High gain observer for a class of implicit systems

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1 High gain observer for a class of implicit systems Hassan HAMMOURI Laboratoire d Automatique et de Génie des Procédés, UCB-Lyon 1, 43 bd du 11 Novembre 1918, Villeurbanne, France hammouri@lagepuniv-lyon1fr Nicolas MARCHAND Laboratoire des Signaux et Systèmes Supelec, Plateau de Moulon, 3 rue Joliot-Curie, 9119 Gif sur Yvette, France NicolasMarchand@lsssupelecfr Abstract Under some observability assumptions uniform observability, a high gain observer for a class of implicit dynamical systems is given in this paper Numerically, the computation of trajectories of such implicit systems usually necessitates the use of an optimization algorithm together with an ODE numerical method This complicates the synthesis of an observer The observer design proposed here leads to a classical dynamical system defined on some R N, with N n, n being the dimension of the state space of the implicit system Keywords : Observer, Nonlinear Systems, Implicit Systems 1 Problem statement In this paper, the class of implicit systems of the following form will be considered: ẋ = f x, ρ + p u if i x, ρ ϕx, ρ = y = hx, ρ 1 where x, ρ R n R d, the f i s and ϕ are assumed to be sufficiently smooth and: is full rank x, ρ M 2 x,ρ where M is the set of zeros of ϕ: M := { x, ρ R n R d, st ϕx, ρ = } 3 Clearly, from condition 2, M becomes a smooth manifold The computation of trajectories of such systems usually needs the use of optimization techniques either to explicit ϕ, or to make sure that ϕx, ρ effectively vanishes This can for instance take the form of an ODE routine to integrate the first part of the system together with a optimization routine to solve the second part of the problem These systems are often encountered in various fields such as the control of nonholonomic mechanical systems or of chemical reactors Assumption 2 is known to be a characterization of the implicity called Hessenberg index one [1] see also the implicit function theorem Now, set z := x ρ, system 1 becomes equivalent to: ż = z + u i F i z y = hz z M 4 with, for i p: f i x, ρ F i z := 1 5 f i x, ρ x,ρ x,ρ System 4 is called uniformly observable observable independently on the input if for any input u defined on any interval [, T ] and for every initial states x x, there exists a time t [, T ], such that hxt; x, u hxt; x, u where xt; x, u and xt; x, u are respectively the trajectories of system 4 with input u and initial conditions x and x For such systems, it has been shown [2, 3, 4] that the map Φ : z hz, L F hz,, L n 1 hz is a local diffeomorphism almost everywhere which transforms system 4 in the following canonical form: with: ζ = Aζ + f ζ + y = Cζ u i fi ζ 1 A = 1 6

2 C = 1 f =,,, f n T f ij ζ = f ij ζ 1,, ζ j It is proved in [3], that the following system: ˆζ = Aˆζ + f ˆζ+ u i fi ˆζ S 1 θ CT C ˆζ y is an exponential observer for system 6 as soon as the fields F i are global lipschitz with a lipschitz constant depending only upon the upper bound u From a purely mathematical point of view, the observer for system 4 can be obtained as follows: let T φ be the tangent map from the tangent space T M into T R n = R n R n, the observer for 4 takes the form: n ẑ =ẑ+ u if iẑ Tẑφ 1 S 1 θ C T hẑ y 7 ẑ M In practice, this observer may work only if: i ẑ M that is ϕˆx, ρ = ii there is no parameter uncertainty iii the measurements are not noisy The aim here consists in robustify the above observer so that the initialisation of ẑ can be taken in some tubular neighbourhood of the manifold M This paper is organized as follows In section 2, some assumptions and preliminary results are given required by the observer s construction Section 3 is dedicated to the main result of this paper 2 Assumptions and preliminary results Some notations and preliminary results used in the following to state the main result are given here Assumptions 1 One shall assume that the following assumptions hold for system 1: i There exists two compact sets K K M and a bounded subset U L R +, R p such that for every u U and every associated trajectory z u of system 1 issued from K, one has: z u t K t ii The map: φ : z hz, L F hz,, L n 1 hz T is a diffeomorphism from a bounded open set Ω M containing K into its range iii For j =,, n 1, i = 1,, p, one has on M: dl Fi L j h dl j h dl j 1 h dh = where denotes the exterior product of differential forms Remark 21 Conditions 1ii and 1iii express the uniform observability of system 4 More precisely, ii and iii imply that system 4 restricted to Ω can be steered into the canonical form 6 Hence, the restriction of system 4 to Ω becomes observable independently of the input see [2, 3] The candidate observer proposed here is based on the following construction First, remark that Assumption 1ii together with condition 2 imply the existence of a tubular neighbourhood T M of the manifold M such that the map φ : R n+d R n+d defined for all z = x, ρ by: φz = hz, L F hz,, L n 1 hz, ϕ T z T is a diffeomorphism from T M into its range For all j {1,, n}, consider the following construction: Ω j and K j are subsets of R n defined by: { } Ω j := hz, L F hz,, L j 1 hz, z Ω { } K j := hz, L F hz,, L j 1 hz, z K let χ j : R j [, 1] be C functions such that: i Suppχ j Ω j where Suppχ j denotes the closure of {x; χ j x } ii χ j ξ 1,, ξ j = 1 for ξ 1,, ξ j K j For i =,, p, let the fields F ei be defined by: L F hz φ L n 1 hz F e z= χ nhz,, L n 1 F z hzl n hz z

