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1 This article was downloaded by: [UNAM Ciudad Universitaria] On: 21 June 29 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 129 Registered office: Mortimer House, 3-1 Mortimer Street, London W1T 3JH, UK International Journal of Control Publication details, including instructions for authors and subscription information: Finite-time state observation for non-linear uncertain systems via higher-order sliding modes Jorge Davila a ; Leonid Fridman a ; Alessandro Pisano b ; Elio Usai b a Department of Control, Division of Electrical Engineering, Facultad de Ingenieria, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico, DF, Mexico b Dipartimento di Ingegneria Elettrica ed Elettronica (DIEE), Universitá di Cagliari - Piazza d' Armi, Cagliari, Italy First Published:August29 To cite this Article Davila, Jorge, Fridman, Leonid, Pisano, Alessandro and Usai, Elio(29)'Finite-time state observation for non-linear uncertain systems via higher-order sliding modes',international Journal of Control,82:8,16 1 To link to this Article: DOI: 118/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material

2 International Journal of Control Vol 82, No 8, August 29, 16 1 Finite-time state observation for non-linear uncertain systems via higher-order sliding modes Jorge Davila a *, Leonid Fridman a, Alessandro Pisano b and Elio Usai b a Department of Control, Division of Electrical Engineering, Facultad de Ingenieria, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico, DF, Mexico; b Dipartimento di Ingegneria Elettrica ed Elettronica (DIEE), Universita di Cagliari Piazza d Armi, Cagliari, Italy (Received 1 June 28; final version received 29 October 28) This article deals with the problem of finite-time state estimation for a class of non-linear systems possibly affected by modelling uncertainties and/or unknown inputs The proposed method, based on the high-order sliding mode control approach, does not require the system to be transformed to any normal form, which can be difficult to achieve in the presence of model uncertainties The sufficient conditions for observability are derived in terms of certain geometric restrictions imposed on the system s vector fields Methods for the approximate and exact reconstruction of the unknown inputs are given and simulation results are provided and commented Keywords: high-order sliding modes; non-linear observers; uncertain systems 1 Introduction Control under uncertainty conditions is a major topic in modern control theory The uncertainty may manifest itself both as an uncertainty of the mathematical model and as a lack of information about the system state The idea of using a dynamical system to generate estimates of the system states was proposed by Luenberger for linear systems (Luenberger 196) In spite of the extensive development of robust control techniques, sliding mode control (SMC) remains a key choice for handling bounded uncertainties/disturbances and unmodelled dynamics in both control and estimation problems During the last decade, SMC techniques have been widely used to design observers (sliding mode observers) suitable for robust state estimation also in the presence of unknown inputs The problem of robust state observation was actively studied in recent years using sliding modes and the variable structure systems theory (see, for example, Edwards and Spurgeon (1998), Utkin, Guldner and Shi (1999), Haskara and Ozguner (2), Barbot, Djemai, and Boukhobza (22), Ahmed-Ali, Floret, and Lamnabhi-Lagarrigue (2), Davila, Fridman, and Levant (2), Fridman, Levant, and Davila (2)) The corresponding implementation issues were extensively studied in Edwards, Spurgeon, and Tan (22) and Poznyak (23) The sliding mode observation has such attractive features as insensitivity (more than robustness) with respect to unknown inputs; and also offers the possibility of reconstructing the unknown input terms via using the equivalent output injection (Davila et al 2; Davila, Fridman, and Poznyak 26), which is the counterpart, in the field of sliding mode observation, of the equivalent control method allowing the replacement of the discontinuous infinitefrequency switching terms with an equivalent smooth signal Recent works on non-linear systems observation are based on a transformation to some canonical form and successive estimation of the state vector