High-gain observers for a class of 2 × 2 quasilinear hyperbolic systems with 2 different velocities

11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 210–215

High-gain observers for a class of 22 × 22 High-gain observers for a class of × High-gain observers for a class of 2 × 2 quasilinear hyperbolic with High-gain forsystems a class of 2 ×222 quasilinearobservers hyperbolic systems with quasilinear hyperbolic systems with 2 different velocities quasilinear hyperbolic systems with 2 different velocities different velocities different velocities ∗ ∗ Gildas Besan¸ Constantinos Kitsos con ∗∗∗ Christophe Prieur ∗∗∗ ∗ Constantinos Kitsos ∗∗ Gildas Besan¸ con ∗∗ Christophe Prieur ∗∗ Constantinos Kitsos Gildas Besan¸ con Christophe Prieur ∗ ∗ ◦ ∗ ∗ Constantinos Kitsos Gildas Besan¸ con Christophe Prieur ◦ , GIPSA-lab, ∗ Univ. Grenoble Alpes, CNRS, Grenoble INP 38000 ∗ ◦ Univ. Grenoble Alpes, CNRS, Grenoble INP , GIPSA-lab, 38000 ◦ ∗ ∗Grenoble, Univ. Grenoble CNRS, Grenoble INP◦ , GIPSA-lab, 38000 FranceAlpes, (e-mails: {konstantinos.kitsos, gildas.besancon, gildas.besancon, ∗Grenoble, France (e-mails: {konstantinos.kitsos, Univ. Grenoble Alpes, CNRS, Grenoble INP◦ , GIPSA-lab, 38000 christophe.prieur }@gipsa-lab.grenoble-inp.fr). Grenoble, France (e-mails: {konstantinos.kitsos, gildas.besancon, }@gipsa-lab.grenoble-inp.fr). Grenoble,christophe.prieur France (e-mails: {konstantinos.kitsos, ◦ Institute of Engineering Univ. Grenoble Alpesgildas.besancon, christophe.prieur }@gipsa-lab.grenoble-inp.fr). ◦ Institute Institute of of Engineering Engineering Univ. Univ. Grenoble Grenoble Alpes Alpes ◦ christophe.prieur }@gipsa-lab.grenoble-inp.fr). ◦ Univ. ◦ Institute Institute of of Engineering Engineering Univ. Grenoble Grenoble Alpes Alpes ◦ Institute of Engineering Univ. Grenoble Alpes

