Pauline Bernard
email: pauline.bernard@minesparis.psl.eu

Ricardo G Sanfelice
email: ricardo@ucsc.edu

Semiglobal High-Gain Hybrid Observer for a Class of Hybrid Dynamical Systems with Unknown Jump Times

We study the problem of observer design for hybrid dynamical systems in the challenging setting where the times at which jumps occur are unknown or not detected precisely. We remark that when the solutions of interest are known to remain in a compact set and admit a uniform dwell-time and when the flows are strongly differentially observable, a sufficiently fast high-gain observer can be designed to estimate the state during flow, but using the output of the system near its jump times is counterproductive. We thus propose to "disconnect" the high-gain observer when its estimate gets close to the jump set. More precisely, the proposed observer generates an estimate that, during flow, is obtained via the high-gain observer and, around jump times, is obtained by integrating forward the flow map of the system, until reaching the jump set. Under appropriate assumptions around the jump set of the system, we show that this observer guarantees local uniform asymptotic stability of an appropriately defined zero-error set. Then, we develop a method to turn any such local hybrid observer into a semiglobal hybrid observer. This observer operates sequentially by first employing a continuous-time high-gain observer, and then, after a finite amount of time, solely determined by the current estimate, employing the available local hybrid observer. The capabilities and performance of the proposed hybrid observer are illustrated on a hybrid spiking neuron model.

I. INTRODUCTION

A. Background

T HE problem of designing observers for general hybrid systems presenting both continuous-time and discretetime behaviors is still largely unsolved, mainly due to the fact that the system jump times, that is, the times at which discrete events occur in the system solution, generally depend on its initial condition, which is unknown in the context of observer design. When the system jump times are known or can be detected immediately, it is natural to design an observer that is synchronized with the system, namely, a hybrid observer whose jumps are triggered at the same time as those of the system. This has been done under assumptions on the time elapsed between successive jumps (e.g., upper/lower bounds, reverse/average dwell-time) in a large variety of contexts, including impulsive (possibly switched) systems [START_REF] Alessandri | Switching observers for continuous-time and discrete-time linear systems[END_REF], [START_REF] Medina | State estimation for linear impulsive systems[END_REF], [START_REF] Shim | Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems[END_REF], [START_REF] Tanwani | Comments on "observability of switched linear systems: Characterization and observer design[END_REF], sampled continuous-time systems [START_REF] Ferrante | State estimation of linear systems in the presence of sporadic measurements[END_REF], [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF], [START_REF] Sferlazza | Time-varying sampleddata observer with asynchronous measurements[END_REF], and general hybrid systems [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], [START_REF] Ríos | State estimation for linear hybrid systems with periodic jumps and unknown inputs[END_REF], [START_REF] Tran | Coupling flow-and jumpbased observers for hybrid systems with nonlinear maps and known jump times[END_REF], among others. Because the observer jumps at the same time as the system, both observer and system solutions are defined on the same (hybrid) time domain, which facilitates the analysis of the estimation error and the design of an observer. A similar feature is exploited in [START_REF] Tanwani | Observer design for unilaterally constrained Lagrangian systems: A passivity-based approach[END_REF] for hybrid mechanical systems modeled as constrained measure differential inclusions where it is assumed that the position triggering the constraint (and thus discontinuities in solutions) is measured and directly used in the observer as a constraint. Unfortunately, exact synchronization between the system and the observer is usually difficult to achieve in practice, due to noisy/delayed jump detection. Robustness with respect to delays in the jump triggering has been studied in [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], but only practical stability outside the delay intervals may be expected. In other contexts, it may even be impossible to detect the jumps of the system from the available measurements.

When the observer jumps are not triggered at the same time as those of the system, the mismatch of time domains between the system and the observer solutions makes the formulation of observability and observer design challenging [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF]. In the context of switched (mostly linear) systems, the design of mode location observers able to estimate the switching signal (i.e, the jump times), either by running parallel observers and monitoring the residual output error [START_REF] Balluchi | The design of dynamical observers for hybrid systems: Theory and application to an automotive control problem[END_REF], [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF], via LMI design [START_REF] Xiang | Robust observer design for nonlinear uncertain switched systems under asynchronous switching[END_REF], [START_REF] Zammali | Switching signal estimation based on interval observer for a class of switched linear systems[END_REF], or via optimization to find the switching signal matching most the output signal over a time window [START_REF] Shankar | State estimation for asynchronously switched sampled-data systems[END_REF] has been widely explored. But very few observer results exist for general hybrid dynamical systems [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF] when the system jump times are unknown. Exceptions are [START_REF] Kim | State estimation and tracking control for hybrid systems by gluing the domains[END_REF], [START_REF] Menini | State immersion observers for mechanical systems with impacts[END_REF], where the existence of a change of coordinates transforming the jump map into the identity map is studied, thus allowing the use of a continuous-time observer in those new coordinates. However, there is no systematic design of such a change of coordinates. Also in [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF], an observer with nonsynchronized jumps is designed for billiard-type systems, but the knowledge of the system jump times is still required to trigger the jumps.

B. Contributions

Motivated by the shortcomings mentioned above, we propose a systematic observer design for a class of general hybrid systems, that can be done without relying on the knowledge or detection of the jumps of the hybrid dynamical system. For starters, in the preliminary version of this work [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF], we proposed a local hybrid observer for a class of hybrid dynamical systems modeled as in [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF], with linear flow, jump, and output maps, assuming the flow dynamics are observable and solutions of interest admit a uniform dwell-time and evolve within a compact set. We showed that, while it is tempting to use a sufficiently fast observer during flow and trigger the jumps when its estimate reaches the jump set, using the system output around the jump times to define the innovation term is actually counterproductive. Indeed, an arbitrarily small mismatch between the jump times of the system and of the observer typically leads to large estimation errors, even in nominal (noise-free) conditions. Under appropriate assumptions on the behavior of solutions around the jump set, we proposed a hybrid observer that -via an innovation termproperly injects the measurements during continuous evolution of the hybrid system and, around the jump times, produces an (open-loop) estimate until it reaches the jump set.

In this paper, we make the following contributions: 1) We show that the local design of [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF] can be extended to handle the larger class of nonlinear hybrid systems with strongly differentially observable flow dynamics, for which a sufficiently fast continuous-time high-gain observer [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] can be used during flow, still assuming solutions of interest are known to remain in a certain compact set and admit a uniform dwell-time. Because, unlike in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF], the highgain observer is generally not in the system coordinates an additional state is added to the observer and the hybrid logic appropriately adapted. 2) Unlike in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF] where only local asymptotic convergence to an appropriate zero-error set is presented, we prove here local uniform asymptotic stability. 3) We then propose a novel general method enabling to adapt any local asymptotic observer designed for this class of systems into a semiglobal one. The idea is to run a preliminary high-gain observer and "switch" to the local hybrid observer at an appropriate time. But this "switching" time has to be chosen with care in order to ensure that the estimation error is sufficiently small at that time, no matter when the system may have jumped in the meantime. We thus provide an algorithm to choose this "switching time" based on the value of the high-gain estimate and show semiglobal convergence of the obtained hybrid observer. Finally, the performance of such a design are illustrated in simulations for different scenarios of initial conditions and compared to a more standard synchronous observer design with delayed jump detection, on an example featuring a spiking neuron.

C. Notation and Preliminaries

We denote R (resp. N) the set of real numbers (resp. integers), R ≥0 := [0, +∞), R >0 := (0, +∞), and N >0 =: N \ {0}. For x in R n and A subset of R n , |x| A denotes the distance from x to A and ∂A the boundary of A. For a symmetric real matrix P , λ(P ) (resp. λ(P )) stands for its smallest (resp. largest) eigenvalue. For a C 1 map V : R n → R, L F V (x) := ⟨∇V (x), F(x)⟩ denotes the Lie derivative along the vector field F. For a differential equation ẋ = f (x) with f locally Lipschitz, Ψ f (x, τ ) denotes the value at time τ of the solution initialized at x. We recall that a C 1 map T : O ⊆ R n → R m with m ≥ n is an immersion on O if ∂T ∂x (x) is full-rank for all x ∈ O. We consider hybrid dynamical systems of the form (1) introduced in Section II-A, as defined in [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF], whose solutions are defined on socalled hybrid time-domains. A subset E of R ≥0 × N is a compact hybrid time-domain if E = jm-1 j=0 ([t j , t j+1 ], j) for some finite sequence of times t 0 ≤ t 1 ≤ . . . ≤ t jm , and it is a hybrid time domain if for any

(t m , j m ) ∈ E, E ∩ [0, t m ] × {0, 1, . . . , j m } is a compact hybrid time domain.
For a solution (t, j) → x(t, j) (see [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF]Definition 2.6]), we denote dom x its domain, dom t x (resp. dom j x) its projection on the time (resp. jump) component, and I j := {t ∈ dom t x : (t, j) ∈ dom x} the jth interval of flow. We say that x is maximal if there does not exist any other solution x ′ defined on dom x ′ ⊃ dom x and agreeing with x on dom x. We say that x is t-complete if dom t x is unbounded and that it has a dwell-time τ m > 0 if it flows at least τ m units of time in between consecutive jumps. Finally, as defined in [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF], [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF],

x r is a j-reparametrization of x if there exists a function ρ : N → N verifying ρ(0) = 0, ρ(j + 1) -ρ(j) ∈ {0, 1}, and such that x r (t, j) = x(t, ρ(j)) for all (t, j) ∈ dom x r . If in addition dom x = (t,j)∈dom x r (t, ρ(j)), then it is a full j-reparametrization.

II. PROBLEM STATEMENT

A. Framework

We consider a hybrid system of the form [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF]

H ẋ = f (x) x ∈ C x + = g(x) x ∈ D , y = h(x) (1)
with state x ∈ R nx and output y ∈ R ny , flow and jump maps f and g locally Lipschitz, output map h, flow and jump sets C and D. For this broad class of hybrid dynamical systems, denoted H = (C, f, D, g, h), we are interested in estimating the state of H when its solutions are initialized in a bounded subset X 0 ⊂ C ∪ D. We denote S H (X 0 ) the set of maximal solutions of H with initial condition in X 0 .

If the system jump times were known or immediately detected as in [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], [START_REF] Bernard | Observers for hybrid dynamical systems with linear maps and known jump times[END_REF], [START_REF] Ríos | State estimation for linear hybrid systems with periodic jumps and unknown inputs[END_REF], one could design an observer for (1) of the form

Ĥ ξ ∈ F(ξ, y) when H flows ξ + ∈ G(ξ, y) when H jumps , x = T (ξ) (2)
that is synchronized with the system, for some maps F, G : R n ξ × R ny → R n ξ and T : R n ξ → R nx to be chosen such that x asymptotically reconstructs the system state x. The advantage of such a setting is that the observer and system solutions are defined on the same domain, which facilitates the analysis of the estimation error. Unfortunately, exact synchronization between the system and the observer is usually difficult to achieve in practice, due to noisy/delayed jump detection. Motivated by this issue, in this paper, we design an observer whose jumps are triggered based on its own estimate of the system state, rather than an exogenous signal. More precisely, we aim for a hybrid observer of the form

Ĥ ξ ∈ F(ξ, y) (ξ, y) ∈ C ξ + ∈ G(ξ, y) (ξ, y) ∈ D , x = T (ξ) (3)
for some maps F, G, T and sets C, D to be designed. In this setting, y = h(x) is a hybrid signal defined on the domain of the system solution dom x, which typically differs from the domain of the observer solutions given by (3).

To cope with this mismatch, following [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF], [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF], solutions to (3) are defined as pairs (ξ, y r ), where dom ξ = dom y r and y r is a j-reparametrization of y. More precisely, jumps in the domain of the input y (namely, system jumps) trigger a jump in the domain of the solution to (3), with the observer state ξ either reset identically (trivial jump) or in G(ξ, y) depending on the logic defined in [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF], [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF]. Similarly, ξ may jump using G at times that are not jump times of the system solution x, so such trivial jumps are also added in the domain of y leading to its reparametrization y r . This purely analytical process enables us to build a j-reparametrization of the system solution x and its output y on the same domain as the observer solution. The j-reparametrization of two hybrid arcs on a common hybrid time domain is illustrated on Figure 1, following the algorithm proposed in [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF]. However, this artificial addition of trivial jumps for analysis purposes does not change the fact that the system and observer solutions "truly" jump using g and G, respectively, at different times. Consequently, even when x "tends" to x asymptotically, x may always be slightly ahead/behind x around the jump times, which typically prevents the estimation error x-x to converge to zero asymptotically around the jump times -the so-called hybrid peaking phenomenon in the tracking context [START_REF] Biemond | Tracking control for hybrid systems with state-triggered jumps[END_REF], [START_REF] Sanfelice | An embedding approach for the design of state-feedback tracking controllers for references with jumps[END_REF].

As in [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF], we consider more general notions of convergence of (x, ξ) to appropriate observation sets A that are as close as possible to the ideal zero-error set

A ideal = {(x, ξ) ∈ R nx × R n ξ : T (ξ) = x}
where x = x. Our goal is formulated as follows.