using distinct techniques as high-gain observers (Gauthier, Hammouri and Othman 1992), the dissipativity approach (Rocha-Cozatl, and Moreno 2), and high-order sliding modes (Floquet and Barbot 26; Ahmed-Ali, Kenne, and Lamnabhi- Lagarrigue 2; Fridman, Shtessel, Edwards, and Yan 28) In this work, we suggest an approach to finite-time state observation based on high-order sliding modes which allows the finite-time exact state reconstruction of the state vector without requiring the system being subject to any transformation devoted to normal form reduction The treatment is first developed assuming perfectly known system dynamics, and is then generalised by stating the conditions ensuring the finite-time state reconstruction in the presence of model uncertainties For a specific form of the model uncertainties, namely a scalar unknown input multiplied by a known constant distribution matrix, methods for the approximate and exact reconstruction of the unknown input are described *Corresponding author jadavila@unammx ISSN 2 19 print/issn online ß 29 Taylor & Francis DOI: 118/

3 International Journal of Control 16 This article is structured as follows Section 2 deals with the problem formulation and presents the main assumption underlying the feasibility of the proposed procedure Sections 3 and study the state estimation problems for the case of perfectly known system dynamics and uncertain system dynamics, respectively Section describes the suggested design for the observer discontinuous input injection term Section 6 presents two methods for reconstructing the unknown input A thoroughly discussed example of application via computer simulation is given in Section Section 8 resumes the achieved results and presents some concluding remarks 2 State observation for non-linear systems Consider a non-linear uncertain system _x ¼ f ðxþþ f ~ ðxþ ð1þ y ¼ hðxþ with the state vector x 2XR n and the scalar output y 2YR The vector fields f(x):x!r n and h(x):x!y represent the known nominal part of the system dynamics, while f ~ ðxþ : X!R n is assumed to be uncertain The aim is that of designing a finite-time converging observer for the above system We consider the following observer structure: _^x ¼ f ð ^xþþgð^xþu ð2þ ^y ¼ hð ^xþ with observed state vector ^x 2 R n and observed output variable ^y 2 R The equations are understood in the Filippov sense (Filippov 1988) in order to provide for the possibility to use discontinuous signals in the observers Note that Filippov solutions coincide with the usual solutions, when the right-hand sides are continuous It is assumed also that all considered correction terms allow the existence and extension of solutions to the whole semi-axis t The vector field g() and the corrective term u 2 R (observer input) will be designed along this article Under a certain observability condition that is going to be specified, we shall select the vector function g() in such a way that the observer output has full relative degree n with respect to the observer input u With reference to a generic scalar function q with vector argument z defined on an open set R n, q(z):r n! R, denote n Define the following n-th order square matrix: 2 3 dhðzþ dl f ðzþ hðzþ MðzÞ ¼ 6 dlf n 2 ðzþ hðzþ dlf n 1 ðzþ hðzþ where L f(z) h(z) is sometimes called the Lie derivative (Isidori 1996) of h(z) along f(z) and is defined as L f ðzþ hðzþ ¼ð@hðzÞ=@zÞf ðzþ and the k-th derivative of h(z) along f(z) is defined as L k f ðzþ hðzþ ¼ ð@l k 1 f ðzþ hðzþ=@zþf ðzþ Let the following assumption be satisfied Assumption 1: Matrix M(z) in (3) is non-singular for every possible value of z Assumption 1 is a structural assumption on the nominal part of the plant to be observed Let us select the vector gð ^xþ in (2) as the unique solution of equation We get Mð ^xþgð ^xþ ¼½,,,1Š T : gð ^xþ ¼M 1 ð ^xþ½,,,1š T, ie gð ^xþ is the last column of matrix M 1 ð ^xþ Note that gð ^xþ is well defined, according to Assumption 1 In light of () the observer input/output dynamics is ^y L f ð ^xþ hð ^xþ d dt 6 _^y ^y ðn 1Þ ¼ 6 L 2 f ð ^xþ hð ^xþ þ u 6 hð ^xþ 1 L ðnþ f ð ^xþ which means that the observer output ^y has full relative degree n with respect to the observer input u Define the output and state observation errors 2 3 e 1 6 e y ¼ ^y y, e ¼ ^x x ¼ 6 e 2 e n : Clearly, the output error e y ¼ ^y y possesses the same relative degree n with respect to u ð3þ ðþ ðþ ð6þ

