An algebraic framework for the design of nonlinear observers with unknown inputs

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1 An algebraic ramework or the design o nonlinear observers with unknown inputs Jean-Pierre Barbot, Michel Fliess, Thierry Floquet To cite this version: Jean-Pierre Barbot, Michel Fliess, Thierry Floquet. An algebraic ramework or the design o nonlinear observers with unknown inputs. 46th IEEE Conerence on Decision and Control, 7, New Orleans, United States. 7. <inria-17366> HAL Id: inria Submitted on 15 Sep 7 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not. The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés.

2 An algebraic ramework or the design o nonlinear observers with unknown inputs Jean-Pierre BARBOT, Michel FLIESS and Thierry FLOQUET Abstract The observability properties o nonlinear systems with unknown inputs are characterized via dierentially algebraic techniques. State variables and unknown inputs are estimated thanks to a new algebraic numerical dierentiator. It is shown through an academic example and a concrete casestudy that the proposed scheme can be applied to systems that ail to ulill some usual structural assumptions. I. INTRODUCTION In this note, the problem o designing unknown input observers or nonlinear MIMO systems is discussed rom a dierentially algebraic standpoint. The design o an observer or a multivariable system partially driven by unknown inputs has been widely studied in the literature. Indeed, speciic applications o asymptotic and inite time observers or systems with unknown inputs can be ound in a large number o ields such as: robust reconstruction o state variables or systems subject to exogenous disturbances, parameter identiication, ault detection and identiication or secure communication. The design o observers or nonlinear systems is a challenging problem even or accurately known systems that has received a considerable amount o attention. In most approaches, nonlinear coordinate transormations are employed to transorm the nonlinear system into block triangular observer canonical orms 1. Then, high gain [17], backstepping [8], or sliding mode observers [], [31], [4] can be designed. Some o them are concerned with applications in the ield o ault detection and identiication, in particular the nonlinear Fundamental Problem o Residual Generation FPRG [15], [18], [3]. Actually, structural conditions have to be met such that the nonlinearities and the unknown inputs act only on the last dynamics o each triangular orm. This unknown input-output injection quasi-linearization problem is similar to the eedback linearization problem or MIMO systems see [3], Chapter 5. In particular, this implies that The authors are supported by the Projet ALIEN, INRIA-Futurs. J.P. Barbot is with Équipe Commande des Systèmes ECS, ENSEA, 6 Avenue du Ponceau, 9514 Cergy, France. barbot@ensea.r M. FLiess is with Équipe MAX, LIX CNRS, UMR 7161, École polytechnique, 9118 Palaiseau, France. Michel.Fliess@polytechnique.edu T. Floquet is with LAGIS CNRS, UMR 8146, École Centrale de Lille, BP 48, Cité Scientiique, Villeneuve-d Ascq, France Thierry.Floquet@ec-lille.r 1 In the case o linear systems, easily veriiable system theoretic conditions, which are necessary and suicient or the existence o a linear unknown input observer, have been established see e.g. [] and the reerence therein. One possible statement o these conditions is that the transer unction matrix between the unmeasurable input and the measured outputs must be minimum phase and o relative degree one. some coupling matrix between the outputs and the unknown inputs should be invertible. This assumption is also known as the observability matching condition. One can reer to [16] or a ormulation o this condition in terms o observability indices and the unknown input characteristic indexes or relative degrees. In this paper, a dierentially algebraic approach, which is extending a module-theoretic setting or linear systems in [5], is considered or the design o unknown input observers. A necessary and suicient condition or their existence is given in terms o input-output invertibility and zero dynamics. Note that the design o unknown input observers is rarely seen