Abstract: Abstract: In In this this work work we we extend extend recently recently proposed proposed observer observer designs designs based based on on high-gain high-gain to to a first order quasilinear hyperbolic system of laws. This of Abstract: In this extend recently proposed designs on high-gain to a more more general general firstwork orderwe quasilinear hyperbolic systemobserver of balance balance laws.based This class class of systems systems Abstract: Inan this work we extend recently proposed observer designs based onvelocities high-gain to is written in observable form with two states, two different characteristic and a more general first order quasilinear hyperbolic system of balance laws. This class of systems is written in anfirst observable form withhyperbolic two states, two of different characteristic velocities and a more general order quasilinear system balance laws. This class of systems distributed measurement. The exponential stability of the related observation error is fully is written in an observable form with two states, two different characteristic velocities and distributed measurement. The exponential stability of related observation error is written in an observable form with two states, two different characteristic and distributed measurement. The exponential stability of the the related observation velocities error is is fully fully established by means of analysis. established by means of Lyapunov-based Lyapunov-based analysis. distributed by measurement. The exponential stability of the related observation error is fully established means of Lyapunov-based analysis. © 2019, IFACby(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. established means observers, of Lyapunov-based analysis. Keywords: high-gain quasilinear hyperbolic systems of balance laws, Lyapunov Keywords: high-gain observers, quasilinear hyperbolic 1 Keywords: observers, quasilinear hyperbolic systems systems of of balance balance laws, laws, Lyapunov Lyapunov 1 analysis, stability. 1 exponential exponential stability. analysis, H Hhigh-gain 1 1 Keywords: observers, quasilinear hyperbolic systems of balance laws, Lyapunov exponential stability. analysis, Hhigh-gain analysis, H 1 exponential stability. 1. INTRODUCTION commute with the matrix of the characteristic velocities. 1. INTRODUCTION commute with matrix the velocities. 1. INTRODUCTION commute with the the matrix of of stability the characteristic characteristic For considering problems in For this this reason, reason, considering stability problems velocities. in Bastin Bastin 1. INTRODUCTION commute with the matrixand of stability the characteristic velocities. and Coron [2016], Coron Bastin [2015] and in Prieur For this reason, considering problems in Bastin and Coron [2016], Coron and Bastin [2015] and in Prieur The classical high-gain observer design for finite-dimensional The classical high-gain observer design for finite-dimensional and Foralthis reason, considering stability problems in Bastin et [2014] (Proposition 2.1) and other approaches of Coron [2016], Coron and Bastin [2015] and in Prieur nonlinear systems has gained great academic interest et al [2014] (Proposition 2.1) and other approaches of The classical high-gain observer design for finite-dimensional and Coron [2016], Coron and Bastin [2015] and in Prieur nonlinear systems has gained great academic interest et al [2014] (Proposition 2.1) and other approaches of these authors, either such a commutativity property is The classical high-gain observer design for finite-dimensional through the last decades. They apply to a large class these authors, either such a commutativity property is nonlinear systems has gained great academic interest et al [2014] (Proposition 2.1) and other approaches of through the last decades. They apply to a large class these authors, either such a commutativity property assumed to be satisfied, or a diagonal Lyapunov matrix is nonlinear systems has gained great academic interest assumed to be satisfied, or a diagonal Lyapunov matrix is through the last decades. They apply to a large class of cases corresponding to uniformly observable systems these either such commutativity property of cases the corresponding to uniformly observable systems chosen.authors, In our the matrix cannot assumed be case, satisfied, or aadiagonal Lyapunov matrix is through decades. TheyGauthier apply to class Gauthier and Bornard [1981], et [1992]. It Into our case, the Lyapunov Lyapunov matrix cannot commute commute of cases corresponding to uniformly observable systems Gauthier andlast Bornard [1981], Gauthier et aal al large [1992]. It chosen. assumed to be satisfied, or a diagonal Lyapunov matrix is with the matrix of the characteristic velocities, since it chosen. In our case, the Lyapunov matrix cannot commute of cases corresponding to uniformly observable systems has been extensively studied in the literature remains the matrix of the characteristic velocities, since it Gauthier and Bornard [1981], Gauthier et aland [1992]. It with chosen. In our case, the Lyapunov matrix cannot commute has been extensively studied in the literature and remains with the matrix of the characteristic velocities, since it is required to satisfy silmutaneously a matrix Lyapunov Gauthier and Bornard [1981], Gauthier et aland[1992]. It is required to satisfy silmutaneously a matrix Lyapunov widely considered, see Khalil [2017] and references therein. has been extensively studied in the literature remains with the matrix of the characteristic velocities, since it widely considered, see Khalil [2017] and references therein. is required to satisfy silmutaneously a matrix Lyapunov equation and the involved stabilizable matrix, which dehas been extensively studied literature and remains equation and the involved stabilizable matrix, which dewidely considered, Khalilet [2017] and references therein. In recent paper Kitsos al [2018], we this In the the recent papersee Kitsos etin althe [2018], we extended extended this equation is required to the satisfy silmutaneously a matrix, matrix Lyapunov scribes the linear part of balance does not involved dewidely considered, see Khalil [2017] and references therein. approach to a class of hyperbolic systems, for which first scribes theand linear part of the thestabilizable balance laws, laws, does which not allow allow In the recent paper Kitsos et al [2018], we extended this approach to a class of hyperbolic systems, for which first equation and the involved stabilizable matrix, which dediagonal stability to hold. Due to this technical limitation, scribes the linear part of the balance laws, does not allow In the recent et al [2018], we for extended this diagonal stability to hold. Due to this technical limitation, results on high-gain design have proposed approach apaper class Kitsos ofobserver hyperbolic systems, which first scribes the linear part of the balance does not allow results onto high-gain observer design have been been proposed diagonal stability to hold. Due to [2018], this laws, technical limitation, in the first approach Kitsos et al the characteristic approach to a class of hyperbolic systems, for which first for a case uniformly observable systems, writin the first approach KitsosDue et al the characteristic results on high-gain observer design have been proposed stability to hold. to [2018], thisIn for a particular particular case of uniformly observable systems, writvelocities were considered Kitsos et al in the first approach Kitsosidentical. et al [2018], the characteristic results on n design have been velocities were considered identical. Intechnical Kitsos etlimitation, al [2019] [2019] for ofobserver uniformly observable systems, writ- diagonal ten an × quasi-linear hyperbolic system of balance ten aas asparticular an nhigh-gain ×n ncase quasi-linear hyperbolic system ofproposed balance in the first approach Kitsos et al [2018], the characteristic we considered a more general problem for a 2 × 2 system, velocities were considered identical. In Kitsos et al [2019] for a particular case of uniformly observable systems, writlaws and considering distributed measurements. In a more we considered a more general problem for a 2 × 2 system, ten as an n × n quasi-linear hyperbolic system of balance laws and considering distributed measurements. In a more velocities were identical. Inwere Kitsos [2019] where space derivatives of the problem output in the we considered aconsidered more general for ainjected 2 et × 2alsystem, ten asand an n ×Kitsos n quasi-linear hyperbolic system of abalance recent work al we where space derivatives of output were in the laws considering distributed measurements. we considered a moreasgeneral forin ainjected 2order × 2 system, recent work Kitsos et et al [2019], [2019], we considered consideredIn aaa more more where space derivatives of the the problem output were injected inconthe observer’s equations correction terms, to laws and considering distributed measurements. In more general case, where a 2 × 2 system is written in an observobserver’s equations as correction terms, in order to conrecent work Kitsos et al [2019], we considered a where space derivatives of the output were injected in the general case, where a 2 × 2 system is written in an observfront