4 166 J Davila et al Define also the output error vector ", containing the output error e y and its first n 1 derivatives: " 1 e y " 2 _e y " ¼ 6 ¼ 6 : ðþ " n e ðn 1Þ y Let us analyse in next two sections the error dynamics for the nominal dynamics ( ~ f ¼, Section 3) and for the uncertain dynamics (1) (Section ) 3 Nominal system Consider the following simplified, perfectly known, system: _x ¼ f ðxþ ð8þ y ¼ hðxþ: The main result of this section is summarised in Lemma 1 Lemma 1: Consider system (8) and let Assumption 1 be satisfied Consider the observer (2), () Then the following implication holds locally: Proof of Lemma 1: is given by " ¼, e ¼ : ð9þ The observation error dynamics _e ¼ f ð ^xþ f ðxþþgð ^xþu e y ¼ hð ^xþ hðxþ: ð1þ ð11þ Let us derive the expression for the successive derivatives of the output observation error e y up to the order n Considering (), it yields e ð y jþ ¼ L j f ð ^xþ hð ^xþ Lj f ðxþhðxþ, 1 j n 1 ð12þ ey ðnþ ¼ L n f ð ^xþ hð ^xþ Ln f ðxþhðxþþu: ð13þ Considering that x ¼ ^x e, Equations (11) (13) allows for constructing explicitly the following mapping: " ¼ ðe, ^xþ ¼col i ðe, ^xþ, 1 i n ð1þ whose entries can be rewritten component-wise as follows: 1 ðe, ^xþ ¼hð ^xþ hð^x eþ ð1þ 2 ðe, ^xþ ¼L f ð ^xþ hð ^xþ L f ð ^x eþ hð ^x eþ ð16þ n ðe, ^xþ ¼L n 1 f ð ^xþ hð ^xþ Ln 1 f ð ^x eþ hð ^x eþ: ð1þ If e ¼ then all components of vector " are identically zero irrespectively of ^x, ie the mapping fulfils the condition ð, ^xþ ¼ In order to prove the reversed implication, let us consider the inverse mapping It must be shown that e ¼ 1 ð", ^xþ: ð18þ 1 ð, ^xþ ¼, 8 ^x: ð19þ In other words, the lemma holds provided that the mapping ðe, ^xþ is locally bijective in the neighbourhood of e ¼ Considering the Jacobian matrix Jðe, ^xþ ¼ð@ðe, ^xþþ=, the bijectivity of the mapping corresponds to the following condition: det Jð, ^xþ 6¼, 8 ^x: ð2þ It is now demonstrated that condition (2) is implied by Assumption 1 Since x ¼ ^x e, it makes sense to denote x ¼ xðe, ^xþ Let us differentiate the Jacobian matrix Jðe, ^xþ By (11) (13) it derives 1 ðe, j ðe, ^xþ ¼ ¼ ð j 1Þ ^xþ 1 f ðxþ ^xþ, 2 j n: ð21þ Substituting x ¼ ^x e into (21) and (22), noticing ^xþ ¼ 1, and putting e ¼ in the resulting equations, 1 ðe, ^xþ ¼ ð22þ j ðe, ^xþ 1 f ð ^xþ hð ^xþ, ^x 2 j n ð23þ which can be resumed compactly as Jðe, ^xþ e¼ ¼ Mð ^xþ: ð2þ Thus, Assumption 1 implies the condition (2), which proves the lemma œ It has been shown that, under Assumption 1, the observer (2) and () can reconstruct the state of the system (8) exactly and in finite time, provided that the observer input u is selected in such a way that the vector " is steered to zero in finite time Last lemma is satisfied for any uniformly observable system (Gauthier et al 1992) This condition is also satisfied, in a trivial form, by any Lagrange s system, and also, by any system which can be written in the Cauchy form

5 International Journal of Control 16 Uncertain system Now we consider the uncertain system (1) along with the observer (2) and () The uncertainty term f ~ is not modelled in the observer dynamics We give an additional, necessary and sufficient, condition guaranteeing the preservation of implication (9) Let the following assumption hold Assumption 2: Functions f, ~ f, and h are such that 2 3 L ~fðxþ hðxþ 2 3 L ~fðxþ L f ðxþ hðxþ 6 ¼ 6 : ð2þ L ~fðxþ L n 2 f ðxþ hðxþ The main result of this section is summarised in the following Lemma 2 Lemma 2: Consider system (1) and let Assumptions 1 and 2 be satisfied Consider the observer (2) and () Then the implication (9) holds Proof of Lemma 2: The observation error dynamics is now _e ¼ f ð ^xþ f ðxþ ~ f ðxþþgð ^xþu e y ¼ hð ^xþ hðxþ: ð26þ ð2þ Let us derive the perturbed, uncertain, mapping " ¼ ~ðe, ^xþ ¼col ~ i ðe, ^xþ, 1 i n : ð28þ The first component ~ 1 ¼ e y is not affected by the uncertainty, thus, ~ 1 ðe, ^xþ ¼ 1 ðe, ^xþ The second component ~ 2 ¼ _e y is ~ 2 ðe, ^xþ ¼L f ð ^xþ hð ^xþ L f ðxþ hðxþ L ~f ðxþ hðxþ: ð29þ From Assumption 2, the last term in the right-hand side of (29) is zero, then ~ 2 ðe, ^xþ ¼L f ð ^xþ hð ^xþ L f ðxþ hðxþ ¼ 2 ðe, ^xþ: ð3þ The third component ~ 3 ¼ e y is ~ 3 ðe, ^xþ ¼L 2 f ð ^xþ hð ^xþ L2 f ðxþ hðxþ L ~ f ðxþ L f ðxþ hðxþ: ð31þ Again, from Assumption 2 the last term in the right-hand side of (31) is zero, which means that ~ 3 ðe, ^xþ ¼L 2 f ð ^xþ hð ^xþ L2 f ðxþ hðxþ ¼ 3ðe, ^xþ: ð32þ By iterating the procedure until