as a let invertibility problem. In general, one usually tries to compute let inverse systems in input reconstruction problems [1] where observing the whole state variables is not necessarily required and algorithms that provide such inverse systems are given or instance in [6], [7], [39]. The approach given in this paper speciically shows that the restrictive structure o observers, imposed by the matching condition, can be weakened i additional independent output signals are generated by means o dierentiators o the available measurements. For this, a new type o non-asymptotic observers, also based on dierential algebra concepts and already developed in [13], is recalled. The given procedure is illustrated via an academic example and an application to secure communication. II. AN ALGEBRAIC SETTING FOR NONLINEAR OBSERVABILITY 3 A. Dierential algebra Commutative algebra, which is mainly concerned with the study o commutative rings and ields, provides the right tools or understanding algebraic equations see, e.g., [1]. Dierential algebra, which was mainly ounded by Ritt [35] and Kolchin [5], extends to dierential equations concepts and results rom commutative algebra 4. 1 Basic Deinitions: A dierential ring R see, e.g., [5] and [4] will be here a commutative ring 5 o characteristic See [34] or a survey about let invertibility o nonlinear control systems, see [3], [37], [38], [45] or pioneering works on the existence the construction o an inverse system in the linear case, and see [4], [39] or generalizations to nonlinear systems. We ollow here the dierentially algebraic setting [11] where necessary and suicient conditions were expressed in terms o the dierential output rank. 3 See also [13], [14], [36], [4] or more details on the dierential algebraic approach to nonlinear systems. 4 Algebraic equations are dierential equations o order. 5 See, e.g., [1], [4] or basic notions in commutative algebra.

3 zero, which is equipped with a single derivation d dt : R R such that, or any a,b R, d dt d dt a + b = ȧ + ḃ, ab = ȧb + aḃ. where da dt = ȧ, dν a dt = a ν, ν. A dierential ield is ν a dierential ring which is a ield. A constant o R is an element c R such that ċ =. A dierential ring resp. ield o constants is a dierential ring resp. ield which only contains constants. The set o all constant elements o R is a subring resp. subield, which is called the subring resp.subield o constants. Field extensions: All ields are assumed to be o characteristic zero. A dierential ield extension K/k is given by two dierential ields k, K, such that k K. Write k S the dierential subield o K generated by k and a subset S K. Assume also that the dierential ield extension K/k is initely generated, i.e., there exists a inite subset S K such that K = k S. An element a o K is said to be dierentially algebraic over k i, and only i, it satisies an algebraic dierential equation with coeicients in k: there exists a non-zero polynomial P over k, in several indeterminates, such that Pa,ȧ,...,a ν =. It is said to be dierentially transcendental over k i, and only i, it is not dierentially algebraic. The extension K/k is said to be dierentially algebraic i, and only i, any element o K is dierentially algebraic over k. An extension which is not dierentially algebraic is said to be dierentially transcendental. The ollowing result is playing an important rôle: Proposition.1: The extension K/k is dierentially algebraic i, and only i, its transcendence degree is inite. A set {ξ ι ι I} o elements in K is said to be dierentially algebraically independent over k i, and only i, the set {ξ ι ν ι I,ν } o derivatives o any order is algebraically independent over k. I a set is not dierentially algebraically independent over k, it is dierentially algebraically dependent over k. An independent set which is maximal with respect to inclusion is called a dierential transcendence basis. The cardinalities, i.e., the numbers o elements, o two such bases are equal. This cardinality is the dierential transcendence degree o the extension K/k; it is written di tr deg K/k. Note that this degree is i, and only i, K/k is dierentially algebraic. B. Nonlinear systems 1 Generalities: Let k be a given dierential ground ield. A nonlinear