SINGLE OUTPUT DEPENDENT OBSERVABILITY NORMAL FORM

Author manuscript, published in "SIAM Journal on Control and Optimization 46, 6 (2007) 2242-2255"

SINGLE OUTPUT DEPENDENT OBSERVABILITY NORMAL FORM G. ZHENG1 , D. BOUTAT2 AND J.P. BARBOT1

ˆ INRIA RHONE ALPES, 655 AVENUE DE L’EUROPE, 38334 ST ISMIER CEDEX, FRANCE. [email protected] 2 LVR/ENSI, 10 BOULEVARD DE LAHITOLLE, 18020 BOURGES, FRANCE. [email protected] 1 ECS/ENSEA, 6 AVENUE DU PONCEAU, 95014 CERGY-PONTOISE, PROJECT ALIEN, INRIA-FUTUR, FRANCE. 1 BIPOP,

[email protected]

inria-00179184, version 1 - 13 Oct 2007

Abstract. This paper gives the sufficient and necessary conditions which guarantee the existence of a diffeomorphism in order to transform a nonlinear system without inputs into a canonical normal form depending on its output. Moreover we extend our results to a class of systems with inputs.

1. Introduction. Since Luenberger’s work [9], the design of an observer for observable linear systems with linear outputs has been a well-known concept. In order to use the same observer for nonlinear systems, the so-called observability linearization problem for nonlinear systems was born. The sufficient and necessary conditions which guarantee the existence of a diffeomorphism and of an output injection to transform a single output nonlinear system without inputs into a linear one with an output injection were firstly addressed in [12]. Then, for a multi-output nonlinear system without inputs, the linearization problem was partially solved in [13]. The complete solution to the linearization problem was given in [16]. Another approach was introduced for the analytical systems in [11] by assuming that the spectrum of the linear part must lie in the Poincar´e domain and it was generalized in [14] by assuming that the spectrum of the linear part must lie in the Siegel domain. These assumptions are not in generally fulfilled. Other approaches using quadratic normal forms were given in [1] and [3]. All these approaches enable us to design an observer for a larger class of nonlinear systems. Meanwhile, other researchers worked on designing nonlinear observers directly, such as high-gain observers [6], [4], [7]. Nevertheless, even if the conditions which guarantee the linearization method to design an observer were not generically fulfilled, this method still remained important for the nonlinear observer design first because it works well for non-analytic systems, and second because it could be used in the adaptive theory and also be useful for the observation of systems with unknown inputs. All these reasons explain why researchers carry on investigating this matter. In [10], the author gave the sufficient and necessary geometrical conditions to transform a nonlinear system into a so-called output-dependent time scaling linear canonical form. While [5] gave the dual geometrical conditions of [10]. In this paper, as an extension of [17], we will propose a method to deduce the geometrical conditions which are sufficient and necessary to guarantee the existence of a local diffeomorphism z = φ (x) which transforms the locally observable dynamical system ½

x˙ = f (x), y = h(x),

(1.1)

where x ∈ U ⊂ IR n , f : U ⊂ IR n → IR n and h : U ⊂ IR n → IR are sufficiently smooth, into the following form ½

where

z˙ = A(y)z + β(y), y = zn = Cz,

(1.2)

    A(y) =   

0 ··· α1 (y) · · · .. .. . . 0 ··· 0 ···

0 0 .. .

0 0

0 0 .. .





β1 (y) β2 (y) .. .

       , β(y) =  ···    βn−1 (y) αn−2 (y) 0 0  0 αn−1 (y) 0 βn (y)

    ,  

and αi (y) 6= 0 for y ∈ ]−a, a[ and a > 0. This kind of linearization is called Single Output Dependent Observability normal form (SODO normal form). For dynamical systems in the form (1.2), we may for example apply the following high gain observer [2]: ½

zˆ˙ = A(y)ˆ z + β(y) − Γ−1 (y) Rρ−1 C T (C zˆ − y), 0 = −ρRρ − A¯T Rρ − Rρ A¯ + C T C.

where Γ (y) is the n × n diagonal matrix Γ (y) = diag

i=1

n × n matrix defined as follows

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·n−1 Q

   A¯ =  

0 ··· 1 ··· .. . . . . 0 ···

αi (y),

0 0 .. .

0 0 .. .

1

0

n−1 Q i=2

(1.3)

¸ αi (y), · · · , αn−1 (y), 1 and A¯ is the

   . 