the n-th component ~ n ðe, ^xþ, and exploiting the Assumption 2 at each step, it follows that ie the same mapping of the nominal case dealt with in Lemma 1 is achieved From this point on the proof will follow the same steps as that of Lemma 1, so that condition ~ 1 ð, ^xþ ¼ holds for any ^x and Lemma 2 is proven œ Observer input design Lemmas 1 and 2 can be fruitfully exploited for state observation purposes if we are able to find an appropriate observer input u which can steer the vector " to zero in finite time Clearly, any controller which stabilises the observation error dynamics for the uncertain plant (1) will be also working for the nominal plant dynamics (8) Thus only the uncertain case will be studied in the following The observation output error dynamics takes the following Brunovsky chainof-integrators canonical form: 8 _" 1 ¼ _" 2 >< _" 2 ¼ _" 3 ð3þ >: _" n ¼ ~ nþ1 ðe, ^xþþu, where ~ nþ1 ðe, ^xþ ¼L n f ð ^xþ hð ^xþ Ln f ð ^x eþ hð ^x eþ L ~fð ^x eþ Lf n 1 ð ^x eþ hð ^x eþ: ð3þ Make the following boundedness assumption Assumption 3: There is known constant such that function ~ nþ1 ðe, ^xþ satisfies j ~ nþ1 ðe, ^xþj : ð36þ A recently proposed method based on the so-called arbitrary-order sliding mode approach (Levant 2) is now outlined in order to provide for the finite-time stabilisation of system (3) (36) 1 Quasi-continuous controller The quasi-continuous arbitrary-order sliding mode controller was suggested in Levant (2, 26) in order to stabilise (3) (36) in finite time Let i ¼ 1,, n 1 and denote i,n ¼ e ðiþ y,n ¼ e y, N,n ¼je y j, ð3þ,n ¼,n =N,n ¼ sign e y, ð38þ þ in ðn iþ=ðn iþ1þ i 1,n i 1,n, ð39þ N i,n ¼je ðiþ y jþ in ðn iþ=ðn iþ1þ i 1,n, ðþ ~ðe, ^xþ ¼ðe, ^xþ, ð33þ i,n ¼ i,n =N i,n, ð1þ

6 168 J Davila et al where 1,, n 1 are positive numbers The quasicontinuous n-sliding controller is u ¼ n 1,nðe y, _e y,, e ðn 1Þ y Þ: ð2þ It was shown in Levant (2, 26) that provided that the tuning parameters 1,, n 1, are chosen sufficiently large in the given order then the control law defined by (3) (2) stabilises the system (3) (36) in finite time Note that control defined by (3) (2) is globally bounded (juj) and continuous everywhere but the origin of the n-dimensional error space, from which the name of the controller was derived Following are reported examples of second- and third-order quasicontinuous controllers: u ¼ _e y þje y j 1=2 sign e y j_e y jþje y j 1=2, u ¼ e y þ 2ðj_e y jþje y j 2=3 Þ 1=2 ð_e y þje y j 2=3 sign e y Þ j e y jþ2ðj_e y jþje y j 2=3 Þ 1=2 : ð3þ Taking into account Lemma 2, the following theorem can be stated Theorem 1: Consider system (1) and let Assumptions 1, 2 and 3 be satisfied Consider the observer (2) and () with the observer input u designed according to (3) (2) Then, provided that the tuning parameters 1,, n 1, are chosen sufficiently large in the given order, the state estimation error e ¼ ^x x converge to zero in finite time In light of the convergence properties of the quasicontinuous controller (Levant 2) the proof of Theorem 1 is a trivial consequence of Lemma 2 2 Output error derivatives estimation The above n-th order quasi-continuous controller requires the availability of the successive derivatives of the output estimation error up to the order n 1 In order to reconstruct such derivatives exactly and in finite time, the well-known arbitrary-order sliding mode differentiator by Levant (23) can be used The n-th order differentiator can be expressed in the following non-recursive form: _z ¼ v ¼ z 1 jz e y ðtþj n=ðnþ1þ signðz e y ðtþþ, _z 1 ¼ v 1 ¼ z 2 1 jz 1 v j ðn 1Þ=n signðz 1 v Þ, _z i ¼ v i ¼ z i i jz i v i 1 j ðn iþ=ðn iþ1þ signðz i v i 1 Þ, _z n ¼ n signðz n v n 1 Þ ðþ for suitable positive constant coefficients i to be chosen recursively large in the given order (Levant 23) Under the assumption that a constant C exists such that je ðnþ y jc, the following equalities are true in the absence of measurement noise after a finite time transient process: jz i e ðiþ y ðtþj ¼, i ¼,, n : ðþ Clearly, for the considered closed-loop system (3) (36) the C constant exists and it is overestimated by C ¼ þ The separation and robustness results relevant to the combined use of the above differentiator and any n-sliding homogenous controller were discussed in Levant (23) It was demonstrated by Levant (23) that nonidealities like measurement noise and finite frequency commutation cause a bounded