this difficulty. The contribution of the present paper observer’s equations as correction terms, in order to conrecent work Kitsos al2from [2019], considered more front this difficulty. The contribution of the present paper general case, where aet2 × system iswewritten an aobservable differentiating from Kitsos et al in [2018] where able form, form, differentiating Kitsos et al [2018] where observer’s equations as correction in present order conis twofold: First, employing aa terms, triangular lineartopaper coorthis difficulty. contribution of the general case, where a 2 ×identical 2from system is written an observlinear coorKitsos et al in [2018] where front we needed to consider chatacteristic velocities. is twofold: First, by byThe employing triangular able form, differentiating front this difficulty. The contribution of the present paper we needed to consider identical chatacteristic velocities. linear coordinates transformation technique, we avoid the restrictive is twofold: First, by employing a triangular able form, differentiating from Kitsos et al for [2018] where dinates transformation technique, we avoid the restrictive There exist some results on observer design hyperbolic we needed to consider identical chatacteristic velocities. is twofold: byspace employing a we triangular coorThere exist some results on observer design hyperbolic dinates transformation technique, avoid thelinear injection of output’s derivatives, in to confront we needed to consider chatacteristic velocities. injection of First, output’s space derivatives, in order order torestrictive confront There exist some resultsidentical on observer design for for hyperbolic systems in the literature, mainly considering the full state dinates transformation technique, we avoid the restrictive systems in the literature, mainly considering the full state the problem of distinct characteristic velocities. Second, injection of output’s space derivatives, in order to confront There some results on design Amongst for the problem of distinct characteristic velocities. Second, we we systems the literature, mainly considering thehyperbolic full state vector on boundaries as measurement. oth1 vector exist oninthe the boundaries asobserver measurement. Amongst oth- injection of output’s space derivatives, in order to confront 1 prove a stronger result of H exponential stability of the the problem of distinct characteristic velocities. Second, we 1 systems in the literature, mainly considering the Hasan full state ers, one can refer to Di Meglio et al [2013] and et prove a stronger result of H exponential stability of the vector on the boundaries as measurement. Amongst oth1 1 problem of distinct velocities. Second, we ers, oneoncan refer to Di Meglio et al [2013] Amongst and Hasan et the observer error, contrary to the exponential stability result prove a stronger resultcharacteristic of H exponential stability of the vector the boundaries as measurement. othal [2016] for the backstepping design, to Besan¸ on et al observer error, contrary to the ers, one can refer to Di Meglio et al [2013] and ccHasan et 1 exponential stability result prove a stronger result of H exponential stability of the al [2016] for the backstepping design, to Besan¸ on et al observer error, contrary to the exponential stability result ers, one can refer Di Meglio et alto[2013] and in the the supremum supremum norm norm (Kitsos (Kitsos et et al al [2018]). [2018]). al [2016] for the to backstepping design, to Besan¸ cHasan on[2013] et et al in [2006] for a discretization approach, Castillo et al observer error, contrary to the exponential stability result [2006] for aafor discretization approach, to Castillo et al [2013] in the supremum norm (Kitsos et al [2018]). al [2016] the backstepping design, to Besan¸ c on et al [2006] for discretization approach, to Castillo et al [2013] for direct infinite-dimension-based Lyapunov techniques problem is illustrated in details in Section 2. The for direct infinite-dimension-based Lyapunov techniques The in the supremum norm (Kitsos et al [2018]). The problem is illustrated in details in Section 2. [2006] forBesan¸ ainfinite-dimension-based discretization approach, to Castillo et ettechniques al [2016] [2013] The (see also ccon et al [2013]) or to Nguyen al problem is illustrated infull details in for Section 2. The The for direct Lyapunov theoretical analysis and the proofs the observer theoretical analysis and the full proofs for the observer (see also Besan¸ on et al [2013]) or to Nguyen et al [2016] for direct infinite-dimension-based Lyapunov techniques The problem isdevelop illustrated infull details inSection Section 2.where The optimization methods. For semigroup-based methods (see also Besan¸ c on et al [2013]) or to Nguyen et al [2016] design that we are presented in 3, theoretical analysis and the proofs for the observer for methods. For methods design that analysis we develop in Section 3, where (see also Besan¸ con etChristophides al [2013]) or to Nguyen et al [2016] for optimization optimization methods. For semigroup-based semigroup-based methods theoretical andare thepresented full proofs for the observer see Curtain [1982], and Daoutidis [1996] Theorem 2 constitutes the main result. In addition, design that we develop are presented in Section 3, where see Curtain [1982], Christophides and Daoutidismethods [1996] Theorem 2 constitutes the main result. In addition, an an for For semigroup-based that develop are in Section 3, where and Schaum al [2015]. Theorem 2 we constitutes the presented main result. In addition, an see Curtain [1982], Christophides and Daoutidis [1996] design example illustrates the nature of our methodology. Some andoptimization Schaum et et almethods. [2015]. example illustrates the nature of our methodology. Some see Curtain [1982], Christophides and Daoutidis [1996] Theorem 2 constitutes the main result. In addition, example illustrates the nature of our methodology. Some and Schaum et al [2015]. conclusions and perspectives are discussed in Section 4. conclusions and perspectives are discussed in Section 4.an The present aims and et al [2015]. example illustrates the natureare ofdiscussed our methodology. The Schaum present paper paper aims at at providing providing sufficient sufficient conditions conditions conclusions and perspectives in SectionSome 4. n for observer for class The present paper aims design at providing conditions Notation: n , |x| denotes its usual EuFor aa perspectives given x ∈ R n for high-gain high-gain observer design for aa sufficient class of of quasilinear quasilinear conclusions and are discussed in Section 4. Notation: For given x ∈ R , |x| denotes its usual Eun m×n n , |x| denotes The presentsystems paper aims at providing sufficient conditions hyperbolic of balance laws. of the Notation: For aFor given x ∈ R its usual Eu-,, for high-gain observer design for a The classdifficulty of quasilinear m×n clidean norm. aa given constant matrix A ∈ R m×n clidean norm. For given constant matrix A ∈ R hyperbolic systems of balance laws. The difficulty of the n m×n T m×n for high-gain observer design forfact a The class of quasilinear Notation: For a given x ∈ R , |x| denotes its usual Eu-, present approach comes from the that the Lyapunov hyperbolic systems of balance laws. difficulty of the T A denotes its transpose, |A| := sup {|Ax| , |x| = 1} is clidean norm. For a given constant matrix A ∈ R T present approach comes from the fact that the Lyapunov A its For transpose, |A| :=A+A supT {|Ax| , |x| =∈1}Rm×n is its its, T denotes T hyperbolic systems of we balance laws. of the present comes from the factThe thatdifficulty theexistence Lyapunov clidean norm. a given constant matrix A stability analysis that employ requires the of T {|Ax| denotes its transpose, |A| := sup , |x| = 1} is its T stabilityapproach analysis that we employ requires the existence of A A+AT stands for its syminduced norm and Sym(A) = A+A T 2 T {|Ax| induced norm and Sym(A)|A|=:=A+A stands its sympresent approach comes the requires fact that theexistence Lyapunov A+A A denotes its transpose, sup , |x|for = 1} is its aa positive definite symmetric Lyapunov matrix involved in 2 stability analysis wefrom employ the of metric 2 positive definitethat symmetric Lyapunov matrix involved in part. By eig(A) we denote the minimum eigenvalue induced norm and Sym(A) = stands for its sym2 T minimum eigenvalue part. By eig(A) we denote the 2 stability analysis that we employ requires the existence of metric A+A the chosen Lyapunov functional, which has to additionally a positive definite symmetric Lyapunov matrix involved in inducedpart. norm Sym(A) = the stands foreigenvalue its symmetric Byand eig(A) we denote minimum the chosendefinite Lyapunov functional, which has to additionally 2 a positive symmetric Lyapunov matrix involved in metric part. By eig(A) we denote the the chosen Lyapunov functional, which has to additionally minimum eigenvalue the chosen Lyapunov functional, which hasoftoAutomatic additionally 2405-8963 © 2019 2019, IFAC IFAC (International Federation Control) Copyright © 290 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 290 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 290 10.1016/j.ifacol.2019.11.780 Copyright © 2019 IFAC 290

