LYAPUNOV-SCHMIDT METHOD DEDICATED TO THE OBSERVER ANALYSIS AND DESIGN

An analysis of observation error dynamics and an observer design are presented using the so-called Lyapunov-Schmidt method, firstly in order to analyze the solution’s bifurcation of the observation error dynamics, and secondly in order to propose a robust observer with respect to such solution’s bifurcation. Two numerical examples based on the Galerkin method show the pertinence of the proposed approach.


Introduction
In this paper, the (so-called) Lyapunov-Schmidt method is used in order to analyze the solution's bifurcation of the observation error dynamics [3].This method relies on Fredholm operators of finite index defined in Banach spaces which are immerged in appropriate Hilbert spaces in order to use two projections allowing to transform an infinite dimensional problem into a finite one.Following this, a bifurcation equation is established, solving this equation gives us the number of solutions of the problem (i.e., the number of solutions of the observation error dynamics).Obviously, a solution bifurcation is less known than the stability [1,2,5,12], controllability [8,6,7] and observability [4] bifurcations.Nevertheless, it is of prior importance to know if the real solution is or is not in the neighborhood of the solution obtained with a linear approximation.Roughly speaking, we call solution bifurcation the nonexistence or the existence of two solutions in the vicinity of the linear one.This phenomena may appear as a purely mathematical interest, but obviously, it is also often an important problem in control systems theory.For example, it is usual to design an observer for nonlinear system on the basis of a linear approximation.It is then expected that the error dynamics have a solution closed to the linear one.This is not obvious, especially when considering the fact that the initial value of the nonlinear system is totally or partially unknown.In this paper, an analysis of the observer design is proposed on the basis of the Lyapunov-Schmidt approach.
2 Lyapunov-Schmidt method and observer analysis Moreover, an academic example of observability analysis in the case of unidimensional nonlinear systems is given.After that, we highlight the technological interest of the proposed approach by considering an analysis and an observer design for a two dimensional Lagrangian system.
The paper is organized as follows: in Section 2, a solution's bifurcation is studied using Lyapunov-Schmidt method.An observer analysis and the related design are introduced in Section 3. In Section 4, two numerical examples using the Galerkin method [11] are developed in order to highlight the efficiency of the proposed analysis and the interest of the approach.

Problem analysis using Lyapunov-Schmidt method
Lyapounov-Schmidt method [10] essentially allows to carry out a local study of nonlinear system's solution in the neighborhood of an equilibrium point near the linear one.It is important to note that a function solving the problem and depending on perturbation parameters is determined on an open set of initial conditions.

Problem statement.
Let us consider the following system: ė = A(t)e + γ(e,χ,t), ( where t ∈ J = [0,T], e : J → M ⊂ R n is the state and M is a nonempty open set of R n .χ : J → Ω ⊂ R p is an external signal and Ω is a nonempty open set of R p .A : J −→ R n×n is a continuous application.γ : M × Ω × J → R n is an application regrouping all the nonlinearities and satisfying the following assumptions.
Assumption 2.1.γ is C 2 with respect to e.
T 0 γ(0,0,t)dt = T 0 (∂γ/∂e)(0,0,t)dt = 0. e(0) is assumed to be free (i.e., not fixed) in the neighborhood of e = 0. Hereafter our purpose is to analyze the number of solutions for the system (2.1), in the neighborhood of a nontrivial solution e L of the related linear system defined as ė = A(t)e. (2.2)

Abstract problem formulation.
Let C 0 n := C 0 (J,R n ) be the set of continuous functions defined on the time interval J and having values in R n .Moreover, C 0 n is provided with the uniform convergence norm given by where • 0 is a norm defined on R n and so Similarly, let C 1 n = C 1 (J,R n ) be the set of C 1 functions defined on the time interval J and having values in R n , the norm for C 1 n is defined as follows: and thus (C 1 n , • ) is also a Banach space [2].X ⊂ C 0 (J,Ω) denotes the set of continuous external signal χ defined on the time interval J and having values in Ω.
In our application to observer analysis, χ is the output signal.(χ may be a known or unknown input perturbations, this in function of the considered problem.) In order to compare the linear solution of (2.2) with the nonlinear one of (2.1), it is natural to recall the following results.
Proposition 2.3 [9].A general solution of the linear equation (2.2) is given by where φ(t) is the fundamental matrix associated to the linear system (2.2).
Proposition 2.4 [9].A function e ∈ C 1 n is a solution of the nonlinear problem (2.1) if and only if it is a solution of the following equation: (2.6)

Proof.
The function e L given by (2.5) being a solution of the linear problem (2.2), thus any solution of the nonlinear problem (2.1) may be written in the form of (2.6) [9].
Remark 2.5.The initial condition of the nonlinear system (2.1) (or (2.6)) noted by e(0) is the same as the one of the linear system (2.2) (or (2.5)) noted by e L (0), nevertheless, in order to avoid any confusion both notations are kept.The linear operator £ is now defined as follows: and NG denotes the Nemystskii nonlinear operator [10] related to γ: Using these notations, we obtain the following proposition.
Proposition 2.6.Equation (2.1) is equivalent (equivalent means that each solution of (2.9) is also solution of (2.1) and reciprocally) to The proof is directly deduced from (2.7) and (2.8).

