Unknown inputs observers for a class of nonlinear systems

Transcription

1 Unknown inputs observers for a class of nonlinear systems M Triki,2, M Farza, T Maatoug 2, M M Saad, Y Koubaa 2,B Dahhou 3,4 GREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN 6 Bd Maréchal Juin, 4050 Caen Cedex, France 2 ENIS, Département de Génie électrique, BP W, 3038 Sfax, Tunisia 3 Universit e de Toulouse; UPS, INSA, INP, ISAE; LAAS; F-3077 Toulouse France 4 LAAS-CNRS; Universit e de Toulouse, 7 avenue du Colonel Roche, F-2077 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 explicitly given Moreover, its tuning is reduced to the choice of two real numbers The performances of the proposed observer are demonstrated in simulation through an illustrative example Key words Nonlinear system, High gain observer, Unknown inputs Introduction Over the last twenty years, many researches have focused on the observer design for linear systems with unknown inputs [9, 0, 8, 5, 2] 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 [], 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 relatively recent work [4], 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 [3, 4, 6,, 3] In this paper, one considers a class of uniformly observable MIMO systems involving nonlinear inputs that intervene in a nonlinear manner Then, one proposes a full order! " " # $ %!! & " # $ % " % % ## ' " %! * + #!!, -,,,

2 2 Unknown inputs observers for a class of nonlinear / 0 29 high gain observer 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 Section 4 is devoted to a simulation example in order to highlight the performance of the proposed observer 2 Problem formulation Consider MIMO systems of the form: { ẋt = fut, vt, xt y = Cx = x x f u, v, x, x 2 x 2 f 2 u, v, x, x 2, x 3 with x = ; fu, v, x = ; x q f q u, v, x f q u, v, x C = [I n, 0 n n 2, 0 n n 3,, 0 n n q ] 2 k= where the state x IR n with x k IR n k, k =,, q and p = n n 2 n q, q n k = n; the input w = u, v W the set of bounded absolutely continuous functions with bounded derivatives from IR + into W a compact subset of IR s ; the output y IR p and fu, v, x IR n with f k u, x IR n k One shall suppose that the subvector u U IR s m of the input w is known while the remaining subvector v V IR m is unknown The objective then consists in synthesizing an observer to simultaneously estimate the vector of unknown inputs vt 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: H The state xt, the control ut and the unknown inputs v are bounded, ie xt X, ut U and v V where X IR n, U IR s m and V IR m are compacts sets

3 IJ-STA, Volume 4, N, July, H2 There exist α f, β f > 0 such that for all k {,, q }, x IR n, u, v U V, αf 2 k T k I n k+ u, v, x xk+ x k+ u, v, x β2 f I nk+ One also assumes that for k q, for all u, v U V, the map x k+ f k u, v, x,, x k, x k+ from IR n k+ into IR n k is one to one H3 The output x can be partitioned as follows: x = x x 2 with x IR m, x 2 IR p m and m m < p Such a partition induces the following one f u, v, x, x 2 f u, v, x, x 2 = f2 u, v, x, x 2 that has to satisfy the following two conditions: i There exist α v, β v > 0 such that x IR n, u, v U V αvi 2 T m v u, v, x, x 2 v u, v, x, x 2 βvi 2 m One also assumes that for all u, x, x 2 U IR n +n 2, the map v f u, v, x, x 2 from IR m into IR m is one to one ii Rank x 2 u, v, x, x 2 v u, v, x, x 2 2 x 2 u, v, x, x 2 2 = n 2 + m v u, v, x, x 2 for all x, x 2 IR n+n2, u, v U V H4 The time derivative of the unknown input vt is a completely unknown function, εt, which is uniformly bounded that is sup εt β ε where β ε > 0 is a real number t 0 For clarity purposes, one introduces the following notations: x 2 x x = x 2 ; x x x 2 = ; x 2 = v ; C = [I m 0]; C 2 = [I p m 0]; x q f C = diagc, C 2 fu, u, x x = f 2 ; f f u, v, x, x 2 u, x = ; u, x 0