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of a matrix A. For a function f (·), we use the difference ˆ − f (ξ), parametrized operator given by ∆ξˆ [f ] (ξ) := f (ξ) 1 ˆ For f ∈ C by Df we denote its Jacobian. For by ξ. a continuous (C 0 ) map [0, 1]  x → ξ(x) ∈ Rn we adopt the notation ξ0 := max{|ξ(x)| , x ∈ [0, 1]}. For a continuously differentiable (C 1 ) map [0, 1]  x → ξ(x) ∈ Rn we adopt the notation ξ1 := ξ0 + ξx 0 . For a function ξ ∈ H 1 ([0, 1]; Rn ) the definition of the H 1    1/2 1 . By B(δ) we |ξ|2 + |ξx |2 dx norm is ξH 1 := 0   denote the set B(δ) := {ξ ∈ C 1 [0, +∞) × [0, 1]; R2 : ξ(t, ·)1 ≤ δ, ∀t ≥ 0}. 2. CLASS OF SYSTEMS AND MAIN OBSERVER RESULT

211

The following global Lipschitzness assumption is crucial for the stability analysis of the observer error equation. A2. There exists Lipschitz constant Lf > 0, such that for all ξ ∈ R2 , |Df (ξ)| ≤ Lf . There exists Lipschitz constant Lf  > 0, such that for all ξ, ξˆ ∈ R2 , |∆ξˆ [Df ] (ξ)| ≤ Lf  |ξˆ− ξ|. In order to be able to adopt a high-gain observer methodology, we need to first perform the following invertible linear transformation, in order to obtain a more appropriate form: ζ = Tξ (3a)   1 0 (3b) T := a2 1

with a2 to be defined later. Let us consider the first-order quasilinear hyperbolic system described by the following equations on a time and space domain Π := [0, +∞) × [0, 1]: ξt (t, x) + Λ(ξ1 (t, x))ξx (t, x) = Aξ(t, x) + f (ξ(t, x)) (1a) T

where [ξ1 ξ2 ] = ξ : [0, +∞) × [0, 1] → R2 is the state. Consider also distributed measurement y : [0, +∞) × [0, 1] that is available at the output, given by y(t, x) = Cξ(t, x) (1b) We assume that   01 A= , C = [1 0] , 00   f1 (ξ1 ) Λ(ξ1 ) := diag {λ1 (ξ1 ), λ2 (ξ2 )} , f (ξ) = f2 (ξ1 , ξ2 )

with λ1 (ξ1 ), λ2 (ξ1 ) > 0, ∀ξ1 ∈ R. We observe that the system satisfies some triangular structure (as in Kitsos et al [2018]), which illustrates an analogy to the finitedimensional case. We consider initial and boundary conditions of the form (2a) ξ(0, x) =: ξ 0 (x), x ∈ [0, 1] ξ(t, 0) = H (ξ1 (t, 1)) , t ∈ [0, +∞) (2b) Considering system’s dynamics, we assume the following regularity.   R2 , i = 1, 2, A0. The involved mappings λi ∈ C 1 R;     f ∈ C 1 R2 ; R2 , H ∈ C 1 R; R2 , ξ 0 ∈ C 1 [0, 1]; R2 .