3 and for 1 i p: χ 1hzL Fi hz χ 2hz, L F hzl Fi L F hz φ F ei z= χ nhz,, L n 1 F z hzl Fi L n 1 hz z Clearly, F ei is uniquely defined and coincide with F i on K The extended system is then defined by: ż = F e z + u i F ei z Remarks 22 y = hz z M 8 i Let u U and z u be a trajectory of system 4 issued from K, then z u is also a trajectory of 8 and conversely ii The F ei s are global lipschitz fields of R n+d Now, consider the following change of coordinates ς = ξ η := φz = hz, LF hz,, L n 1 hz, ϕ T z T defined on T M ; then system 8 takes the following form: where ξ = Aξ + f e ξ, η + η = y = Cξ u i fei ξ, η A and C are defined by: 1 A = 1 C = 1 f e is defined by: f e ς := L n hφ 1 ς 9 from Assumption 1ii and the above construction, the f ei s are such that i {1,, p}, j {1,, n}, ς φm, one has: f eij ς = f eij ς 1,, ς j 1 3 Main result With the above notations and definitions, our main result can be stated: Theorem 31 Assume that system 1 fulfills Assumptions 1, then: i for every compact ˆK of R n+d, there exists a real positive number θ and a positive definite matrix Ω so that for any initialisation ˆς = ˆξ, ˆη in ˆK T M, the following system: ˆξ = Aˆξ + f e ˆξ, ˆη + u i fei ˆξ, ˆη S 1 θ CT C ˆξ 11 y ˆη = Ωˆη is an exponential observer for system 4, where S θ is defined by: θs θ + A T S θ + S θ A = C T C 12 ii In the original set of coordinates, observer 11 takes the form: ˆx = f e ˆx, ˆρ + u i f ei ˆx, ˆρ ˆρ = + 1 φ S 1 φ Remarks 31 θ CT hˆx, ˆρ y 1 [Ωϕˆx, ˆρ [ f e ˆx, ˆρ+ u i f ei ˆx, ˆρ 1 ] ] S 1 θ CT hˆx, ˆρ y 13 i Observer 13 only requires the jacobian of diffeomorphism φ Consequently, the above scheme does not require to explicit the solution ρ of ϕx, ρ = ii The dynamic of ˆη of equation 11 can be seen as a continuous time transcription of an optimization Newtown s algorithm The above scheme mixes a classical high gain observer with an optimization routine Hence, such an approach can be successfully used with any other exponential observer