Indeed, here the output of system (1.2) is considered as an input of (1.3). Setting e = z − zˆ, the observation error can be obtained as follows: ¡ ¢ e˙ = A (y) − Γ−1 (y) Rρ−1 C T C e, And the convergence of such observer is proved in [2], thus in section 4 we simply highlight the design of such observer for systems in the form (1.2). Moreover, we generalize our result to a class of systems with inputs. Then, some useful corollaries are discussed in order to deal with affine systems and the so-called left invertibility problem. This paper is organized as follows. The next section addresses notations and technical results which are key to prove our main result. In section 3, we present our method to deduce the geometrical conditions for a nonlinear system without inputs in order to transform it into a SODO normal form. Section 4 is devoted to the generalization of our results to a class of systems with inputs. In the same section, some practical particular cases are studied, including the state affine systems and the left invertibility problem. Throughout this paper, examples are discussed in order to highlight our theoretical results. 2. Notations and technical results. Throughout this article, Li−1 f h for 1 ≤ i ≤ n denotes the th (i − 1) Lie derivative of output h in the direction of f, and set θi = dLi−1 f h as its differential. Assume ¡ ¢T that system (1.1) is locally observable, thus θ = θ1 , · · · , θn is a basis of the cotangent bundle T ∗ U of U . Then, we also consider the vector field τ1 defined in [12] as follows ½ θi (τ1 ) = 0, f or 1 ≤ i ≤ n − 1, (2.1) θn (τ1 ) = 1, and by induction we define k−1

τk = (−1)

adk−1 (τ1 ) , f or 2 ≤ k ≤ n. f

It is clear that {τ1 , · · · , τn } is a basis of the tangent bundle T U of U .

(2.2)

Let us recall a famous result from [12]. Theorem 2.1. The following conditions are equivalent i) There exist a diffeomorphism and an output injection which transform system (1.1) into normal form (1.2) with αk (y) = 1 for 1 ≤ k ≤ n − 1. ii) [τi , τj ] = 0 for 1 ≤ i, j ≤ n. If for some 1 ≤ k ≤ n − 1 the functions αk (y) in the form (1.2) are not constant, then ii) of Theorem 2.1 is not fulfilled. Consequently, the rest of this section is devoted to use [τi , τn ] in order to determine all the functions αi (y) for 1 ≤ i ≤ n − 1. Lemma 2.2. For a system in the form (1.2) we have for 1 ≤ k ≤ n − 1, ¡ ¢ k τk = π1k ∂z∂k + Akk−1 (zn ) zn−1 + ηk−1 (zn ) ∂z∂k−1 à ! k−2 n−1 n−1 P P P ∂ k + Aki (zn ) zn−k+i + Tj,l (zn ) zj zl ∂z i (2.3) i=1 j=n−k+i+1 l=j à ! k−2 n P P [3] ∂ + ηik (zn ) zj + Ozn (zn−k+i+1 , · · · , zn−1 ) ∂z , i i=1

j=n−k+i+1

k where πn = 1 and πk−1 = πk αk−1 for 2 ≤ k ≤ n, ηik (zn ) and Tj,l (zn ) are some smooth functions of zn ,

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[3]

Ozn (zn−k+2 , · · · , zn−1 ) represents the residue higher than order 2 with coefficient which is function of zn and    m k−1 0 0 Y X π π k−i+1  k k−m  k αk−j  πn−k+i , (2.4) Sk−i,m−k+i+1 Aki (zn ) = (−1) Sk−i,1 2i + 2 πi πk−m m=k−i+1

j=k−i+1

k k where Sk−i,1 and Sk−i,m−k+i+1 are defined as follows k−1 k−1 k k Sj,1 = 1, Sj,l = Sj−1,l + Sj,l−1 , f or 2 ≤ k ≤ n, 1 ≤ j ≤ k − 1 and 1 ≤ l ≤ k − j.

(2.5)

k k and S0,l = Si,0 = 0.

Proof. For a system in the (1.2) form, equation (2.1) gives τ1 = ³ 0 ´ π π0 obtain τ2 = π12 ∂z∂ 2 + π12 πn−1 zn−1 + π12 βn ∂z∂ 1 , and 1

1 ∂ π1 ∂z1 ,

then, we use equation (2.2) to

1

¶ µ 0 ¶ ¶ 1 ∂ π20 π1 π20 ∂ τ3 = + 2 πn−1 zn−1 + α1 + 2 βn + π3 ∂z3 π2 π12 π2 ∂z2 à ! µ 0 ¶0 π10 π1 ∂ 2 − πn−2 zn−2 + πn−1 πn−1 zn−1 π12 π12 ∂z1 Ãõ ! ! ¶0 π10 π10 ∂ 0 0 − . πn−1 βn + πn−1 βn zn−1 + 2 πn−1 βn−1 + βn βn 2 π1 π1 ∂z1 µµ

