G Q B Tran

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

Lorenzo Marconi
email: lorenzo.marconi@unibo.it

Gia Quoc

Bao Tran

Observer design for hybrid systems with linear maps and known jump times

This chapter unifies and develops recent developments in observer design for hybrid systems with linear dynamics and output maps, whose jump times are known. We define and analyze the (pre-)asymptotic detectability and uniform complete observability of this class of systems, then present two different routes for observer design. The first one relies on a synchronized Kalman-like observer that gathers observability from both flows and jumps. The second one consists in decomposing the state into parts with different observability properties and coupling observers estimating each of these parts, possibly exploiting an extra fictitious measurement coming from the combination of flows and jumps. These observers are based on a Linear Matrix Inequality (LMI) or the Kazantzis-Kravaris/Luenberger (KKL) paradigm. A comparison of these methods is presented in a table at the end.

Introduction

Consider a hybrid system with linear maps

H 𝑥 = 𝐹𝑥 + 𝑢 𝑐 (𝑥, 𝑢 𝑐 ) ∈ 𝐶 𝑦 𝑐 = 𝐻 𝑐 𝑥 𝑥 + = 𝐽𝑥 + 𝑢 𝑑 (𝑥, 𝑢 𝑑 ) ∈ 𝐷 𝑦 𝑑 = 𝐻 𝑑 𝑥 (1)
where 𝑥 ∈ R 𝑛 𝑥 is the state, 𝐶 and 𝐷 are the flow and jump sets, 𝑦 𝑐 ∈ R 𝑛 𝑦,𝑐 and 𝑦 𝑑 ∈ R 𝑛 𝑦,𝑑 are the outputs known during the flow intervals and at the jump times respectively, 𝑢 𝑐 ∈ R 𝑛 𝑥 and 𝑢 𝑑 ∈ R 𝑛 𝑥 are known exogenous input signals, as well as the dynamics matrices 𝐹, 𝐽 ∈ R 𝑛 𝑥 ×𝑛 𝑥 and the output matrices 𝐻 𝑐 ∈ R 𝑛 𝑦,𝑐 ×𝑛 𝑥 , 𝐻 𝑑 ∈ R 𝑛 𝑦,𝑑 ×𝑛 𝑥 which are all known and possibly time-varying. Models of the form (1) include not only hybrid systems with linear maps described in the setting of [START_REF] Sanfelice | Hybrid Feedback Control[END_REF], but also switched and/or impulsive systems with linear maps where the active mode is seen as an exogenous signal making (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ) time-varying (see [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] Tanwani | Observability for Switched Linear Systems: Characterization and Observer Design[END_REF] among many other ones), and continuous-time systems with sporadic or multi-rate sampled outputs where the "jumps" correspond to sampling events, 𝐽 = Id, 𝑢 𝑑 = 0, 𝑦 𝑐 = 0, and 𝑦 𝑑 the outputs available at the sampling event [START_REF] Raff | Observers with Impulsive Dynamical Behavior for Linear and Nonlinear Continuous-Time Systems[END_REF][START_REF] Ferrante | State Estimation of Linear Systems in the Presence of Sporadic Measurements[END_REF][START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF]. See [START_REF] Sanfelice | Hybrid Feedback Control[END_REF][START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF] for some examples of those classes of systems set in the framework of [START_REF] Alessandri | Switching Observers for Continuous-time and Discrete-time Linear Systems[END_REF]. The goal of this chapter is to present in a unified and more complete way recent advances concerning the design of an asymptotic observer for system [START_REF] Alessandri | Switching Observers for Continuous-time and Discrete-time Linear Systems[END_REF], assuming that its jump times are known or detected. In practice, we may be interested in estimating only certain trajectories of "physical interest", initialized in some set X 0 ⊂ R 𝑛 𝑥 and with exogenous terms (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ) in some set U of interest. We then denote S H (X 0 , U) as the set of those maximal solutions of interest. Because we look for an asymptotic observer, we assume maximal solutions are complete as stated next.

Assumption 1 Given X 0 and U, each maximal solution in S H (X 0 , U) is complete.

Since the jump times of system H in (1) are known, it is natural to strive for a synchronized asymptotic observer of the form

Ĥ        ẑ = Ψ 𝑐 ( ẑ, 𝑦 𝑐 , 𝑢 𝑐 ) when H flows ẑ+ = Ψ 𝑑 ( ẑ, 𝑦 𝑑 , 𝑢 𝑑 )
when H jumps x = Υ( ẑ, 𝑦 𝑐 , 𝑦 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ) [START_REF] Allan | Nonlinear Detectability and Incremental Input/Outputto-State Stability[END_REF] where ẑ ∈ R 𝑛 𝑧 is the observer state (with 𝑛 𝑧 ≥ 𝑛 𝑥 in general), Υ : R 𝑛 𝑧 × R 𝑛 𝑦,𝑐 × R 𝑛 𝑦,𝑑 × R 𝑛 𝑥 × R 𝑛 𝑥 → R 𝑛 𝑥 is the observer output map, Ψ 𝑐 : R 𝑛 𝑧 × R 𝑛 𝑦,𝑐 × R 𝑛 𝑥 → R 𝑛 𝑧 and Ψ 𝑑 : R 𝑛 𝑧 ×R 𝑛 𝑦,𝑑 ×R 𝑛 𝑥 → R 𝑛 𝑧 are respectively the observer flow and jump maps designed such that each maximal solution (𝑥, ẑ) to the cascade H -Ĥ initialized in X 0 × R 𝑛 𝑧 with inputs in U is complete and verifies lim 𝑡+ 𝑗→+∞ (𝑡 , 𝑗) ∈dom 𝑥 |𝑥(𝑡, 𝑗)x(𝑡, 𝑗)| = 0.

(

) 3
The knowledge of the jump times is not only used to trigger the observer jumps at the same time as those of the system, but it can also be used to design the observer maps Ψ 𝑐 and Ψ 𝑑 . The way this information is exploited varies depending on whether these maps rely on:

• Gains that are computed offline (for example via matrix inequalities), according to all possible lengths of flow intervals in between jumps, i.e., they depend on each individual flow length, not the particular sequence of them. This is the path taken by [START_REF] Alessandri | Switching Observers for Continuous-time and Discrete-time Linear Systems[END_REF][START_REF] Moarref | Observer Design for Linear Multi-rate Sampled-data Systems[END_REF][START_REF] Ferrante | State Estimation of Linear Systems in the Presence of Sporadic Measurements[END_REF][START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF][START_REF] Ríos | State Estimation for Linear Hybrid Systems with Periodic Jumps and Unknown Inputs[END_REF][START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF][START_REF] Tran | Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]; • Or, gains that are computed online along the time domain of each solution of interest, i.e., they depend on the sequence of flow lengths in each particular solution. This is the path taken by all Kalman-like approaches in [START_REF] Medina | State Estimation for Linear Impulsive Systems[END_REF][START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF][START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF].

In the former case, the design requires some information about the possible duration of flow intervals between successive jumps in each solution of interest (at least after a certain time) as defined next. In the context of switched (resp. sampled) systems, this corresponds to information about the possible switching (resp. sampling) rates.

Definition 1 (Set of flow lengths of a hybrid arc)

For a closed subset I of [0, +∞) and some 𝑗 𝑚 ∈ N, we say that a hybrid arc (𝑡, 𝑗) ↦ → 𝑥(𝑡, 𝑗) has flow lengths within I after jump 𝑗 𝑚 if

• 0 ≤ 𝑡 -𝑡 𝑗 (𝑥) ≤ sup I for all (𝑡, 𝑗) ∈ dom 𝑥;

• 𝑡 𝑗+1 (𝑥) -𝑡 𝑗 (𝑥) ∈ I holds for all 𝑗 ∈ N with 𝑗 ≥ 𝑗 𝑚 if sup dom 𝑗 𝑥 = +∞, and for all 𝑗 ∈ { 𝑗 𝑚 , 𝑗 𝑚 + 1, . . . , sup dom 𝑗 𝑥 -1} otherwise.

In brief, I contains all the possible lengths of the flow intervals between successive jumps, at least after some time. The first item is to bound the length of the flow intervals not covered by the second item, namely possibly the first ones before 𝑗 𝑚 , and the last one, which is dom 𝑡 𝑥 ∩ [𝑡 𝐽 ( 𝑥) , +∞) where 𝑡 𝐽 ( 𝑥) is the time when the last jump happens (when defined). If I is unbounded, the system may admit (eventually) continuous solutions and the observer should correct the estimate at least during flows, while 0 ∈ I means the hybrid arc can jump more than once at the same time instance or have flow lengths going to zero (including (eventually) discrete and Zeno solutions) and the observer should reduce the estimation error at least at jumps.

From there, one may design either:

• A flow-based observer with an innovation term during flows only, exploiting the observability of the full state during flows from 𝑦 𝑐 when 0 ∉ I [1, 6]; • A jump-based observer with an innovation term at jumps only, exploiting the detectability of the full state via the combination of flows and jumps from 𝑦 𝑑 available at the jumps only when I is bounded [START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF][START_REF] Medina | State Estimation for Linear Impulsive Systems[END_REF][START_REF] Ferrante | State Estimation of Linear Systems in the Presence of Sporadic Measurements[END_REF][START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF][START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF]; • An observer with innovation terms during both flows and jumps, exploiting the observability from both 𝑦 𝑐 and 𝑦 𝑑 and the combination of flows and jumps: this is done via a hybrid Kalman-like approach in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], or via an observability decomposition in [START_REF] Tran | Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], or Lyapunov-based LMIs in [START_REF] Ríos | State Estimation for Linear Hybrid Systems with Periodic Jumps and Unknown Inputs[END_REF][START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF].