4 4 Unknown inputs observers for a class of nonlinear / 0 22 f2 u, v, x, x 2 f 2 u, v, x, x 2, x 3 f 2 u, x = ; ε = 0 ε ; ε = ε 0 f q u, v, x Using these notations, system can be written as follows: x t = f ut, xt + εt x 2 t = f 2 ut, xt 3 y = C x = x x 2 or equivalently in the more following condensed form { xt = fut, xt + εt y = C x 4 One shall consider the following injective map Φ : R n+m R n+m x z v x 2 x 2 x q z2 x 2 x 2 x q with { z = x z 2 = f u, v, x, x 2 According to Assumption i of H3 and using the implicit function Theorem, the unknown input v can be considered as a function of u, x, x 2 Taking the derivative with respect to x 2 of each side of the second equation of 2 gives: This means that 0 = x 2 u, v, x, x 2 + v u, v, x, x 2 v x 2 u, x, x 2 v x 2 u, x, x 2 + = v u, v, x, x 2 x 2 u, v, x, x 2

5 IJ-STA, Volume 4, N, July, To summarize, the nonlinear system which will be considered with view to observer synthesis can be written under the following form: ż = z2 ż2 = x u, v, x, x 2 f u, v, x, x 2 + x 2 u, v, x, x 2 f 2 u, v, x, x 2, x 3 + u u, v, x, x 2 ut + 5 v u, v, x, x 2 εt y = z ẋ 2 = f2 u, vu, x, x 2, x, x 2 ẋ 2 = f 2 u, vu, x, x 2, x, x 2, x 3 6 ẋ q = f q u, vu, x, x 2, x,, x q y 2 = x 2 Notice that subsystem 5 can be written in the following condensed form: { ż = A z + ϕ u, u, z, x 2, x 3 + ε 2 t y = C z = z with z = z z2, A = ϕ u, u, z, x 2, x 3 = with 0 I m 0 0 is the 2m 2m anti-shift matrix and 0 IR 2m, εt = ε 2 t 0 ϕ 2u, u, z, x 2, x 3 7 IR 2m ϕ 2u, u, z, x 2, x 3 = x u, v, x, x 2 f u, v, x, x 2 + x 2 u, v, x, x 2 f 2 u, v, x, x 2, x 3 + u u, v, x, x 2 ut ε 2 t = v u, v, x, x 2 εt 8 Now, one shall introduce a coordinate transformation which shall put subsystem 6 under an appropriate form for the observer synthesis Before the presentation of such transformation one shall point out some properties satisfied by this subsystem According to Hypothesis H2, one has Rank ẋk x u, v, x xk+ = n k+ for k = 2,, q 9 k+ u, v, x = Rank k

6 6 Unknown inputs observers for a class of nonlinear / Let us show that Indeed, on one hand, one has Rank ẋ 2 x 2 u, v, x = n 2 0 ẋ 2 x 2 u, v, x = 2 x 2 u, v, x, x v u, v, x, x 2 v x 2 u, x, x 2 = 2 x 2 u, v, x, x 2 2 v u, v, x, x 2 + v u, v, x, x 2 x 2 u, v, x, x 2 On the other hand, consider the following full rank matrix, I m 0 P u, x = 2 v u, v, x, x 2 + v u, v, x, x 2 2 I p m Since rankp u, x = p for all u, x U X and p n 2 + m according to ii of H3, one has: n 2 + m = Rank P u, x x 2 u, v, x, x 2 v u, v, x, x 2 2 x 2 u, v, x, x 2 2 v u, v, x, x 2 = Rank x 2 u, v, x, x 2 v u, v, x, x 2 ẋ 2 u, v, x 0 x2 ẋ = Rank 2 u, v, x + Rank x2 v u, v, x, x 2 ẋ = Rank 2 u, v, x + m x2 This leads to 0 Now, since properties 9 and 0 are satisfied by subsystem 6, one can consider the transformation introduced in [7, 4] which puts 6 under a canonical form that characterized a subclass of uniformly observable systems Before giving this transformation and for clarity purposes, one introduces the following notation: f u, x, x 2 = f 2 u, vu, x, x 2, x, x 2 3