The following assumption is essential to assert the wellposedeness of our system, along with the minimal observer design requirement of ”forward completeness” and, furthermore, it imposes boundedeness of the classical solutions in the C 1 -norm, which is essential in the design of our observer. For more detailed presentation, the reader can refer to Bastin and Coron [2016], Li [1985] and references therein, where sufficient conditions for the well-posedeness of quasilinear hyperbolic systems of balance laws are given.

A1. There compact set  exists nonempty  M ⊂ C 1 [0, 1]; R2 , such that, for any initial conditions ξ 0 ∈ M, satisfying zero-order and one-order compatibility conditions, problem (1a), (2) admits a unique classical   solution in C 1 [0, +∞) × [0, 1]; R2 . Moreover, for any ξ 0 in the above-mentioned class, there exists δ > 0, such that ξ(t, ·)1 ≤ δ, ∀t ∈ [0, +∞). Also, λ1 (Cξ) ≥ λ2 (Cξ), ∀ξ ∈ B(δ). 291

Now system (1a), (1b), (2b) is rewritten as follows: ¯ 1 (t, x))ζx (t, x) = Aζ(t, x) + M ζ(t, x) ζt (t, x) + Λ(ζ (4a) +T f (T −1 ζ(t, x)) y(t, x) = Cζ(t, x) (4b) ζ(t, 0) = T H (ζ1 (t, 1)) (4c) defined on the domain Π, where M is given by   −a2 0 M := (5) −a22 a2

and ¯ 1 ) := T C T C (λ1 (ζ1 ) − λ2 (ζ1 )) + λ2 (ζ1 ) I2×2 Λ(ζ

(6)

Consider now K ∈ R2 and P a symmetric and positive definite matrix satisfying 2Sym (P (A + KC)) = −I2×2 (7) which is always feasible, due to the observability of the pair (A, C). Define also diagonal matrix Θ, given by   Θ := diag θ, θ2 (8) where θ > 1 is the candidate high-gain constant of the observer, which will be selected later. Then, set a2 := a2,0 /a1,0 (9a)   a1,0 := θ−1 ΘP −1 C T (9b) a2,0 Notice that a2 is written in the form a2 = θ¯ a2 (10) where a ¯2 is independent of θ and depends only on components of P .

Let us now introduce our candidate observer dynamics defined on the domain Π and its boundary conditions for system (4), as follows: ˆ x) + M ζ(t, ˆ x) ¯ x))ζˆx = Aζ(t, ζˆt (t, x) + Λ(y(t,      2a2 ˆ x) + T f (T −1 ζ(t, ˆ x)) y(t, x) − C ζ(t, − ΘK − a22 (11a) ˆ ζ(t, 0) = T H (y(t, 1)) , t ≥ 0 (11b) The following lemma guarantees the existence of unique global classical solutions for our candidate observer. We invoke Kmit [2008], where an analogous result is proven under Lipschitzness properties of the dynamics. It is easy

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to check that our candidate observer under the transformation ζ := T −1 ζˆ is written in a well-posed characteristic form and satisfies semilinear hyperbolic laws. Assumptions A0 - A2 in conjunction with the previously mentioned comments (details are left to the reader) are compatible with the sufficient conditions of Theorem 2.1 in Kmit [2008] and, thereby similar global existence result is established for our observer system, as given in the following result. Lemma 1. Under Assumptions A0, A2 and considering y ∈ C 1 ([0, +∞) × [0, 1]; R), with y globally bounded in the C 1 -norm, the problem described by (11) on domain Π and ˆ x), ∀x ∈ [0, 1], satisfying zeroinitial conditions ζˆ0 := ζ(0, order and one-order compatibility conditions (see Bastin and Coron [2016] for details on compatibility conditions) admits a unique classical  solution on Π, i.e., there exists a unique solution ζˆ ∈ C 1 [0, +∞) × [0, 1]; R2 .

We are now in a position to present our main result on the observer design. Theorem 2. Consider system (1a) - (2), defined on Π with output (1b) and suppose that Assumptions A0, A1 and A2 hold. Let also P ∈ R2×2 be a positive definite symmetric matrix and K a vector both satisfying (7). Then, for θ > 1, system (11) with initial condition ζˆ0 ∈ ˆ x) = ζˆ0 (x), satisfying zero-order C 1 ([0, 1]; R2 ), with ζ(0, and one-order compatibility conditions, is a well-posed high-gain observer for ζ = T ξ, in the sense that it admits a unique classical solution in Π on the one hand, providing an estimate for the state of system (1a) - (2) for choice of θ large enough on the other hand. More precisely, for any κ > 0, there exist θ > 1, such that the following estimate is satisfied: ˆ ·)H 1 ≤ le−κt ξ 0 − T −1 ζˆ0 H 1 , t ≥ 0 ξ(t, ·) − T −1 ζ(t, (12) for some l > 0 depending on θ. This theorem states that for system (1a) - (2) we have a systematic high-gain observer design providing an estimate of its full state, with a convergence rate adjustable via the high gain θ. 3. OBSERVER CONVERGENCE PROOF This section is dedicated to the proof of Theorem 2. We define the linearly transformed error by    := Θ−1 ζˆ − ζ

and we obtain the following equations defined on Π: t + Λ1 (y)x = θ (A + KC)    (13) +θ¯ a2  + Θ−1 T ∆T −1 ζˆ [f ] T −1 ζ (t, 0) = 0