error in the estimated derivatives and, as a result, a bounded loss of accuracy for the controller that uses the noisy derivative estimates (Levant 23, 2, 26) 6 Non-linear systems with unknown inputs We now study an important special instance of the general uncertain dynamics (1) with the uncertainty being represented by some bounded Lipschitz unknown input term v(x, t) 2 R premultiplied by a known distribution column vector D of appropriate dimension, ie: _x ¼ f ðxþþdvðx, tþ y ¼ hðxþ: ð6þ It is assumed that v(x, t) allows unique solutions for Equation (6) in the Filippov sense (Filippov 1988) for all t For the system (6) we specialise the previous results to solve the problem of finite-time state observation and we address the problems of the approximate and exact reconstruction of the unknown input The reconstruction of unknown inputs can be performed only in case the system is perfectly known (no modelling uncertainties are allowed) 61 State observation The finite-time observer for system (6) takes the same form (2) and () as before, where f(), h() are exact copies of the functions in (6) The uncertain term Dv(x, t) is not modelled in the observer dynamics To take advantage of Lemma 2, let us specialise Assumption 2 into an explicit set of conditions involving the unknown input distribution column vector D Simple computations lead to the following

7 International Journal of Control 169 Assumption 2A: dl i f ðxþhðxþd ¼, i ¼,, n 2: Lemma 3: Consider system (6) satisfying Assumptions 1, 2A and 3 Design the observer as (2) and (), with the observer input u set according to the quasi-continuous controller described in Theorem 1 Then, the system state is recovered exactly and in finite time 62 Approximate unknown-input reconstruction Consider that the unknown input is scalar, ie v(x, t) 2 R By making reference to the class of uncertain systems (6), it is found that the observation output error dynamics takes the Brunovsky canonical form (3) (3) In steady state, all entries of vectors " and e are identically zero, while _" n, directly affected by the discontinuous control, is zero in the average (or Filippov ) sense (Filippov 1988) Thus we are in position to exploit one of the main features of sliding mode observers, the equivalent output injection principle The expression of _" n is _" n ¼ L n f ð ^xþ hð ^xþ Ln f ðxþ hðxþ Ln 1 f ðxþ hðxþdvðx, tþþu: ðþ Starting from the moment at which the exact state reconstruction is achieved, Equation () simplifies as follows: _" n ¼ L n 1 f ðxþhðxþdvðx, tþþu: ð8þ Assumption : The column vector D is such that dl n 1 f ðxþ hðxþd 6¼ : ð9þ The convergence of e to zero ensures that L n f ð ^xþ hð ^xþ L n f ðxþhðxþ ¼ Then, the observer input u will take the value of the equivalent output injection u eq, ie u eq ¼ L n 1 f ð ^xþ hð ^xþd vðx, tþ ðþ which derives from imposing the zeroing of _" n (equivalent control method; see Utkin et al (1999)) If u eq were available then the following simple relationship would trivially allow for the reconstruction of the unknown input v(x, t): 1 vðx, tþ ¼ L n 1 f ð ^xþ hð ^xþd u eq: ð1þ Since u is actually discontinuous, the equivalence u ¼ u eq holds only in the Filippov sense (Filippov 1988) so that the recovery of the equivalent output injection u eq from the discontinuous control signal u requires filtration Let us define the following equivalent control estimate ^u eq : _^u eq ¼ u ^u eq : ð2þ Theoretically, u is a discontinuous switching signal commuting at infinite frequency in the steady state In practice, the commutation frequency will be high but finite, due to sensors and actuators bandwidth restrictions Thus, the solution of (2) will depend on the finite value f sw of the switching frequency of u It is known (Utkin et al 1999) that lim! f sw!1 ^u eq ¼ u eq : ð3þ Furthermore, with sufficiently large f sw, and by neglecting exponentially decaying terms, it holds that the reconstruction error depends on the filter time constant as follows: j ^u eq u eq joðþ ðþ which means that a constant k can be found such that, after a finite transient, the reconstruction error is bounded as follows: j ^u eq u eq jk: ðþ Thus, by implementing filter (2) one can achieve an estimate of the unknown input by means of (1) with the equivalent control estimate ^u eq used in place of u eq 63 Exact unknown-input reconstruction The method presented in the previous section allows