Problem statement:

Given H = (C, f, D, g, h) and X 0 ⊂ C ∪ D, design an observation set A ⊂ R nx × R n ξ , hybrid data (C, F, D, G, T
) and an initializations set Ξ 0 ⊂ R n ξ such that for any maximal solution x to H with x(0, 0) ∈ X 0 and any maximal solution (ξ, y r ) to Ĥ defined by (3) with input y = h(x) and with ξ(0, 0) ∈ Ξ 0 , there exists a full j-reparametrization (x r , ξ r ) of x and ξ such that dom x r = dom ξ r and

lim t+j→∞ |(x r , ξ r )(t, j)| A = 0 .

B. Assumptions

The following assumptions describe the class of hybrid systems considered in this study.

Assumption 2.1: Given H = (C, f, D, g, h) and X 0 ⊂ C ∪ D, there exist τ m > 0 and a compact subset X of C ∪ D such that any solution x ∈ S H (X 0 ) is t-complete with dwell-time τ m and remains in X at all times.

The uniform dwell-time condition enables our design to rely on an arbitrarily fast continuous-time observer. Under wellposedness, the existence of such a dwell-time is guaranteed if g(D) ∩ D = ∅ using [START_REF] Sanfelice | Invariance principles for hybrid systems with connections to detectability and asymptotic stability[END_REF]Lemma 2.7] and the fact that the solutions evolve in the compact set X . We next assume that a "high-gain" continuous-time observer ż = F ℓ (z, y) is available, allowing to estimate (arbitrarily fast) a certain function T of the state during flow.

Assumption 2.2: Given H = (C, f, D, g, h), there exist λ > 0, ℓ 0 ≥ 0, rational functions c and c, an open set O containing cl(C ∪ D), an injective immersion T : O → R nz , and for all ℓ > ℓ 0 , maps

F ℓ : R nz × R ny → R nz and V ℓ : O × R nz → R such that c(ℓ)|z -T (x)| 2 ≤ V ℓ (x, z) ≤ c(ℓ)|z -T (x)| 2 ∀(x, z) ∈ X × R nz (4a) L F ℓ V ℓ (x, z) ≤ -ℓλ V ℓ (x, z) ∀(x, z) ∈ (C ∩ X ) × R nz (4b) with F ℓ (x, z) = (f (x), F ℓ (z, h(x))).
The subscript ℓ highlights the dependency of V ℓ and F ℓ with respect to the gain ℓ. This gain controls the decay rate in (4b), which can be chosen arbitrarily large. The following two examples show that Assumption 2.2 holds when (f, h) is a linear observable pair and, more generally, when (f, h) is strongly differentially observable.

Example 2.3 (Linear observable pair): Following [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF], assume the flow and output maps of H are defined by f (x) = Ax and h(x) = Hx with the pair (A, H) observable. According to [22, (14)-( 17)-( 18)], by duality, there exist a linear change of coordinates V ∈ R nx×nx and a linear change of outputs Γ ∈ R ny×ny transforming (A, H) into an observable form, i.e.,

VAV -1 = A + BH , ΓHV -1 = H with A := blkdiag(A 1 , . . . , A ny ), B ∈ R nx×ny , H := blkdiag(H 1 , . . . , H ny ), A i =   0 0 ... 0 1 0 . . . . . . . . . 0 1 0 0 0 ... 0 1 0   ∈ R ni×ni , H i = ( 0 ... 0 1 ) ∈ R 1×ni
and n i integers such that ny i=1 n i = n x . Consider vectors K i such that A i -K i H i is Hurwitz, and for ℓ > 0, define L i (ℓ) := diag(ℓ ni-1 , . . . , ℓ, 1). Then, take F ℓ defined by

F ℓ (z, y) = Az -V -1 (B + ℓL(ℓ)K)Γ(Hz -y) (5)
where K := blkdiag(K 1 , . . . , K ny ), L := blkdiag(L 1 , . . . , L ny ). Consider a positive definite matrix P ∈ R nx×nx such that (A -KH) ⊤ P + P (A -KH) ≤ -λP for some λ > 0. Then, for any C, D, g, (4) holds with T = Id, Fig. 1: Jump reparametrization of two hybrid arcs x a and x b on a common hybrid time domain so that x r a (t, j) = x a (t, ρ a (j)) and x r b (t, j) = x b (t, ρ a (j)).

V ℓ (x, z) = (x -z) ⊤ V ⊤ L(ℓ) -1 P L(ℓ) -1 V (x -z) , ℓ 0 = 0 , c(ℓ) = λ(V ⊤ P V) ℓ 2(n-1) , c(ℓ) = λ(V ⊤ P V) ,
where n = max n i . □

Example 2.4 (Strongly differentially observable pair): Assume that H has n y = 1, and its flow/output pair (f, h) is strongly differentially observable of order n z on an open set O ⊂ R nx containing cl(C ∪ D), i.e., the map T : R nx → R nz defined by

T (x) = (h(x), L f h(x), ..., L nz-1 f h(x)) (6)
is an injective immersion on O. Consider a map Φ : R nz → R nz of the form

Φ(z) = sat •L nz f h • T inv (z) ,
where T inv is a locally Lipschitz left-inverse of T verifying T inv • T = Id on X , which is guaranteed to exist due to the properties of T , and sat is a bounded C 1 map verifying sat = Id on the compact set L nz f h(X ). Then, a high-gain observer [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] can be built for the flow dynamics, with

F ℓ (z, y) = A z + BΦ(z) + ℓL(ℓ)K(y -z 1 ) , (7)
A =     0 1 ... 0 0 0 1 . . . . . . . . . . . . . . . 0 1 0 ... ... 0 0     ∈ R nz×nz , B = 0 . . . 0 1 ∈ R nz L(ℓ) = diag(1,
ℓ, ℓ 2 , . . . , ℓ nz-1 ), and K such that A -KH is Hurwitz with H = (1, 0, . . . , 0). Standard high gain computations [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] show that conditions (4) then hold for the Lyapunov function

V ℓ (x, z) = (T (x) -z) ⊤ L(ℓ) -1 P L(ℓ) -1 (T (x) -z) ,
with P a positive definite matrix such that (A -KH) ⊤ P + P (A -KH) ≤ -λ 0 P for some λ 0 > 0, c(ℓ) = λ(P ) ℓ 2(nz -1) , c(ℓ) = λ(P ), λ > 0, and ℓ larger than a threshold ℓ 0 > 0 depending on the Lipschitz constant of Φ. Note that the same tools can be used for multioutput triangular normal forms [START_REF] Hammouri | High gain observer for structured multi-output nonlinear systems[END_REF]. □ Remark 2.5: Assumption 2.2 requires the full state of the system to be instantaneouly observable during flow. Thanks to the dwell-time of Assumption 2.1, this observability property allows to estimate the state arbitrarily fast during flow and to properly trigger the observer jumps in spite of unknown jump times. Since the jump dynamics of the observer are a copy of those of the plant, this observer design does not exploit the potential observability information encoded at the jumps. Handling the mismatch of jump times in a broader context where the interaction between both flow and jump dynamics is necessary to bring observability as in [START_REF] Shim | Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems[END_REF], [START_REF] Tran | Coupling flow-and jumpbased observers for hybrid systems with nonlinear maps and known jump times[END_REF] is still an open question. In fact, Assumption 2.2 requires an exponentially converging observer, which typically needs strong differential observability and Lipschitzness of the nonlinearities. Homogeneous designs [START_REF] Bernard | Observers for a non-Lipschitz triangular form[END_REF] could be considered instead, under weaker regularity conditions, but the Lipschitz-based analysis carried out in this paper would then need to be adapted. Finally, note that the (physical) knowledge of the compact set X of Assumption 2.1 is typically needed to tune the saturation functions and the gain ℓ. Gain adaptation strategies as in [START_REF] Lei | A global observer for autonomous systems with bounded trajectories[END_REF] could probably be exploited to remove this assumption.

In what follows, we consider a strict compact superset X m of X in O, with X defined in Assumption 2.1 and O in Assumption 2.2. Because T is an injective immersion on O, it admits 1 a Lipschitz left-inverse T inv : R nz → R nx such that

T inv • T (x) = x ∀x ∈ X m . (8)
It is shown in [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF] that when the jump times of the system are known or detected instantaneously, a possible observer consists of Ĥ defined in [START_REF] Balluchi | The design of dynamical observers for hybrid systems: Theory and application to an automotive control problem[END_REF], with the high-gain observer F := F ℓ given by Assumption 2.2 during flow, a jump map G of the form

G (z) := sat •T • g • T inv (z) , (9)
with sat a C 1 bounded map verifying sat = Id on the compact set T • g(X ), and an estimate given by x = T inv (z). Indeed, for ℓ sufficiently large compared to the Lipschitz constant of G and the dwell-time τ m , one can show that the (exponential) decrease of the Lyapunov function V ℓ during flow wins over its (polynomial) increase at jumps and the estimation error; thus, it asymptotically converges to zero.

Still relying on the dwell-time and the available high-gain continuous-time observer, the goal of this paper is to design a hybrid observer that, unlike (2), does not rely on the detection of the system jumps. More precisely, rather than detecting when x ∈ D, which is not always possible, we propose to use the information that x := T inv (z) ∈ D to trigger the observer jumps, at least when the initial estimation error is sufficiently small. The construction of a local hybrid observer solving this problem is presented in Section III and the uniform asymptotic stability of its estimation error is proved in Section IV. Then, in Section V, we propose a general procedure to build a semiglobal hybrid observer from such a local design. Their performance is illustrated in simulations in Section VI.

III. LOCAL HIGH-GAIN HYBRID OBSERVER

An initial approach for the design of a local observer could be to use the flow map F ℓ given by Assumption 2.2 during flow, and simply trigger the jumps of the observer when x ∈ D, with the jump map G defined in [START_REF] Bernard | Observers for hybrid dynamical systems with linear maps and known jump times[END_REF]. Indeed, if the estimation error sufficiently decreases during flow, one can expect that the observer jumps will occur close in time to those of the system and somehow the observer will synchronize and converge. However, around the observer jump times, because the system typically jumps slightly sooner or later than the observer, the input y feeding the observer flow map might actually constitute a disturbance and hinder the convergence of the observer. More precisely, assume that x and x are both close to D and x jumps first. Then, the observer input y after the jump could steer x away from D so as to allow x to catch up with x via flow, which could cause x to miss its jump. The same reasoning holds in the reverse case where x jumps slightly ahead of x, and where the use of y would force x to track the value of x before it jumps, instead of simply waiting for x to catch up via a jump.

This issue is dealt with in [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] by making x follow a mirrored image of x with respect to D during the jump time mismatches. But this approach solves the problem in a very 1 See for instance [START_REF] Bernard | Observer design for nonlinear systems[END_REF]Lemma A.12]. Since T is an injective immersion and Xm is compact, there exists c T > 0 such that for all (xa,

x b ) ∈ Xm × Xm, |xa -x b | ≤ c T |T (xa)-T (x b )|.
Then, the inverse map T -1 of T is defined and Lipschitz on the closed set T (Xm). According to [24, Theorem 1], it admits a global Lipschitz extension T inv that agrees with T -1 on T (Xm).

D D δ 1 D δ 0 ż = F ℓ (z, y) x = T inv (z) q = 2 ẋ = f (x) q = 1 ẋ = f (x) q = 0 x x x+ = g(x) 0 ≤ τ ≤ ∆ τ = ∆ ż = F ℓ (z, y) x = T inv (z) q = 2 z + = T (x)
Fig. 2: Hybrid mechanism in observer (13), with system trajectory in black, observer trajectory in blue/yellow, with blue (resp. yellow) representing observer flow (resp. jumps), in the case where x remains in the set where Π(x) = x.

particular setting of billiard systems, where g•g is the identity, and more importantly, the knowledge of the system jump times is necessary to decide whether x should follow x or its mirrored image. In the problem tackled in this paper, the system jump times are unknown (and g • g is not the identity), so the paradigm of [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] cannot be applied.

A. Open-Loop Estimation around Jump Times

Following the approach presented in our preliminary work in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF], we propose to "disconnect" the high-gain observer when x is nearby the jump set D. More precisely, we propose a hybrid mechanism that lets x flow in "open-loop" according to f until it naturally reaches D, and only reconnects the correction term in the observer a small amount of time ∆ later, in a way that ensures the system has also jumped in the meantime. This process is illustrated in Figure 2. For this strategy to work, we assume that (i) the system eventually reaches D when entering a certain neighborhood of D (see (P1) below), (ii) the system necessarily jumps from D (see (P2)), and iii) the system takes at least 0 < τ 0 m < τ m units of time to reach that neighborhood again (see (P3)), with τ m given by Assumption 2.1. Similar conditions are used in [START_REF] Galeani | Robust trajectory tracking for a class of hybrid systems: An internal model principle approach[END_REF] in the context of trajectory tracking.