(14)

where 1 −1 T P C C (λ1 (y) − λ2 (y)) + λ2 (y)I2×2 (15) a1,0 Observe from (15) that P Λ1 (y) is symmetric for any symmetric matrix P , a fact that will allow an integration by parts that we will perform later. Λ1 (y) :=

At this point, let us introduce the operator     K : C 1 [0, +∞) × [0, 1]; R2 → C 0 [0, +∞) × [0, 1]; R2×2

292

defined by K[ξ] := Λ1 (Cξ)Λ−1 1 (Cξ)C

∂ ξ ∂t

(16)

Define also ˆ

K1ζ :     C 1 [0, +∞) × [0, 1]; R2 → C 0 [0, +∞) × [0, 1]; R2×2 ˆ

K2ζ :     C 1 [0, +∞) × [0, 1]; R2 → C 0 [0, +∞) × [0, 1]; R2 (parametrized by ζˆ and acting on ξ = T −1 ζ) by ˆ

ˆ −1 Θ (17a) K1ζ [ξ] := K[ξ] + Θ−1 T Df (T −1 ζ)T   ˆ K2ζ [ξ] := −K[ξ]Θ−1 T ∆T −1 ζˆ [f ] T −1 ζ   ∂ ξ (17b) +Θ−1 T ∆T −1 ζˆ [Df ] T −1 ζ ∂t Next by temporarily assuming that  is of class C 2 , we derive the following hyperbolic equations for t : ˆ

tt + Λ1 (y)tx = K1ζ [ξ]t + θ¯ a2 t

ˆ

+θ(A + KC)t − θK[ξ](A + KC) + K2ζ [ξ] (18) (19) t (t, 0) = 0 Remark 3. Notice that whenever ξ ∈ B(δ), due to continuity and positiveness of λi , i = 1, 2 and furthermore the fact that ξ(t, ·)0 ≤ δ, ∀t ≥ 0, 0 < supξ∈B(δ)) (λi (Cξ)), inf ξ∈B(δ)) (λi (Cξ)) < +∞, i = 1, 2. In addition, whenever ξ ∈ B(δ), as a result of the hyperbolic dynamics (1a), we easily calculate constant δ1 > 0, such that ξt (t, ·)0 = T −1 ζt (t, ·)0 ≤ δ1 , ∀t ≥ 0 (20) By virtue of the above arguments, the fact that θ > 1, (20), continuity and global Lipschitzness (Assumptions A0, A2) of the involved mappings, we can easily calculate positive constants γi , i = 1, . . . , 6, such that whenever ξ ∈ B(δ), are satisfied for all  the following inequalities  ζˆ ∈ C 0 [0, +∞) × [0, 1]; R2 , t ≥ 0, x ∈ [0, 1]: ˆ x))T −1 Θ| ≤ γ1 (21a) |Θ−1 T Df (T −1 ζ(t,  −1  −1 (21b) |Θ T ∆T −1 ζˆ [f ] T ζ(t, x) | ≤ γ2 |(t, x)|   |Θ−1 T ∆T −1 ζˆ [Df ] T −1 ζ(t, x) | ≤ γ3 |(t, x)| (21c) |Λ1 (ξ1 (t, x))| ≤ γ4 (21d) (21e) γ5 ≤ |Λ1 (ξ1 (t, x))| ≤ γ6 Note that all the above constants γ1 , . . . , γ6 depend only on the Lipschitz constants of Assumption A2, constants δ, δ1 , elements of matrices P and Λ and are independent of the high gain θ, due to the triangular structures of the nonlinear mappings. We can similarly obtain constants γ7 , γ8 > 0, such that for all (t, x) ∈ Π, we get ˆ

ˆ

|K1ζ [ξ](t, x)| ≤ γ7 := γ1 + δ1 γ4 /γ5 ,

|K2ζ [ξ](t, x)| ≤ γ8 |(t, x)|; γ8 := δ1 γ4 γ2 /γ5 + γ3 δ1 (22) To prove the error exponential stability, let us define functional W : C 1 ([0, 1]; R2 ) → R by  1   W[] := e−µθ¯a2 x T P  + ρ0 T (23) t P t dx 0

where ρ0 ∈ (0, 1] is a constant (to be chosen appropriately), P ∈ R2×2 is a positive definite symmetric matrix satisfying (7) for appropriate K and µ > 0 will be chosen appropriately later.

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By invoking Lemma 1 and Assumption A1, which establish global unique classical solutions for observer system (11) and system (1a), (2) respectively, we are now in a position to define W : [0, +∞) → R by W (t) := W[](t), t ≥ 0 (where we use the notation (t) := (t, x), ∀x ∈ [0, 1]). ˙ along the classical soCalculating the time-derivative W lutions of (13) - (14), (18) - (19), we get  1 ˙ = e−µθ¯a2 x W 0   T T T (24) × T t P  +  P t + ρ0 tt P t + ρ0 t P tt dx In the following, we omit the arguments (t, x) of the mappings inside the integrals. After substituting the dynamical equations (13) and (18) into the above equation and applying integration by parts, ˙ can be written in the following form: W ˙ = T1 + T2 + T3 + T4 (25) W where  T1 := −e−µθ¯a2 (1)T P Λ1 (y(1))(1)  +ρ0 t (1)T P Λ1 (y(1))t (1) (26a)  1  T2 := a2 T P Λ1 (y) − µθ¯ e−µθ¯a2 x −µθ¯ a 2 ρ0  T t P Λ1 (y)t 0   (26b) +T P Λ1 (y)yx  + ρ0 T t P Λ1 (y)yx t dx  1    T3 := 2 e−µθ¯a2 x T P Θ−1 T ∆T −1 ζˆ [f ] T −1 ζ 0