7 IJ-STA, Volume 4, N, July, Now, consider the following injective map Φ 2 : IR n+m IR 2m+qp m z z2 x 2 x 2 x q z z2 z 2 z 2 2 z 2 q with z 2 = x 2 z2 2 = f u, x, x 2 zk 2 = f x 2 u, x, x 2 for k = 3,, q k 2 i= i x i+ u, v, x,, x i+ f k u, v, x,, x k+ One can show see [7, 4] for more details that the transformation Φ 2 puts subsystem 6 under the following form: { ż 2 = A 2 z 2 + ϕ 2 u, u, z, z 2 y 2 = C 2 z 2 = z 2 5 with z 2 = z2 z 2 q, A 2 = 0 I qp m is the qp m qp m anti-shift matrix, C 2 = [I qp m 0] is p m qp m rectangular matrix and ϕ 2 u, u, z, z 2 is ϕ 2 u, u, z, z 2 ϕ 2 2u, u, z, z, 2 z 2 2 triangular with respect to z 2, ie ϕ 2 u, u, z, z 2 = ϕ 2 q u, u, z, z, 2, zq 2 ϕ 2 qu, u, z, z 2 3 Observer synthesis Consider again the overall system constituted by system 7 and 5, ie { ż = A z + ϕ u, u, z + εt y = C z = z { ż 2 = A 2 z 2 + ϕ 2 u, u, z, z 2 y 2 = C 2 z 2 = z 2 6

8 8 Unknown inputs observers for a class of nonlinear / which can also be written in the more following condensed form: ż = Az + ϕu, u, z + εt 7 where z ϕ u, u, z 0 z = z 2 ; ϕu, u, z = ϕ 2 ; εt = ; u, u, z εt In the case where ε = 0, system 6 is under a form introduced in [2] that characterized a large class of uniformly observable systems As a result, one can use the observer proposed in [2] in order to estimate z and z 2 However, due to the presence of ε, one can show that the estimation error does not converge to zero but can be made as small as desired The equation of the observer specialize as follows: { ẑ = A ẑ + ϕ u, u, ẑ θk C ẑ y ẑ 2 = A 2 ẑ 2 + ϕ 2 u, u, ẑ, ẑ 2 2 θk 2C 2 ẑ 2 8 y 2 where ẑ = ẑ ẑ2 ẑ 2 IR 2m, ẑ, ẑ2 IR m,ẑ 2 2 = IRqp m, ẑi 2 IRp m ẑ k = z k output injection for k =, 2 Cq I p m 2I m C 2 q I K =, K 2 = p m I m with Ck q = Cq q I p m θ = diag θ q I m, where θ > 0 is the observer design parameter ẑ 2 ẑ 2 q q! k!q k!, k =,, q θ 2q I m and 2 θ = diag θ I p m, θ 2,, θ q I p m Observer 8 can be written in the following more condensed form: where z = ẑ ϕ u, u, ẑ ẑ 2 ; ϕu, u, ẑ = ϕ 2 u, u, ẑ ẑ = Aẑ + ϕu, u, ẑ θkcẑ y 9 A = diaga, A 2 ; K = diagk, K 2 ; C = diagc, C 2 ; θ = diag θ, 2 θ;

9 IJ-STA, Volume 4, N, July, Since, the injective map Φ 2 Φ brought system 4 under form form 7, observer 9 can be written in the original coordinates as follows: + ˆ xt = fut, Φ2 ˆ xt x u, v, x Φ u, v, x θkcẑ y 20 x However, according to the particular structure of the jacobian transformation, Φ 2 x u, v, x Φ x u, v, x, one shall exhibit in the sequel another observer which does not need to compute the inverse of the overall jacobian transformation but only the inverse of the diagonal of this jacobian Indeed, let us consider the following two bloc diagonal matrices: Λ = diag I m, v u, v, x, x 2 Λ 2 = diag I p m, f 2 q f f u, x, u, x u, v, x,, x2 x2 x2 x 2 u, x k u, v, x xk+ Λ = diagλ, Λ 2 One has : Φ 2 x u, v, x Φ x Φ2 u, v, x = Λu, v, x + k= x u, v, x Φ u, v, x Λu, v, x x = Λu, v, x + Γ u, v, x 2 with Γ u, v, x = Φ2 x u, v, x Φ u, v, x Λu, v, x x One can show that the matrix Γ is lower triangular matrix with a main diagonal equal to zero As a result, one can use the following observer in the original coordinates [4]: ˆ xt = fut, ˆ xt Λu, v, x + θkcẑ y 22 Observer 22 can be written in the following developed form: { ˆx t = fut, ˆvt, x t, ˆx 2 2θ q ˆx x ˆv = θ 2q v u, ˆv, x, ˆx 2 ˆx x ẋ 2t f2 u, ˆv, x, ˆx 2 ẋ 2 t = f 2 u, ˆv, x, ˆx 2, ˆx 3 ẋ q t f q u, ˆv, x, ˆx 2,, ˆx q Λ 2 u, ˆv, x, ˆx 2,, ˆx q 2 θk 2C 2ˆx 2 y

