. Xiong, Y. and M. Saif (2003). Unknown disturbance inputs estimation based on a state functional observer design. Automatica 39, 1389–1398.

  ˆ¯˙ 1 = f¯1 (u, s, x ˆ¯11 , x ˆ¯21 , x ˆ¯22 ) + G(u, s)x ˆ¯1 x    + −1 −1 T  ˆ¯1 − y1 ) − θ1 Λ1 (u, s)∆1 (θ1 )S C1 (C¯1 x  (23) ˆ¯˙ 2 = f¯2 (u, s, x ˆ¯11 , x ˆ¯2 ) + g¯2 (u, s, x ˆ¯11 , x ˆ¯2 ) x   + −1 1 ˆ2  − θ2 Λ2 (u, s, x ¯1 , x ¯ )∆2 (θ2 )    ˆ¯21 + γ12 (u, s)x ˆ¯11 − y2 ) S2−1 C2T (x µ 1¶ µ 1¶ ˆ¯ ˆ x x ¯1 n+m 1 ˆ¯ = ˆ¯ = where x ∈ IR , x ∈ ˆ1 x ¯ˆ2 x ¯  22  ˆ¯1 x  ˆ¯11 ∈ IRm1 , x ˆ¯12 ∈ IRm , x ˆ¯2 =  ...  IRm1 +m , x  ∈ x ¯ˆ2q ˆ¯21 ∈ IRn−m1 ; x ˆ¯2k ∈ IRnk , k = 2, . . . , q. IRn−m1 , x Now, according to the expression of T (u, s) (equation (17)), one can easily show that system (23) can be written in the original coordinates x under form (14). To end the proof, it suffices to demonstrate theorem 1 by considering system (18) on one hand and system (22) on the other hand. Indeed, for = 1, 2, set ei (t) = T zˆi (t) − z i (t), e¯i = ∆i (θi )ei , Vi (¯ ei ) = e¯i Si e¯i and let V (¯ e1 , e¯2 ) = V1 (¯ e1 ) + V2 (¯ e2 ) be the Lyapunov candidate function. Using classical computations (see e.g.(Gauthier et al., 1992; Hammouri and Farza, 2003; Farza et al., 2004)), one can show that: p p c3 βε p V˙ 1 ≤ −(θ1 − c1 )V1 + c2 θ2q−1 V1 V2 + V1 θ1 p p V˙ 2 ≤ −(θ2 − c4 )V1 + c5 θ2 V1 V2 (24) where ci , i = 1, . . . , 5 are positive constant parameters which do not depend on θ1 nor θ2 and βε is the upper bound of kε(t)k as given in Hypothesis (H5). Combining inequalities of (24) and taking θ2 ≥ 1, one obtains: V˙ ≤ −(θ1 − c1 )V1 − (θ2 − c4 )V2 + p p c3 βε p cθ2q−1 V1 V2 + V1 where c = c2 + c5 . θ1

102