This chapter deals with the third case, where the full state is not necessarily instantaneously observable during flows and not observable from the jump output only. It unifies and extends the work of [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF][START_REF] Tran | Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]. More precisely, in Section 2, we start with an observability analysis allowing us to exhibit necessary conditions and sufficient conditions for observer design: first through hybrid Gramian conditions, and then via an observability decomposition. The latter decomposes the state in two parts: the first one is instantaneously observable through the flow output 𝑦 𝑐 , while the second one must be detectable from an extended jump output featuring the available jump output 𝑦 𝑑 and an additional fictitious one, describing how the non-observable states impact the observable ones at jumps and become visible through 𝑦 𝑐 . While the Gramian-based analysis has led in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF] to a systematic hybrid Kalman-like design, we show how the observability decomposition lets us design observers made of:

• A high-gain flow-based observer of the state components that are instantaneously observable from 𝑦 𝑐 ;

• A jump-based observer for the remaining components, derived from a discrete LMI-based (resp. KKL-based) observer in Section 3 (resp. Section 4), thus revisiting and extending [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF].

Notations: Let R (resp. N) denote the set of real numbers (resp. natural numbers, i.e., {0, 1, 2, . . .}) and N >0 = N \ {0}. We denote R 𝑚×𝑛 (resp. S 𝑛 >0 ) as the set of real (𝑚 × 𝑛)-(resp. symmetric positive definite (𝑛 × 𝑛)-) dimensional matrices. Given a set 𝑆, int(𝑆) denotes its interior. Let ℜ(𝑧) and ℑ(𝑧) be the real and imaginary parts of the complex variable 𝑧, respectively. Denote Id as the identity matrix of appropriate dimension. Let | • | be the Euclidean norm and ∥ • ∥ the induced matrix norm. Let 𝐴 ⊥ be the orthogonal complement of matrix 𝐴 satisfying 𝐴𝐴 ⊥ = 0 and such that 𝐴 ⊤ 𝐴 ⊥ is invertible, and 𝐴 † be the Moore-Penrose inverse of 𝐴 [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF]. For a solution (𝑡, 𝑗) ↦ → 𝑥(𝑡, 𝑗) of a hybrid system, we denote dom 𝑥 its time domain [START_REF] Goebel | Hybrid Dynamical Systems: Modeling Stability, and Robustness[END_REF], dom 𝑡 𝑥 (resp. dom 𝑗 𝑥) the domain's projection on the ordinary time (resp. jump) component, and for 𝑗 ∈ dom 𝑗 𝑥, 𝑡 𝑗 (𝑥) the unique time such that (𝑡 𝑗 (𝑥), 𝑗) ∈ dom 𝑥 and (𝑡 𝑗 (𝑥), 𝑗 -1) ∈ dom 𝑥, and T 𝑗 (𝑥) := {𝑡 ∈ dom 𝑡 𝑥 : (𝑡, 𝑗) ∈ dom 𝑥} (for hybrid systems with inputs, see [START_REF] Sanfelice | Hybrid Feedback Control[END_REF]). The mention of 𝑥 is omitted when no confusion is possible. A solution 𝑥 to a hybrid system is complete if dom 𝑥 is unbounded and Zeno if it is complete and sup dom 𝑡 𝑥 < +∞. Let diag(𝜆 1 , 𝜆 2 , . . . , 𝜆 𝑛 ) be the diagonal matrix with entries 𝜆 𝑖 , 𝑖 = 1, 2, . . . , 𝑛. Occasionally, ★ denotes the symmetric part, i.e., ★ ⊤ 𝑃 = 𝑃 ⊤ 𝑃 or sometimes ★ ⊤ 𝑄𝑃 = 𝑃 ⊤ 𝑄𝑃. Last, let sat 𝑠 be a saturation function with level 𝑠, i.e., sat 𝑠 (𝑀) = 𝑀 if ∥ 𝑀 ∥ ≤ 𝑠 and sat 𝑠 is bounded otherwise.

Detectability and observability analysis

The existence of an asymptotic and synchronized observer (2) for system (1) requires system (1) to be asymptotically detectable, in the following sense.

Definition 2 ((Pre-)asymptotic detectability with known jump times)

System (1) with known jump times, initialized in X 0 , and with inputs in U is pre-asymptotically detectable if any complete solutions 𝑥 𝑎 and 𝑥 𝑏 in S H (X 0 , U) with the same inputs (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ), such that dom 𝑥 𝑎 = dom 𝑥 𝑏 , and whose flow outputs 𝑦 𝑎,𝑐 , 𝑦 𝑏,𝑐 and jump outputs 𝑦 𝑎,𝑑 , 𝑦 𝑏,𝑑 satisfy

𝑦 𝑎,𝑐 (𝑡, 𝑗) = 𝑦 𝑏,𝑐 (𝑡, 𝑗), ∀𝑡 ∈ int(T 𝑗 (𝑥 𝑎 )), ∀ 𝑗 ∈ dom 𝑗 𝑥 𝑎 , (4a)
𝑦 𝑎,𝑑 (𝑡 𝑗 , 𝑗 -1) = 𝑦 𝑏,𝑑 (𝑡 𝑗 , 𝑗 -1), ∀ 𝑗 ∈ dom 𝑗 𝑥 𝑎 , 𝑗 ≥ 1, (4b)
verify lim

𝑡+ 𝑗→+∞ (𝑡 , 𝑗) ∈dom 𝑥 𝑎 |𝑥 𝑎 (𝑡, 𝑗) -𝑥 𝑏 (𝑡, 𝑗)| = 0. ( 5
)
If in addition, all solutions in S H (X 0 , U) are complete, then we have asymptotic detectability. The set X 0 (resp. U) may be omitted if the property holds for any initial condition in X 0 = 𝐶 ∪ 𝐷 (resp. any input).

Remark 1 In Definition 2, when all the flow intervals have non-empty interior (i.e., int(T 𝑗 (𝑥 𝑎 )) ≠ ∅ for all 𝑗 ∈ dom 𝑗 𝑥 𝑎 ), condition (4a) is equivalent to

𝑦 𝑎,𝑐 (𝑡, 𝑗) = 𝑦 𝑏,𝑐 (𝑡, 𝑗), ∀(𝑡, 𝑗) ∈ dom 𝑥 𝑎 , (6)
by the continuity of 𝑡 ↦ → 𝑦 𝑐 (𝑡, 𝑗) during flows for all 𝑗 ∈ dom 𝑗 𝑥 𝑎 . On the contrary, when a solution admits consecutive jumps, condition (4a) is required only on the flow intervals with a non-empty interior since it holds vacuously on the other ones.

In other words, the equality of 𝑦 𝑐 is only required when the system is flowing.

The necessity of asymptotic detectability is typically obtained as follows. First, by definition of a synchronized asymptotic observer initialized in X 0 with inputs in U, all solutions in S H (X 0 , U) must be complete. Then, pick a pair of solutions (𝑥 𝑎 , 𝑥 𝑏 ) as in Definition 2 verifying (4). A solution x produced by observer (2) fed with outputs (𝑦 𝑎,𝑐 , 𝑦 𝑎,𝑑 ) shares the same time domain and must converge asymptotically to 𝑥 𝑎 ; but, x is also a solution to observer (2) fed with outputs (𝑦 𝑏,𝑐 , 𝑦 𝑏,𝑑 ) according to (4), so that it must also converge asymptotically to 𝑥 𝑏 . It follows that 𝑥 𝑎 and 𝑥 𝑏 necessarily converge asymptotically to each other, thus giving us asymptotic detectability. Note that compared to [START_REF] Bernard | On Notions of Detectability and Observers for Hybrid Systems[END_REF], Definition 2 is restricted to pairs of complete solutions with the same time domain because the knowledge of the jump times is used to trigger the jumps of the observer so that only complete solutions with the same time domain and the same outputs are required to converge asymptotically to each other.

For observer design, one typically requires stronger observability assumptions depending on the class of observers and the required observer properties [START_REF] Bernard | Observer Design for Continuous-time Dynamical Systems[END_REF]. Observability typically means that the equality of the outputs in (4), possibly over a large enough time window, implies that the solutions 𝑥 𝑎 and 𝑥 𝑏 are actually the same or said differently, there does not exist any pair of different solutions with the same time domain and the same outputs in the sense of (4). Actually, in the context of observer design, a more relevant property is the ability to determine uniquely the current state from the knowledge of the past outputs over a certain time window Δ > 0, which is typically called backward distinguishability [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF] or constructibility [START_REF] Kailath | Linear Systems[END_REF]. In other words, for all (𝑡, 𝑗) in the domain such that 𝑡 + 𝑗 ≥ Δ, the equality of the outputs (𝑦 𝑐 , 𝑦 𝑑 ) along 𝑥 𝑎 and 𝑥 𝑏 at all past times (𝑡 ′ , 𝑗 ′ ) in the domain such that 0 ≤ (𝑡 + 𝑗) -(𝑡 ′ + 𝑗 ′ ) ≤ Δ implies that 𝑥 𝑎 (𝑡, 𝑗) = 𝑥 𝑏 (𝑡, 𝑗) (see later in Section 2.1). For continuous-time systems, this property is equivalent to observability over a time window because of the uniqueness of solutions in forward and backward time. However, they cease to be equivalent in discrete-time or hybrid systems when the jump maps are not invertible: a system could be constructible without being observable (see [START_REF] Kailath | Linear Systems[END_REF]Section 2.3.3] for a detailed discussion on those notions).