More precisely, consider a projection map Π :

R nx → cl(C ∪ D) verifying Π • Π = Π, Π(X m ) ⊂ X m and for which there exists a p ≥ 1 such that |x -Π(x)| ≤ a p |x -x| ∀(x, x) ∈ X × R nx . (10)
In particular, [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF] implies that Π = Id on X . This projection allows to reset x in cl(C ∪ D) at the observer jumps, and properly define the flow/jump sets, ensuring in particular that, given δ > 0 and for all

x ∈ R nx , Π(x) / ∈ D δ implies |x| D > δ, where D δ = {x ∈ cl(C ∪ D) : |x| D ≤ δ} . Assumption 3.1: Given H = (C, f, D, g), X defined in Assumption 2.1 and X m verifying (8), there exist δ 0 > 0 and compact sets X ′ m , X ′′ m such that X ⊂ X ′′ m ⊂ X ′ m ⊂ X m with inf x∈X |x| ∂X ′′ m > 0 (11)
and the following hold:

(P1) for any x ∈ D δ0 ∩ X ′′ m , there exists τ D ≥ 0 such that • Ψ f (x, τ D ) ∈ D, • Ψ f (x, t) ∈ (D δ0 ∩ X ′ m ) \ D for all t ∈ [0, τ D )
, where Ψ f is the flow operator along the vector field f . In addition, the map T :

D δ0 ∩X ′′ m → R ≥0 which associates τ D to each x ∈ D δ0 ∩ X ′′ m , is Lipschitz. (P2) No flow in D δ0 is possible for ẋ = f (x) starting from D ∩ X ′ m , namely there does not exist any solution t → x(t) such that x(0) ∈ D ∩ X ′ m and x(t) ∈ D δ0 on [0, ϵ] for some ϵ > 0. (P3) g(D ∩ X ′ m ) ∩ D δ0 = ∅ and there exists τ 0 m > 0 such that every solution t → x(t) of ẋ = f (x) with x(0) ∈ g(D ∩ X ′ m ) is defined over the interval [0, τ 0 m ], with x(t) ∈ X m and Π(x(t)) / ∈ D δ0 for all t ∈ [0, τ 0 m ].
Note that only the largest compact set X m and the parameters δ 0 and τ 0 m need to be known for observer design. The existence of the intermediary compact sets X ′ m and X ′′ m is only required for analysis in order to use the Lipschitz constants of the nonlinear maps along the estimate trajectory. In particular, the strict inclusion of X into X ′′ m given by ( 11) allows to say that if the estimation error x -x is sufficiently small, x ∈ X implies x ∈ X ′′ m . Also, the reason why we need to distinguish the compact sets X ′ m and X ′′ m is that there may not exist an invariant compact set by the flow ẋ = f (x) in (P1). Indeed, we only know by Assumption 2.1 that solutions initialized in X 0 remain in X but there is no reason why other solutions, in particular x flowing with f , should remain in a given compact set. So we assume solutions starting in D δ0 ∩ X ′′ m remain in a (possibly larger) compact set X ′ m until they reach D, and then remain in a possibly even larger X m after their jump while they flow on the interval [0, τ 0 m ]. Note that Assumption 3.1 was shown in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF] to hold for the model of a bouncing ball, modulo a regularization of its jump set D encoding the absence of Zeno. It is also shown to hold for a spiking neuron model in Section VI.

Remark 3.2: Sufficient conditions on the data f , C, and D ensuring the Lipschitzness of the time-to-impact function T in (P1) are given in a more general context in [START_REF] Maghenem | Lipschitzness of minimal-time functions in constrained continuous-time systems with applications to reachability analysis[END_REF] and references therein. Actually, when there exists a continuously differentiable function ϖ :

R nx → R such that the map T is characterized at each x ∈ D δ0 ∩ X ′′ m by ϖ(Ψ f (x, T(x))) = 0 with Ψ f (x, T(x)) ∈ D ∩ X ′
m , the continuous differentiability of T is guaranteed by the implicit function theorem under the transversality condition

∂ϖ ∂x (x)f (x) ̸ = 0 ∀x ∈ D ∩ X ′ m . (12)
Note that (12) also ensures that no flow is possible in D ∩ X ′ m , namely (P2) holds. Alternatively, conditions involving the tangent cone of the flow set at points in C ∩ D ∩ X ′ m and the flow map can be formulated to assure this property.

B. Hybrid Observer Construction

Suppose Assumption 3.1 holds with δ 0 and τ 0 m , and pick

0 < δ 1 < δ 0 and 0 < ∆ < τ 0 m 2 .
To implement the observation strategy explained in the previous section, we define our hybrid observer with state (z, χ, τ, q), where z and χ are used to construct the estimate x of the system state x, τ is a resettable timer, and q ∈ {0, 1, 2} is a logic variable that describes the "operating mode" of the observer. The hybrid observer operates as follows (see also Figure 2):

• When q = 2, z flows governed by the high-gain map F ℓ and

x is obtained by inverting T , namely x = T inv (z). In this mode, the estimate x approaches x over the flow interval for sufficiently large values of ℓ. On the other hand, the states χ and τ are unused and, hence, remain constant. For simplicity, τ is forced to be always 0 when q = 2. • When Π(x) reaches D δ1 , the observer jumps to mode q = 1 with χ initialized at Π(x), so as to store the current estimate of x. During this mode q = 1, χ flows governed by f and the estimate x is given by χ. On the other hand, z and τ are unused and remain constant, with still τ = 0. • When χ reaches D, which we know happens in finite flow time thanks to (P1), χ is reset to g(χ) and the mode changes to q = 0. Indeed, no flow is possible from D in C. During this mode q = 0, the timer τ counts flow time and χ flows governed again by f for ∆ units of time. The estimate x is still given by χ and z is still unused, so it is kept constant. • When the timer expires, namely when τ = ∆, the observer jumps back to mode q = 2, with z updated to T (Π(χ)).

Naturally, since q is a logic variable, it is constant during flow in each mode. In the following, we refer to the period of time where z flows governed by F ℓ and q = 2 as the high-gain phase, and to the period of time where χ flows governed by f with q ∈ {0, 1} as the open-loop phase. This observation strategy is modeled by the following hybrid system Ĥ:

    ż χ τ q     =                      F ℓ (z,y) 0 0 0 if (z, χ, τ, q) ∈ C 2 0 f (χ) 0 0 if (z, χ, τ, q) ∈ C 1 0 f (χ) 1 0 if (z, χ, τ, q) ∈ C 0 (13a)     z + χ + τ + q +     =                    z Π(Tinv(z)) 0 1 if (z, χ, τ, q) ∈ D 2 z g(χ) 0 0 if (z, χ, τ, q) ∈ D 1 T (Π(χ)) χ 0 2 if (z, χ, τ, q) ∈ D 0 (13b)
with estimate given by

x = T (z, χ, q) := T inv (z) if q = 2 χ if q ∈ {0, 1} (13c)
the (disjoint) flow sets defined by

C 2 = C z δ1 × R nx × {0} × {2} C 1 = R nz × D δ0 × {0} × {1} C 0 = R nz × R nx × [0, ∆] × {0}
and the (disjoint) jump sets by

D 2 = D z δ1 × R nx × {0} × {2} D 1 = R nz × D × {0} × {1} D 0 = R nz × C δ0 × [∆, +∞) × {0}
where

C δ0 = {x ∈ R nx : Π(x) ∈ cl(R nx \ D δ0 )} (14a) C z δ1 = {z ∈ R nz : Π(T inv (z)) ∈ cl(R nx \ D δ1 )} (14b) D z δ1 := {z ∈ R nz : Π(T inv (z)) ∈ D δ1 } . ( 14c
)
Of course, the system H evolves simultaneously with the observer Ĥ, with jumps that are not necessarily synchronized with those of the observer. However, as long as the estimation error x -x is sufficiently small, the hybrid observer Ĥ in (13) guarantees the following properties: i) When the observer flows in mode

q = 2, | Π(T inv (z))| D = | Π(x)| D ≥ δ 1 so x / ∈ D
and system H is also flowing, with y evolving continuously; ii) When the observer enters mode

q = 1, | Π(T inv (z))| D = | Π(x)| D = δ 1 , so x ∈ D δ0 and from (P1)-(P2),
x jumps in a near future, some time during the open-loop phase where q ∈ {1, 0}; iii) Once x has jumped, the observer has time to finish the openloop phase with q ∈ {1, 0} and to start again the high-gain phase with q = 2, before x reenters D δ0 (according to (P3) and the fact that ∆ < τ 0 m /2). Item iii) ensures that the estimation error has time to decrease via the use of the high-gain observer F ℓ before another openloop phase starts.

It is interesting to note that the definition of the flow and jump sets ensure a certain robustness of implementation because 0 < δ 1 < δ 0 . Indeed, at the end of a high-gain phase in C 2 , a jump occurs in D 2 when Π(T inv (z)) reaches D δ1 . At this point, χ is reset to Π(T inv (z)), thus it belongs to D δ1 , and flow is allowed for χ in the strictly larger set D δ0 by definition of C 1 . Similarly, a jump cannot happen before χ has reached D by definition of D 1 . In other words, at the beginning of the open-loop phase, χ is initialized "δ 0 -δ 1 "-away from the boundary of the flow set and "δ 1 "-away from the jump set, which leads to a robustness margin equal to min{δ 1 , δ 0 -δ 1 }.

Remark 3.3: Note that, compared to [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF], an additional state χ ∈ R nx is used in the open-loop phases with q ∈ {1, 0}, because the state z of the high-gain observer is generally in other coordinates with possibly n z > n x and cannot be used during that time. If z is directly in the x-coordinates, with T inv = Id, then χ can be removed as in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF].

C. Locally Asymptotically Stable Observer

With O and T given by Assumption 2.2, let us consider

A = A 2 ∪ A 10 ∪ A 1 ∪ A 0 ( 15
)
where

A 2 = {(x, z, χ, q) ∈ O × R nz × R nx × {2} : z = T (x)} A 10 = {(x, z, χ, q) ∈ R nx × R nz × R nx × {1, 0} : χ = x} A 1 = {(x, z, χ, q) ∈ g(D) × R nz × D × {1} : x = g(χ)} A 0 = {(x, z, χ, q) ∈ D × R nz × g(D) × {0} : χ = g(x)}
which, according to [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], are such that for each x ∈ X , with X defined in Assumption 2.1, and x defined in (13c),

(x, z, χ, q) ∈ A 2 ∪ A 10 =⇒ x = x (x, z, χ, q) ∈ A 1 =⇒ x ∈ D and x = g(x) (x, z, χ, q) ∈ A 0 =⇒ x ∈ D and x = g(x)
The sets A 2 and A 10 correspond to a zero estimation error, while the sets A 1 and A 0 correspond to x being one jump right ahead or behind of x. Unless exact synchronization of the system and observer jump times is achieved, including A 1 and A 0 cannot be avoided in an asymptotic analysis of a hybrid observer, since such errors are inevitable arbitrarily close to the jump times (the so-called hybrid peaking phenomenon in [START_REF] Biemond | Tracking control for hybrid systems with state-triggered jumps[END_REF], [START_REF] Sanfelice | An embedding approach for the design of state-feedback tracking controllers for references with jumps[END_REF]).

The following theorem shows that for ℓ sufficiently large, A is locally uniformly asymptotically stable for the interconnection of H and Ĥ.

Theorem 3.4: Suppose Assumptions 2.1, 2.2, and 3.1 hold with X 0 , ℓ 0 , δ 0 , τ 0 m and X ′′ m . Pick 0 < δ 1 < δ 0 and 0 < ∆ < τ 0 m 2 . Then, there exists ℓ * ≥ ℓ 0 such that for all ℓ > ℓ * , there exist β ℓ ∈ KL and ϵ ℓ > 0 such that for any x ∈ S H (X 0 ), any maximal solution ϕ := (z, χ, τ, q) to Ĥ defined by [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] with input y = h(x) and

ϕ(0, 0) ∈ C 2 ∪ R nz × (D δ1 ∩ X ′′ m ) × {0} × {1} such that |(x, z, χ, q)(0, 0)| A < ϵ ℓ (16)
is t-complete and there exists a full j-reparametrization x r of x such that for all (t, j) ∈ dom ϕ, |(x r (t, j), z(t, j), χ(t, j), q(t, j)

)| A ≤ β ℓ (|(x, z, χ, q)(0, 0)| A , t + j) . ( 17
)
In other words, [START_REF] Hammouri | High gain observer for structured multi-output nonlinear systems[END_REF] says that during the high-gain phases where q = 2, z asymptotically converges to T (x) (captured by A 2 ), and during the open-loop phases where q ∈ {0, 1}, χ either asymptotically converges to x (captured by A 10 ), or is a jump ahead/behind x during the jump time mismatches (captured by A 1 and A 0 ). However, thanks to T being Lipschitz, it can be seen in the proof that the length of those time mismatches asymptotically goes to zero.