π10 α1 π12

Then by an induction, for 3 < k ≤ n, we get τk =

¡ ¢ ∂ 1 ∂ k + Akk−1 (zn ) zn−1 + ηk−1 (zn ) πk ∂zk ∂zk−1   k−2 n−1 n−1 X X X ∂ k Aki (zn ) zn−k+i + + Tj,l (zn ) zj zl  ∂z i i=1 j=n−k+i+1 l=j   k−2 n X X ∂  + , ηik (zn ) zj + Oz[3]n (zn−k+i+1 , · · · , zn−1 ) ∂z i i=1 j=n−k+i+1

where  π0 k−i+1  k Aki (zn ) = (−1) Sk−i,1 2i + πi

k−1 X m=k−i+1

 0 π k−m  k Sk−i,m−k+i+1 2 πk−m

m Y

 αk−j  πn−k+i ,

j=k−i+1

with the coefficients Sik given by the rule (2.5). In order to determine the αi (y) for 1 ≤ i ≤ n − 1, we impose that ½ ∂ 0, for 1 ≤ i ≤ n − 1, h ◦ φ−1 = 1, when i = n. ∂zi Now, we are ready to state a set of differential equations which enables us to compute functions αi for 1 ≤ i ≤ n − 1. Proposition 2.3. If there exists a diffeomorphism which transforms system (1.1) into form (1.2) then [τk , τn ] = λk (y)τk + G[1] n + R, f or 1 ≤ i ≤ n − 1,

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where G[1] n =

k−2 Xµ i=1

1 k T zn−k+i πk k,n−k+i

¶

∂ 1 k ∂ + T zk , ∂zi πk k,k ∂z2k−n

and R=

k−2 X

 

i=1

n X

j=n−k+i+1

 ∂ η¯ik (zn ) + Oz[2]n (zn−k+i+1 , · · · , zn−1 ) ∂zi

and λk (y) = diag{δ1k (y), · · · , δik (y), · · · , δkk (y), 0, · · · , 0}, f or 1 ≤ i ≤ k − 1,

(2.6)

0

(Ak ) π0 where δkk = Ank + πkk and δik = Ani − Ann−k+i − Aik for 1 ≤ i ≤ k − 1, and Aki is given in (2.4). i Proof. According to equation (2.3), for 1 ≤ k ≤ n − 1 we have µ ¶ 1 ∂ πk0 n [τk , τn ] = Ak + πk πk ∂zk ÃÃ ! ¡ k ¢0 ! k−2 X A 1 1 k ∂ ∂ i k + Ani − Ann−k+i − Aki zn−k+i + Tk,n−k+i zn−k+i + Tk,k zk k π ∂z π ∂z Ai k i k 2k−n i=1   k−2 n X X ∂  + η¯ik (zn ) + Oz[2]n (zn−k+i+1 , · · · , zn−1 ) . ∂zi i=1 j=n−k+i+1

0

π0

Set λk (y) = diag{δ1k (y), · · · , δik (y), · · · , δkk (y), 0, · · · , 0}, where δkk = Ank + πkk and δik = Ani − Ann−k+i − for 1 ≤ i ≤ k − 1, then [τk , τn ] = λk (y)τk + G[1] n + R.

(Aki ) Ak i

(2.7)

[1]

Remark 1. In equation (2.7), λk (y) could be uniquely determined since Gn might be separated according to the coefficients of second-order terms in τn .

Finally, the following result enables us to determine all the functions αi (y) for all 1 ≤ i ≤ n − 1. Proposition 2.4. If there exists a diffeomorphism which transforms system (1.1) into form (1.2), then i for 1 ≤ i ≤ n − 2, and αn−1 = πn−1 , where αi = ππi+1  £R ¡ ¢ ¢ ¤ R¡ i i+1 ¯ n−1 dy , f or 1 ≤ i ≤ n − 2, exp − δin−1 −¶δi+1 dy − B  πi = ci exp i µ µδin−1 ¶ R δn−1 −A¯nn−1 dy ,  πn−1 = cn−1 exp 2

(2.8)

¯ k = 0 and for 1 ≤ i, k ≤ n − 1 and 1 ≤ i ≤ n − 1 with B 1 ¯ik = B

k−1 X

k Sk−i,m−k+i+1

m=k−i+1

0 πk−m . πk−m

(2.9)

