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Some Comments on Observability Normal Form and Step-By-Step Sliding Mode Observer

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Some Comments on Observability Normal Form and Step-By-Step Sliding Mode Observer SOME COMMENTS ON OBSERVABILITY NORMAL FORM AND STEP-BY-STEP SLIDING MODE OBSERVER G. ZHENG1, L. BOUTAT-BADDAS2, J.P. BARBOT1 AND D. BOUTAT3 1 EQUIPE COMMANDE DES SYSTEMES` (ECS), ENSEA, 6 AV. DU PONCEAU, 95014 CERGY CEDEX. fZHENG,[email protected] 2CRAN-CNRS, UHP NANCYI, IUT DE LONGWY, 186, RUE DE LORRAINE, 54400 COSNES-ET-ROMAIN. [email protected] 3 LVR, ENSI-BOURGES, UNIVERSITE´ D’ORLEANS,´ 10, BD. LAHITOLLE, 18020 BOURGES CEDEX. [email protected] Abstract. Classical methods to design an observer for nonlinear system are almost based on the system either linearly observable or detectable. In this paper, an example is provided to point out, for the linearly unobservable and undetectable system, with the help of observability normal form and sliding mode observer, it is also possible to design an observer. In addition, other applications of observability normal form, especially on how to improve the robustness of ciphering in a secure data transmission, are highlighted. Key Words: Observability Normal Form, Sliding Mode Observer, Observability Bifurcation 1. Introduction. For most systems (mechanic, electric, economic and so on), not all state variables can be measured directly, so the problem of estimating the unmeasurable state variables is very important in control theory, and also in other domains. For linear systems, it has been extensively studied. But for nonlinear sys- tems, the theory of observers is not nearly as neither complete nor successful as it is for linear systems. In addition, in many cases and also in the synchronization problem of chaotic system, the linear approximation of a nonlinear system sometimes is nei- ther observable nor detectable on a submanifold. This problem is studied for example in cryptography in continuous-time by L. Boutat-Baddas [1] and in discrete-time by I. Belmouhoub [4]. In the thesis of L. Boutat-Baddas [3] she proposed to use the form of Poincar´e[5] to study the observability bifurcation, and this approach is a dual approach introduced by W. Kang and A. Krener for the study of the control- lability bifurcation. In order to highlight the efficiency of the proposed method, we will use the observability normal form to deal with a simple example which is not linearly detectable. Then we try to design an observer using the intrinsic properties of variable structure of the step by step sliding mode observers to take account of the passage through the unobservable manifold. It should be noticed that this type of observers has more advantages for guaranteeing the convergence of the observation error dynamic in finite time if the system is observable and verified certain conditions [12]. This paper is organized as follows: In section 2 observability normal form is recalled. Section 3 illustrates how to design an observer for a simple example with singularity. The final section discusses other applications of observability normal form, including solving the left invertibility problem, increasing the robustness of ciphering in a secure data transmission. 2. Recall of Observability Normal Form. In this section, for simplification we just recall quadratic observability normal form for a nonlinear system with one real linear unobservable mode (see [3] for general observability normal form and see [6] for 1 exhaustive demonstrations). Consider a SISO (single-input single-output) system: (2.1) »˙ = f(») + g(»)u y = C» = h(») where the vector fields f; g : U ½ Rn ! Rn are assumed to be real analytic, such that @f f(0) = 0; so the pair (»e = 0; u = 0) is an equilibrium point. Supposed A = @t (0), B = g(0), and thus system (2.1) can be expanded into the following form around this equilibrium point via Taylor expansion at order 2: z˙ = Az + Bu + f [2] (z) + g[1](z)u + O[3](z; u)(2.2) y = Cz [2] [2] [2] [2] T [1] [1] [1] [1] T where f (z) = [f1 (z) f2 (z) ::: fn (z)] and g (z) = [g1 (z) g2 (z) ::: gn (z)] . [2] [1] Here, fi (z) and gi (z) for all i 2 [1; n], are homogeneous polynomials of degree 2 and 1 in z respectively. We suppose that the pair (A; C) is the rank of n ¡ 1, so system (2.1) has one unobservable real mode, then this system can be transformed to the following form: : ˜[2] [1] [3] z˜ = Aobsz˜ + Bobsu + f (z) +g ˜ (z)u + O (z; u) n¡1 P [2] [1] [3] (2.3) z˙n = ®zn + ®izi + bnu + fn (z) + gn (z)u + O (z; u) i=1 y = Cobsz˜ 0 1 a1 1 0 ¢ ¢ ¢ 0 B . C B . C B a2 0 1 ¢ ¢ ¢ . C T T T B . C wherez ˜ = [z1; :::; zn¡1] , z = [˜z ; zn] , Aobs = B . .. .. C, B . 