ˆ



ˆ

ζ ζ T +ρ0 T t P K2 [ξ] + ρ0 t Sym(P K1 [ξ])t   T dx +θα ¯ 2  P  + ρ0  T t P t  1 T4 := θ e−µθ¯a2 x

(26c)

213

where, after utilizing (7), Σ : B(δ) → C 0 [0, +∞) × [0, 1]; R4×4 is given by   −ρ0 (A + KC)T KT [ξ]P I2×2 Σ[ξ] := −ρ0 P K[ξ](A + KC) ρ0 I2×2 (32) We can easily verify that Σ[ξ]  0 (by solving a simple LMI) for every ξ ∈ B(δ) if γ2 ρ0 < 2 2 2 5 (33) γ4 δ |P | |A + KC|2 It turns out that for every choice of matrices P and K satisfying equation (7), there always exists a ρ0 (sufficiently small), such that the above inequality is satisfied and this fact renders Σ[ξ] positive. Consequently, for appropriate choice of ρ0 , there exists σ > 0, such that σ T4 ≤ −θ W (34) |P | Now, choose µ, such that 2 µ> (35) inf ξ∈B(δ) λ2 (Cξ) with the right-hand side being well-defined by Remark 3.1. Combining equations (27), (29), (30), (34), (35) with (25), we deduce ˙ ≤ (−θω1 + ω2 )W (36) W |P | ω2 := α + (γ8 + 2 max(γ2 , γ7 )) eig(P ) . We 2 ˙ obtained the estimate (36) of Wp for  of class C , but the proof so far implies that the result does not depend on the C 2 -norms. Therefore, by invoking density arguments, the results remain valid with  only of class C 1 .

where ω1 :=

σ |P | ,

Applying the comparison lemma to (36), we get

0

× 2T Sym(P (A + KC)) + 2ρ0 T t Sym(P (A + KC))t  T T −ρ0 t P K[ξ](A + KC) − ρ0  (A + KC)T KT [ξ]P t dx (26d) It turns out from the above equations that (27) T1 ≤ 0 Next, observe that the term T2 can be bounded as follows: a2 inf λ2 (Cξ))W T2 ≤ (α − µθ¯ ξ∈B(δ)  1   µθ¯ a2 e−µθ¯a2 x (λ1 (y) − λ2 (y)) 21 + ρ0 (∂t 1 )2 dx − a1,0 0 (28) |δγ4 where α := |P eig(P ) and by virtue of Assumption A1, we finally obtain T2 ≤ (α − µθ¯ a2 inf λ2 (Cξ))W (29) ξ∈B(δ)

By exploiting (21e), (22), T3 can be bounded as follows:  1 e−µθ¯a2 x T3 ≤ 0    ¯2W × 2|P | γ2 ||2 + ρ0 γ8 |||t | + ρ0 γ7 |t |2 dx + 2θα   |P | + 2θα ¯2 W (30) ≤ (γ8 + 2 max(γ2 , γ7 )) eig(P ) The term T4 can be rewritten in the following form:     1  T T  −µθ¯ a2 x T4 := −θ e dx (31)  t Σ[ξ]  t 0 293

(37) W (t) ≤ e−(θω1 −ω2 )t W (0), ∀t ≥ 0 Now, one can select the high-gain θ, such that (38) θ > max(1, θ0 ); θ0 := ω2 /ω1 and, therefore, for sufficiently large θ we achieve to obtain θω1 − ω2 > 0. In order to derive an estimation of the H 1 norm, we can observe that a relationship of the following form can be deduced from the dynamics (13): γ5 x (t, ·)L2 ≤ t (t, ·)L2 ≤ γ6 x (t, ·)L2 + (θ|A + KC| + θ|¯ a2 | + γ2 )(t, ·)L2 (39) Performing trivial inequalities and using the abovementioned relation, we can derive a constant c > 0 dependent on θ, such that for all t ≥ 0,  |P | − ω1 (θ−θ0 )t 0 −1/2 (t, ·)H 1 ≤ cρ0 e 2  H 1 (40) eig(P ) where 0 (x) := (0, x) is the initial condition of the error. In the above derivations we have used the inequality ρ0 eig(P )(t, ·)2H 1 ≤ W (t) ≤ eµ¯a2 θ |P |(t, ·)2H 1 , t ≥ 0 (41) Now, it is clear that for any κ > 0, one can choose θ = ω21 κ + θ0 , so as to get an estimation error as in (12), with l being dependent on θ. Hence, we designed an exponential in the H 1 -norm highgain observer which convergences to zero after an initial time t0 and with tunable convergence rate κ, dependent