Proof. Define

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Bik =

πi0 ¯ik . +B πi

(2.10)

According to equation (2.4), for 1 ≤ i, k ≤ n − 1, ¡ k ¢0 ¡ k ¢0 π0 Bi Ai π0 = − i + n−k+i . k k πi πn−k+i Ai Bi As δkk = Ank +

0 πk πk ,

hence ¡

δin−1

=

Ani

−

An1+i

−

Bin−1

¢0

Bin−1

π0 π0 1+i + i − 1+i = δii − δ1+i − πi π1+i

µ

πi0 ¯ n−1 +B i πi

¶0 µ 0 ¶ πi n−1 ¯ / + Bi . πi

which yields ·Z µ πi = ci exp

Z exp

¡

¶ ¸ ¢ 1+i ¯ n−1 dy , f or 1 ≤ i ≤ n − 2, δii − δin−1 − δ1+i dy − B i

¯ n−1 is defined in (2.9) and ci ∈ R, ci 6= 0. where B i n−1 0 π0 n−1 ¯n , where A¯n = P S n πn−m , then + A As δn−1 = 2 πn−1 n−1 n−1 1,m πn−m n−1 m=2

ÃZ Ã πn−1 = cn−1 exp

n−1 δn−1 − A¯nn−1 2

!

! dy .

Remark 2. For system (1.2), if we set αi = s(y) for 1 ≤ i ≤ n − 1, then n−1

n−1 δn−1 = Ann−1 +

0 0 X πn−i πn−1 π0 = 2 n−1 + . πn−1 πn−1 π i=2 n−i

By the definition of πi for 1 ≤ i ≤ n − 1, we have πk = sn−k for 1 ≤ k ≤ n − 1, therefore n−1

n−1 δn−1 =2

where ln =

n(n−1) 2

s0 X s0 s0 + i = ln , s s s i=2

+ 1. In such a way, we obtain the same result as the one stated in [10].

3. Main result. If there exists a diffeomorphism which transforms system (1.1) into form (1.2), then equation (2.8) of Proposition 2.4 gives all αi for 1 ≤ i ≤ n − 1. Therefore, let us consider a new family of vector fields defined as follows: 1 [e τi , f ], f or 1 ≤ i ≤ n − 1. αi

τe1 = π1 τ1 and τei+1 =

(3.1)

Set 

0

    θ(e τ1 , · · · , τen ) =    

0 .. . .. . π1

0 .. .

···

0

1

πn−1

···

··· .. .

π2 e ln,2

··· ···

··· ···

e l2,n .. . .. . e ln,n

···

     e  := Λ,   

where e lk,j = θk (e τj ) f or 2 ≤ k ≤ n and n − k + 2 ≤ j ≤ n.

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Consider the following Rn -valued form ω e −1 θ := (ω1 , ω2 , · · · , ωn )T , ω=Λ

(3.2)

where, for 1 ≤ s ≤ n, we have n X

ωs =

rs,m θm .

(3.3)

m=1

Then, the following algorithm gives all the components of ω. Algorithm 1. f or 1 ≤ j ≤ n, rn,j = · · · = rn−j+2,j = 0 and rn−j+1,j = 1. f or 2 ≤ k ≤ n − 1 and 1 ≤ j ≤ n, k P e rn−k,j = − lk,n−k+i−(j−1) rn−k+i−(j−1),j , i=2

and then, equation (3.3) becomes: ωs =

n−s+1 P m=1

rs,m θm .

Theorem 3.1. The following conditions are equivalent 1) There exists a diffeomorphism which transforms system (1.1) into a SODO normal form (1.2). 2) There exists a family of functions αi (y) for 1 ≤ i ≤ n − 1 such that the family of vector fields τei for 1 ≤ i ≤ n defined in (3.1) satisfies the following commutativity conditions [e τi , τej ] = 0, f or 1 ≤ i, j ≤ n.

(3.4)

3) There exists a family of functions αi (y) for 1 ≤ i ≤ n − 1 such that the Rn -valued form ω defined in (3.2) satisfies the following condition dω = 0.