Note also that the basin of attraction in [START_REF] Hamed | Observer-based feedback controllers for exponential stabilization of hybrid periodic orbits: Application to underactuated bipedal walking[END_REF], which is characterized by ϵ ℓ , may shrink as ℓ increases. This problem is solved in Section V by running a preliminary continuoustime observer with independent gains, able to decrease the estimation error as much as necessary before starting Ĥ with a given ℓ.

Remark 3.5: The analysis of the estimation error heavily relies on items i), ii), and iii) below ( 14) and, thus, necessitates a sufficiently small initial error, guaranteeing that x is only one jump ahead or behind x. One may proceed with initialization of (z, χ, τ, q) as follows. If we know that at the initial time, x is not about to jump or has not just jumped (namely x(0, 0) is not close to either D or g(D)), one may initialize (z, χ, τ, q) to q(0, 0) = 2, τ (0, 0) = 0 and z(0, 0) = T (x 0 ) with x0 ∈ X \ D δ1 such that the estimation error x0 -x(0, 0) is sufficiently small to have ( 16) hold. On the other hand, if we know that x(0, 0) is in D δ0 or close to g(D), one should initialize (z, χ, τ, q) to q(0, 0) = 1, τ (0, 0) = 0 and χ(0, 0) ∈ D δ1 ∩ X such that either χ(0, 0) -x(0, 0) or g(χ(0, 0)) -x(0, 0) is sufficiently small according to [START_REF] Hamed | Observer-based feedback controllers for exponential stabilization of hybrid periodic orbits: Application to underactuated bipedal walking[END_REF].

IV. PROOF OF THEOREM 3.4 In this section, we impose Assumptions 2.1, 2.2, and 3.1. We start by showing that as long as the high-gain Lyapunov function V ℓ is sufficiently small during high-gain phases, solutions to Ĥ are t-complete and alternating between highgain phases and open-loop phases. This is done by following a solution and considering all possible cases (see Lemma 4.1). Then, we study the evolution of the estimation error through one cycle of high-gain and open-loop phases, and finally, by iterating over those cycles, prove that a sufficient large gain and small initial estimation error ensure the Lyapunov function remains small enough and A is asymptotically stable.

Consider

0 < ε < min{δ 1 , δ 0 -δ 1 } such that X ε := {x ∈ R nx : |x| X ≤ ε} ⊂ X ′′ m ,
with X defined in Assumption 2.1 (possible according to [START_REF] Biemond | Tracking control for hybrid systems with state-triggered jumps[END_REF]).

Then, for all

(x, χ) ∈ R nx × R nx , (a) If x ∈ D and |x -χ| ≤ ε, then |χ| D < δ 1 (b) If |χ| D ≤ δ 1 and |x -χ| ≤ ε, then |x| D ≤ δ 0 . (c) If x ∈ X and |x -χ| ≤ ε, then χ ∈ X ε ⊂ X ′′ m . Denoting a inv the Lipschitz constant of T inv , let v ℓ := c(ℓ) ε a p a inv 2 (18)
where a p and c(ℓ) are defined in [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF] and Assumption 2.2 respectively. Then, from (4a), [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], and [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF]

, V ℓ (x, z) ≤ v ℓ with (x, z) ∈ X × R nz implies |x -Π(T inv (z))| ≤ a p |x -T inv (z)| ≤ a p |T inv (T (x)) -T inv (z)| ≤ a p a inv |z -T (x)| ≤ ε .
Therefore, items (a), (b), and (c) hold when V ℓ (x, z) ≤ v ℓ for χ = Π(T inv (z)) and, since a p ≥ 1, also for χ = T inv (z).

A. t-Completeness of Observer Solutions

Let us start by showing that, given a solution x ∈ S H (X 0 ), any observer solution ϕ := (z, χ, τ, q) verifying V ℓ (x, z) ≤ v ℓ during the high-gain phases is t-complete, with the mode q sequentially taking values 2 → 1 → 0 or 1 → 0 → 2 depending on the initial condition. Note that by definition of solutions to hybrid systems with hybrid input y given in [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF], jumps of the input (and thus here of the system H) can trigger artificial trivial jumps in the observer solution, which leads us to consider in the next lemma a sub-parametrization ϕ sub which describes the behavior of ϕ without those trivial jumps.

Lemma 4.1: Consider x ∈ S H (X 0 ) and a maximal solution ϕ = (z, χ, τ, q) of Ĥ with input y = h(x) such that for all (t, j) ∈ dom ϕ such that q(t, j) = 2 and all j ′ ∈ N such that (t, j ′ ) ∈ dom x, V ℓ (x(t, j ′ ), z(t, j)) ≤ v ℓ with v ℓ defined in [START_REF] Izhikevich | Simple model of spiking neurons[END_REF]. Then, ϕ is t-complete, and is a full j-reparametrization of a hybrid arc ϕ sub verifying either

q sub (t, j) = 2 -j (mod 3) ∀(t, j) ∈ dom ϕ sub (19a) if ϕ(0, 0) ∈ C 2 , or q sub (t, j) = 1 -j (mod 3) ∀(t, j) ∈ dom ϕ sub (19b)
otherwise.

Proof: See Appendix A.

B. Evolution of Estimation Error through a Cycle

q = 2 → 1 → 0 → 2
Consider the positive maps W :

O×R nz ×R nx ×{0, 1, 2} → R ≥0 and U : R nx × R nz × R nx × {0, 1} → R ≥0 defined by W (x, z, χ, q) = V ℓ (x, z) if q = 2 |χ -x| 2 if q ∈ {0, 1} U (x, χ, q) = |g(χ) -x| + |x| g(D) + |χ| D if q = 1 |χ -g(x)| + |x| D + |χ| g(D) if q = 0 which verify W (x, z, χ, q) = 0 ⇐⇒ (x, z, χ, q) ∈ A 2 ∪ A 10 , (20a)
U (x, χ, q) = 0 ⇐⇒ (x, z, χ, q) ∈ A 1 ∪ A 0 . (20b)
In high-gain phases, when q = 2, we know from Assumption 2.2 that V ℓ and thus W decrease exponentially at rate ℓλ. The following lemma describes the evolution of the estimation error through an open-loop phase. This error is measured by W or U depending on the period: U is used when x is one jump ahead or behind χ, i.e., during the interval of time with jump mismatch, and W is used the rest of the time. The next lemma roughly says that if the observer starts an open-loop phase with V ℓ sufficiently small, it will go through modes 1, 0 and then back to 2, and V ℓ will have grown at most by a c(ℓ) c(ℓ) through this process. Besides, x will have jumped only once in the meantime. Then, either ϕ flows in mode 2 for all time, or it reaches another open-loop phase with a dwell-time of at least τ 0 m -2∆, so that V ℓ grows at most by a c(ℓ) c(ℓ) e -ℓλ(τ 0 m -2∆)

over the full cycle "open-loop + high-gain" phases.

Lemma 4.2: Consider x ∈ S H (X 0 ) and a maximal solution ϕ = (z, χ, τ, q) of Ĥ with input y = h(x). Consider a transition from mode 2 to 1, namely a jump j ∈ dom j ϕ such that q(t j , j -1) = 2 and q(t j , j) = 1. Assume

V ℓ (x(t j , j ′ ), z(t j , j -1)) < min v ℓ , c(ℓ) a 2 p a 2 inv ∆ 2 a 2 τ
with j ′ ∈ N such that (t j , j ′ ) ∈ dom x, v ℓ defined in [START_REF] Izhikevich | Simple model of spiking neurons[END_REF], a inv the Lipschitz constant of T inv , a p defined in [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF], and a τ the Lipschitz constant of T on X ′′ m ∩ D δ0 given by (P1). Then, j + 3 ∈ dom j ϕ and x jumps exactly once in the time interval (t j , t j+3 ), i.e., j ′ is unique such that

(t j , j ′ ) ∈ dom x , (t j+3 , j ′ + 1) ∈ dom x .
More precisely, q(t, j) = 1 for all t ∈ [t j , t j+1 ], q(t, j +2) = 0 for all t ∈ [t j+2 , t j+3 ], q(t j+3 , j + 3) = 2, and there exist a 1 , a 0 , a 01 > 0 (independent from ℓ, x and ϕ) such that

|χ(t, j) -x(t, j ′ )| ≤ a 1 |χ(t j , j) -x(t j , j ′ )| ∀t ∈ [t j , t j+1 ] |χ(t, j + 2) -x(t, j ′ + 1)| ≤ a 0 |χ(t j , j) -x(t j , j ′ )| ∀t ∈ [t j+2 , t j+3 ]
and2

• either x jumps before ϕ reaches D 1 , so j + 1 is a trivial jump in ϕ, q(t, j + 1) = 1 for all t ∈ [t j+1 , t j+2 ], and during the interval with jump mismatch U (x(t, j ′ + 1), χ(t, j + 1), q(t, j + 1))

≤ a 01 |χ(t j , j) -x(t j , j ′ )| ∀t ∈ [t j+1 , t j+2 ]
• or x jumps after ϕ has reached D 1 so j + 2 is a trivial jump in ϕ, q(t, j + 1) = 0 for all t ∈ [t j+1 , t j+2 ], and during the interval with jump mismatch U (x(t, j ′ ), χ(t, j + 1), q(t, j + 1))

≤ a 01 |χ(t j , j) -x(t j , j ′ )| ∀t ∈ [t j+1 , t j+2 ]
Moreover, there exists a > 0 (independent from ℓ, x and ϕ) such that

V ℓ (x(t j+3 , j ′ + 1), z(t j+3 , j + 3)) ≤ a c(ℓ) c(ℓ) V ℓ (x(t j , j ′ ), z(t j , j -1)) , (21)
with c(ℓ), c(ℓ) given by Assumption 2.2. Finally, if in addition,

V ℓ (x(t j , j ′ ), z(t j , j -1)) < 1 a c(ℓ) c(ℓ) v ℓ , (22)
then, either q(t, j +3) = 2 for all t ≥ t j+3 , or j +4 ∈ dom j ϕ with q(t, j + 3) = 2 for all t ∈ [t j+3 , t j+4 ], q(t j+4 , j + 4) = 1, and

t j+4 -t j+3 ≥ τ 0 m -2∆ > 0 so that V ℓ (x(t j+4 , j ′ + 1), z(t j+4 , j + 3)) ≤ a c(ℓ) c(ℓ) e -ℓλ(τ 0 m -2∆) V ℓ (x(t j , j ′ ), z(t j , j -1)) . ( 23
)
See Appendix [?] for a proof.

C. Iterating Cycles

Exploiting exponential growth over polynomial growth, let us pick ℓ sufficiently large such that

µ ℓ := ae -ℓλ(τ 0 m -2∆) c(ℓ) c(ℓ) < 1 (24a)
and v 1 sufficiently small such that

v 1 < c(ℓ) a 2 p a 2 inv min ∆ 2 a 2 τ , 1 a c(ℓ) c(ℓ) ε 2 (24b)
Then, if we ensure that the observer verifies V ℓ (x, z) < v 1 before each transition from high-gain to open-loop phase, we ensure that i) V ℓ (x, z) < v ℓ after the open-loop phase (once it has switched back to q = 2) thanks to (24b) and [START_REF] Izhikevich | Simple model of spiking neurons[END_REF], and that ii) V ℓ (x, z) < v 1 again at the next transition from high-gain to open-loop phase, namely when q = 2 → 1, thanks to (24a).

Then, these transitions can be iterated by repetitively using Lemma 4.2. Therefore, denoting v k the value of V ℓ before the kth transition q = 2 → 1, we have v k ≤ µ k-1 ℓ v 1 . If q(0, 0) = 2, choosing |(x, z, χ, q)(0, 0)| A sufficiently small ensures that |(T (x) -z)(0, 0)|, and thus V ℓ , are initially smaller than v 1 . Because V ℓ decreases in high-gain phase when q = 2, the observer necessarily verifies V ℓ (x, z) < v 1 at its first transition to open-loop q = 2 → 1. On the other hand, if q(0, 0) = 1, by assumption χ(0, 0) ∈ D δ1 ∩ X ′′ m and choosing |(x, z, χ, q)(0, 0)| A sufficiently small ensures that |(x-χ)(0, 0)| and thus W is initially sufficiently small, so that all the previously described steps are still valid and V ℓ is also smaller than v 1 at its first transition to open-loop q = 2 → 1.