Hybrid observability Gramian

Consider a pair of solutions 𝑥 𝑎 and 𝑥 𝑏 in S H (X 0 , U) with the same inputs (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ) such that dom 𝑥 𝑎 = dom 𝑥 𝑏 := D. Then, for all hybrid times ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) ∈ D × D, we have

𝑥 𝑎 (𝑡, 𝑗) -𝑥 𝑏 (𝑡, 𝑗) = Φ 𝐹,𝐽 ((𝑡, 𝑗), (𝑡 ′ , 𝑗 ′ )) (𝑥 𝑎 (𝑡 ′ , 𝑗 ′ ) -𝑥 𝑏 (𝑡 ′ , 𝑗 ′ )), (7)
where Φ 𝐹, 𝐽 is a hybrid transition matrix defined as

Φ 𝐹, 𝐽 ((𝑡, 𝑗), (𝑡 ′ , 𝑗 ′ )) = 𝜙 𝐹 (𝑡, 𝑡 𝑗+1 ) 𝑗 ′ +1 𝑘= 𝑗 𝜙 𝐹 (𝑡 𝑘+1 , 𝑡 𝑘 )𝐽 (𝑡 𝑘 , 𝑘 -1) 𝜙 𝐹 (𝑡 𝑗 ′ +1 , 𝑡 ′ ), (8)
if 𝑡 ≥ 𝑡 ′ and 𝑗 ≥ 𝑗 ′ , and, if the jump matrix 𝐽 is invertible at the jump times,

Φ 𝐹, 𝐽 ((𝑡, 𝑗), (𝑡 ′ , 𝑗 ′ )) = 𝜙 𝐹 (𝑡, 𝑡 𝑗 ) 𝑗 ′ 𝑘= 𝑗+1 𝜙 𝐹 (𝑡 𝑘-1 , 𝑡 𝑘 )𝐽 -1 (𝑡 𝑘 , 𝑘 -1) 𝜙 𝐹 (𝑡 𝑗 ′ , 𝑡 ′ ), (9)
if 𝑡 ≤ 𝑡 ′ and 𝑗 ≤ 𝑗 ′ , with the domain of 𝐹 and 𝐽 inherited from D, where 𝜙 𝐹 denotes the continuous-time transition matrix associated with 𝐹, i.e., describing solutions to 𝑥 = 𝐹𝑥. By summing and integrating squares, it follows that the equality of the outputs (𝑦 𝑐 , 𝑦 𝑑 ) along 𝑥 𝑎 and 𝑥 𝑏 between time (𝑡 ′ , 𝑗 ′ ) ∈ D and a later time (𝑡, 𝑗) ∈ D is equivalent to

(𝑥 𝑎 (𝑡 ′ , 𝑗 ′ ) -𝑥 𝑏 (𝑡 ′ , 𝑗 ′ )) ⊤ G (𝐹,𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) (𝑥 𝑎 (𝑡 ′ , 𝑗 ′ ) -𝑥 𝑏 (𝑡 ′ , 𝑗 ′ )) = 0, (10) or
, assuming the invertibility of 𝐽 at the jump times,

(𝑥 𝑎 (𝑡, 𝑗) -𝑥 𝑏 (𝑡, 𝑗)) ⊤ G 𝑏𝑤 (𝐹, 𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) (𝑥 𝑎 (𝑡, 𝑗) -𝑥 𝑏 (𝑡, 𝑗)) = 0, (11)
where G (𝐹, 𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) (resp. G 𝑏𝑤 (𝐹,𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗))) is the observability Gramian (resp. backward observability Gramian) between those times as defined next.

Definition 3 (Observability Gramians)

The observability Gramian and backward observability Gramian of a quadruple (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ) defined on a hybrid time domain D, between time (𝑡 ′ , 𝑗 ′ ) ∈ D and a later time (𝑡, 𝑗) ∈ D, are defined as [START_REF] Ferrante | State Estimation of Linear Systems in the Presence of Sporadic Measurements[END_REF] and, when 𝐽 is invertible at the jump times,

G (𝐹, 𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) = ∫ 𝑡 𝑗 ′ +1 𝑡 ′ ★ ⊤ Ψ 𝑐 ((𝑠, 𝑗 ′ ), (𝑡 ′ , 𝑗 ′ ))𝑑𝑠 + 𝑗-1 ∑ 𝑘= 𝑗 ′ +1 ∫ 𝑡 𝑘+1 𝑡 𝑘 ★ ⊤ Ψ 𝑐 ((𝑠, 𝑘), (𝑡 ′ , 𝑗 ′ ))𝑑𝑠 + 𝑗-1 ∑ 𝑘= 𝑗 ′ ★ ⊤ Ψ 𝑑 ((𝑡 𝑘+1 , 𝑘), (𝑡 ′ , 𝑗 ′ )) + ∫ 𝑡 𝑡 𝑗 ★ ⊤ Ψ 𝑐 ((𝑠, 𝑗), (𝑡 ′ , 𝑗 ′ ))𝑑𝑠,
G 𝑏𝑤 (𝐹, 𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) = ∫ 𝑡 𝑗 ′ +1 𝑡 ′ ★ ⊤ Ψ 𝑐 ((𝑠, 𝑗 ′ ), (𝑡, 𝑗))𝑑𝑠 + 𝑗-1 ∑ 𝑘= 𝑗 ′ +1 ∫ 𝑡 𝑘+1 𝑡 𝑘 ★ ⊤ Ψ 𝑐 ((𝑠, 𝑘), (𝑡, 𝑗))𝑑𝑠 + 𝑗-1 ∑ 𝑘= 𝑗 ′ ★ ⊤ Ψ 𝑑 ((𝑡 𝑘+1 , 𝑘), (𝑡, 𝑗)) + ∫ 𝑡 𝑡 𝑗 ★ ⊤ Ψ 𝑐 ((𝑠, 𝑗), (𝑡, 𝑗))𝑑𝑠, (13)
respectively, where

Ψ 𝑐 ((𝑠, 𝑘), (𝑡, 𝑗)) = 𝐻 𝑐 (𝑠, 𝑘)Φ 𝐹,𝐽 ((𝑠, 𝑘), (𝑡, 𝑗)), (14a) Ψ 𝑑 ((𝑡 𝑘+1 , 𝑘), (𝑡, 𝑗)) = 𝐻 𝑑 (𝑡 𝑘+1 , 𝑘)Φ 𝐹,𝐽 ((𝑡 𝑘+1 , 𝑘), (𝑡, 𝑗)), (14b)
with all the jump times determined from D.

According to [START_REF] Cox | Isolating Invisible Dynamics in the Design of Robust Hybrid Internal Models[END_REF], we deduce that the observability between times (𝑡 ′ , 𝑗 ′ ) and (𝑡, 𝑗), namely the ability to reconstruct the initial state 𝑥(𝑡 ′ , 𝑗 ′ ) from the knowledge of the future output until (𝑡, 𝑗), is equivalent to the positive definiteness of the observability Gramian over this period. On the other hand, when 𝐽 is invertible at the jump times, the ability to reconstruct the current state 𝑥(𝑡, 𝑗) from the knowledge of the past output until (𝑡 ′ , 𝑗 ′ ), i.e. the backward distinguishability or constructibility, is characterized by the positive definiteness of the backward observability Gramian over this period according to [START_REF] Jr | Conditions for Asymptotic Stability of the Discrete Minimum-variance Linear Estimator[END_REF]. Actually, in that case, both notions are actually equivalent but the backward observability Gramian tends to appear more naturally in the analysis of observers.

Remark 2 Note that unlike for purely continuous-or discrete-time linear systems, the inputs (𝑢 𝑐 , 𝑢 𝑑 ) involved in the hybrid dynamics (1) may impact the observability properties since they may change the domain of the solutions and thus the Gramian.

For observer design, the invertibility of the Gramian is typically assumed to be uniform, leading to the following hybrid uniform complete observability, extending the classical UCO condition of the Kalman and Bucy's filter [START_REF] Kalman | New Results in Linear Filtering and Prediction Theory[END_REF].

Definition 4 (Uniform complete observability (UCO))

The quadruple (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ) defined on a hybrid time domain D is uniformly completely observable with data (Δ, 𝜇) if there exists Δ > 0 and 𝜇 > 0 such that for all ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗))

∈ D × D verifying (𝑡 -𝑡 ′ ) + ( 𝑗 -𝑗 ′ ) ≥ Δ, we have G (𝐹, 𝐽 ,𝐻 𝑐 ,𝐻 𝑑 ) ((𝑡 ′ , 𝑗 ′ ), (𝑡, 𝑗)) ≥ 𝜇 Id . (15)
In [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], this condition is stated with G 𝑏𝑤 replacing G because the former appears directly in the analysis. They are actually equivalent, assuming the uniform invertibility of 𝐽 at the jump times and the boundedness of 𝐹 and 𝐽. In [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], this UCO condition is exploited to design a systematic hybrid Kalman-like observer of the form

               x = 𝐹 x + 𝑢 𝑐 + 𝑃𝐻 ⊤ 𝑐 𝑅 -1 𝑐 (𝑦 𝑐 -𝐻 𝑐 x) 𝑃 = 𝜆𝑃 + 𝐹𝑃 + 𝑃𝐹 ⊤ -𝑃𝐻 ⊤ 𝑐 𝑅 -1 𝑐 𝐻 𝑐 𝑃 when H flows x+ = 𝐽 x + 𝑢 𝑑 + 𝐽𝐾 (𝑦 𝑑 -𝐻 𝑑 x) 𝑃 + = 𝛾 -1 𝐽 (𝐼 -𝐾 𝐻 𝑑 )𝑃𝐽 ⊤ when H jumps (16a) with 𝐾 = 𝑃𝐻 ⊤ 𝑑 (𝐻 𝑑 𝑃𝐻 ⊤ 𝑑 + 𝑅 𝑑 ) -1 , (16b)
where 𝜆 ≥ 0 and 𝛾 ∈ (0, 1] are design parameters, 𝑅 𝑐 ∈ S 𝑛 𝑦,𝑐 >0 and 𝑅 𝑑 ∈ S 𝑛 𝑦,𝑑 >0

are (possibly time-varying) weighting matrices that are positive definite and are uniformly upper-and lower-bounded. In [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], it is shown that the estimation error:

• Converges asymptotically to zero under UCO and the boundedness of the system matrices along each of the considered solutions in S H (X 0 , U); • Is exponentially stable with an arbitrarily fast rate and robustly stable (as defined in [START_REF] Allan | Nonlinear Detectability and Incremental Input/Outputto-State Stability[END_REF] but for hybrid systems) after a certain time, under UCO and the boundedness of the system matrices uniformly across the considered solutions in S H (X 0 , U), when additionally 𝜆 is sufficiently large and 𝛾 is sufficiently small.