(3.5)

Proof. Assume that there exists a diffeomorphism which transforms system (1.1) into form (1.2), then we compute αi (y) for 1 ≤ i ≤ n − 1 from equation (2.8) in Proposition 2.4. Thus, it is easy to show ∂ for 2 ≤ i ≤ n. that τ1 = π11 ∂z∂ 1 which yields that τe1 = ∂z∂ 1 and then, by construction we obtain τei = ∂z i Consequently, we have [e τi , τej ] = 0 for 1 ≤ i, j ≤ n. Reciprocally, assume that there exist αi > 0 for 1 ≤ i ≤ n − 1 such that [e τi , τej ] = 0 for 1 ≤ i, j ≤ n, then it is well-known ([8], [15]) that we can find a local diffeomorphism φ = z such that φ∗ (e τi ) = As φ∗ (e τi ) =

∂ ∂zi

∂ . ∂zi

is constant, hence ∂ ∂ φ∗ (f ) = φ∗ ([e τi , f ]) = αi φ∗ (e τi+1 ) = αi , ∂zi ∂zi+1

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∂ thus ∂z φ∗ (f ) = αi ∂z∂i+1 for 1 ≤ i ≤ n − 1. Consequently, by integration we obtain: φ∗ (f ) = A(y)z + β(y). i Moreover, as dh ◦ τei = 0 for 1 ≤ i ≤ n − 1 and dh ◦ τen = 1, we obtain h ◦ φ−1 = zn . Finally, in order to prove that in Theorem (3.1) Condition 2) is equivalent to Condition 3), it is sufficient to prove that equation (3.4) is equivalent to equation (3.5). Recall that for any two vector fields X, Y, we have

dω(X, Y ) = LX (ω(Y )) − LY (ω(X)) − ω([X, Y ]). Setting X = τei and Y = τej , we obtain dω(e τi , τej ) = Lτei ω(e τj ) − Lτej ω(e τi ) − ω([e τi , τej ]). As ω(e τj ) and ω(e τi ) are constant, then we have dω(e τi , τej ) = −ω([e τi , τej ]). Because ω is an isomorphism and (e τi )1≤i≤n is a basis of T U, then equation (3.4) is equivalent to equation (3.5). Remarks 1. i) The Rn -valued form ω can be viewed as an isomorphism T U n → U × IR n which brings each τei to the canonical vector basis ∂z∂ I . Moreover, dω = 0 means that there is a local diffeomorphism φ : U → U such that ω is the tangent map of φ. ii) The diffeomorphism φ(x) = z is determined by ω = φ∗ (x), which can be given locally as follows Z zi = φi (x) = ωi + φi (0) f or 1 ≤ i ≤ n, γ

where γ is a smooth path from 0 to x lying in a neighborhood V0 ⊆ U of 0. The following simple example is studied in order to illustrate Theorem 3.1. Example 1. Let us consider the following system  γ(y)    x˙ 1 = 1+x4 x1 x3 ,  β(y)   x˙ 2 = 1+x x1 , 4 x ˙ = µ(y)x 3 2,    x ˙ = γ(y)x  4 3,   y=x . 4 which gives  θ1 = dx4 ,    0   θ2 = γdx3 + γ x3 dx4 , ¢ ¡ 0 0 θ3 = γµdx2 + 2γ 0 γx¡3 dx3 + (γµ)¢ x2 + (γ 0 γ) x23 dx4 ,  0 β  dx + 2γ 0 µ + (γµ) γx3 dx2 θ4 = γµ 1+x   ¡ 0 4 1 ¢  0 0 + 2γ γµx2 + γ (γµ) x2 + 3γ (γ 0 γ) x23 dx3 + O[2] (x1 , x2 , x3 ) θ1 .

(3.6)

Then we have τ1 =

1+x4 ∂ γµβ ∂x1 .

Consequently we obtain

 τ2 =       τ3 =      

(γµβ)0 1 ∂ ∂ γµ ∂x2 + (1 + x4 ) γ (γµβ)2 x3 ∂x1 , ³ ´ 0 (γµβ) (γµ)0 (γµβ)0 ∂ ∂ 1 ∂ γµ (1 + x4 ) (γµβ) γx2 ∂x 2 x3 ∂x + 2 + β 2 γ ∂x3 − (γµ) (γµβ) 2 1 ³ 0 ´ ³ ´ (γµ)0 (γµβ)0 (γµ)0 (γµβ)0 γ ∂ ∂ ∂ τ4 = ∂x4 + γ + (γµ) + (γµβ) x3 ∂x3 − (γµ) + 2 (γµβ) x2 ∂x2 ³ ´ 0 ∂ 1 + (γµβ) x1 ∂x + R1,4 (z3 , z2 ) τ1 + R2,3 (z32 )τ2 . + 1+x γµβ 4 1