D. Asymptotic stability of A

For each system jump, the observer solution jumps four times: three times for its own transitions between modes 2 → 1 → 0 → 2, with its state reset using the observer jump map (13b), and once trivially at the system jump with its state reset via the identity map. For each j ∈ dom j ϕ, let us thus denote

k(j) := ⌊ 1 4 (j + 3)⌋ , if q(0, 0) = 2 ⌊ 1 4 j⌋ , if q(0, 0) = 1.
At each hybrid time (t, j) ∈ dom ϕ, the integer k(j) represents the number of transitions from high-gain to open-loop phase, i.e., q = 2 → 1, that have occurred in the observer solution since the start. In the following, we denote x r the jreparametrization of x such that dom x r = dom ϕ, containing trivial jumps whenever ϕ jumps and x does not. According to [START_REF] Kim | State estimation and tracking control for hybrid systems by gluing the domains[END_REF] and by continuity, there exists K-maps ρ, ρ ′ such that on the compact set containing the solutions,

ρ ′ (W (x, z, χ, q)) ≤ |(x, z, χ, q)| A2∪A10 ≤ ρ(W (x, z, χ, q)) ρ ′ (U (x, χ, q)) ≤ |(x, z, χ, q)| A0∪A1 ≤ ρ(U (x, χ, q)) and ρ ′ (min{W (x, z, χ, q), U (x, χ, q)}) ≤ |(x, z, χ, q)| A ≤ ρ min{W (x, z, χ, q), U (x, χ, q)}
If k(j) = 0 for all j ∈ dom j ϕ, it means no transition from mode 2 to 1 occurs, namely the solution is eventually continuous in mode 2. Exponential decrease of W during this high-gain phase allows to conclude. Now, assume instead there exists j 1 ∈ N such that k(j 1 ) = 1, namely the jump index marking the first transition between q = 2 and q = 1, and denote w 1 := W (x r (t j1 , j 1 -1), z(t j1 , j 1 -1), χ(t j1 , j 1 -1), 2) = V ℓ (x r (t j1 , j 1 -1), z(t j1 , j 1 -1)), the value of the Lyapunov function right before this transition. By applying Lemma 4.2 iteratively, on each cycle of "open-loop + highgain" phases happening after hybrid time (t j1 , j 1 ), we deduce that there exists π ℓ ≥ 1 (depending on ℓ, a, a 0 , a 1 , a 01 ) such that for all (t, j) ∈ dom ϕ with k(j) ≥ 1, one of the following holds:

• W (x r (t, j), z(t, j), χ(t, j), 1) ≤ π ℓ µ k(j)-1 ℓ w 1 during the first period of the open-loop phase where q = 1 and neither x nor χ has jumped, • U (x r (t, j), z(t, j), χ(t, j), q(t, j)) ≤ π ℓ µ k(j)-1 ℓ v 1 during the jump time mismatch, • W (x r (t, j), z(t, j), χ(t, j), 0) ≤ π ℓ µ k(j)-1 ℓ w 1 in the last period of the open-loop phase where q = 0 and both x and χ have jumped,

• W (x r (t, j), z(t, j), χ(t, j), 2) ≤ π ℓ e -λℓ(t-tj-1) µ k(j)-1 ℓ w 1 if q(t, j) = 2. Those upper bounds ensure asymptotic convergence to A since the mode q = 2 is persistently visited, µ ℓ < 1 and t is unbounded by t-completeness of solutions. However, in order to show asymptotic stability as in [START_REF] Hammouri | High gain observer for structured multi-output nonlinear systems[END_REF], we need a KL-bound with respect to both t + j and the initial condition. First, concerning the initial condition, note that if q(0, 0) = 2, we have j 1 = 1 and w 1 ≤ e -λℓt1 W (x(0, 0), z(0, 0), χ(0, 0), 2) .

Otherwise, if q(0, 0) = 1, we have j 1 = 3, and

w 1 ≤ π ℓ e -λℓ(t3-t2) W (x(0, 0), z(0, 0), χ(0, 0), 1)
On the other hand, concerning t + j, we need to be more precise about the exponential decrease during each high-gain phase. In particular, the bounds by µ k(j)-1 ℓ only account for a decrease of W by e -ℓλτ ′ m during each high-gain phase with q = 2 (see the definition of µ ℓ in (24a)), while a high-gain phase could last for more than τ ′ m amount of time. In order to obtain a KL-bound, we need to reestablish this unaccounted decrease. Because each open-loop phase where q ∈ {0, 1} lasts at most 2∆, and because τ ′ m amount of flow is taken into account in µ ℓ at each high-gain phase where q = 2, extra exponential decrease should be accounted for whenever t ≥ (k(j) + 1)(2∆ + τ ′ m ) so that we actually have |(x r (t, j), z(t, j), χ(t, j), q(t, j)

)| A ≤ ρ π 2 ℓ µ k(j)-1 ℓ e -λℓ max{t-(k(j)+1)(2∆+τ ′ m ),0} ρ ′-1 (|(x, z, χ, q)(0, 0)| A )
which gives the result.

V. SEMIGLOBAL HYBRID ASYMPTOTIC OBSERVER

With the high-gain hybrid observer formulated in Section III assuring local uniform asymptotic stability of the set A for the interconnection between H and Ĥ, in this section we provide a hybrid observer that, through the use of a continuoustime high-gain observer, which we call preliminary highgain observer, enlarges the region of attraction and leads to a semiglobal result. The resulting observer, which we refer to as semiglobal hybrid observer, sequentially switches from the preliminary high-gain observer to the local hybrid observer when (16) holds. A key challenge in enlarging the basin of attraction is the presence of jumps of H when the preliminary high-gain observer is used, which may prevent the state estimate from converging to the state of H. To overcome this issue, we exploit the properties in Assumption 2.1, as we describe below. Also, since the system is no longer in its initialization set X 0 at the switching time where the local observer is launched, and since this switching time will be chosen smaller than τ m , we introduce an additional initialization set X ′ 0 . Assumption 5.1: There exists a compact subset X ′ 0 of X such that any solution x ∈ S H (X 0 ) remains in X ′ 0 for (t, j) ∈ dom x such that t ∈ [0, τ m ].

Since the construction of the semiglobal hybrid observer we provide works for any observer that assures local convergence to the set A, we design it for a general local hybrid observer. To this end, we assume the existence of a local hybrid observer

Ĥ ξ ∈ F(ξ, y) (ξ, y) ∈ C ξ + ∈ G(ξ, y) (ξ, y) ∈ D , x = T (ξ) (25)
for H, e.g, through the design proposed in Section III. Similar to the properties in Theorem 3.4, we assume that Ĥ induces the following local convergence property; see Section II-A for the definition of solutions to [START_REF] Medina | State estimation for linear impulsive systems[END_REF] with hybrid input y. Assumption 5.2: Consider A ⊂ R nx × R n ξ . Let X ′ 0 come from Assumption 5.1. There exists a set Ξ 0 ⊂ R n ξ and ϵ > 0 such that for any x ∈ S H (X ′ 0 ), any maximal solution (ξ, y r ) to Ĥ defined by [START_REF] Medina | State estimation for linear impulsive systems[END_REF] with input y = h(x) and with ξ(0, 0) ∈ Ξ 0 such that x(0, 0) := T (ξ(0, 0)) verifies

|x(0, 0) -x(0, 0)| < ϵ , (26)
is t-complete and there exists a full j-reparametrization (x r , ξ r ) of x and ξ such that dom x r = dom ξ r and lim t+j→∞ |(x r , ξ r )(t, j)| A = 0.

In other words, Ĥ is a local observer relative to A in the sense of [START_REF] Bernard | On notions of detectability and observers for hybrid systems[END_REF].

Example 5.3: Assume any solution x ∈ S H (X ′ 0 ), with X ′ 0 from Assumption 5.1, still verifies Assumption 2.1, namely is t-complete with dwell-time τ m and remains in X at all times. If Assumptions 2.2 and 3.1 hold, we apply Theorem 3.4 from X ′ 0 . Then, Ĥ defined by [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] with state ξ = (z, χ, τ, q) and a fixed sufficiently large gain ℓ, is a local observer for H. Indeed, Assumption 5.2 is verified for A defined in [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF], with

Ξ 0 = C 2 ∪ R nz × (D δ1 ∩ X ′′ m ) × {0} × {1}
for X ′′ m from Assumption 3.1, and ϵ sufficiently small in ( 26) for ( 16) to hold for this given ℓ.

A. Preliminary High-Gain observer and Switching Logic

Building upon the local observer in [START_REF] Medina | State estimation for linear impulsive systems[END_REF] satisfying Assumption 5.2, we design an observer that ensures semiglobal asymptotic convergence of the estimation error. Given an arbitrary compact set Z 0 ⊂ R nz , we propose the following time-driven logic:

1) Run a preliminary high-gain observer

żp = F ℓp (z p , y) , x = T inv (z p ) (27)
on the time interval [0, t p ], initialized in Z 0 , with F ℓp and T inv defined in Assumption 2.2 and (8);

2) After time t p , launch the local observer Ĥ defined in ( 25), initialized at ξ 0 ∈ Ξ 0 such that T (ξ 0 ) = x(t p ), with t p chosen in a way that ensures the estimation error x-x provided by ( 27) at time t p , is sufficiently small to verify [START_REF] Menini | State immersion observers for mechanical systems with impacts[END_REF], so that Ĥ can be launched from an appropriate initialization guaranteeing asymptotic convergence to A. In the following, we consider a map Ξ : R nx → Ξ 0 such that for any x ∈ R nx , ξ 0 := Ξ(x) verifies T (ξ 0 ) = x.

The observer [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF] guarantees that the Lyapunov function V ℓp defined in Assumption 2.2 decreases arbitrarily fast for a sufficiently large gain ℓ p , while the system H flows. However, jumps of H could make V ℓp increase as [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF] is not designed to account for the effects of those jumps. And because the jump times of H are unknown, we cannot adapt t p to the first jump of H as it may happen any time. Fortunately, under Assumption 2.1, we know that H can jump only once in the interval [0, τ m ]. Therefore, choosing t p ≤ τ m ensures that only one jump of H should be handled in the error analysis on the interval [0, t p ).

A first idea could be to choose ℓ p sufficiently large for V ℓp to decrease by a sufficient amount over a time window smaller than τ m and fix t p = τ m . The problem is when the system H jumps right before τ m : V ℓp may increase at the jump and may not have enough time to decrease again before Ĥ is launched at time τ m . A second idea could be to choose ℓ p sufficiently large for V ℓp to decrease by a sufficient amount over a time window smaller than τm 2 and then authorize an immediate switch to Ĥ as soon as Π(x) ∈ D δ1 in the time interval τm 2 , τ m , or at time τ m at the latest if this does not happen. Indeed, the fact that Π(x) ∈ D δ1 announces a system jump if the estimation error x -x is sufficiently small. But this strategy raises a similar issue as above if H jumps right before τm 2 , since Π(x) could happen to be near D right after τm 2 without having x-x sufficiently small to launch Ĥ. Actually, in this latter case, it would have been better to stick to the first strategy and wait till time τ m in order for the estimation error to decrease again after the system jump.

Due to those reasons, we propose to combine both strategies and, considering τ 1 < τ 2 in (0, τ m ), switch to Ĥ either

• at time τ m , if the event "Π(x) ∈ D δ1 " happens in the time window [τ 1 , τ 2 ] or does not happen at all over the window [0, τ m ]; or, • as soon as Π(x) ∈ D δ1 if such an event happens after τ 2 . To make this strategy more precise, and following the definition of T in (P1), let ∆ max be an upper bound of T on D δ0 ∩X , namely an upper bound of the time elapsed between the time where a solution x of H enters D δ0 ∩X and the time it reaches D. The existence of ∆ max is guaranteed by (P1). Since T vanishes on D and, by definition of D δ0 , ∆ max can be chosen to verify ∆ max ≤ a τ δ 0 , where a τ is the Lipschitz constant of T, so that ∆ max < τ m for sufficiently small δ 0 > 0. Then, consider 0 < δ 1 < δ 0 and times τ 1 , τ 2 ∈ [0, τ m ], such that 0 < τ 1 < τ 2 < τ 2 + ∆ max < τ m , i.e., a solution to H entering D δ0 ∩ X before τ 2 necessarily reaches D before τ 2 + ∆ max < τ m . We show that for ℓ p sufficiently large, the switching time t p to Ĥ can be chosen in [τ 2 , τ m ] based on when Π(x) ∈ D δ1 , with x provided by [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF]. The strategy is described in Algorithm 1 (for simplicity, we index the solutions to [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF] on continuous time only, even though its input y is an hybrid signal).