Bifurcation Analysis.
When e(0) = 0, the linear system (2.2) admits the null function as a trivial solution and consequently the Cauchy problem corresponding to the nonlinear system (2.1) (or equivalently (2.9)) has one and only one solution in the neighborhood of e = 0.It is well known that for fixed e 0 = 0, we obtain one and only one solution of (2.9) by elementary translation in the vicinity of 0. Nevertheless, if e(0) is free, the existence and uniqueness of the solution are not guaranteed: this case is the most interesting one, because it is due to the fact that the Kernel of the operator £ is of dimension n and some solution's bifurcations may appear.
In order to apply the Lyapounov-Schmidt method, we must verify that £ has relevant features.
Proposition 2.7.(i) £ is a linear continuous and bounded operator.
Proof.(i) Results from the definition of £.
(ii) Ker £ is spanned by ϕ i (t), i = 1,...,n, thus dimKer £ = n and from the Fredholm alternative [10], problem £e = NG(e,χ) admits a solution if and only if where Now, for the sake of notation's compactness, ℵ refers to NG(e,χ) and the projection P 1 is defined as follows: Therefore, for all ℵ ∈ C 0 n : Remark 2.8.The restriction of £ denoted by £ : H 1 → Im £ is invertible and the following operator: n , where Id designs the identity operator.
Consequently and by Fredholm alternative, it comes to the following lemma.
6 Lyapunov-Schmidt method and observer analysis It is now possible to deduce the following lemma.
Lemma 2.11.The function e is a solution of (2.17) if and only if e = n i=1 α i ϕ i + v, where ) Proof.Lemma 2.11 results directly from the application of the projections P 0 , P 1 , K, and the Fredholm alternative to (2.17).

Auxiliary equation analysis.
The correction function v modifies the linear solution in order to take into account the nonlinear behavior of the system.This function can be rewritten in the neighborhood of e = 0 as In order to analyze the existence of solutions of (2.18a), let us introduce a new operator: This operator is defined by replacing (2.19) in (2.18a).The analysis of the auxiliary equation is thus reduced to the analysis of the equation G(w,α,χ,e(0)) = 0.
From Assumptions 2.1 and 2.2, it can be deduced that G satisfies the following statements: (i) G(0,0,0,0 , where D w is the Frechet partial derivative of G relatively to w.Now, using the implicit function theorem, the following proposition is obtained.Proposition 2.13.There exist a neighborhood 0) , the auxiliary equation ( 2.18a) admits one D. Benmerzouk and J. P. Barbot 7 and only one solution w * (α,χ,e(0)) ∈ V w such that w * (0,0,0) =0 if and only if (α,χ,e(0)) is a solution of bifurcation equation (2.18b).Remark 2.14.As w * is uniquely determined in V w , the number of solutions of (2.17) is exactly determined by the number of α = (α 1 ,...,α n ) solutions of the bifurcation equation (2.18b).This is due to the fact that each solution of (2.17) can be written as In fact, this formulation of our problem transforms the resolution of the infinite dimensional problem (2.17) into the resolution of two equations: the first one is the auxiliary equation and is in infinite dimension but has one and only one solution w * and the second one is resolved in R n and thus is in finite dimension (n).

Bifurcation equation analysis.
As it was proved in the previous subsection that (2.18a) admits one and only one solution w * , the resolution of the system (2.18) is reduced to the bifurcation equation analysis (2.18b), where w * is substituted to w.Thus system (2.18) admits at least one solution if and only if α satisfies the bifurcation (2.18b) which may be rewritten for i = 1,...,n as In the analysis of this system of the n previous equations, at least two cases occur.Here, only two cases corresponding to Assumptions 2.1 and 2.2 are considered.
Therefore, the problem (2.17) admits one and only one solution near e L which is denoted e * and given by Consequently, there is no bifurcation.
Case 2. the condition of the first case is not satisfied, that is So, the resolution of bifurcation equations in R n is based on numerical methods (see [11], e.g.).
In order to detail all the technical points of the proposed approach, some results concerning the case n = 1 are presented hereafter.