+ R1,3 τ1 ,

A straightforward computation gives 0

0

0

0

(γµβ) (γµβ) γ0 (γµ) (γµβ) , δ22 = −2 , δ33 = 2 + + , γµβ γµβ γ γµ γµβ · 0 0 ¸0 · 0¸ (γµβ) (γµβ) (γµβ) 3 δ1 = 4 − / , (γµβ) (γµβ) (γµβ) µ µ 0 0 0¶ 0 0 ¶0 µ 0 0¶ γ (γµ) (γµβ) (γµ) (γµβ) (γµ) (γµβ) δ23 = − 2 + +3 − + / + . γ γµ γµβ γµ γµβ γµ γµβ

inria-00179184, version 1 - 13 Oct 2007

δ11 = 2

According to equation (2.8) in Proposition 2.4, we obtain  ¢ ¢ ¤ £R ¡ R¡ π1 = c1 exp h ³exp δ11 − δ13 − δ22 dy dy =   ´ c1 γµβ, i  ¢ R R¡ 2 π10 3 3 π2 = c2 exp exp δ2 − δ2 − δ3 dy − π1 dy = c2 γµ, ³R ³ ³ ´´ ´   π10 π20 1 3  δ − − π3 = c3 exp dy = c3 γ. 3 2 π1 π2 Thus α1 =

π1 π2

=

c1 c2 β,

α2 =

π2 π3

=

τe1 = c1 (1 + x4 )

c2 c3 µ

and α3 =

π3 π4

= c3 γ, so the new vector fields are

∂ ∂ x1 ∂ ∂ ∂ , τe2 = c2 , τe3 = c3 , τe4 = + . ∂x1 ∂x2 ∂x3 ∂x4 1 + x4 ∂x1

It is clear that [e τi , τej ] = 0 for all 1 ≤ i, j ≤ 4. Therefore, according to Theorem 3.1, system (3.6) can be transformed into SODO normal form (1.2). Moreover as   0 0 0 1  0  0 γ γ 0 x3 e= 0 2 , 0 Λ 0 0  0 γµ ¢ 2γ γx3 (γµ) x2 + 2 (γ γ) x3  ¡ 0 0 0 0 x1 γµβ 2γ µ + (γµ) γx3 2γ 0 γµx2 + γ (γµ) x2 + 6γ (γ 0 γ) x23 γ (1+x µβ + R )2 4

[2]

where R = Ox4 (x1 , x2 , x3 ), a straightforward computation gives ³ ³ ´ ³ ´ e −1 θ = d x1 , d x2 , d x3 , ω=Λ c1 (1+x4 ) c2 c3

dx4

´T

.

As ω = dφ, thus the diffeomorphism which transforms system (3.6) into SODO normal form (1.2) is φ(x) = z = (

x1 x2 x3 , , , x4 )T . c1 (1 + x4 ) c2 c3

with which system (3.6) could be transformed into  z˙1 = 0,    z˙ = c1 β(y)z , 1 2 c2 c2 z ˙ = µ(y)z ,  3 2 c3   z˙4 = c3 γ(y)z3 . So far, in this paper, we have only considered systems without inputs. The next section is devoted to systems that are also driven by an input term.

4. Extension to systems with inputs. Consider a system with inputs in the following form ½ x˙ = f (x) + g(x, u), (4.1) y = h(x), where x ∈ U ⊂ IR n , f : U ⊂ IR n → IR n , g : U × IR m → IR n , h : U ⊂ IR n → IR are analytic functions and for x ∈ U, g(x, 0) = 0. For system (4.1), the SODO normal form along its output trajectory y(t) is as follows ½ z˙ = A(y)z + β(y) + η(y, u), (4.2) y = zn = Cz, £ ¤T where A(y) and β(y) are given in (1.2) and η(y, u) = η1 (y, u), η2 (y, u), · · · , ηn (y, u) . Theorem 4.1. System (4.1) can be transformed into SODO normal form (4.2) by a diffeomorphism if and only if i) one of conditions in Theorem 3.1 is fulfilled. ii) [g, τei ] = 0 for 1 ≤ i ≤ n − 1. Proof. From Theorem 3.1, we can state that there exists a diffeomorphism φ such that