Algorithm 1 Semiglobal observer strategy

Pick 0 < τ 1 < τ 2 < τ m such that τ 2 + ∆ max < τ m . Pick z p (0) ∈ Z 0 . q w ← 0, t p ← τ m , t ← 0. while t < t p do
Run [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF] with input y = h(x) and output x.

if Π(x(t)) ∈ D δ1 and q w = 0 then if t ∈ [τ 1 , τ 2 ) then q w ← 1 else if t ∈ [τ 2 , τ m ] then t p ← t. end if end if end while Pick ξ(t p , 0) ∈ Ξ 0 such that T (ξ(0, 0)) = Π(x(t p )). while t ≥ t p do
Run Ĥ in [START_REF] Medina | State estimation for linear impulsive systems[END_REF] with input y = h(x) and output x. end while

B. Semiglobal Hybrid Observer Construction

The solutions produced by Algorithm 1 are generated by a hybrid observer Ĥsg , defined below in [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF], with state ξ sg = (z p , τ p , q p , q w , ξ), where z p is the state of the preliminary observer [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF], ξ is the state of the local observer Ĥ, τ p ∈ [0, τ m ] is a timer, q w ∈ {0, 1} is a warning state used to determine the time t p to switch to Ĥ, and q p ∈ {0, 1} is used to indicate whether the observer is in preliminary mode (q p = 1) or local mode (q p = 0). Actually, in order to check the robustness of our algorithm, we also include the solutions obtained by allowing either q w ← 1 or t p ← t at the frontier time τ 2 in Algorithm 1. This allows to obtain an outer semicontinuous jump map with a closed jump set. Given the local and preliminary observers [START_REF] Medina | State estimation for linear impulsive systems[END_REF], [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF], the dynamics of the observer Ĥsg are thus defined by

      żp τp qp qw ξ       =                  F ℓp (zp,y) 1 0 0 0   if ξ sg ∈ C sg,1   0 0 0 0 F (ξ,y)   if ξ sg ∈ C sg,0 (28a)
      z + p τ + p q + p q + w ξ +       ∈                                            zp τp 0 0 Ξ(Π(Tinv(zp)))   if ξ sg ∈ D sg,1 \ D w sg,1   zp τp qp 1 ξ   if ξ sg ∈ D w sg,1 \ D sg,1   zp τp qp 1 ξ   ∪   zp τp 0 0 Ξ(Π(Tinv(zp)))   if ξ sg ∈ D sg,1 ∩ D w sg,1   zp τp qp qw G(ξ,y)   if ξ sg ∈ D sg,0 (28b
) with estimate given by

x = T sg (ξ sg ) := T inv (z p ) if q p = 1 T (ξ) if q p = 0 (28c)
the flow and jump sets defined by

C sg,1 = R nz × [0, τ m ] × {1} × {1} × R n ξ ∪ R nz × [0, τ 1 ] × {1} × {0} × R n ξ ∪ C z δ1 × [τ 1 , τ m ] × {1} × {0} × R n ξ C sg,0 = R nz × [0, τ m ] × {0} × {0, 1} × C D sg,1 = R nz × {τ m } × {1} × {0, 1} × R n ξ ∪ D z δ1 × [τ 2 , τ m ] × {1} × {0} × R n ξ D w sg,1 = D z δ1 × [τ 1 , τ 2 ] × {1} × {0} × R n ξ D sg,0 = R nz × [0, τ m ] × {0} × {0, 1} × D
where C and D are defined in [START_REF] Medina | State estimation for linear impulsive systems[END_REF] and C z δ1 and D z δ1 in [START_REF] Galeani | Robust trajectory tracking for a class of hybrid systems: An internal model principle approach[END_REF]. Solutions Ĥsg initialized in Z 0 × {0} × {1} × {0} × R n ξ flow in C sg,1 with q p = 1 and q w = 0 until either

• the timer τ p reaches τ m ; or • Π(x) ∈ D δ1 at some time in [τ 1 , τ m ].
In the former case, the solution is in D sg,1 \ D w sg,1 so that q p is reset to 0, marking the end of the interval over which the preliminary observer is used, ξ is reset to Ξ(Π(x)), and the solution then evolves according to Ĥ. In the latter case, either

• τ p ∈ [τ 1 , τ 2 ), the solution is in D w sg,1 \ D sg,1
and, after the jump, q p remains at 1 and q w is equal to 1; or

• τ p ∈ (τ 2 , τ 3 ], the solution is in D sg,1 \ D w sg,1
, q p is reset to 0, marking the end of the preliminary observer mode; or

• τ p = τ 2 , the solution is in D sg,1 ∪ D w
sg,1 and we have a choice between the former two items. In the first item, after q w has been reset to 1, the preliminary observer evolves continuously with the solution in C sg,1 until the timer τ p reaches τ m . When this happens, the solution is in D sg,1 , so q p is reset to 0 and Ĥ is launched from Ξ(Π(x)).

C. Semiglobal Result

The hybrid observer Ĥsg guarantees the following semiglobal property for the set A.

Theorem 5.4: Suppose Assumptions 2.1, 2.2, 3.1, 5.1 and 5.2 hold with δ 0 sufficiently small such that ∆ max < τ m , where

∆ max is an upper bound of T. Consider a compact subset Z 0 of R nz and 0 < τ 1 < τ 2 < τ m such that τ 2 + ∆ max < τ m .
Then, there exists ℓ * > ℓ 0 such that for all ℓ p > ℓ * and for any x ∈ S H (X 0 ), any solution ϕ = (z p , τ p , q p , q w , ξ) to Ĥsg defined in [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF]

and initialized in Z 0 × {0} × {1} × {0} × R n ξ is t-complete and verifies lim t+j→∞ |(x r , ξ r )(t, j)| A = 0 (29)
for some full j-reparametrizations x r and ϕ r = (z r p , τ r p , q r p , q r w , ξ r ) of x and ϕ respectively such that dom x r = dom ϕ r .

Note that this result ensures asymptotic convergence, but not stability. In particular, if Ĥ is chosen as in [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF], the stability provided by Theorem 3.4 ensures stability of |(x, ξ)| A with respect to its value at time t p , but not with respect to its initial condition. Indeed, the preliminary observer [START_REF] Ping | Robust estimation algorithm for both switching signal and state of switched linear systems[END_REF] running on [0, t p ] may miss up to one jump of x. In other words, |(x, ξ)(0, 0)| A = 0 does not imply |(x, ξ)| A remains zero. In order to ensure semiglobal stability, other observers must be designed able to follow the jumps of H from the start.

On the other hand, the observer data is chosen verifying the hybrid basic conditions, namely with closed sets, and outer semicontinuous locally bounded maps (see [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF]Assumption 6.5]). Due to the absence of stability with respect to the initial error, we cannot directly claim robustness through [START_REF] Goebel | Hybrid Dynamical Systems : Modeling, Stability and Robustness[END_REF]Theorem 7.21]. However, the observer sequentially uses robust elements (robust preliminary high-gain observer, hysteresis in the choice of t p , robust local observer) which provide robustness as observed in simulations in Section VI, although of course, this design does not escape from the well-known performance limitations of high-gain observers in the presence of output noise.

Proof: (of Theorem 5.4) Consider x ∈ S H (X 0 ) and ϕ a solution to [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] initialized in Z 0 × {0} × {1} × {0} × R n ξ with input y = h(x). The solution ϕ evolves as detailed above the statement of Theorem 5.4, with first a preliminary observer mode (q p = 1) and then a local observer mode (q p = 0).

1) Behavior of solutions and sufficient condition. Let us denote t p ∈ [τ 1 , τ m ] the time at which the preliminary observer stops and the associated jump j p ∈ {1, 2} verifying q p (t p , j p -1) = 1 , q p (t p , j p ) = 0 .

If j p = 2, it means that Π(x) has been in D δ1 in [τ 1 , τ 2 ],
leading to the warning state q w being reset to 1 and t p necessarily equal to τ m . On the other hand, if j p = 1, it means that no warning state has been used (i.e., q w (t, 0) = 0 for all t ∈ [0, t p ]) and t p ∈ [τ 2 , τ m ]. Then, for all t ≥ t p and all j ≥ j p such that (t, j) ∈ dom ϕ, (t, j) → ξ(t, j) evolves according to the local observer dynamics Ĥ. By Assumption 2.1, the system solution x jumps at most once on [0, t p ] and x(t p , j ′ ) ∈ X , for j ′ ∈ {0, 1} such that (t p , j ′ ) ∈ dom x. Since ξ is initialized to Ξ(Π(x(t p , j p ))) at the beginning of the local observer mode, Assumptions 5.1 and 5.2 imply the result if

|Π(x(t p , j p )) -x(t p , j ′ )| < ϵ . (30)
2) Lyapunov upper-bounds until switching time. Without loss of generality, we assume 0 < ϵ < min{δ 1 , δ 0 -δ 1 } in [START_REF] Menini | State immersion observers for mechanical systems with impacts[END_REF]. Then, similarly to Section IV, for all

(x, χ) ∈ R nx ×R nx , (a') If x ∈ D and |x -χ| ≤ ϵ, then |χ| D < δ 1 . (b') If |χ| D ≤ δ 1 and |x -χ| ≤ ϵ, then |x| D ≤ δ 0 . Still denoting a inv the Lipschitz constant of T inv , let v ℓp := c(ℓ) ϵ a p a inv 2 (31)
where a p and c(ℓ) are defined in [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF] and in Assumption 2.2, respectively. Then, for all (x, z) ∈ X × R nz ,

V ℓp (x, z) < v ℓp =⇒ |x -Π(T inv (z))| ≤ a p |x -T inv (z)| < ϵ .
In particular, it is enough to show that

V ℓp (x(t p , j ′ ), z p (t p , j p )) < v ℓp (32)
to ensure [START_REF] Sanfelice | Invariance principles for hybrid systems with connections to detectability and asymptotic stability[END_REF] holds.

Let us study the evolution of

V ℓp (x, z p ) on [0, t p ]. Define r 0 = max (x,z)∈X0×Z0 |T (x)-z| , r 1 = max x∈X |T (g(x))-T (x)| .
Before the first jump of x, i.e. for all t ∈ [0, min{t 1 (x), τ m }], and for j ∈ {0, 1} such that (t, j) ∈ dom ϕ, V ℓp (x(t, 0), z p (t, j)) ≤ e -ℓpλt V ℓp (x 0 (0, 0), z p (0, 0))

≤ c(ℓ p )e -ℓpλt |z p (0, 0) -T (x(0, 0))| 2 ≤ c(ℓ p )r 2 0 e -ℓpλt (33a)
Then, if t 1 (x) < t p , i.e., if the first jump of the system happens before t p , x jumps to g(x) while z p does not jump. Since for all (x, z) ∈ X × R nz ,

V ℓp (g(x), z) ≤ c(ℓ p )|z -T (g(x))| 2 ≤ c(ℓ p ) 2 (|T (x) -T (g(x))| 2 + |z -T (x)| 2 ) ≤ c(ℓ p ) 2 r 2 1 + 1 c(ℓ p ) V ℓp (x, z)
we have for all t ∈ [t 1 (x), t p ], and for j ∈ {0, 1} such that (t, j) ∈ dom ϕ,

V ℓp (x(t, 1), z p (t, j)) ≤ c(ℓ p ) 2 r 2 1 + c(ℓ p ) c(ℓ p ) r 2 0 e -ℓpλt1(x) e -ℓpλ(t-t1(x)) ≤ c(ℓ p ) 2 r 2 1 e -ℓpλ(t-t1(x)) + c(ℓ p ) c(ℓ p ) r 2 0 e -ℓpλt . (33b)
Denoting τ 3 := τ 2 +∆ max < τ m , exploiting exponential decay over polynomial growth, pick ℓ p sufficiently large such that

c(ℓ p )r 2 0 e -ℓpλτ1 < v ℓp (34a) c(ℓ p ) 2 r 2 1 e -ℓpλ min{τ2-τ1,τm-τ3} + c(ℓ p ) c(ℓ p ) r 2 0 e -ℓpλτ2 < v ℓp (34b)
3) Solution-based proof of [START_REF] Sferlazza | Time-varying sampleddata observer with asynchronous measurements[END_REF], no matter the first jump time t 1 (x) of the system solution x. We have the following cases:

• If t 1 (x) ≤ τ 1 , (32)
holds according to (33b) and (34b) since

t p ∈ [τ 2 , τ m ] and thus t p -t 1 (x) ≥ τ 2 -τ 1 . • If τ 1 < t 1 (x) ≤ τ 2 ,
we have according to (33a) and (34a),

V ℓp (x(t, 0), z p (t, 0)) < v ℓp
for any τ 1 ≤ t ≤ t 1 (x), and thus, according to item (a'), there exists

t * ∈ [τ 1 , t 1 (x)) ⊆ [τ 1 , τ 2 ) such that Π(T inv (z p (t * , 0))) ∈ D δ1 .
According to the behavior of solutions described above (see also Algorithm 1), t p = τ m and (32) holds at time t p thanks to (33b) and (34b) since t p -t 1 (x) ≥ τ m -τ 2 > τ m -τ 3 .

• If τ 2 ≤ t 1 (x) ≤ τ m , still through (33a), (34a), V ℓp (x(t, 0), z p (t, 0)) < v ℓp for any τ 1 ≤ t ≤ t 1 (x), and with item (a'), there exists a minimal time t * ∈ [τ 1 , t 1 (x)) ⊆ [τ 1 , τ m ] such that Π(T inv (z p (t * , 0))) ∈ D δ1 . We distinguish two cases: • If t 1 (x) ≥ τ m , then t p ≤ t 1 (x) and ( 32) holds according to (33a) and (34a).

-If t * ∈ [τ 1 , τ 2 ],
Therefore, in all cases, [START_REF] Sferlazza | Time-varying sampleddata observer with asynchronous measurements[END_REF] holds and the result is proved. The parameters τ i in Ĥsg can be chosen arbitrarily as long as τ m -τ 2 > ∆ max , namely, in sight of Assumption 3.1, as long as any solution entering D δ0 before τ 2 reaches D and jumps before τ m . But of course, the smaller τ 1 , τ 2 -τ 1 and τ m -τ 3 , the larger ℓ p must be taken according to [START_REF] Shim | Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems[END_REF].