Bifurcation analysis, case n
where Let us recall that the purpose is to analyze the number of solutions of (2.27) in the neighborhood of a nontrivial solution e L of the related linear system: (2.29)

Abstract problem formulation.
For the sake of notation simplicity, C 0 and C 1 , respectively, stand for C 0 0 (J,R) and C 1 0 (J,R).X ⊂ C 0 0 (J,Ω) is the set of continuous external signal χ.Some results are first recalled.Proposition 2.16 [9].A general solution of linear (2.29)The linear operator L is defined as and the Nemystskii nonlinear operator [10] N is given by  Let us denote (similarly to the case n > 1) N(e,χ) = ℵ and let us define the projection P 1 as (2.39) Therefore, for all ℵ ∈ C 0 : Remark 2.20.The restriction of L denoted by L : Consequently and using the Fredholm alternative, the following lemma is obtained.
Lemma 2.21.The function e = KP 1 ℵ is a solution of the following system: and each solution of the system (2.34) is given by e = αe L + kP 1 ℵ, where α ∈ R and kP 1 ℵ ∈ (Id −P 0 )C 1 .
Using this lemma, it comes to the following one.
Lemma 2.22.Equation (2.34) is equivalent (equivalent in the meaning that each solution of (2.34) is a solution of (2.41) and reciprocally) to (2.41) Applying the projections P 0 , P 1 , k, and the Fredholm alternative to (2.41), we obtain the following lemma.

Lemma 2.23. The function e is a solution of (2.41) if and only if e
(2.42b)

Auxiliary equation analysis.
Recalling that the correction function v modifies the linear solution so as to take into account the nonlinear behavior of the system, it can be rewritten in the neighborhood of e = 0 as (2.43) D. Benmerzouk and J. P. Barbot 11 In order to analyze the existence of solutions of (2.42a), a new operator (noted also by G as for the case n > 1) is introduced: Analyzing the auxiliary equation is equivalent to analyzing the equation G(w,α,χ,e(0)) = 0. From Assumptions 2.1 and 2.2, it can be deduced that G satisfies the following statements: (i) G(0,0,0,0 Now, using the implicit function theorem, it comes to the following proposition.
Remark 2.25.As w * is uniquely determined in V w , the number of solutions of (2.41) is exactly determined by the number of α solutions of the bifurcation equation (2.42b).The resolution of bifurcation equation will be done in R.

Bifurcation equation analysis.
It was highlighted in the previous subsection that the system (2.42) admits a solution if and only if α satisfies the bifurcation equation and in the scalar case, (2.42b) can be rewritten as follows: For the sake of simplicity, I stands for I 1 .
It can be noticed that this method using the study of some functions signs is specific to the case n = 1, this justifies the addition of Assumption 2.15.
In order to synthesize all the previous results, the following that we call singular operator is introduced: ∂ 2 I ∂α 2 0,χ,e(0) . (2.52) The bifurcation set Bif := {(χ, e(0)) ∈ v χ × v e(0) : S(χ,e(0)) = 0} characterizes the transient manifold such that the solution number changes with respect to χ and e(0).And naturally, we set : S χ,e(0) < 0 ,  This theorem has many direct and important applications in systems theory, for example, the robustness and the validity analysis of linear control by gain schedule applied to nonlinear systems.A natural dual case of the previous one is the case of linear observer applied to nonlinear systems.Hereafter, a very simple academic application is first given.In this example, all the computational aspects of this analysis will be detailed.Moreover, a nonlinear corrective term is proposed in order to overcome the bifurcation submanifold.Finally, the study of reduced observer design for practical and well-known Lagrangian system ends the observer analysis section in order to highlight the practical engineering aspect of our analysis.