inria-00179184, version 1 - 13 Oct 2007

φ∗ (f ) = A(y)z + β(y). For 1 ≤ i ≤ n − 1, because φ∗ (e τi ) =

∂ ∂zi

is constant, hence we have ∂ φ∗ (g) = φ∗ ([g, τei ]) = 0. ∂zi

Therefore φ∗ (g) = η(y, u). Thus, we obtain the form (4.2). Remark 3. If g(x, u) = g1 (x)u1 + · · · + gm (x)um , and also both conditions i) and ii) of Theorem 4.1 are fulfilled, then η(y, u) = B1 (y)u1 + · · · + Bm (y)um . Let us now study some special cases of the term η(y, u). Corollary 4.2. Assume that conditions i) and ii) of Theorem 4.1 are fulfilled, a) if [g, τen ] = 0, then η(y, u) = η(u). b) if g(x, u) = g1 (x)u1 + · · · + gm (x)um and [gk , τei ] = 0, f or 1 ≤ i ≤ n and 1 ≤ k ≤ m, then η(y, u) = B1 u1 + · · · + Bm um , where Bi are constant vector fields. Example 2. Let us consider the following system  γ(y)  x1 x2 + x˙ 1 = 1+x  3   µ(y) x˙ 2 = 1+x3 x1 ,  x˙ 3 = γ(y)x2 + u,    y=x . 3

x1 1+x3 u,

(4.3)

16

14

12

10

e1

8

6

4

2

0

−2

−4

0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 4.1. Observation error between z1 and zˆ1

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A straightforward computation gives: µ 0¶ 1 + x3 ∂ 1 ∂ (γµ) ∂ τ1 = , τ2 = + (1 + x3 ) x2 , γµ ∂x1 γ ∂x2 γµ2 ∂x1 µ ¶ µ ¶ 0 0 ∂ (µγ) γ0 ∂ 1 (γµ) ∂ τ3 = + + x2 + − x1 . ∂x3 (µγ) γ ∂x2 1 + x3 γµ ∂x1 Then, we obtain 0

δ11 = 0, δ22 = 2

0

γ0 (µγ) (µγ) γ0 + , δ12 = − −2 − γ (µγ) (µγ) γ

µ

0 ¶0

(µγ) (µγ)

µ /

0¶

(µγ) (µγ)

.

Then, according to (2.8), we have £R ¡ ¢ ¢ ¤ R¡ ( π1 = c1 exp ³ ³exp³ δ11 − ´´ δ 2 − δ 2 dy dy = c1 γµ, R 1 2 π10 1 ´2 π2 = c2 exp dy = c2 γ. 2 δ 2 − π1 ∂ which yields α1 (y) = cc12 µ (y) and α2 (y) = c2 γ (y) . Therefore, we obtain τe1 = c1 (1 + x3 ) ∂x , τe2 = 1 x1 ∂ ∂ ∂ c2 ∂x2 and τe3 = ∂x3 + 1+x3 ∂x1 . x1 ∂ ∂ + ∂x = τe3 then [g, τe1 ] = [g, τe2 ] = 0 and system (4.3) is transformed into As g = 1+x 3 ∂x1 3

 z˙1 = 0,    z˙ = c1 µ (y) z , 2 1 c2 z ˙ = c γ (y) z  2 2 + u,   3 y = z3 .

(4.4)

by the following diffeomorphism φ(x) = z = (

x1 x2 , , x3 )T . c1 (1 + x3 ) c2

Following the proposed high gain observer in the form (1.3), the corresponding observer for the system (4.4) can be designed as follows:  . ρ3  z3 − z3 ) ,  zˆ. 1 = − γµ (ˆ 2 c1 zˆ2 = c2 µ (y) zˆ1 − 3 ργ (ˆ z3 − z3 ) ,  .  zˆ3 = c2 γ (y) zˆ2 − 3ρ (ˆ z3 − z3 ) + u,

5

4

3

e2

2

1

0

−1

−2

0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 4.2. Observation error between z2 and zˆ2 0.5

0

e3

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−0.5

−1

−1.5

−2

0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 4.3. Observation error between z3 and zˆ3

where ρ is the tunable gain. For a more specific but simple simulation, choose c1 = c2 = 1, u (t) = 1, µ (y) = 1 + y 2 , and γ (y) = 2 + cos(y). Its simulation results are presented in Fig. 4.1, Fig. 4.2 and Fig. 4.3 which respectively present the convergence of system’s states and their estimations. In addition, in order to solve the left invertibility problem, the Observability Matching Condition (OMC) for system (4.1) with m = 1 is as follows ½ Lg Li−1 f h = 0, ∀x ∈ U, 1 ≤ i ≤ n − 1, Lg Ln−1 h 6= 0. f Corollary 4.3. Assume conditions i) and ii) of Theorem 4.1 are fulfilled and the OMC is verified then £ ¤T η(y, u) = η1 (y, u), 0, · · · , 0 .

Remark 4. The OMC for system (4.1) with m = 1 is equivalent to g ∈ span{e τ1 }. We give another example in order to highlight Corollary 4.3. Example 3. Consider the following system  x˙ 1 = u,    x˙ = µ(y)x + µ(y)x2 + x2 u, 2

1

x2 x˙ 3 = γ(y) 1+x ,   1  y = x3 .