Example 1 Consider a pendulum equipped with an IMU and bouncing on a vertical wall, with angular position 𝜽 ∈ R 3 . The IMU contains a gyroscope measuring its angular velocity 𝝎 ∈ R 3 and an accelerometer measuring its proper acceleration (linear acceleration minus gravity) 𝒚 𝒂 ∈ R 3 in the IMU frame, modulo an unknown constant bias 𝒃 𝒂 ∈ R 3 . We assume its tilt 𝒕 ∈ R 3 is measured at the jump times (when the mass impacts the wall) and that the linear velocity 𝒗 ∈ R 3 in the sensor frame can be deduced from the gyroscope measurement via kinematics [START_REF] Vigne | MOVIE: A Velocity-Aided IMU Attitude Estimator for Observing and Controlling Multiple Deformations on Legged Robots[END_REF]. We also assume the velocity magnitude is reduced by an unknown constant restitution coefficient 𝑐 ∈ (0, 1] at each impact. With these in mind, we model the system in hybrid form with state 𝑥 = (𝒕, 𝒗, 𝒃 𝒂 , 𝑐) as (see [START_REF] Vigne | MOVIE: A Velocity-Aided IMU Attitude Estimator for Observing and Controlling Multiple Deformations on Legged Robots[END_REF] for the flow dynamics) [START_REF] Medina | State Estimation for Linear Impulsive Systems[END_REF] where

           𝒕 = -[𝝎] × 𝒕 𝒗 = -[𝝎] × 𝒗 + 𝒚 𝒂 -𝒃 𝒂 + 𝑔𝒕 𝒃 𝒂 = 0 𝑐 = 0 𝑦 𝑐 = 𝒗,            𝒕 + = 𝒕 𝒗 + = -𝑐𝒗 𝒃 + 𝒂 = 𝒃 𝒂 𝑐 + = 𝑐 𝑦 𝑑 = (𝒗, 𝒕),
[𝝎] × = 0 -𝜔 3 𝜔 2 𝜔 3 0 -𝜔 1 -𝜔 2 𝜔 1 0
and 𝑔 is the gravitational acceleration, with the flow and jump sets depending on the wall configuration. This system takes the form [START_REF] Alessandri | Switching Observers for Continuous-time and Discrete-time Linear Systems[END_REF] where 𝐹 =

-[𝝎] × 0 0 0 𝑔 Id -[𝝎] × -Id 0 0 0 0 0 0 0 0 0 , 𝐽 = Id 0 0 0 0 0 0 -𝑦 𝑑,𝒗 0 0 Id 0 0 0 0 1 1, 𝑢 𝑐 = 0 𝒚 𝒂 0 0
, and 𝑢 𝑑 = 0, with 𝑦 𝑑,𝒗 the 𝒗-component of 𝑦 𝑑 . We would like to estimate both 𝒃 𝒂 and 𝑐. Assuming the system is persistently not at rest (with external excitation if 𝑐 < 1), it is observable over each period of time containing at least one jump and a flow interval, because

• 𝒕 is available at jumps and its dynamics are independent of the other state components, making it observable, independently of the input signal [𝝎] × ; • 𝒗 and 𝒃 𝒂 are both instantaneously observable during flows from 𝑦 𝑐 once 𝒕 is known, also independently of [𝝎] × ; • 𝑐 is observable at jumps by seeing 𝒗 as a known input, because 𝒗 is measured.

It follows that if the pendulum velocity is uniformly lower-bounded (thanks to an appropriate input in the mechanical system, whose effects are in fact contained in 𝒚 𝒂 in the IMU frame), the observability Gramian computed over a time window Δ larger than the maximal length of flow intervals would be uniformly positive definite. Then, assuming the boundedness of 𝐹 and 𝐽, a Kalman-like observer [START_REF] Kalman | New Results in Linear Filtering and Prediction Theory[END_REF] fed with (𝑦 𝑐 , 𝑦 𝑑 , 𝝎, 𝒚 𝒂 ) can be designed for system [START_REF] Medina | State Estimation for Linear Impulsive Systems[END_REF]. Now, consider the case where there is an unknown constant bias 𝒃 𝒈 ∈ R 3 in the gyroscope measurement. As a result, [𝝎] × in system ( 17) is replaced by [𝝎 𝒎 -𝒃 𝒈 ] × , where 𝝎 𝒎 is the biased measurement. Assume an estimate b𝒈 of 𝒃 𝒈 is available such that b𝒈 -𝒃 𝒈 asymptotically vanishes. Then, the dynamics (17) may be re-written as

𝑥 = F𝑥 + (𝐹 -F)𝑥, (18)
where

[𝝎 𝒎 -b𝒈 ] × replaces [𝝎 𝒎 -𝒃 𝒈 ] × in F.
Consider the previous Kalman-like observer, but designed with the known F instead of 𝐹. According to our observability analysis above, the observability of the quadruple (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ) does not depend on 𝐹 and therefore still holds for ( F, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ). The robust stability of the Kalmanlike observer [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]Theorem 3] then ensures that the error converges asymptotically to zero because: (i) The UCO condition holds with F replacing 𝐹, and (ii) The "disturbance" (𝐹 -F)𝑥 vanishes asymptotically thanks to b𝒈 converging to 𝒃 𝒈 and the boundedness of 𝑥. In other words, we only need to find an asymptotic estimate of 𝒃 𝒈 and feed it to the hybrid Kalman-like observer of 𝑥. For that, notice that the pendulum position 𝜽 verifies the hybrid dynamics

𝜽 = 𝝎 𝒎 -𝒃 𝒈 𝒃 𝒈 = 0 𝑦 ′ 𝑐 = 0, 𝜽 + = 𝜽 𝒃 + 𝒈 = 𝒃 𝒈 𝑦 ′ 𝑑 = 𝜽, (19)
with the flow and jump sets depending on the wall configuration. In other words, 𝜽 has continuous-time dynamics, but with sampled measurement at each impact obtained via the impact condition. Because system [START_REF] Moore | Coping with Singular Transition Matrices in Estimation and Control Stability Theory[END_REF] has linear maps and only the jump output, we can design a jump-based observer with a constant gain using LMIs on the equivalent discrete-time system sampled at the jumps (see [START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF]Corollary 5.2]). All in all, still if the pendulum velocity is uniformly lower-bounded away from zero and (𝐹, 𝐽) upper-bounded, an observer of 𝑥 is obtained by the cascade of a jumpbased observer of 𝒃 𝒈 and a Kalman-like one for system [START_REF] Medina | State Estimation for Linear Impulsive Systems[END_REF] fed with

[𝝎 𝒎 -b𝒈 ] × instead of [𝝎] × .
In what follows, we attempt to analyze more precisely the observability/detectability of the system by decomposing the state according to the different sources of observability. Beyond a finer comprehension, this allows for the design of observers when UCO is not satisfied, for instance under mere detectability properties, or to design observers of smaller dimensions through decoupling (see Table 1 for a comparison).

Observability decomposition

In the case where the full state is instantaneously observable during flows via the flow output 𝑦 𝑐 and the system admits an average dwell time, a high-gain flow-based observer (using only 𝑦 𝑐 ) can be designed (see [START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF]Section 4]); and when the full state is observable from the jump output 𝑦 𝑑 only, a jump-based observer based on an equivalent discrete-time system can be designed if the jumps are persistent (see [START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF]Section 5]). We are thus interested here in the case where observability rather comes from the combination of flows and jumps and/or the combination of (𝑦 𝑐 , 𝑦 𝑑 ). The idea of the decomposition is thus to isolate state components that are instantaneously observable during flows from 𝑦 𝑐 , from other ones that become visible thanks to 𝑦 𝑑 or the combination of flows and jumps. It follows that both the flow and jump outputs may need to be fully exploited to reconstruct the state and that neither (eventually) continuous nor discrete/Zeno trajectories are allowed: both flows and jumps need to be persistent at least after a certain time, unlike in Section 2.1, as assumed next.