Problem statement.
Considering the following nonlinear system: with χ ∈ R is the state, y ∈ R is the output, and ā and c are some constants different from 0 and functions Γ [2] (•) and h [2] (•) are at least quadratic in χ.
14 Lyapunov-Schmidt method and observer analysis The linear approximation of (3.1) is χ = āχ, For system (3.2),there exist many observer schemes.Hereafter, the classical Luemberger observer is considered: It is well known that the observation error e = χ − χ with respect to the linear approximation is given by ė = ( ā − kc)e := ae. (3.4) Consequently, the observer gain k ∈ R is computed such that e is exponentially stable.Unfortunately, the real system is generally not the linear one (3.2) but the nonlinear one (3.1).Consequently, the real observation error dynamic is From Theorem 2.26, the question of the relevance to design a linear observer based on the linear approximation (3.2) for the nonlinear system (3.1) is of prior importance.In practice, the gain k is increased in order to crush the nonlinearity's effect.In fact, this strategy has two negative effects: first of all, increasing k is equivalent to decrease the filtering effects of the observer and secondly this does not guarantee to avoid the solution's bifurcation.In order to avoid this problem, the following observer scheme is proposed: where d ∈ R is an acting parameter and θ [2] is a quadratic correcting term (only depending on known variables).Thus, the following new observation error dynamics is obtained: where γ(e,χ,d,t is the nonlinear term of (3.7) (θ [2] will be designed with respect to Theorem 2.26).This design D. Benmerzouk and J. P. Barbot 15 is realized in order to overcome the bifurcation and to locally obtain a unique solution of (3.7) close to e L for any e(0).In order to take into account the correction term d, the related nonlinear operator N is modified as where N(e,χ,d)(t) = γ(e,χ,d,t).Consequently, the following corollary holds.
Corollary 3.2.Equation (3.7) is equivalent (equivalent in the meaning that each solution of (3.9) is solution of problem (3.7) and reciprocally) to Let us note that the projections P 0 , P 1 , and k are defined as in Section 2 but relatively to the operator N. Lemma 2.22 thus becomes the following corollary.
Corollary 3.3.Equation (3.9) is equivalent (equivalent in the meaning that each solution of (3.10) is a solution of (3.9) and reciprocally) to (3.10) Similarly, Lemma 2.23 becomes the following corollary.
Corollary 3.4.The function e is a solution of (3.10) if and only if e = αe L + v, where Naturally, from the same analysis as in Section 2 (reported in Appendix A), a new singularity operator is defined over vχ × ve(0 The bifurcation set is modified as Bif := {(χ, e(0),d) ∈ v3 × v4 × v5 : s(χ,e(0),d) = 0} and W − and W + are defined as This allows us to propose the following theorem.
Theorem 3.5.Under Assumptions 2.1, 2.2, 2.15 relatively to γ, there exist a neighborhood In order to reach the bifurcation set Bif when the dynamics are originally in W + for example, an action on d is used to obtain one and only one solution of the observation error (3.7) near the linear one.
As the function s(χ,e(0),•) is continuous on J, the following assumption is introduced.

Bifurcation computation.
It can be noticed that γ satisfies Assumptions 2.1, 2.2, and 2.15.Thus, from Theorem 2.26, a solution's bifurcation may appear.The aim is to D. Benmerzouk and J. P. Barbot 17 determine, if there are zero, one, or two solutions closed to e L , the solution of the linear corresponding to the system ė = −0.3e.The Galerkin method [11] is an efficient tool for computing solutions when the implicit function theorem is claimed for solving formal equations.Let us introduce the following notations: (i) e h , the approximation of the state e given by e hL (t) = (1 − 0.3t)e(0), (ii) χ h , the approximation of the entry χ given by χ h (t) = 0.2 − 3t, (iii) v h , the approximation of v given by v h (0) = (1 − α)e(0).Consequently, after a simple computation, the approximation of w, w h at order one is given by w h (t) = −3αat, where a is a real parameter.
In the first step, the coefficient a which characterizes w * is computed.The computation is based on the resolution of the approximation at order one of the approximated auxiliary equation given by G a w h ,α,χ h ,e(0) := G w h ,α,χ h ,e(0) = 0.      where θ is the angular position, ω is the angular velocity, Γ is the nonnull input computed torque, J is the inertia, F and k are disturbance parameters, and y is the output of the system (4.16).We can rewrite (4.16) as where Ω = Γ/J, = F/J, and K = k/J.So, denoting x 1 = θ and x 2 = ω, the system (4.17) can be rewritten as follows: Now, considering that the linear estimation x 2 of x 2 is given by the following reduced observer, it comes that So, the dynamics of the observation error e = x 2 − x 2 satisfy the following equation:

Bifurcation correction.
In order to eliminate the no solution's case, a new observer is proposed: where d is a real parameter.
The new observer error is

Conclusion
In this paper, the analysis of the solution's bifurcation has been dealt with special attention pointed to the observation error dynamics.Contrarily to the classical ones (stability, controllability, and observability bifurcations), this type of bifurcation is not well known.Nevertheless, considering a robustness problem, solution's bifurcation is of practical importance.This fact has been highlighted in the case of linear observer design for nonlinear Lagrangian systems.This type of bifurcation is also of the great interest in the study of population dynamics, specially when a parameter changes and causes an explosion of demography due to a chain reaction.