1

1+x1

(4.5)

1 ∂ 1 x2 ∂ A straightforward computation gives τ1 = γµ ∂x1 + γµ 1+x1 ∂x2 . From equation (2.8)in Proposition 2.4, x2 ∂ ∂ we can determine α1 (y) = cc21 µ (y) and α2 (y) = c2 γ (y). Thus, we have τe1 = c1 ∂x + c1 1+x , τe2 = 1 1 ∂x2 ∂ ∂ c2 (1 + x1 ) ∂x2 and τe3 = ∂x3 . As g ∈ span{e τ1 }, then the OMC condition is fulfilled, therefore system (4.5) could be transformed by the following diffeomorphism

µ φ(x) = z =

x1 x2 , , x3 c1 c2 (1 + x1 )

¶T .

into

inria-00179184, version 1 - 13 Oct 2007

 z˙1 = cu1 ,    z˙2 = cc12 µ (y) z1 ,  z˙ = c2 γ (y) z2 ,   3 y = z3 . 5. Conclusion. In this paper, we have put forward the geometrical conditions which allow us to determine whether a nonlinear system can be transformed locally into the SODO normal form by means of a diffeomorphism and of an output injection. In our main result we state two equivalent ways to check these conditions. In the first one, we have used Lie brackets commutativity and the second one was based on the one-forms. Moreover, an extension of our results is stated for a class of nonlinear systems with inputs. Acknowledgments. The authors are deeply grateful to the anonymous reviewers for valuable comments and helpful suggestions that enable us to improve the presentation of this paper. We are also deeply grateful to Professor Vincent Maki for many corrections. REFERENCES [1] L. Boutat-Baddas, D. Boutat, J-P. Barbot and R. Tauleigne, Quadratic Observability normal form, in Proc. of IEEE CDC (2001). [2] K. Busawon, M. Farza and H. Hammouri, A simple observer for a class of nonlinear systems, Appl. Math. Lett, Vol. 11, No. 3, pp. 27-31, (1998). [3] S. Chabraoui, D. Boutat, L.Boutat-Baddas and J.P. Barbot, Observability quadratic characteristic numbers, in Proc. of IEEE CDC (2001). [4] J. P. Gauthier and I. Kupka, Deterministic observation theory and applications, Cambridge University Press, 2001. [5] M. Guay, Observer linearization by output diffeomorphism and output-dependent time-scale transformations, NOLCOS’01 Saint Petersburg, Russia ,(2001) pp. 1443-1446. [6] H. Hammouri and J.P. Gauthier , Bilinearization up to the output injection, System and Control Letters, 11 , (1988) pp. 139–149. [7] H. Hammouri and J. Morales, Observer synthesis for state-affine systems, in Proc. of IEEE CDC (1990). [8] A. Isidori, Nonlinear control systems, 2nd edtion, Berlin: Springer-Verlag,(1989). [9] D. Luenberger, An introduction to observers, in IEEE Transactions on Automatic Control 16 (6), (1971) pp. 596-602. [10] W. Respondek, A. Pogromsky and H. Nijmeijer, Time scaling for observer design with linearization error dynamics, in IEEE Transactions on Automatic Control 3, (1989) pp. 199-216. [11] N. Kazantzis and C. Kravaris, Nonlinear observer design using Lyapunov’s auxiliary theorem, in Systems & Control Letters, Vo 34, (1998) pp. 241-247 1998. [12] A. Krener and A. Isidori, Linearization by output injection and nonlinear observer, in Systems & Control Letters, Vo 3, (1983) pp. 47-52. [13] A. Krener and W. Respondek, Nonlinear observer with linearizable error dynamics in SIAM J. Control and Optimization, Vol 30, No 6, (1985) pp. 197-216. [14] A. Krener and M. Q. Xiao, Nonlinear observer design in the Siegel domain through coordinate changes in Proc of the 5th IFAC Symposium, NOLCOS01, Saint-Petersburg, Russia, (2001) pp. 557-562. [15] H. Nijmeijer and A.J. Van Der Schaft, Nonlinear Dynamical Control Systems Springer (1996). [16] X.H. Xia and W.B. Gao, Nonlinear observer design by observer error linearization in SIAM J. Control and Optimization, Vol 27,(1989) pp. 199–216. [17] G. Zheng, D. Boutat and J.P. Barbot, Output Dependent Observability Linear Normal Form, in Proc. of IEEE CDC (2005).