only the approximate recovery of the unknown input It is proposed here a modification to the observer structure that allows its exact, finite time, recovery Let the unknown input v(x, t) 2 R be a Lipschitz signal Under this condition, the unknown input can be reconstructed exactly and in finite time by the following procedure The key point is to modify the observer input selection Instead of the control defined by (3) (2), the following augmented quasi-continuous (n þ 1)-sliding controller should be implemented: _u ¼ n,nþ1ðe y, _e y,, e ðnþ y Þ: ð6þ It was shown in Levant (2, 26) that provided that the tuning parameters 1,, n, are chosen sufficiently large in the given order then the above control law stabilises the system (3) (36) in finite time The difference from the previous case is that the actual observer control u is now continuous everywhere, including the origin of the error space

8 1 J Davila et al This means that the equivalence u ¼ u eq should be no longer understood in the sense of Filippov, but, on the contrary, there is finite t ¼ t* such that u ¼ u eq, t t : ðþ Thus, low-pass filtration is no longer necessary and an exact reconstruction of the unknown input becomes possible from the relationship (1) with the actual control u replacing u eq There are two main consequences underlying the exact reconstruction of the unknown input First, the stronger smoothness requirement on the unknown input Second, most critical, the n-th order output is required The latter implies that a sliding mode differentiator of increased order is necessary to provide for the exact reconstruction of the unknown input, as compared with the scheme described in Section 2 error derivative e ðnþ y Simulation example The well-known Ro ssler chaotic dynamics is considered (Ro ssler 196) a third-order non-linear autonomous system which can be written as follows: _x 1 ¼ x 2 x 3 _x 2 ¼ x 1 þ ax 2 _x 3 ¼ b þ x 3 ðx 1 cþ, ð8þ ð9þ ð6þ where a, b and c are constant parameters set as a ¼ b ¼ 2 and c ¼ 6 Let the initial conditions be x() ¼ [ ] T This choice for the system parameters and initial conditions gives rise to the chaotic strange attractor that is depicted in Figure 1 x x Figure 1 Chaotic behaviour of Ro ssler system in the state space 1 x The chaotic behaviour of the Ro ssler circuit guarantees that the states will be bounded, and this upper bound can be estimated In this case the upper bound is considered as 2 Let the measured output be y ¼ x 2 and let the system model be temporarily considered fully known The matrix Mð ^xþ in (3) takes the form 2 3 dðhð ^xþþ Mð ^xþ ¼6 dðl f hð ^xþþ ¼ 6 1 a : dðl 2 f hð ^xþþ a a Since det M ¼ 1, Assumption 1 is globally satisfied Vector gð ^xþ, that must be chosen according to () as the last column of matrix M 1, results in The observer dynamics is gð ^xþ ¼½,, 1Š T : _^x1 ¼ ^x 2 ^x 3 _^x2 ¼ ^x 1 þ a ^x 2 _^x3 ¼ b þ ^x 3 ð ^x 1 cþ u: ð61þ ð62þ ð63þ The observer initial conditions are set customarily to zero, ^xðþ ¼½ Š T, and the observer control u is selected according to the third order quasi-continuoussliding mode controller: u ¼ 1 e y þ 2ðj_e y jþje y j 2=3 Þ 1=2 ð_e y þje y j 2=3 sign e y Þ j e y jþ2ðj_e y jþje y j 2=3 Þ 1=2 ð6þ with e y ¼ ^x 2 y, and with the output error derivative estimate ^_e y provided by the differentiator () with n ¼ 2, ¼, 1 ¼ 6, 2 ¼ 22 and ^_e y ¼ z 1 The actual and estimated state variables are shown in Figure 2 The convergence to zero of e y and its first two derivatives is shown in Figure 3 1 Observation in the presence of model uncertainties Now the assumption of perfectly knowing the system dynamics is dispensed with The conditions for robust state observation under model uncertainties are developed by applying the previously given Lemma 2 Let us establish which kind of model uncertainties and disturbances can be tolerated by the proposed scheme without losing the finite time exact state observation