input-output system is a initely generated dierential extension K/k, where K contains the sets u = u 1,...,u m and y =,...,y p o control and output variables, and such that the extension K/k u is dierentially algebraic. Assume or simplicity s sake that the control variables are independent, i.e., that u is a dierential transcendence basis o the extension k u /k. Remark.1: Remember that dierential algebra considers algebraic dierential equations, i.e., dierential equations which only contain polynomial unctions o the variables and their derivatives up to some inite order. This is o course not always the case in practice. In the example o Section IV, or instance, appears the transcendental unction sin θ. As already noted in [14], we recover algebraic dierential equations by introducing tan θ. State-variable representation: We know, rom proposition.1, that the transcendence degree o the extension K/k u is inite, say n. Let x = x 1,...,x n be a transcendence basis. Any derivative ẋ i, i = 1,...,n, and any output variable y j, j = 1,...,p, are algebraically dependent over k u on x: A i ẋ i,x = B j y j, x = i = 1,...,n j = 1,...,p where A i k u [ẋ i, x], B j k u [y j,x], i.e., the coeicients o the polynomials A i, B j depend on the control variables and their derivatives up to some inite order. Remark.: Note the dierence with the usual statevariable representation ẋ = Fx, u y = Hx 3 Invertibility and zero dynamics: System K/k is said to be let resp. right invertible [11] i, and only i, the dierential transcendence degree o the extension k y /k is equal to m resp. p. It is easy to check that let invertibility is equivalent to the act that the extension K/k y is dierentially algebraic. I the system is square, i.e., m = p, let and right invertibilities coincide and the system is said to be invertible. Call the extension K/k y the zero, or the inverse, dynamics compare with [3]. This dynamics is said to be trivial i, and only i, the extension K/k y is algebraic. The next result is clear: Proposition.: An input-output system with a trivial zero dynamics is let invertible. 4 Dierential latness: System K/k is said to be dierentially lat see, e.g., [14], [36], [4] i, and only i, the algebraic closure K o K is equal to the algebraic closure o a purely dierentially transcendental extension o k. It means in other words that there exists a inite subset z = {z 1,...,z m } o K pure such that z 1,...,z m are dierentially algebraically independent over k, z 1,...,z m are algebraic over K, any system variable is algebraic over k z 1,...,z m. z is a lat, or linearizing, output. For a lat dynamics, it is known that the number m o its elements is equal to the number o independent control variables. Assume that K/k is an input-output system. The next property is clear: Proposition.3: An input-output system is lat and its output is a lat output i, and only i, it is square with a trivial zero dynamics. C. Observability and reconstructors 1 Generalities: According to [8], [9], system K/k is said to be observable i, and only i, the extension K/k u,y is algebraic. It means in plain words that observability is 1

4 equivalent to the ollowing act: Any system variable, a state variable or instance, may be expressed as a unction o the input and output variables and o their derivatives up to some inite order. Call this unction a reconstructor o the corresponding system variable. This new deinition o observability is roughly equivalent see [8], [9] or details 6 to its usual dierential geometric counterpart due to [19]. Unknown inputs: A system variable ξ K is said to be observable with unknown inputs i, and only i, it belongs to the algebraic closure o k y. More generally, the system K/k is said to be observable with unknown inputs i, and only i, the algebraic closure o k y in K coincides with K. It means in other words that any system variable, a state variable or an input variable or instance, can be expressed as unction o the output variables and their derivatives up to some inite order. Call this unction the input-ree reconstructor or the ast input-ree observer o the reconstructor. The ollowing property ollows at once rom Propositions. and.3: Proposition.4: An input-output system is observable with unknown