10 0 4 Example Unknown inputs observers for a class of nonlinear / Consider the following dynamical system: ẋ = x 2 x 3 + v 3 + v tanhx x 3 ẋ 2 = x 3 + x 2 v x 2 ẋ 3 = x 4 x 3 + v +v 2 ẋ 4 = cosx 4 + vsinv y = [x x 2 ] T 25 where x = [x x 2 x 3 x 4 ] T IR 4 with x i IR, vt is the unknown input 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: It is easy to see that system 25 is under form with: vt = cost 26 x = [x x 2 ] T ; x 2 = x 3 ; x 3 = x 4 ; Concerning the partition of x needed in hypothesis H4, one can consider the following one the only possible partition in this example: x = x and x 2 = x 2 Now, one can easily check hypotheses H to H4 and an observer under form can be used in order to achieve the required estimations 4 simulation Results An observer of the form has been used in order to estimate x 3, x 4 and v This observer has been simulated using data issued from simulation In figure, the true time evolutions of x 3, x 4 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 θ is equal to 20 The initial conditions for the model and the observer are: x 0 = ˆx 0 = ; x 2 0 = ˆx 2 0 = 2; x 3 0 = ; x 4 0 = ; ˆx 3 0 = ˆx 4 0 = ˆv0 = 0 The obtained results clearly show the good agreement between the estimated and simulated variables Recall that the expression of the unknown input equation 26 introduced for simulation purposes is ignored by the observer

11 IJ-STA, Volume 4, N, July, TIME EVOLUTION OF x 3 SIMULATED ESTIMATED TIME EVOLUTION OF x 4 SIMULATED ESTIMATED TIME TIME 5 TIME EVOLUTION OF v SIMULATED ESTIMATED TIME Fig Estimation of x 3, x 4 and v

12 2 Unknown inputs observers for a class of nonlinear / Conclusion: A class of nonlinear systems which is nonlinearly parameterized with respect to unknown inputs has been considered with view to observer design Under appropriate sufficient conditions that ensure the observability of all the state as well as that of the unknown inputs, a high gain observer was synthesized in order to simultaneously estimate the whole state and the the unknown inputs Simulation results was given in order to highlighted the the theory Références M Corless and J Tu State and Input Estimation for a Class of Uncertain Systems Automatica, 346: , M Darouach, M Zasadzinski, and SJ Xu Full-Order Observer for Linear Systems with Unknown Inputs IEEE Trans on Aut Control, 393: , M Farza, M M Saad, FL Liu, and B Targui Generalized observers for a class of nonlinear systems Int Journal of Modelling, Identification and Control, 2:24 32, M Farza, M M Saad, and L Rossignol Observer design for a class of MIMO nonlinear systems Automatica, 40:35 43, Y Guan and M Saif A Novel Approach to the Design of Unknown Inputs Observers IEEE Trans on Aut Control, 365: , 99 6 QP Ha and H Trinh State and input simultaneous estimation for a class of nonlinear systems Automatica, 40: , H Hammouri and M Farza Nonlinear observers for locally uniformly observable systems ESAIM J on Control, Optimisation and Calculus of Variations, 9: , M Hou and P C Müller Design of observers for linear systems with unknown inputs IEEE Trans on Aut Control, AC-37:87 875, C D Johnson On observers for linear systems with with unknown and inaccessible inputs International Journal of Control, 4:825 83, J Kudva, N Viswanadham, and A Ramakrishna Observers for linear systems with unknown inputs IEEE Trans on Aut Control, AC-25:3 5, 980 FL Liu, M Farza, and M M Saad Nonlinear observers for state and unknown inputs estimation Int Journal of Modelling, Identification and Control, 2:33 48, FL Liu, M Farza, M M Saad, and H Hammouri Observer design for a class of uniformly observable mimo nonlinear systems with coupled structure In Proc of the 7th IFAC World Congress, Seoul, Korea, Y Xiong and M Saif Sliding Mode Observer for Nonlinear Uncertain Systems IEEE Trans on Aut Control, 462: , Y Xiong and M Saif Unknown disturbance inputs estimation based on a state functional observer design Automatica, 39: , 2003