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on the selection of θ. The higher the values θ attains, the faster the observation error converges to zero. Remark 4. Let us remark here, that in (12), constant l depends exponentially on θ, contrary to the classical highgain observer design results, where it should only have polynomial dependence. In different formulation, we can rewrite (12) as ˆ ·)H 1 ≤ ¯le−κ(t−t0 ) ξ 0 − T −1 ζˆ0 H 1 , t ≥ 0 ξ(t, ·) − T −1 ζ(t, (42) ¯ where, as we showed in the proof, l depends polynomially on θ and µ¯ a2 θ (43) t0 := ω1 (θ − θ0 ) denotes the time from which the exponential converges starts and, by definition, it depends only on the minimum value of λ2 (·). It is worthwile to remark that, the previous limitation leads to a slightly weaker result, compared to our previous approaches Kitsos et al [2018], Kitsos et al [2019], although here, we confronted the problem with distinct characteristic velocities by avoiding the restrictive limitation of injecting output’s spatial derivatives, as in Kitsos et al [2019]. Future studies will be dedicated to this. To better illustrate the nature of the high-gain observer design, we use an example. Example 1. Consider system ∂t ξ1 + 0.1(2 + cos(ξ1 ))∂x ξ1 = ξ2 + sin(ξ1 ), (44a) ∂t ξ2 + 0.1(2 + sin(ξ1 ))∂x ξ2 = sin(ξ2 − ξ1 ), (44b) ∀(t, x) ∈ Π, with distributed measurement y = ξ1 (44c) and boundary conditions of the form ξ(t, 0) = 0, t ≥ 0 (44d) Consider initial conditions ξ10 (x) = x2 , ξ20 (x) = −x2 /2, x ∈ [0, 1]. System (44) is of the form (1a) with boundary conditions described by (2). More particularly, Λ(ξ  1) =  sin(ξ1 ) , diag (2 + cos(ξ1 ), 2 + sin(ξ1 )), f (ξ) = sin(ξ2 − ξ1 ) H(·) = 0. All Assumptions A0 - A2 that we have assumed for system (1a) are satisfied for the choice of these initial conditions. We choose vector gain K = (−2, −1)T and after calculating all the essential constants that are used in Theorem 2, we can proceed to the observer design. We appropriately choose the high gain constant θ being equal to 50. We also calculate a2 by (9) in order to perform the transformation T as in (3) into the system ζ and we can, therefore, obtain the high-gain observer dynamics for ζ as in (11). Finally, we choose arbitrary observer initial conditions (in accordance with the compatibility conditions) ζˆ10 (x) = ζˆ20 (x) = 0, x ∈ [0, 1].

Fig. 1. Time and Space Evolution of Solution ξ1 (system’s output)

Fig. 2. Time and Space Evolution of Observer Error T −1 ζˆ1 − ξ1

In Figure 1 the solution ξ1 is shown and in Figures 2 and 3 we illustrate the estimation error functions for both states, which exhibit exponential convergence to zero, as predicted by Theorem 2. 4. CONCLUSION In this paper we designed a high-gain observer for a class of observable hyperbolic systems with distributed measurement. This result constitutes an extension of the high-gain

294

Fig. 3. Time and Space Evolution of Observer Error T −1 ζˆ2 − ξ2

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observer design for finite-dimensional systems to a class of hyperbolic systems and, also, an extension of our previous works towards this direction, as we considered here distinct characteristic velocities. We proved the exponential decay of the observer error in the H 1 - norm step by step by first choosing an appropriate Lyapunov functional. The extension of this methodology to more general cases of hyperbolic systems and weakening some of our assumptions, in order this methodology to apply to real systems, like chemical reactors, is subject to our future approaches. REFERENCES G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications. Springer International Publishing, 2016. G. Besan¸con, D. Georges, and Z. Benayache, “Sur la commande en dimension finie d’une classe de syst`emes non lin´eaires de dimension infinie,” Conf. Internationale Francophone d’Automatique, Bordeaux, France, 2006. G. Besan¸con, B. Pham, and D. Georges, “Robust state estimation for a class of convection-diffusion-reaction systems,” IFAC Workshop on Control of Systems Governed by partial Differential Equations, Paris, France, 2013. F. Castillo, E. Witrant, C. Prieur, and L. Dugard, “Boundary Observers for Linear and Quasi-Linear Hyperbolic Systems with Application to Flow Control,” Automatica, vol. 49, no. 11, pp. 3180–3188, 2013. P. D. Christofides and P. Daoutidis, “Feedback control of hyperbolic pde systems,” AIChE Journal, vol. 42, no. 11, pp. 3063–3086, 1996. J.-M. Coron and G. Bastin, “Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm,” SIAM J. Control Optim., vol. 53, no. 3, pp. 1464–1483, 2015. R. Curtain, “Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input,“ IEEE Transactions on Automatic Control, vol. 27, no. 1, pp. 98–104, 1982. F. Di Meglio, M. Krstic, and R. Vasquez, “A backstepping boundary observer for a class of linear first-order hyperbolic systems,” European Control Conf., Zurich, Switzerland, pp. 1597–1602, 2013. J. P. Gauthier and G. Bornard, “Observability for any u(t) of a class of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 26, no. 4, pp. 922–926, 1981. J. P. Gauthier, H. Hammouri, and S. Othman, “A Simple Observer for Nonlinear Systems Applications to Bioreactors,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 875–880, 1992. A. Hasan, O. M. Aamo, and M. Krstic, “Boundary observer design for hyperbolic PDE-ODE cascade systems,“ Automatica, vol. 68, pp.75–86, 2016 H.K. Khalil, “High-gain observers in nonlinear feedback control,”, Advances in Design and Control. SIAM, 2017. C. Kitsos, G. Besan¸con, and C. Prieur “High-gain observer design for a class of hyperbolic systems of balance laws,” IEEE Conference on Decision and Control, Miami, USA, 2018. C. Kitsos, G. Besan¸con, and C. Prieur “A high-gain observer for a class of 2x2 hyperbolic systems with C1 exponential convergence,” IFAC Workshop on Control 295

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