Assumption 2 There exists 𝑗 𝑚 ∈ N such that solutions have flow lengths within a compact set I ⊆ [𝜏 𝑚 , 𝜏 𝑀 ] where 𝜏 𝑚 > 0 after jump 𝑗 𝑚 . Assumption 2 means that, for all solutions 𝑥 ∈ S H (X 0 , U), the hybrid arc (𝑡, 𝑗) ↦ → (𝑥(𝑡, 𝑗), 𝑡 -𝑡 𝑗 ) is solution after some time to the hybrid system

H 𝜏            𝑥 = 𝐹𝑥 + 𝑢 𝑐 𝜏 = 1 (𝑥, 𝜏) ∈ 𝐶 𝜏 𝑦 𝑐 = 𝐻 𝑐 𝑥 𝑥 + = 𝐽𝑥 + 𝑢 𝑑 𝜏 + = 0 (𝑥, 𝜏) ∈ 𝐷 𝜏 𝑦 𝑑 = 𝐻 𝑑 𝑥 (20a)
with the flow and jump sets

𝐶 𝜏 = R 𝑛 𝑥 × [0, 𝜏 𝑀 ], 𝐷 𝜏 = R 𝑛 𝑥 × I, (20b)
where 𝜏 ∈ R is a timer keeping track of the time elapsed since the previous jump. Note that H 𝜏 admits (after the first 𝑗 𝑚 jumps) a larger set of solutions than S H (X 0 , U) since the information of the flow and jump sets are replaced by the knowledge of flow lengths in I only (as long as the inputs (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ) are defined along the time domains of those extra solutions). But, as discussed in Section 1, when the observer contains gains that are computed offline, based on the knowledge of the possible flow lengths only, it is actually designed for H 𝜏 instead of H and it is thus the detectability/observability of H 𝜏 that is relevant. In that case, the design depends only implicitly on the sets X 0 , U, 𝐶, and 𝐷 through the choice of I satisfying Assumption 2.

In view of observer design and motivated by [START_REF] Cox | Isolating Invisible Dynamics in the Design of Robust Hybrid Internal Models[END_REF], we start by proposing a change of variables decomposing the state 𝑥 of H 𝜏 into components associated with different types of observability. In order to guarantee the existence of the decomposition, we assume in the next section that the flow pair (𝐹, 𝐻 𝑐 ) is constant. However, the subsequent results of this chapter still hold with (𝐹, 𝐻 𝑐 ) varying, as long as the transformation into the decomposition form exists and is invertible uniformly in time as explained in Remark 3.

Observability from 𝒚 𝒄 during flows

Assume (𝐹, 𝐻 𝑐 ) is constant. Let the (flow) observability matrix be

O := row(𝐻 𝑐 , 𝐻 𝑐 𝐹, . . . , 𝐻 𝑐 𝐹 𝑛 𝑥 -1 ), (21)
and assume it is of rank

𝑛 𝑜 := dim Im O < 𝑛 𝑥 . Consider a basis (𝑣 𝑖 ) 1≤𝑖 ≤𝑛 𝑥 of R 𝑛 𝑥 such that (𝑣 𝑖 ) 1≤𝑖 ≤𝑛 𝑜 is
D 𝑜 := 𝑣 1 . . . 𝑣 𝑛 𝑜 ∈ R 𝑛 𝑥 ×𝑛 𝑜 , (22a)
D 𝑛𝑜 := 𝑣 𝑛 𝑜 +1 . . . 𝑣 𝑛 𝑥 ∈ R 𝑛 𝑥 ×𝑛 𝑛𝑜 , (22b)
which by definition satisfies for all 𝜏 ≥ 0,

OD 𝑛𝑜 = 0, 𝐻 𝑐 𝑒 𝐹 𝜏 D 𝑛𝑜 = 0. ( 23
)
We denote V := D -1 which we decompose consistently into two parts V =:

V 𝑜 V 𝑛𝑜 ,
so that V 𝑜 𝑥 represents the part of the state that is instantaneously observable during flows (see [START_REF] Chen | Linear System Theory and Design[END_REF]Theorem 6.O6]).

A first idea could be to stop the decomposition here and design a sufficiently fast high-gain observer for V 𝑜 𝑥 while estimating the rest of the state V 𝑛𝑜 𝑥 through 𝑦 𝑑 and detectability. However, as noticed in [START_REF] Cox | Isolating Invisible Dynamics in the Design of Robust Hybrid Internal Models[END_REF]Proposition 6], the fact that V 𝑜 𝑥 and V 𝑛𝑜 𝑥 possibly interact with each other during flows prevents achieving stability by further pushing the high gain. The case of such a decomposition where V 𝑜 𝑥 and V 𝑛𝑜 𝑥 evolve independently during flows is exploited in a more general context in [START_REF] Tran | Coupling Flow and Jump Observers for Hybrid Systems with Known Jump Times[END_REF]. Here, because the maps are linear, we can go further and solve this possible coupling by more efficiently decoupling the state components as follows.

Indeed, the estimation of any state that is not instantaneously observable during flows needs to take into account the combination of flows and jumps. That is why it is relevant to exhibit explicitly this combination via the change of coordinates

𝑥 ↦ → 𝑧 = 𝑧 𝑜 𝑧 𝑛𝑜 = V𝑒 -𝐹 𝜏 𝑥 = V 𝑜 V 𝑛𝑜 𝑒 -𝐹 𝜏 𝑥, (24)
whose inverse transformation is

𝑥 = 𝑒 𝐹 𝜏 D𝑧 = 𝑒 𝐹 𝜏 (D 𝑜 𝑧 𝑜 + D 𝑛𝑜 𝑧 𝑛𝑜 ), (25)
and which, according to [START_REF] Sanfelice | Hybrid Feedback Control[END_REF], transforms

H 𝜏 into                        𝑧 𝑜 = 𝐺 𝑜 (𝜏)𝑢 𝑐 𝑧 𝑛𝑜 = 𝐺 𝑛𝑜 (𝜏)𝑢 𝑐 𝜏 = 1 𝑧 + 𝑜 = 𝐽 𝑜 (𝜏)𝑧 𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏)𝑧 𝑛𝑜 + V 𝑜 𝑢 𝑑 𝑧 + 𝑛𝑜 = 𝐽 𝑛𝑜𝑜 (𝜏)𝑧 𝑜 + 𝐽 𝑛𝑜 (𝜏)𝑧 𝑛𝑜 + V 𝑛𝑜 𝑢 𝑑 𝜏 + = 0, (26a)
with the flow and jump sets

R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × [0, 𝜏 𝑀 ], R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × I, (26b)
and the measurements

𝑦 𝑐 = 𝐻 𝑐,𝑜 (𝜏)𝑧 𝑜 , 𝑦 𝑑 = 𝐻 𝑑,𝑜 (𝜏)𝑧 𝑜 + 𝐻 𝑑,𝑛𝑜 (𝜏)𝑧 𝑛𝑜 , (26c)
where

𝐺 𝑜 (𝜏) = V 𝑜 𝑒 -𝐹 𝜏 , 𝐺 𝑛𝑜 (𝜏) = V 𝑛𝑜 𝑒 -𝐹 𝜏 , 𝐽 𝑜 (𝜏) = V 𝑜 𝐽𝑒 𝐹 𝜏 D 𝑜 , 𝐽 𝑜𝑛𝑜 (𝜏) = V 𝑜 𝐽𝑒 𝐹 𝜏 D 𝑛𝑜 , 𝐽 𝑛𝑜𝑜 (𝜏) = V 𝑛𝑜 𝐽𝑒 𝐹 𝜏 D 𝑜 , 𝐽 𝑛𝑜 (𝜏) = V 𝑛𝑜 𝐽𝑒 𝐹 𝜏 D 𝑛𝑜 , 𝐻 𝑐,𝑜 (𝜏) = 𝐻 𝑐 𝑒 𝐹 𝜏 D 𝑜 ,
𝐻 𝑑,𝑜 (𝜏) = 𝐻 𝑑 𝑒 𝐹 𝜏 D 𝑜 , and 𝐻 𝑑,𝑛𝑜 (𝜏) = 𝐻 𝑑 𝑒 𝐹 𝜏 D 𝑛𝑜 . This idea of bringing at the jumps the whole combination of flows and jumps is similar to the so-called equivalent discrete-time system exhibited in [START_REF] Bernard | Observer Design for Hybrid Dynamical Systems with Approximately Known Jump Times[END_REF] for jump-based observer designs. Notice that by definition and thanks to linearity, the observability decomposition through V ensures that the flow dynamics of 𝑧 𝑜 and 𝑦 𝑐 are totally independent of 𝑧 𝑛𝑜 , which only impacts 𝑧 𝑜 at jumps. In other words, the whole dependence of the observable part on the non-observable part via flows and jumps has been gathered at the jumps. Besides, 𝑧 𝑜 is by definition instantaneously observable from 𝑦 𝑐 . More precisely, for any 𝛿 > 0, there exists 𝛼 > 0 such that the observability Gramian of the continuous-time pair (0, 𝐻 𝑐,𝑜 (𝜏)) satisfies

∫ 𝑡+ 𝛿 𝑡 𝐻 ⊤ 𝑐,𝑜 (𝑠)𝐻 𝑐,𝑜 (𝑠)𝑑𝑠 = ∫ 𝑡+ 𝛿 𝑡 D ⊤ 𝑜 𝑒 𝐹 ⊤ 𝑠 𝐻 ⊤ 𝑐 𝐻 𝑐 𝑒 𝐹𝑠 D 𝑜 𝑑𝑠 ≥ 𝛼 Id, ∀𝑡 ≥ 0.
(27) Indeed, this Gramian corresponds to the observability Gramian of the pair (𝐹, 𝐻 𝑐 ) projected onto the observable subspace. This condition is thus related to the uniform complete observability of the continuous-time pair (0, 𝐻 𝑐,𝑜 (𝜏)) in the Kalman literature [START_REF] Kalman | New Results in Linear Filtering and Prediction Theory[END_REF] (continuous-time version of the one in Definition 4), but here with an arbitrarily small window 𝛿. Since 𝑧 𝑜 is observable via 𝑦 𝑐 , we propose to estimate 𝑧 𝑜 sufficiently fast during flows to compensate for the interaction with 𝑧 𝑛𝑜 at jumps. Then, intuitively from system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF], information about 𝑧 𝑛𝑜 may be drawn from two sources: the jump output 𝑦 𝑑 and the part of 𝑧 𝑛𝑜 impacting 𝑧 𝑜 at jumps, namely 𝐽 𝑜𝑛𝑜 (𝜏)𝑧 𝑛𝑜 , which may become "visible" in 𝑧 𝑜 , via 𝑦 𝑐 during the following flow interval. This is illustrated in Example 2 below. Actually, we show next in Section 2.4 that the detectability of 𝑧 𝑛𝑜 comes from these two sources of information only.