9 International Journal of Control 11 1 State x 1 (continuous line) and ˆx 1 (dashed line) Output estimation error e y (t) State x 2 (continuous line) and ˆx 2 (dashed line) State x 3 (continuous line) and ˆx 3 (dashed line) Figure 2 State reconstruction for the perfectly known Ro ssler system For third-order systems, Assumption 2 reduces to the conditions dðhðxþþ fðxþ ~ ¼ ð6þ dðl f ðxþ hðxþþ fðxþ ~ ¼, ð66þ which for the considered output function takes the form 2 1 ~ 3 f 1 ðxþ 6 ~f 1 a 2 ðxþ ¼ f ) ~ 1 ðxþ ¼ ~f 2 ðxþ ¼: ~f 3 ðxþ The above conditions mean that the first two observer equations should match the form of the system model exactly, while an arbitrary model uncertainty in the third observer equation can be tolerated Thus it can be implemented the following simple observer: _^x1 ¼ ^x 2 ^x 3 _^x2 ¼ ^x 1 þ a ^x 2 _^x3 ¼ u: ð6þ Using the same parameters for the quasi-continuous controller and for the differentiator the state First derivative of the output estimation error ė y (t) Second derivative of the output estimation error ë y (t) Figure 3 The output error e y and its first and second derivative reconstruction presented in Figure is achieved, which confirms the correctness of the presented analysis 2 Observation in the presence of unknown inputs Consider now the system affected by an unknown input as described in (6) Assumption is satisfied if the next equalities hold: dðhðxþþdv ¼ dðl f ðxþ hðxþþdv ¼ : For the considered Ro ssler system, such equalities can be rewritten as 1 Dv ¼ : 1 a Notice that the distribution matrix D T ¼½ 1Š satisfies the above condition, thus the following controlled Ro ssler system is considered: _x 1 ¼ x 2 x 3 _x 2 ¼ x 1 þ ax 2 _x 3 ¼ b þ x 3 ðx 1 cþþvðtþ, ð68þ ð69þ ðþ

10 12 J Davila et al 2 State x 1 (continuous line) and ˆx 1 (dashed line) 1 State x 1 (continuous line) and ˆx 1 (dashed line) State x 2 (continuous line) and ˆx 2 (dashed line) State x 3 (continuous line) and ˆx 3 (dashed line) Figure State reconstruction for the uncertain case where v(t) is an uncertain exogenous input term which is taken as vðtþ ¼sinð 1 tþþ:sinð23 tþþ:1sinð2 tþ in the considered simulation test The observer takes the form _^x1 ¼ ^x 2 ^x 3 _^x2 ¼ ^x 1 þ a ^x 2 _^x3 ¼ b þ ^x 3 ð ^x 1 cþ u, ð1þ ð2þ ð3þ where u is given by (6) The state reconstruction is shown in Figure The approximate unknown-input reconstruction by means of (1) is presented in Figure 6 The exact state reconstruction by means of the method described in Section 63 is now presented The input u of the observer (1) (3) is now computed according to the following: 3, ¼ e ð3þ y þ 3 e y þðj_e y jþ:j~e y j 3= Þ 1=3 ð_e y þ :je y j 3= sign e y Þ j e y jþðj_e y jþ:je y j 3= Þ 2=3 1=2, N 3, ¼jey ð3þ jþ3 j e yjþðj_e y jþ:je y j 3= Þ 2=3 1=2, _u ¼ 3, ðþ N 3, State x 2 (continuous line) and ˆx 2 (dashed line) State x 3 (continuous line) and ˆx 3 (dashed line) Figure State reconstruction in the presence of the unknown input Unknown input (dashedline), reconstruction (continuous line) Figure 6 Approximate unknown input reconstruction with e y ¼ ^x 2 y, and with the output error derivative estimate ^_e y provided by the differentiator () with n ¼ 3, ¼ 3() 1/, 1 ¼ 2() 1/3, 2 ¼ 1() 1/2, 3 ¼ 11() and ^_e y ¼ z 1, ^ e y ¼ z 2, ^e ð3þ y ¼ z 3

11 International Journal of Control 13 1 State x 1 (continuous line) and ˆx 1 (dashed line) Unknown input (dashedline), reconstruction (continuous line) State x 2 (continuous line) and ˆx 2 (dashed line) State x 3 (continuous line) and ˆx 3 (dashed line) Figure State reconstruction in the presence of the unknown input by extended quasi-continuous method The unknown input reconstruction is made through the relationship ^vðx, tþ ¼ L n 1 f ð ^xþ 1 u, ðþ hð ^xþd which specialises as follows for the considered example ^vðx, tþ ¼ u: ð6þ The finite-time reconstruction of the state variables and of the unknown input are shown in Figures and 8, respectively Figure 9 compares the errors (difference between actual and estimated unknown input) in the two cases of the approximate and exact reconstruction methods, showing the higher accuracy of the latter 8 Conclusions An approach to finite-time state estimation for a class of non-linear uncertain systems has been proposed The method starts from constructing a non-linear Luenberger-like observer for the nominal part of the system dynamics in such a way that the observer output has full relative degree with respect to the Figure 8 Unknown input reconstruction by extended quasi-continuous method Unknown Input reconstruction errors: comparison Figure 9 Comparison between the unknown input estimation error using the approximate (dashed line) and exact (continuous line) reconstruction methods observer input injection signal, and continues by enforcing a high-order sliding mode in the observation error dynamics Necessary and sufficient geometric structural restrictions on the model uncertainties are given which guarantee the finitetime exact reconstruction of the system state also when some part of the original system dynamics is not modelled in the observer equations Methods for the approximate and exact reconstruction of unknown inputs acting on the system are presented Simulation results confirmed the effectiveness of the presented methods