inputs i, and only i, its zero dynamics is trivial. I moreover the system is square, it is lat and its output is a lat output. III. NUMERICAL DIFFERENTIATION 7 A. Polynomial time signals Consider the real-valued polynomial unction x N t = N ν= xν tν ν! R[t], t, o degree N. Rewrite it in the well known notations o operational calculus see, e.g., [44]: N x ν X N s = s ν+1 ν= We know utilize d ds, which corresponds in the time domain to the multiplication by t. Multiply both sides by dα ds s N+1, α α =, 1,...,N. The quantities x ν, ν =, 1,...,N are given by the triangular system o linear equations: d α s N+1 N X N ds α = dα ds α x ν s N ν ν= The time derivatives, i.e., s µ dι X N ds, µ = 1,...,N, ι ι N, are removed by multiplying both sides o Equation by s N, N > N. B. Analytic time signals Consider a real-valued analytic time unction deined by the convergent power series xt = ν= xν tν ν!, where t < ρ. Approximate xt in the interval, ε, < ε ρ, by its truncated Taylor expansion x N t = N ν= xν tν ν! o order N. Introduce the operational analogue o xt, i.e., Xs = x ν ν s. Denote ν+1 6 The dierential algebraic and the dierential geometric languages are not equivalent. We cannot thereore hope or a one-to-one bijection between deinitions and results which are expressed in those two settings. 7 See [13], [9] or more details and or related reerences on numerical dierentiation. by [x ν ] en t, ν N, the numerical estimate o x ν, which is obtained by replacing X N s by Xs in Eq.. It can be shown [13] that a good estimate is obtained in this way. C. Noisy signals Assume that our signals are corrupted by additive noises. Those noises are viewed here as highly luctuating, or oscillatory, phenomena. They may be thereore attenuated by low-pass ilters, like iterated time integrals. Remember that those iterated time integrals do occur in Equation ater multiplying both sides by s N, or N > large enough. Remark 3.1: See [1] or a precise mathematical oundation, which is based on nonstandard analysis. A highly luctuating unction o zero mean is then deined by the ollowing property: its integral over a inite time interval is ininitesimal, i.e., very small 8. Remark 3.: Resetting and utilizing sliding time windows permit to estimate derivatives o various orders at any sampled time instant. IV. EXAMPLES In this section, two examples, that do not satisy the conditions required by usual nonlinear unknown input observers, are presented. A. A lat system The ollowing nonlinear system ẋ 1 ẋ ẋ 3 = ẋ 4 y1 y = x sinx x 4 + m 1 x 4 + m 1 x 3 + cosx 3 + m h1 h = x1 x 3 3 where m 1, m are unknown input variables, has relative degree ρ = {,1} with respect to the output y = [,y ] T and the coupling matrix Lg1 L ρ 1 1 h 1x L g L ρ ««1 1 h 1x 1 Γx = L g1 L ρ 1 h x L g L ρ 1 = h x 1 is singular. According to [16], 3-4 cannot be transormed into a canonical block triangular observable orm. Usual observers ail thereore to estimate the state. Easy but lengthy computations, which will not be reported here due to a lack o space, yield x = ẏ 1 x 4 = 1 ẏ ÿ 1 sinẏ 1 m 1 = 1 ẏ + ÿ 1 + sinẏ 1 m = 1 ÿ y 3 1 ÿ 1 cosẏ 1 + y cosy They show according to Section II-B.4 that the system 3-4 is lat and that the couple {,y } is a lat output. 8 This approach applies as well to multiplicative noises see [1]. The assumption on the noises being only additive is thereore unnecessary. 4

5 Moreover the above equations may be interpreted according to Section II-C. as input-ree reconstructors or ast inputree observers. The simulation results given Fig. 1,, 3 and 4 were obtained using ast algebraic dierentiators. Random noises were added on both outputs m and mo 1.5 x and xo Fig. 4. m and its estimate Fig. 1. x and its estimate x4 and x4o switch keying see, e.g., [6], chaotic modulation see, e.g., [43],... Hereater is highlighted the interest o our approach or the chaotic modulation method in the case o multi-input multi-output system. Note that the algebraic viewpoint as well as the use o the algebraic derivative method or the eicient and ast computation o accurate approximations to the successive time derivatives o the transmitted observable output has already been successully developed or the state estimation problem associated with the chaotic