Example 2 Consider a hybrid system of form [START_REF] Alessandri | Switching Observers for Continuous-time and Discrete-time Linear Systems[END_REF] with state 𝑥 = (𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 ), 𝑢 𝑐 = 0, 𝑢 𝑑 = 0, and the matrices

𝐹 = 0 -1 0 0 1 0 0 0 0 0 0 -2 0 0 2 0 , 𝐻 𝑐 = 1 0 0 0 , 𝐽 = 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 , 𝐻 𝑑 = 0 0 1 0 , ( 28
)
with random flow lengths varying in some compact set I ⊂ 0, 𝜋 2 . It can be seen that only 𝑥 1 and 𝑥 2 are instantaneously observable during flows from 𝑦 𝑐 , but 𝑥 3 impacts 𝑥 1 at jumps (or 𝑦 𝑑 ) and 𝑥 4 impacts 𝑥 3 during flows. Therefore, we may hope to estimate the full state. In order to decouple those various impacts and analyze detectability more easily, we proceed with the change of variables [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF]. We obtain

𝑧 𝑜 = cos(𝜏) sin(𝜏) 0 0 -sin(𝜏) cos(𝜏) 0 0 𝑥, 𝑧 𝑛𝑜 = 0 0 cos(2𝜏) sin(2𝜏) 0 0 -sin(2𝜏) cos(2𝜏) 𝑥, (29)
and system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with the matrices

𝐽 𝑜 (𝜏) = 0 0 0 0 , 𝐽 𝑜𝑛𝑜 (𝜏) = cos(2𝜏) -sin(2𝜏) 0 0 , 𝐽 𝑛𝑜𝑜 (𝜏) = 0 0 0 0 , 𝐽 𝑛𝑜 (𝜏) = cos(2𝜏) -sin(2𝜏) sin(2𝜏) cos(2𝜏) , 𝐻 𝑐,𝑜 (𝜏) = cos(𝜏) -sin(𝜏) ,
𝐻 𝑑,𝑜 (𝜏) = 0 0 , and 𝐻 𝑑,𝑛𝑜 (𝜏) = cos(2𝜏)sin(2𝜏) . It can be seen that in this case, the terms 𝐽 𝑜𝑛𝑜 (𝜏)𝑧 𝑛𝑜 and 𝐻 𝑑,𝑛𝑜 (𝜏)𝑧 𝑛𝑜 contain the same information on 𝑧 𝑛𝑜 and both should be able to let us estimate this part.

Remark 3

In what follows, a varying pair (𝐹, 𝐻 𝑐 ) can be considered as long as the transformation into the form ( 26) satisfying [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF] exists and is invertible uniformly in time. This can be done with the transition matrix of 𝐹 replacing the exponential form, if the observable subspace remains the same at all times. In that case, the jump matrices 𝐽 𝑜 , 𝐽 𝑜𝑛𝑜 , 𝐽 𝑛𝑜𝑜 , 𝐽 𝑛𝑜 , 𝐻 𝑐,𝑜 , 𝐻 𝑑,𝑜 , and 𝐻 𝑑,𝑛𝑜 are (discrete) known inputs that are no longer functions of 𝜏 only, but of the (discrete) jump index, which is not considered in this chapter. Similarly, 𝐽 could vary at each jump as long as every related condition in the rest of this chapter holds uniformly in 𝑢 𝑑 .

Detectability analysis

We first provide a more specific characterization of the (pre-)asymptotic detectability of system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] in the case of zero inputs (𝑢 𝑐 , 𝑢 𝑑 ). Indeed, we will see in Theorem 2 that this detectability is relevant to characterize that of the initial system H .

𝑦 𝑐 (𝑡, 𝑗) = 0, ∀𝑡 ∈ int(T 𝑗 (𝑧)), ∀ 𝑗 ∈ dom 𝑗 𝑧, (30a)
𝑦 𝑑 (𝑡 𝑗 , 𝑗 -1) = 0, ∀ 𝑗 ∈ dom 𝑗 𝑧, 𝑗 ≥ 1, ( 30b
)
verifies lim 𝑡+ 𝑗→+∞ (𝑡 , 𝑗) ∈dom 𝑧 𝑧(𝑡, 𝑗) = 0. ( 31
)
Proof First, assume system ( 26) is pre-asymptotically detectable. Let (𝑧, 𝜏) be a complete solution to system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) and with outputs satisfying [START_REF] Tran | Coupling Flow and Jump Observers for Hybrid Systems with Known Jump Times[END_REF]. Notice that the hybrid arc (𝑧 ′ , 𝜏), with dom 𝑧 ′ = dom 𝑧 and 𝑧 ′ constantly zero, is also solution to system (26) (thanks to linearity in the maps and the inputs (𝑢 𝑐 , 𝑢 𝑑 ) being zero). It can be seen that this solution is complete and also satisfies [START_REF] Tran | Coupling Flow and Jump Observers for Hybrid Systems with Known Jump Times[END_REF]. By the pre-asymptotic detectability of system ( 26 (𝑧 𝑎 -𝑧 𝑏 ) (𝑡, 𝑗) = 0, which implies the (pre-)asymptotic detectability of system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF]. □

Note that the equivalence of the incremental detectability as in Definition 2 with the zero detectability as in Lemma 1 is classical for linear continuous-or discretetime systems, but it is not automatic for hybrid systems with linear maps due to the flow/jump conditions. Here, it holds only because:

• The flow and jump conditions in system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] do not depend on 𝑧 but only on 𝜏;

• 𝜏 is determined uniquely after the first jump by the time domain of solutions;

• The inputs (𝑢 𝑐 , 𝑢 𝑑 ) are removed, thus avoiding a restriction of solutions to system (26) due to a mismatch of time domains.

Theorem 1 Assume that 0 ∉ I and I is compact. Then, the following three statements are equivalent:

1. The hybrid system [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF] with zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) and known jump times is asymptotically detectable;

2. The hybrid system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) and known jump times is asymptotically detectable; 3. The discrete-time system defined as

𝑧 𝑛𝑜,𝑘+1 = 𝐽 𝑛𝑜 (𝜏 𝑘 )𝑧 𝑛𝑜,𝑘 , 𝑦 𝑘 = 𝐻 𝑑,ext (𝜏 𝑘 )𝑧 𝑛𝑜,𝑘 , (32)
where 𝐻 𝑑,ext (𝜏 𝑘 ) = 𝐻 𝑑,𝑛𝑜 (𝜏 𝑘 ) 𝐽 𝑜𝑛𝑜 (𝜏 𝑘 ) , with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N, is asymptotically detectable.

Proof First, notice that all maximal solutions to systems [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF] and ( 26) are complete because their dynamics maps are linear and the flow and jump conditions do not depend on the state but only on the timer. Notice also that since 0 ∉ I, consecutive jumps cannot happen, so that condition (4a) is equivalent to (6) following Remark 1.

Because systems [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF] and ( 26) are the same system modulo a uniformly invertible change of variables, 1. and 2. are equivalent. Then, let us prove that 2. implies 3. So assume 2. holds and consider a solution (𝑧 𝑛𝑜,𝑘 ) 𝑘 ∈N to system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] with input (𝜏 𝑘 ) 𝑘 ∈N in I, such that 𝑦 𝑘 = 0 for all 𝑘 ∈ N. We want to show that (𝑧 𝑛𝑜,𝑘 ) 𝑘 ∈N asymptotically goes to zero. For that, we build and analyze the complete solution 𝑧 = (𝑧 𝑜 , 𝑧 𝑛𝑜 ) to system (26) initialized as 𝑧 𝑜 (0, 0) = 0, 𝑧 𝑛𝑜 (0, 0) = 𝑧 𝑛𝑜,0 , and 𝜏(0, 0) = 0 with jumps verifying 𝜏(𝑡 𝑗 , 𝑗 -1) = 𝜏 𝑗-1 ∈ I for all 𝑗 ≥ 1 and zero inputs (𝑢 𝑐 , 𝑢 𝑑 ). It follows2 from the fact that (i) 𝑧 𝑜 and 𝑧 𝑛𝑜 are constant during flows, (ii) 𝑦 𝑐 is independent of 𝑧 𝑛𝑜 , and (iii) 𝑦 𝑘 = 0 for all 𝑘 ∈ N, that for all 𝑗 ∈ N,