0 C @ A an¡2 0 ¢ ¢ ¢ 0 1 an¡1 0 ¢ ¢ ¢ ¢ ¢ ¢ 0 Bobs = [b1; :::; bn¡1], Cobs = [1 0 ¢ ¢ ¢ 0]. The terms Aobs;Bobs; ® and the ®i constitute the residue of order one which represents the normal form of the linear approximation. Definition 2.1. System (2.2) is said to be quadratically equivalent modulo an output injection(QEMOI) to the system x˙ = Ax + Bu + f¯[2] (x) +g ¯[1](x)u + ¯[2](y) + γ[1](y)u + O[3](x; u) (2.4) y = Cx if there exists an output injection ¯[2](y) + γ[1](y)u and a diffeomorphism of the form x = z ¡ Á[2](z), which transform f [2] (z) + g[1](z)u into f¯[2] (x) +g ¯[1](x)u + [2] [1] [2] [2] [2] T [2] [2] [2] T ¯ (y) + γ (y)u, where Á (z) = [Á1 (z); :::; Án (z)] , ¯ (y) = [¯1 (y); :::; ¯n (y)] [1] [1] [1] T and γ (y) = [γ1 (y); :::; γn (y)] represent respectively homogeneous polynomials of degree 2 in z, of degree 2 in y and of degree 1 in u. We can recall the proposition below: Proposition 2.2. [3] System (2.2) is QEMOI to system (2.4), if and only if the following two homological equations are satisfied: [2] @Á[2] ¯[2] [2] [2] i) AÁ (z) ¡ @z Az = f (z) ¡ f (z) + ¯ (z1) @Á[2] [1] [1] [1] ii) ¡ @z B =g ¯ (z) ¡ g (z) + γ (z1) 2 [2] [2] [2] [2] @Á @Á1 (z) @Án T @Ái (z) [2] where @z Az := [ @z Az; :::; @z Az] and @z is the Jacobian matrix of Ái (z) for all i 2 [1; n]. For the proof see [3]. Just like the linear normal form, we choose the output always equal to x1, because the output should rest unchanged, which means the diffeomorphism (x = z ¡ Á[2](z)) [2] should verify Á1 (z) = 0. Thus, the quadratic observability normal form for nonlinear systems with one dimensional linear unobservable mode is as follows: Theorem 2.3. [3] There is a quadratic diffeomorphism and an output injection which transform system (2.3) into the following normal form: Pn x˙ 1 = a1x1 + x2 + b1u + k1ixiu + E1 i=2 . Pn x˙ n¡2 = an¡2x1 + xn¡1 + bn¡2u + k(n¡2)ixiu + En¡2 i=2 Pn Pn x˙ n¡1 = an¡1x1 + bn¡1u + hij xixj + h1nx1xn + k(n¡1)ixiu + En¡1 j¸i=2 i=2 nP¡1 nP¡1 [2] [2] [2] @Án x˙ n = ®nxn + ®ixi + bnu + ®nÁn (x) + ®iÁi (x) ¡ @x˜ Aobsx˜ i=1 i=1 n [2] P +fn (x) + knixiu + En i=2 [2] [1] with Ei = ¯i (y) + γi (y) for all i 2 [1; n]. From the normal form of system½ (2.3) we can deduce that the observability bifur-¾ nP¡1 cation manifold of state xn is Sn = x 2 U j hi;nxi + 2hn;nxn + k(n¡1);nu = 0 , i=1 in which the terms hij xixj (for all n ¸ i ¸ j ¸ 2) and kjixiu (for all n ¸ j ¸ 1 and n ¸ i ¸ 2) are called the resonant terms. And on Sn, there is not any linear or quadratic relation between the derivative of the output and the last component of the state xn. So, firstly if for some index i 2 [1; n], hi;nxi 6= 0, then we can quadratically recover, at least locally, xn. Secondly, if we have some ki;n 6= 0; with a well chosen input u; it is also possible to preserve the observability. Thirdly, if the system is on the observability bifurcation manifold, we can use ®n to analyze the xn’s detectability property: if ®n > 0, xn is undetectable, else if ®n < 0, xn is detectable, else if ®n = 0, we can use the center manifold theory to analyze its property: 3. Observer Design. Each normal form characterizes one and only one equiva- lent class, thus the structural properties of the normal form must be also the same as those of each system in the corresponding equivalence class. In this section in order to highlight the efficiency of the proposed method, we will give a system (3.1) with unobservable dimension 2 which is more difficult than the one considered in Section 2, and simultaneously consider to construct an observer for this system. x˙ 1 = x2 x˙ 2 = ¡x1 + x2x3 (3.1) ¡x4 x˙ 3 = 3 x3 x˙ 4 = 3 with the output y = x1. Obviously, the system (3.1) is linearly unobservable on the directions x3 and x4 and has an observability bifurcation when x2 = 0, so the method 3 based on global or at least local observable system cannot be applied to design an observer. Furthermore, this system has four eigenvalues f§i; §i=3g, which mean this system is not linearly detectable, so Kazanzis and Kravaris’s method ([8],[9],[10]), in which the observable is not a necessary condition, but the detectable is, also doesn’t work.
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