Remark 5.5: Note that if Ĥ defined in [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] with state ξ = (z, χ, τ, q) is chosen as local observer, the implementation of its semiglobal version Ĥsg defined in [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] can be simplified by using the states z and q of Ĥ in place of z p and q p , respectively, with, for instance q ∈ {0, 1, 2, 3}, where q = 3 denotes the "local observer mode".

VI. EXAMPLES

A. Bouncing ball

The local observer [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] was illustrated in [START_REF] Bernard | A local hybrid observer for a class of hybrid dynamical systems with linear maps and unknown jump times[END_REF] on a bouncing ball with state (x 1 , x 2 ) ∈ R 2 and data given by

f (x) = (x 2 , -g) , g(x) = -x , h(x) = x 1 C = {(x 1 , x 2 ) ∈ R 2 : x 1 ≥ 0} (35) D = {(x 1 , x 2 ) ∈ R 2 : x 1 = 0 , x 2 ≤ 0}
In order to satisfy Assumption 3.1 and encode the absence of Zeno behavior in the solutions of interest, the set D was replaced in the observer by D m := {(x 1 , x 2 ) ∈ R 2 : x 1 = 0 , x 2 ≤ -m} for m > 0. The semiglobal observer [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] presented in this paper is implemented in https://github. com/HybridSystemsLab/NonSyncHybridHighGainObserver.

B. Spiking neuron

Consider the parameterized nonlinear model of a spiking neuron presented in [START_REF] Izhikevich | Simple model of spiking neurons[END_REF] given by a hybrid system H as in [START_REF] Alessandri | Switching observers for continuous-time and discrete-time linear systems[END_REF] with state (x 1 , x 2 ) ∈ R 2 and data given by

f (x) = 0.04x 2 1 + 5x 1 + 140 -x 2 + I ext , a(bx 1 -x 2 ) g(x) = (c, x 2 + d) , h(x) = x 1 C = {(x 1 , x 2 ) ∈ R 2 : x 1 ≤ v m } (36) D = {(x 1 , x 2 ) ∈ R 2 : x 1 = v m }
where x 1 is the membrane potential, x 2 is the recovery variable, and I ext represents the (constant) synaptic or injected DC current. The value of the input I ext and the model parameters a, b, c, d, and v m characterize the neuron type and its firing pattern. See [START_REF] Izhikevich | Simple model of spiking neurons[END_REF] for units and physical details. We choose I ext = 10, a = 0.02, b = 0.2, c = -55, d = 4, and v m = 30, leading to an intrinsically bursting neuron. The solutions are known to remain in a physical compact set X and have a uniform dwell-time, thus verifying Assumption 2.1. A solution is plotted on Figure 3.

The map

T (x) = (h(x), L f h(x)) = (x 1 , 0.04x 2 1 +5x 1 +140-x 2 +I ext )
is a diffeomorphism on R 2 , so that the flow dynamics admit a high-gain observer [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF] as detailed in Example 2.4. Since the jump times can be detected from the jumps of the output y = x 1 , it is proposed in [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF] to use an observer of the type (2), with F = F ℓ given by the high-gain observer, G defined in [START_REF] Bernard | Observers for hybrid dynamical systems with linear maps and known jump times[END_REF], and jumps triggered at the same time as those of the system. However, because of the unstable quadratic term 0.04x 2 1 in the flow dynamics, slight delays in the jump detection deteriorate very quickly the estimate around each jump, as illustrated on Figure 4a.

Instead, we would like to implement observer [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF], which does not rely on jump detection and that automatically synchronizes its jumps with those of the system. Because of the quadratic term in f , items (P1) and (P2) of Assumption 3.1 hold for any choice of δ 0 and δ 1 . However, some care should be taken in the choice of ∆ < Fig. 4: Estimation error for system [START_REF] Tanwani | Comments on "observability of switched linear systems: Characterization and observer design[END_REF] with two observers, where F = F ℓ defined in [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF] with ℓ = 4 and K = (1, 1) ⊤ , x(0, 0) = (-55, -6), x0 = (-20, 0), z(0, 0) = T (x 0 ).

a "larger" compact set X m in which the observer trajectory should evolve. Unfortunately, these choices depend on the magnitude of the initial error. Figure 4b shows the results of a simulation with the same initial conditions and same high-gain map F ℓ as in Figure 4a, but this time with observer [START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF], which does not rely on the jump detection and which automatically synchronizes its jumps with those of the system. Compared to Figure 4a, we observe that the mismatch of jump times marked by vertical dashed lines and the estimation error asymptotically converge to 0. A major difference between both designs is that the former is global while the latter is only local. In particular, as mentioned above, for too large initial estimation errors, observer trajectories could explode in finite time during the first open-loop phase (pick χ(0, 0) = (24, 0) in the previous simulation to witness finite-time escape). It may also happen that the observer "misses" a jump of the system and simply catches up afterwards with the high-gain observer without any guarantee from the analysis. In order to avoid this, or do it in a "safe" way that ensures convergence, we run a preliminary continuous-time high-gain observer and launch observer (13) at a well-chosen time t p according to Algorithm 1 given in Section V, namely we implement the semiglobal observer [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF]. Figures 5,6 illustrates3 the behavior of the observer depending on the first jump time of the system and shows that the algorithm choosing t p is effective to ensure asymptotic convergence in each case. When the local observer is launched at time τ m and the system jumps before that time, two transients occur: one at the initial time, and another after the system jump (see Figures 5a-5b). In particular, a possible system jump is successfully detected between τ 1 and τ 2 in Figure 5b), leading to the warning state q w being turned on and the switching time t p fixed at τ m for safety. On the other hand, in Figure 5c), a system jump is successfully anticipated after τ 2 and the local observer directly switched on, thus avoiding another transient.

Finally, the robustness of this design was also tested in presence of output noise in the three previous scenarios. Figure 5d) reproduces the errors plotted on Figures 5a,5b,5c but in presence of output noise and on a longer time horizon. As expected, the noise is amplified through the high-gain observers and non vanishing large errors are observed around the jump times since the jumps of the system and the observer no longer asymptotically synchronize.

VII. CONCLUSION

We have proposed a semiglobal hybrid observer for hybrid dynamical systems with bounded solutions, dwell-time, and strongly differentially observable flow dynamics, whose jump times are unknown. The observer combines a preliminary continuous-time high-gain observer and a local hybrid observer which relies on a high-gain observer of the flow and jumps triggered based on the observer state, in a way that "disconnects" the correction term around the jump times. This novel observer avoids the problems of delayed/noisy detection of the system jump times and its robustness to noise was tested in simulations. Of course, this design heavily relies on exponentially convergent high-gain flow-based observers. The possibility of designing nonsynchronized hybrid observers for larger classes of systems with less regular dynamics or exploiting the observability brought by both flow and jumps as in [START_REF] Shim | Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems[END_REF], [START_REF] Tran | Coupling flow-and jumpbased observers for hybrid systems with nonlinear maps and known jump times[END_REF], remains to be investigated. Moreover, the requirement to pass the observer transient within at most one system jump requires large gains which might make its use unrealistic in the presence of noise. Also, because the switching to the local hybrid observer happens only once, it is not robust to large pointwise disturbances. Actually, if such a disturbance is detected, the observer should be reinitialized from the start. More robust back and forth switching strategies could be studied, with the repeated challenge of detecting when to switch and of handling the unknown system jump times. Finally, similarly to continuous-time systems [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], it would be interesting to see whether such a high-gain observer can be exploited for hybrid output-feedback under uncertainties in the jump times detection, like in bipedal robotics [START_REF] Hamed | Observer-based feedback controllers for exponential stabilization of hybrid periodic orbits: Application to underactuated bipedal walking[END_REF].

APPENDIX

A. Proof of Lemma 4.1

Consider first the case where ϕ(0, 0) ∈ C 2 , i.e., q(0, 0) = 2. -By definition of Ĥ, ϕ(t, 0) ∈ C 2 for all t ∈ I 0 . During that time, since

Π(T inv (z(t, 0))) ∈ cl(C ∪ D) ∩ cl(R nx \ D δ1 ) by definition of Π and C 2 , | Π(T inv (z(t, 0)))| D ≥ δ 1 . Therefore, since V (x(t, 0), z(t, 0
)) ≤ v ℓ by assumption, x(t, 0) / ∈ D by item (a) and both x and ϕ flow simultaneously. Thus, z evolves according to the continuous-time high-gain observer F ℓ with input y = h(x), which, due to x flowing in C \ D, is a continuous signal during this period. On the other hand, the components χ, τ, q of the observer solution ϕ remain constant with mode q = 2. This is the high-gain phase, and defining ϕ sub (t, 0) = ϕ(t, 0) for t ∈ I 0 , we have (19a) for j = 0.

-If I 0 = [0, +∞), then ϕ is t-complete and the proof is concluded. Otherwise, since y is bounded due to the system solution x being bounded, ϕ cannot explode in finite time and according to the previous item, the input y does not jump on I 0 , so necessarily, by definition of C 2 , there exists t 1 such that ϕ(t 1 , 0) ∈ D 2 and ϕ jumps. At this point,

| Π(T inv (z(t 1 , 0)))| D = δ 1 , so χ + = Π(T inv (z(t 1 , 0))) ∈ D δ1 \ D and q + = 1. Therefore, (z + , χ + , τ + , q + ) ∈ C 1 \ D 1 and ϕ(•, 1) necessarily flows. Besides, from item (c), χ + ∈ X ′′ m .
The open-loop phase starts. In this phase, the behavior of ϕ is independent from that of the input y, i.e., that of x. Therefore, we concentrate on the behavior of ϕ, putting aside the jumps of x possibly happening during that phase. Indeed, such jumps may only trigger trivial jumps in ϕ according to [START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF] and do not alter the values taken by ϕ. Thus, ϕ may be analyzed as in an autonomous hybrid system during this phase, which actually corresponds to ϕ sub in the statement of the lemma.

-While ϕ(•, 1) flows in C 1 , the states z, τ, q remain constant.

Since χ(t 1 , 1) ∈ D δ0 ∩ X ′′ m and χ flows with f , we know by (P1) of Assumption 3.1 that χ remains in D δ0 ∩ X ′ m and reaches D ∩ X ′ m in finite time. Besides, ϕ cannot jump before χ has reached D according to the definition of D 1 .

-When χ reaches D ∩ X ′ m , we know by (P2) in Assumption 3.1 that ϕ can no longer flow. Since at this point ϕ(t 2 , 1) ∈ D 1 , ϕ jumps with χ + = g(χ) and q + = 0.

-From there, ϕ(t 2 , 2) ∈ C 0 \ D 0 , with τ = 0 and χ ∈ g(D ∩ X ′ m ), so ϕ can only flow as long as τ ≤ ∆, namely during ∆ units of time. Since ∆ < τ 0 m , χ can indeed flow with f during that time and remains in X m with Π(χ) / ∈ D δ0 according to (P3). -Thus, when τ reaches ∆, we have Π(χ) / ∈ D δ0 , i.e., ϕ(t 3 , 2) ∈ D 0 and since no flow is possible in C 0 when τ = ∆, ϕ jumps with z + = T (Π(χ)), τ + = 0 and q + = 2. Since χ ∈ X m and Π(X m ) ⊂ X m , Π(χ) ∈ X m , and from [START_REF] Bernard | Observer design for hybrid dynamical systems with approximately known jump times[END_REF], Π(T inv (z + )) = Π(Π(χ)) = Π(χ) / ∈ D δ0 so that ϕ(t 3 , 3) ∈ C 2 . This is the end of the open-loop phase, the high-gain phase starts again, and we are back to where the argument started.

On the other hand, by assumption, if ϕ(0, 0) / ∈ C 2 , q(0, 0) = 1 and χ(0, 0) ∈ D δ1 ∩ X ′′ m so that the same reasoning holds, starting from the open-loop phase in the third item.

B. Proof of Lemma 4.2

For convenience, we denote v 1 := V ℓ (x(t j , j ′ -1), z(t j , j -1)), the value of V ℓ before the transition to open-loop phase.

By definition of the observer data, ϕ(t j , j -1) ∈ D 2 with | Π(T inv (z(t j , j -1)))| D = δ 1 and from Assumption 2.1, x(t j , j ′ ) ∈ X . Since v 1 ≤ v ℓ , it follows from items (a), (b) and (c) that x(t j , j ′ ) ∈ (D δ0 \ D) ∩ X and χ(t j , j) = Π(T inv (z(t j , j -1))) ∈ D δ1 ∩ X ′′ m . In particular, t j is not a jump time for x. Let us denote x 1 := x(t j , j ′ ) and χ with z denoting z(t j , j -1) for brevity, a inv the Lipschitz constant of T inv , a p defined in [START_REF] Bernard | Semi-global high-gain hybrid observer for a class of hybrid dynamical systems with unknown jump times[END_REF], and a τ the Lipschitz constant of T on X ′′ m ∩D δ0 . This mode with q = 1 lasts until χ reaches D, namely T(χ 1 ) units of time. so that the interval of time where q = 1 lasts at most a τ δ 1 .