𝑧 𝑜 (𝑡, 𝑗) = 0, ∀𝑡 ∈ [𝑡 𝑗 , 𝑡 𝑗+1 ], 𝑧 𝑛𝑜 (𝑡, 𝑗) = 𝑧 𝑛𝑜, 𝑗 , ∀𝑡 ∈ [𝑡 𝑗 , 𝑡 𝑗+1 ], 𝑦 𝑐 (𝑡, 𝑗) = 0, ∀𝑡 ∈ [𝑡 𝑗 , 𝑡 𝑗+1 ],
𝑧 𝑜 (𝑡 𝑗+1 , 𝑗 + 1) = 𝐽 𝑜𝑛𝑜 (𝜏 𝑗 )𝑧 𝑧 𝑛𝑜,𝑘 = 0, implying 3. Finally, let us prove that 3. implies 2. Consider a complete solution (𝑧, 𝜏) = (𝑧 𝑜 , 𝑧 𝑛𝑜 , 𝜏) to system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) and such that (30) holds. By definition of the jump set, for all 𝑗 ∈ dom 𝑗 𝑧, 𝜏 𝑗 := 𝜏(𝑡 𝑗+1 , 𝑗) ∈ I. Since 0 ∉ I, the solution admits a dwell time and, because we look for an asymptotic property, we may assume without any loss of generality that the solution starts with a flow, possibly overlooking the first part of the domain (with 𝑗 = 0) in case of a jump at time 0. Since 𝑧 𝑜 is instantaneously observable during flows according to [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF], 𝑦 𝑐 = 0 implies that 𝑧 𝑜 is zero during each flow interval. Next, as 0 ∉ I, there is no more than one jump at each jump time so that 𝑧 𝑜 (𝑡, 𝑗) = 0 for all (𝑡, 𝑗) ∈ dom 𝑧. Besides, since I is compact, dom 𝑗 𝑧 = N and from (26a), we then have 𝐽 𝑜𝑛𝑜 (𝜏(𝑡 𝑗 , 𝑗 -1))𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗 -1) = 0 for all 𝑗 ∈ dom 𝑗 𝑧 with 𝑗 ≥ 1. Therefore, since 𝑧 𝑛𝑜 is constant during flows and 𝑦 𝑑 (𝑡 𝑗 , 𝑗 -1) = 0, for all 𝑗 ∈ dom 𝑗 𝑧, 𝑗 ≥ 1, we have for all 𝑗 ∈ N,

𝑧 𝑛𝑜 (𝑡 𝑗+1 , 𝑗 + 1) = 𝐽 𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗))𝑧 𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐽 𝑛𝑜 (𝜏 𝑗 )𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗), 𝐻 𝑑,𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗))𝑧 𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐻 𝑑,𝑛𝑜 (𝜏 𝑗 )𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0, 𝐽 𝑜𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗))𝑧 𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐽 𝑜𝑛𝑜 (𝜏 𝑗 )𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0.
By considering the sequence (𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗)) 𝑗 ∈N solution to system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] □

We thus conclude that (at least when the inputs (𝑢 𝑐 , 𝑢 𝑑 ) are zero) the asymptotic detectability of system [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF] requires 𝑧 𝑛𝑜 to be asymptotically detectable through the output made of the measured output 𝑦 𝑑 and the fictitious one 𝐽 𝑜𝑛𝑜 (𝜏)𝑧 𝑛𝑜 , which describes how 𝑧 𝑛𝑜 impacts 𝑧 𝑜 at jumps. We insist also that the detectability of 𝑧 𝑛𝑜 comes from the combination of flows and jumps and not due to jumps alone since the useful information contained in the flow dynamics and output is gathered at the jumps via the transformation [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF].

It follows from this analysis that the design of an asymptotic observer for H with gains computed offline from the knowledge of I only (without any special consideration of X 0 , U, 𝐶, or 𝐷), namely an observer for system [START_REF] Penrose | A Generalized Inverse for Matrices[END_REF], without considering the possible restrictions of time domains by the inputs (𝑢 𝑐 , 𝑢 𝑑 ), requires the asymptotic detectability of the discrete-time system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]. This is because any time domain with flow lengths in I is a priori possible. This will be done in Section 3 with an LMI-based design. On the other hand, if we consider the more precise problem of observer design with time domains restricted to that of solutions in S H (X 0 , U), we end up with the following sufficient condition for asymptotic detectability of H . Theorem 2 Suppose Assumptions 1 and 2 hold. Then, H initialized in X 0 with inputs in U is asymptotically detectable if for each 𝑥 ∈ S H (X 0 , U), the discrete-time system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] with input (𝜏 𝑘 ) 𝑘 ∈N defined as 𝜏 𝑘 = 𝑡 𝑘+1 (𝑥) -𝑡 𝑘 (𝑥) for all 𝑘 ∈ N is asymptotically detectable.

Proof Pick solutions 𝑥 𝑎 and 𝑥 𝑏 in S H (X 0 , U) with the same inputs (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 , 𝑢 𝑐 , 𝑢 𝑑 ), such that dom 𝑥 𝑎 = dom 𝑥 𝑏 , and with outputs 𝑦 𝑎,𝑐 , 𝑦 𝑏,𝑐 , 𝑦 𝑎,𝑑 , 𝑦 𝑏,𝑑 satisfying (4). By Assumptions 1 and 2, these solutions are complete with dom 𝑗 𝑥 𝑎 = dom 𝑗 𝑥 𝑏 = N. By Assumption 2, for all 𝑗 ≥ 𝑗 𝑚 , 𝜏 𝑗 := 𝑡 𝑗+1 (𝑥 𝑎 ) -𝑡 𝑗 (𝑥 𝑎 ) = 𝑡 𝑗+1 (𝑥 𝑏 ) -𝑡 𝑗 (𝑥 𝑏 ) ∈ I. Since 0 ∉ I, the solutions admit a dwell time after the first 𝑗 𝑚 jumps. Since we look for an asymptotic property, we may discard the first part of the solutions with 𝑗 < 𝑗 𝑚 , and assume without any loss of generality that they start with a flow interval and have a dwell time. Then, condition (4a) is equivalent to (6) following Remark 1. Consider the hybrid signals (𝑧 𝑎 , 𝑧 𝑏 , 𝜏) defined for all (𝑡, 𝑗) ∈ dom 𝑥 𝑎 as

𝜏(𝑡, 𝑗) = 𝑡 -𝑡 𝑗 , 𝑧 𝑎,𝑜 (𝑡, 𝑗) 𝑧 𝑎,𝑛𝑜 (𝑡, 𝑗) = V 𝑜 V 𝑛𝑜 𝑒 -𝐹 𝜏 (𝑡 , 𝑗) 𝑥 𝑎 (𝑡, 𝑗), 𝑧 𝑏,𝑜 (𝑡, 𝑗) 𝑧 𝑏,𝑛𝑜 (𝑡, 𝑗) = V 𝑜 V 𝑛𝑜 𝑒 -𝐹 𝜏 (𝑡 , 𝑗) 𝑥 𝑏 (𝑡, 𝑗).
We see that both (𝑧 𝑎 , 𝜏) and (𝑧 𝑏 , 𝜏) are solutions to system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] (𝑧 𝑎,𝑛𝑜 (𝑡, 𝑗) -𝑧 𝑏,𝑛𝑜 (𝑡, 𝑗)) = 0, implying that H initialized in X 0 with inputs in U is asymptotically detectable according to Lemma 1. □ Unlike Theorem 1, Theorem 2 does not give a necessary condition for detectability (and thus observer design). The reason is that the flow and jump conditions of H are not taken into account in system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]. But it still suggests us to build observers for H under observability/detectability conditions on system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] for the flow length sequences (𝜏 𝑘 ) 𝑘 ∈N appearing in S H (X 0 , U). This will be done in Section 3 and 4 through LMI-or KKL-based design.

Remark 4 Compared to the preliminary work [START_REF] Tran | Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], the observer designs in this chapter do not require an extra (constant) transformation decoupling the part of 𝑧 𝑛𝑜 detectable from 𝑦 𝑑 and the part detectable from the fictitious output 𝐽 𝑜𝑛𝑜 (𝜏)𝑧 𝑛𝑜 . We instead consider an extended output in system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] and the decomposition [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] proposed in this chapter is thus less conservative. For instance, Example 2 can be cast into the decomposition [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF], but fails to fall into the scope of the decomposition of [START_REF] Tran | Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times[END_REF] unless I = 𝜋 2 .

Example 3 Consider the system in Example 2. It is possible to check, by computing the (time-varying) observability matrix of the pair (𝐽 𝑛𝑜 (𝜏 𝑘 ), 𝐻 𝑑,𝑛𝑜 (𝜏 𝑘 )), that system (32) is observable for any sequence (𝜏 𝑘 ) 𝑘 ∈N as long as sin(𝜏 𝑘 + 𝜏 𝑘+1 ) ≠ 0 at some 𝑘 ∈ N, which is the case for 𝜏 𝑘 ∈ I since I ⊂ 0, 𝜋 2 . This implies in particular that system (32) is detectable. Actually, even if 𝐻 𝑑 = 0, i.e., no output is available at jumps, the pair (𝐽 𝑛𝑜 (𝜏 𝑘 ), 𝐽 𝑜𝑛𝑜 (𝜏 𝑘 )) is also observable using the same arguments. This means that 𝑧 𝑛𝑜 is actually observable through the fictitious measurement of 𝑧 𝑜 at jumps. We see from this example that thanks to the flow-jump coupling, by using 𝑧 𝑜 as a fictitious measurement, state components that are not observable during flows from the flow output may become observable via jumps even without any additional measurements at jumps (hidden dynamics), only through the way they impact the observable ones.

LMI-based observer design from discrete quadratic detectability in observability decomposition

Inspired by the detectability analysis of Theorem 1, we propose a first observer design under the following detectability assumption.