12 1 J Davila et al Acknowledgements This work was supported in part by Mexican CONACyT (Consejo Nacional de Ciencia y Tecnologia), grant no 6819, Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnolgica (PAPIIT) UNAM, grant no 11126, Programa de Apoyo a Proyectos Institucionales para el Mejoramiento de la Ensenanza (PAPIME), UNAM, grant PE19, SRE Programa Ejecutivo de Cooperacion Mexico Italia 2 29, DGSCA-DTD-PASPA, Italian MUR Project 93 Real-time Simulation and Control of Combined-Cycle Power Plants and by EU th FWP, grant no PRODI References Ahmed-Ali, T, Floret, F, and Lamnabhi-Lagarrigue, F (2), Robust Identification and Control with Timevarying Parameter Perturbations, Mathematical and Computer Modelling of Dynamical Systems, 1, Ahmed-Ali, T, Kenne, G, and Lamnabhi-Lagarrigue, F (2), Nonlinear Systems Parameter Estimation using Neural Networks: Application to Syncronous Machines, Mathematical and Computer Modelling of Dynamical Systems, 13, Barbot, J, Djemai, M, and Boukhobza, T (22), Sliding Mode Observers, in Sliding Mode Control in Engineering, ser Control Engineering, eds W Perruquetti and J Barbot, New York: Marcel Dekker, pp Davila, J, Fridman, L, and Levant, A (2), Secondorder Sliding-mode Observer for Mechanical Systems, IEEE Transactions on Automatic Control,, Davila, J, Fridman, L, and Poznyak, A (26), Observation and Identification of Mechanical Systems via Second-order Sliding Modes, International Journal of Control, 9, Edwards, C, and Spurgeon, S (1998), Sliding Mode Control, London: Taylor and Francis Edwards, C, Spurgeon, S, and Tan, CP (22), On Development and Applications of Sliding Mode Observers, in Variable Structure Systems: Towards 21st Century, ser Lecture Notes in Control and Information Science, eds X Yu and J Xu, Berlin, Germany: Springer Verlag, pp Filippov, A (1988), Differential Equations with Discontinuous Right-hand Sides, Dordrecht, The Netherlands: Kluwer Academic Publishers Floquet, T, and Barbot, J (26), A Canonical Form for the Design of Unknown Input Sliding Mode Observers, in Advances in Variable Structure and Sliding Mode Control, ser Lecture Notes in Control and Information Science, eds C Edwards, E Fossas and L Fridman, Berlin: Springer Verlag, pp Fridman, L, Levant, A, and Davila, J (2), Observation of Linear Systems with Unknown Inputs via High-order Sliding-modes, International Journal of System Science, 38, 3 91 Fridman, L, Shtessel, Y, Edwards, C, and Yan, X (28), Higher-order Sliding-mode Observer for State Estimation and Input Reconstruction in Nonlinear Systems, International Journal of Robust Nonlinear Control, 18, Gauthier, J, Hammouri, H, and Othman, S (1992), A Simple Observer for Nonlinear Systems Applications to Bioreactors, IEEE Transactions on Automatic Control, 3, 8 88 Haskara, I, and Ozguner, U (2), Stability Analysis of a Sliding Observer Based Robust Output Tracking Control Design for a Nonlinear System, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, pp Isidori, A (1996), Nonlinear Control Systems, London: Springer-Verlag Levant, A (23), High-order Sliding Modes: Differentiation and Output-feedback Control, International Journal of Control, 6, Levant, A (2), Quasi-continuous High-order Slidingmode Controllers, IEEE Transactions on Automatic Control,, Levant, A (26), Homogeneous Quasi-continuous Slidingmode Control, Lecture Notes in Control and Information Sciences, 33, Luenberger, DG (196), Observing the State of a Linear System, IEEE Transactions on Military Electronics, MIL-8(2), 8 Poznyak, A (23), Stochastic Output Noise Effects in Sliding Mode Estimations, International Journal of Control, 6, Rocha-Cozatl, E, and Moreno, J (2), Dissipativity and Design of Unknown Input Observers for Nonlinear Systems, in Proceedings of the 6th IFAC-Symosium on Nonlinear Control Systems, Stuttgart, Germany, pp Ro ssler, O (196), An Equation for Continuous Chaos, Physics Letters A,, 39 8 Utkin, V, Guldner, J, and Shi, J (1999), Sliding Modes in Electromechanical Systems, London: Taylor and Francis