The following mode with q = 0 lasts ∆ units of time. But since the time mismatch between observer and system jump ∆ τ < ∆, the system state x jumps some time before the observer switches back to q = 2.

We are now ready to study the evolution of W , namely the estimation error x -χ during the open-loop phase where q ∈ {0, 1}. For that, we decompose the analysis in three consecutive periods: 1) both x and χ flow with f until one of them reaches D; 2) one jumps and then the other, with a jump time mismatch ∆ τ ; 3) both x and χ flow with f until the observer switches back to q = 2.

As long as neither x nor χ reaches D, both flow on the time interval [t j , t j+1 ] with f locally Lipschitz in X ′ m with q = 1, so that

Ẇ = 2(χ -x) ⊤ (f (χ) -f (x)) ≤ 2|χ -x| |f (χ) -f (x)| ≤ λ f W
where λ f /2 is the Lipschitz constant of f on the compact set X m . This period lasts at most a τ δ 1 , so χ -x grows by at most a 1 := e 1 2 λ f aτ δ1 and denoting (χ ′ 1 , x ′ 1 ) := (χ, x) := (χ(t j+1 , j), x(t j+1 , j ′ )) the value of (χ, x) at the end of this first period, W (x ′ 1 , z, χ ′ 1 , 1) ≤ e λ f aτ δ1 W (x 1 , z, χ 1 , 1) ≤ e λ f aτ δ1 a 2 p a 2 inv c(ℓ) v 1

Then, the mismatch of jump times ∆ τ makes W increase to

|g(Ψ f (x ′ 1 , ∆ τ )) -Ψ f (g(χ ′ 1 ), ∆ τ ) | 2 or |Ψ f (g(x ′ 1 ), ∆ τ ) -g(Ψ f (χ ′ 1 , ∆ τ ))| 2
depending which of x or χ jumps first. Since x ′ 1 ∈ X and Ψ f (x ′ 1 , ∆ τ ) ∈ X by Assumption 2.1, and χ ′ 1 ∈ X ′ m according to (P1), we have

|g(Ψ f (x ′ 1 , ∆ τ )) -Ψ f (g(χ ′ 1 ), ∆ τ ) | ≤ |g(x ′ 1 ) -g(χ ′ 1 )| + |g(Ψ f (x ′ 1 , ∆ τ )) -g(x ′ 1 )| + |Ψ f (g(χ ′ 1 ), ∆ τ ) -g(χ ′ 1 )| ≤ a g |x ′ 1 -χ ′ 1 | + a g |Ψ f (x ′ 1 , ∆ τ ) -x ′ 1 | + ∆t 0 f (χ(s))ds ≤ a g |x ′ 1 -χ ′ 1 | + a g ∆t 0 f (x(s))ds + c f ∆ τ ≤ a g |x ′ 1 -χ ′ 1 | + a g c ′ f ∆ τ + c f ∆ τ ≤ a g |x ′ 1 -χ ′ 1 | + (a g c ′ f + c f )a τ |x 1 -χ 1 |
where a g is the Lipschitz constant of g on X ′ m , s → x(s) denotes the solution flowing with f from x ′ 1 , χ(s) the solution flowing with f from g(χ ′ 1 ), and c f , c ′ f are bounds of f around g(D ∩ X ′ m ) and D ∩ X ′ m respectively. We obtain in the same way the same bound for |Ψ f (g(x ′ 1 ), ∆ τ ) -g(Ψ f (χ ′ 1 , ∆ τ ))|. Actually, during this mismatch period, assuming χ jumps first, i.e. χ ′ 1 ∈ D, we have similarly

|x| D ≤ |x -χ ′ 1 | ≤ |x ′ 1 -χ ′ 1 | + c ′ f ∆ τ |χ| g(D) ≤ |χ -g(χ ′ 1 )| ≤ c f ∆ τ |χ -g(x)| ≤ a g |x ′ 1 -χ ′ 1 | + a g c ′ f ∆ τ + c f ∆ τ which gives U (x, χ, q) ≤ a 01 |x 1 -χ 1 | using |x ′ 1 -χ ′ 1 | ≤ a 1 |x 1 -χ 1 | and ∆ τ ≤ a τ |x 1 -χ 1 |.
Finally, in the last period, both x and χ have jumped once, so necessarily q = 0, τ ∈ [0, ∆], and both system and observer flow according to f during at most ∆ units of time, until the timer τ reaches ∆. When that happens ϕ reaches D 0 , at jump time t j+3 and the solutions have thus reached x(t j+3 , j ′ + 1) and ϕ(t j+3 , j + 2) respectively. During this period, |χ -x| thus grows by at most e 1 2 λ f ∆ which gives a 0 . More precisely, denoting (χ 2 , x 2 ) := (χ(t j+3 , j + 2), x(t j+3 , j ′ + 1)) at that time, we thus have

|x 2 -χ 2 | 2 ≤ e λ f ∆ a g |x ′ 1 -χ ′ 1 | + (a g c ′ f + c f )a τ |x 1 -χ 1 | 2 ≤ 1 2 e λ f ∆ a 2 g |x ′ 1 -χ ′ 1 | 2 + (a g c ′ f + c f ) 2 a 2 τ |x 1 -χ 1 | 2 ≤ 1 2 e λ f ∆ a 2 g e λ f aτ δ1 + (a g c ′ f + c f ) 2 a 2 τ a 2 p a 2 inv c(ℓ) v 1
Then, the observer jumps back to mode q = 2 and z is reset to z 2 := z(t j+3 , j + 3) = T (Π(χ 2 )) with χ 2 ∈ X m according to (P3), and therefore

W (x 2 , z 2 , χ 2 , 2) = V ℓ (x 2 , z 2 ) ≤ c(ℓ)|z 2 -T (x 2 )| 2 ≤ c(ℓ)|T (Π(χ 2 )) -T (x 2 )| 2 ≤ c(ℓ)a 2 T | Π(χ 2 ) -x 2 | 2 ≤ c(ℓ)a 2 T a 2 p |χ 2 -x 2 | 2 ≤ 1 2 c(ℓ) c(ℓ) a 2 T a 4 p e λ f ∆ a 2 g e λ f aτ δ1 + (a g c ′ f + c f ) 2 a 2 τ a 2 inv v 1 ≤ a c(ℓ) c(ℓ) v 1
for a > 0 independent from ℓ, with a T the Lipschitz constant of T on X m . After that, ϕ flows with q = 2 and V ℓ decreases exponentially. Therefore, if additionally v 1 < 1 m units of time have elapsed since the previous jump of x according to (P3). The time between the previous jump of x and the beginning of the high-gain phase with q = 2 being at most ∆ τ + ∆, it follows that the high-gain phase lasts at least τ ′ m > 0 verifying τ ′ m ≥ τ 0 m -∆ τ -∆ ≥ τ 0 m -2∆ > 0 since v 1 was chosen to ensure ∆ τ < ∆. Therefore, after a full cycle, V ℓ grows at most by ae -ℓλτ ′ m c(ℓ) c(ℓ) .

Fig. 3 :

 3 Fig.3: Solution to hybrid system[START_REF] Tanwani | Comments on "observability of switched linear systems: Characterization and observer design[END_REF] with initial condition x(0, 0) = (-55, -6).

τ 0 m 2

 2 with τ 0 m given in (P3). Indeed, the flow dynamics exhibit finite-time escape in openloop, so ∆ should be smaller than the minimal time needed for the flow dynamics to escape in finite-time from (c, x 2 ) in Observer[START_REF] Balluchi | The design of dynamical observers for hybrid systems: Theory and application to an automotive control problem[END_REF] with delay in the jump detection of 0.5 units of time.

  Observer[START_REF] Forni | Follow the bouncing ball : global results on tracking and state estimation with impacts[END_REF] with q(0, 0) = 2, δ 0 = 5, δ 1 = 3, ∆ = 3.

  x(0, 0) = (-55, -6), x(0, 0) = (24, 0): the system jumps before τ 1 and the local observer is launched at time τm. x(0, 0) = (-55, -4),x(0, 0) = (27, 0): the system jumps between τ 1 and τ 2 , the warning state is turned on around t = 6.42 when Π(x) ∈ D δ 1 and the local observer is launched at time τm. x(0, 0) = (-55, -3.8), x(0, 0) = (27, 0): the system jumps after τ 2 , and the local observer is launched right before around time t = 9.61 as soon as Π(x) ∈ D δ 1 . Estimation errors in the three previous scenarios with band-limited white noise of power 0.1 and sample time 0.1.

Fig. 5 :

 5 Fig.5: Estimation errors for system[START_REF] Tanwani | Comments on "observability of switched linear systems: Characterization and observer design[END_REF] with observer[START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] designed for the local observer Ĥ given by (13) with ℓ = ℓ p = 6 and K = (1, 1) ⊤ in F ℓ = F ℓp defined in[START_REF] Bernard | Hybrid dynamical systems with hybrid inputs: Definition of solutions and applications to interconnections[END_REF], δ 0 = 5, δ 1 = 3, ∆ = 3, τ m = 25, τ 1 = τ m /4, τ 2 = τ m /3, and different initial conditions with z p (0, 0) = T (x(0, 0)). The y-axis is chopped on[-40, 20] for better overall view, but the error during the first transient typically goes down to -130.

  x(0, 0) = (-55, -6), x(0, 0) = (24, 0): the system jumps before τ 1 and the local observer is launched at time τm. x(0, 0) = (-55, -4), x(0, 0) = (27, 0): the system jumps between τ 1 and τ 2 , the warning state is turned on around t = 6.42 when Π(x) ∈ D δ 1 and the local observer is launched at time τm.

  x(0, 0) = (-55, -3.8), x(0, 0) = (27, 0): the system jumps after τ 2 , and the local observer is launched right before around time t = 9.61 as soon as Π(x) ∈ D δ 1 .

Fig. 6 :

 6 Fig.6: Logic variables observer[START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] designed for the local observer Ĥ given by (13) with ℓ = ℓ p = 6, K = (1, 1) ⊤ , δ 0 = 5, δ 1 = 3, ∆ = 3, τ m = 25, τ 1 = τ m /4, τ 2 = τ m /3, and different initial conditions with z p (0, 0) = T (x(0, 0)).

  1 := χ(t j , j) the values of x and χ at the beginning of the openloop phase. From (P1), flow according to f will lead both x and χ to D with a time mismatch ∆ τ verifying∆ τ = |T(χ 1 ) -T(x 1 )| ≤ a τ |χ 1 -x 1 | ≤ a τ | Π(T inv (z)) -T inv (T (x 1 ))| ≤ a τ a p |T inv (z) -T inv (T (x 1 ))| ≤ a τ a p a inv |z -T (x 1 )| ≤ a τ a p a inv v 1 c(ℓ) < ∆

  For all x D ∈ D ∩X ′′ m , |T(χ 1 ) -T(x D )| = |T(χ 1 )| ≤ a τ |χ 1 -x D | ≤ a τ δ 1

  ) v ℓ , namely[START_REF] Luenberger | Canonical forms for linear multivariable systems[END_REF] holds, V ℓ remains smaller than v ℓ and x does not jump while ϕ flows in C 2 by item (a). It follows that ϕ flows either for all times, or until reaching t j+4 where another transition to openloop phase occurs. Let us evaluate the minimal length of this high-gain phase. z flows with the high-gain observer at least till | Π(T inv (z))| D = δ 1 , so by item (a), x flows according to f during this period and when | Π(T inv (z))| D = δ 1 , by item (b), x ∈ D δ0 . So x must have had time to flow with f from g(D ∩X ) to D δ0 while staying in X where Π = Id, namely at least τ 0

  then, either t p = τ m or t p = t * , and since x(t * , 0) ∈ D δ0 by item (b'), x jumps at a timet 1 (x) ≤ t * + ∆ max ≤ τ 2 + ∆ max = τ 3by definition of ∆ max . Therefore, if t p = τ m , (32) holds thanks to (33b) and (34b) since t p -t 1 (x) ≥ τ m -τ 3 . And if t p = t * , (32) holds according to (33a) and (34a) since τ 1 ≤ t p ≤ t 1 (x).-If on the other hand, t * ∈ (τ 2 , τ m ], then necessarily t p = t * and again, (32) holds according to (33a) and (34a) since τ 1 ≤ t p ≤ t 1 (x).

x could also jump at the exact time where ϕ reaches D 1 , but in that case it is equivalent to counting two jumps, one after the other, for easiness of presentation, since the jump maps are independent.

Simulations available at https://github.com/ HybridSystemsLab/NonSyncHybridHighGainObserver

Research partially supported by the French National Research Agency (ANR-23-CE48-0013-01), the National Science Foundation (Grants ECS-1710621, CNS-2039054, CNS-2111688), the Air Force Office of Scientific Research (Grants FA9550-19-1-0053, FA9550-19-1-0169, FA9550-20-1-0238), the Army Research Office (Grant W911NF-20-1-0253), and the Banatao Institute at the University of California.

P.