4 Proof : Here, the trajectories of the system 4 are assumed to be into K Assumption 1i This in particular implies that ξt belongs to a bounded subset of R n Let ε := θ ˆξ ξ, where θ is the n n diagonal matrix diag 1 θ,, 1 θ Let V := V n 1 + V 2 be a candidate Lyapunov function with: V 1 := ε T S 1 ε and V 2 := ˆη T ˆη By posing Λ e ξ, η, u := f e ξ, η+ similar calculations as in [3] yields: u i fei ξ, η, V = V 1 + [ V 2 = 2ε T S 1 θa S1 1 CT Cε 14 ] + θ Λ e ˆξ, ˆη, u Λ e ξ,, u 2ˆη T Ωˆη Using 12: V = θv 1 εc T Cε 2ˆη T Ωˆη 15 +2ε T S 1 θ Λ e ˆξ, ˆη, u Λ e ξ,, u The main difficulty that occurs here is that, contrary to the usual high gain observer s proof [3], Λ e does not have any triangular structure elsewhere that on the manifold M Indeed, the estimate ˆξ ˆη can not be assumed to remain on the manifold though ξ does Writing Λ e ˆξ, ˆη, u Λ e ξ,, u = Λ e ˆξ, ˆη, u Λ e ˆξ,, u Λ e ξ,, u+λ e ˆξ,, u equality 15 becomes: V = θv 1 εc T Cε +2ε T S 1 θ Λ e ˆξ,, u Λ e ξ,, u 2ˆη T Ωˆη + 2ε T S 1 θ Λ e ˆξ, ˆη, u Λ e ˆξ,, u Using Schwarz inequality and noting by λ min Ω the smallest eigenvalue of Ω, it gives: V θv 1 2λ minωv V 1 [ 16 [ θλ eˆξ,,u Λ eξ,,u] T S 1[ θλ eˆξ,,u Λ eξ,,u] + [ θλ eˆξ,ˆη,u Λ eˆξ,,u] T S 1[ θλ eˆξ,ˆη,u Λ eˆξ,,u] Similarly as in [3], using: the triangular form 1 the i th component of Λ e ξ,, udepends only upon ξ 1,, ξ i, ] the uniform bound on the controls Assumption 1i, the global lipschitz property with of the fields F ei, one deduces: [ θλ eˆξ,,u Λ eξ,,u] T S 1[ θλ eˆξ,,u Λ eξ,,u] k ε k V1 17 λ min S 1 where k is a positive constant which does not depend on θ Using the mean value theorem and noticing that V 2 = ˆη 2, there exists a continuous function k θ, ˆξ, ˆη, u such that: [ θλ eˆξ,ˆη,u Λ eˆξ,,u] T S 1[ θλ eˆξ,ˆη,u Λ eˆξ,,u] k θ, ˆξ, ˆη, u V 2 18 Using inequalities 17 and 18 in 16 gives: V θ V 1 2λ min ΩV 2 λ min S 1 +k θ, ˆξ, ˆη, u V 1 V2 19 For any compact set ˆK, let: { α := sup V ξ, η, ˆξ, ˆη; ξ, η K, ˆξ, ˆη ˆK } { } ˆK := ˆξ, ˆη R n R d ; V ξ, η, ˆξ, ˆη α, ξ, η K k θ := sup k θ, ˆξ, ˆη, ut ˆξ,ˆη ˆK u U t Then, for any initialisation ˆς = ˆξ ˆη ˆK, it is sufficient to chose θ and Ω such that: θ > λ min S 1 λ min Ω 8 θ k θ 2 λ mins 1 Indeed, with the above choices, one has: V θ min θ V λ minωv 2 λ mins 1 λ mins 1, 2λminΩ This last inequality proves the exponential decrease of V Finally, the form 13 can simply be obtained using φ 1 and noticing that ˆη = ϕˆx, ˆρ, which gives: 1 1 ˆρ = ˆx Ωϕˆx, ˆρ V

5 4 Conclusion In this paper, a high gain observer for a class of implicit systems moving onto a manifold M is proposed The observer is proved to be exponentially convergent in a tubular neighbourhood of the manifold The proposed scheme does not require to explicit the implicit relation defining M or even to use optimization techniques as it is usually needed It would be interesting for further investigations to see if the proposed scheme can be generalized to higher implicity degrees as it can be done for some numerical methods [1] References [1] M U Ascher and R L Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations Society for Industrial & Applied Mathematics, July 1998 [2] J P Gauthier and G Bornard, Observability for any input of a class of nonlinear systems, IEEE Trans on Automatic Control, vol 26, no 4, pp , 1981 [3] J P Gauthier, H Hammouri, and S Othmann, A simple observer for nonlinear systems application to bioreactors, IEEE Trans on Automatic Control, vol 37, no 6, pp , 1992 [4] H Hammouri, K Busawon, and M Farza, Nonlinear observer for local uniform observable systems, internal report, LAGEP - 43 bd du 11 Novembre 1918, Villeurbanne, France, 1997