encryptiondecoding problem in the case o some classes o nonlinear systems [41]. Let us consider the ollowing chaotic system given in [33]: Fig.. x 4 and its estimate ẋ 1 = ax x 1 + x x 3 ẋ = bx 1 + x x 1 x 3 ẋ 3 = cx 3 ex 4 + x 1 x 1 m1 and m1o ẋ 4 = dx 4 + x 3 + x 1 x In order to send the conidential messages m 1 and m, the ollowing transmitter is designed: 4 ẋ 1 = ax x 1 + x x 3 + µ 1 x 1,x,x 3 m ẋ = bx 1 + x x 1 x 3 Fig. 3. m 1 and its estimate ẋ 3 = cx 3 ex 4 + x 1 x + µ x 1, x,x 3 m 1 ẋ 4 = dx 4 + x 3 + x 1 x 3 + µ 3 xm 1 + µ 4 xm B. Application to secure communication Since the work o Pecora and Caroll [3], it is known that the problem o secure communication can be investigated with respect to the synchronization o chaotic systems or usual cryptography methods see, e.g., []. The idea is to use the output o a particular dynamical chaotic system that masks the message to drive the response o a generally identical system that recovers the message so that they oscillate in a synchronized manner. Secure communication by the synchronization o chaotic systems has dierent design methods such as: addition see, e.g., [7], chaotic The chosen outputs are = x 1, y = x and m 1 and m should be seen as two unknown inputs. This system ails to ulil the observability matching condition. Thus, the complexity o the encryption algorithm is increased, because well known observer-based approach can not be used here. In this example, the key similarly to a usual ciphering problem is the parameter vector [ a b c d e ] T, and the µi are chosen in order to guarantee the conidentiality and such that the chaotic behavior o the system is not destroyed. For the conidentiality o the system, it is important or example to veriy that the parameters are not identiiable with known inputs known-plaintext attack.

6 As in Section IV-A, easy calculations yield x 3 = m 1 = by1 + y ẏ 1 µ 1, y, ẏ ẏ 1 a y + y «y a x 4 = 1 e cy a + y + µ, y, ẏ m 1 5 m = ẋ4 + dx 4 ya + y a µ 3, y, ẏ 1, ẏ, ÿ m 1 µ, y, ẏ 1, ẏ, ÿ where y a = x 1 x 3 = b + y ẏ and µ i. = µ i.. The simulation results were obtained with the ollowing parameters a = 4.5, b = 4, c = 13, d =, e = 5, = 4, µ 1 = µ 4 = 1 and µ = µ 3 =. Moreover, due to the observability singularity in =, the numerical dierentiators must be switched o in the vicinity o =. In this case, the previous estimations o the derivatives are used in the set o equations 5. Moreover, due to the delay introduced by the estimation rame some extra delays o s are introduced in 5 in order to compensate or the inluences o the irst one. Finally, a reshaping procedure is used on m to overcome the distortion due to the observability singularities. Figure 5 illustrates the chaotic behaviour o the system in the phase plot while Figures 6-9 show that one can estimate the state and recover the hidden messages. The peaks are due to the observability singularities as it is highlighted in Figure 8 where m 1, the estimate o m 1 and x 1 are given Fig Fig. 7. x 4 in blue and its estimate in red m 1 in blue, its estimate in red and x 1 in green Phase plot x1, x and x Fig. 9. m in blue and its estimate in red Fig. 5. Phase plot sponding conditions or observability and unknown input identiiability conditions were given in an algebraic setting. Then, it was shown that, using a new type o ast algebraic dierentiators, both the state and the unknown inputs can be recovered, even when some structural conditions usually required by other methods are not ulilled. The eiciency and robustness o the proposed scheme with respect to noise measurements were highlighted via an academic example. The practical interest o the method was also demonstrated with an application to secure communication Fig. 6. x 3 in blue and its estimate in red V. CONCLUSION We considered in this paper the problem o the unknown input observer design or nonlinear systems. The corre- REFERENCES [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, [] J.-P. Barbot, T. Boukhobza, M. Djemaï, Sliding mode observer or triangular input orm, IEEE Con. Decision Control, Kobe, [3] R.W. Brockett, M.D. Mesarovic, The reproducibility o multivariable systems, J. Math. Anal. Appl., vol. 11, pp , 1965.

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