Assumption 3 Given 𝜏 𝑀 and I defined in Assumption 2, there exist 𝑄 𝑛𝑜 ∈ S 𝑛 𝑛𝑜 >0 , 𝐿 𝑑,𝑛𝑜 : [0, 𝜏 𝑀 ] → R 𝑛 𝑛𝑜 ×𝑛 𝑦,𝑑 bounded on [0, 𝜏 𝑀 ] and continuous on I, and 𝐾 𝑛𝑜 ∈ R 𝑛 𝑛𝑜 ×𝑛 𝑜 such that

★ ⊤ 𝑄 𝑛𝑜 𝐽 𝑛𝑜 (𝜏) -𝐿 𝑑,𝑛𝑜 (𝜏) 𝐾 𝑛𝑜 𝐻 𝑑,𝑛𝑜 (𝜏) 𝐽 𝑜𝑛𝑜 (𝜏) -𝑄 𝑛𝑜 < 0, ∀𝜏 ∈ I. ( 33
)
We refer the reader to Remark 5 for constructive methods to solve [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF], where it is shown that the solvability of [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF] in 𝑄 𝑛𝑜 and 𝐿 𝑑,𝑛𝑜 (𝜏) 𝐾 𝑛𝑜 is equivalent to that of a reduced LMI involving 𝑄 𝑛𝑜 only. Consistently with Theorem 1, Assumption 3 requires the detectability of the pair 𝐽 𝑛𝑜 (𝜏), 𝐻 𝑑,𝑛𝑜 (𝜏) 𝐽 𝑜𝑛𝑜 (𝜏) for each frozen 𝜏 ∈ I. But it is actually stronger because it further requires 𝑄 𝑛𝑜 and 𝐾 𝑛𝑜 to be independent of 𝜏. It corresponds to a stronger version of the quadratic detectability of system (32) defined in [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]. Actually, the detectability of Assumption 3 allows us to build an observer for any sequence of flow lengths (𝜏 𝑘 ) 𝑘 ∈N ∈ I and thus requires the detectability of the discrete-time pair for any such sequences, which is still consistent with the result of Theorem 1. Note that the reason why 𝐾 𝑛𝑜 is required to be independent of 𝜏 is that it is used to carry out another change of variables in the proof of Theorem 3 below, allowing us to exhibit the fictitious output in the analysis.

LMI-based observer design in the 𝒛-coordinates

Because the flow output matrix 𝐻 𝑐,𝑜 (•) varies and satisfies the observability condition [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF], we design a flow-based Kalman-like observer of 𝑧 𝑜 during flows using 𝑦 𝑐 [START_REF] Besancon | Further Results on High Gain Observers for Nonlinear Systems[END_REF]. Its advantage over a Kalman observer is that it admits a strict Lyapunov function, allowing for direct robust Lyapunov analysis. Besides, it provides a direct relationship between the Lyapunov matrix and the observability Gramian. Then, as suggested by the detectability analysis, 𝑧 𝑛𝑜 should be estimated thanks to both 𝑦 𝑑 and its interaction with 𝑧 𝑜 at jumps via a fictitious output. The latter is not available for injection in the observer, but it becomes visible through 𝑧 𝑜 after the jump, and thus through 𝑦 𝑐 during flows. This justifies correcting the estimate of 𝑧 𝑛𝑜 during flows with 𝑦 𝑐 , via the gain 𝐾 𝑛𝑜 . The dynamics of the observer are then given by

                                   ẑ𝑜 = 𝐺 𝑜 (𝜏)𝑢 𝑐 + 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) (𝑦 𝑐 -𝐻 𝑐,𝑜 (𝜏) ẑ𝑜 ) ẑ𝑛𝑜 = 𝐺 𝑛𝑜 (𝜏)𝑢 𝑐 + 𝐾 𝑛𝑜 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) (𝑦 𝑐 -𝐻 𝑐,𝑜 (𝜏) ẑ𝑜 ) 𝑃 = -𝜆𝑃 + 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) 𝜏 = 1 ẑ+ 𝑜 = 𝐽 𝑜 (𝜏) ẑ𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏) ẑ𝑛𝑜 + V 𝑜 𝑢 𝑑 ẑ+ 𝑛𝑜 = 𝐽 𝑛𝑜𝑜 (𝜏) ẑ𝑜 + 𝐽 𝑛𝑜 (𝜏) ẑ𝑛𝑜 + V 𝑛𝑜 𝑢 𝑑 + 𝐿 𝑑,𝑛𝑜 (𝜏) (𝑦 𝑑 -𝐻 𝑑,𝑜 (𝜏) ẑ𝑜 -𝐻 𝑑,𝑛𝑜 (𝜏) ẑ𝑛𝑜 ) 𝑃 + = 𝑃 0 𝜏 + = 0, (34)
with jumps triggered at the same time as H in the same way as observer (2), 𝑃 0 ∈ S 𝑛 𝑜 >0 , 𝐾 𝑛𝑜 and 𝐿 𝑑,𝑛𝑜 given by Assumption 3, and where 𝜏 ↦ → 𝑅(𝜏) ∈ S 𝑛 𝑦,𝑐 >0 is a positive definite weighting matrix that is defined and is continuous on [0, 𝜏 𝑀 ] to be chosen arbitrarily. The estimate is then recovered by using (25) on ẑ with the global exponential stability (GES) of the estimation error as stated next.

Theorem 3 Under Assumptions 1, 2, and 3, given any 𝑃 0 ∈ S 𝑛 𝑜 >0 , there exists 𝜆 ★ > 0 such that for any 𝜆 > 𝜆 ★ , there exist 𝜌 1 > 0 and 𝜆 1 > 0 such that for any solution 𝑥 ∈ S H (X 0 , U) and any solution ( ẑ, 𝑃, 𝜏) to observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF] with 𝑃(0, 0) = 𝑃 0 , 𝜏(0, 0) = 0, and jumps triggered at the same time as in 𝑥, ( ẑ, 𝑃, 𝜏) is complete and we have

|𝑥(𝑡, 𝑗) -x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(0, 0) -x(0, 0)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀(𝑡, 𝑗) ∈ dom 𝑥, (35)
with x obtained by x = 𝑒 𝐹 𝜏 D ẑ with D defined in [START_REF] Tanwani | Observability for Switched Linear Systems: Characterization and Observer Design[END_REF].

Proof Consider a solution 𝑥 ∈ S H (X 0 , U) and a solution ( ẑ, 𝑃, 𝜏) to observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF] with 𝑃(0, 0) = 𝑃 0 and 𝜏(0, 0) = 0, and jumps triggered at the same time as in 𝑥. By Assumption 1, it is complete and so is ( ẑ, 𝑃, 𝜏). Following [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF], define 𝑧(𝑡, 𝑗) := V𝑒 -𝐹 𝜏 (𝑡 , 𝑗) 𝑥(𝑡, 𝑗),

∀(𝑡, 𝑗) ∈ dom 𝑥, (36)
and consider the error z = ( z𝑜 , z𝑛𝑜 ) = (𝑧 𝑜 -ẑ𝑜 , 𝑧 𝑛𝑜 -ẑ𝑛𝑜 ). Because the flow lengths of 𝑥 are in I by Assumption 2 after the first 𝑗 𝑚 jumps only, the proof consists of two parts: first, we use Lemma 2 to show the exponential convergence of z starting at hybrid time (𝑡 𝑗 𝑚 , 𝑗 𝑚 ) by putting the error dynamics into the appropriate form, and then, we analyze the behavior of the error before (𝑡 𝑗 𝑚 , 𝑗 𝑚 ) using Lemma 3. Consider first the solution ( z, 𝑃, 𝜏) starting from (𝑡 𝑗 𝑚 , 𝑗 𝑚 ). According to Assumption 2 and since the observer's jumps are synchronized with those of H , ( z, 𝑃, 𝜏) is solution to

                               z𝑜 = -𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 z𝑛𝑜 = -𝐾 𝑛𝑜 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 𝑃 = -𝜆𝑃 + 𝐻 ⊤ 𝑐,𝑜 (
R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × R 𝑛 𝑜 ×𝑛 𝑜 × [0, 𝜏 𝑀 ], R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × R 𝑛 𝑜 ×𝑛 𝑜 × I. (37b)
We next perform the change of variables

η = z𝑛𝑜 -𝐾 𝑛𝑜 z𝑜 , (38)
which transforms the error system (37) into

                               z𝑜 = -𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 η = 0 𝑃 = -𝜆𝑃 + 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) 𝜏 = 1 z+ 𝑜 = 𝐽 𝑜 (𝜏) z𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏) η η+ = (𝐽 𝑛𝑜𝑜 (𝜏) -𝐿 𝑑,𝑛𝑜 (𝜏)𝐻 𝑑,𝑜 (𝜏)) z𝑜 + 𝐽 𝜂 (𝜏) η 𝑃 + = 0 𝜏 + = 0, (39)
with the same flow and jump sets where 𝐽 𝑜 (𝜏) = 𝐽 𝑜 (𝜏) + 𝐽 𝑜𝑛𝑜 (𝜏)𝐾 𝑛𝑜 , 𝐽 𝑛𝑜𝑜 (𝜏) = 𝐽 𝑛𝑜𝑜 (𝜏)+𝐽 𝑛𝑜 (𝜏)𝐾 𝑛𝑜 -𝐾 𝑛𝑜 𝐽 𝑜 (𝜏)-𝐾 𝑛𝑜 𝐽 𝑜𝑛𝑜 (𝜏)𝐾 𝑛𝑜 , 𝐻 𝑑,𝑜 (𝜏) = 𝐻 𝑑,𝑜 (𝜏)+𝐻 𝑑,𝑛𝑜 (𝜏)𝐾 𝑛𝑜 , and 𝐽 𝜂 (𝜏) = 𝐽 𝑛𝑜 (𝜏)-𝐿 𝑑,𝑛𝑜 (𝜏)𝐻 𝑑,𝑛𝑜 (𝜏)-𝐾 𝑛𝑜 𝐽 𝑜𝑛𝑜 (𝜏), with the flow set R 𝑛 𝑜 ×R 𝑛 𝑛𝑜 × [0, 𝜏 𝑀 ] and the jump set R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × I. From Assumption 3, 𝐽 𝜂 (𝜏) is Schur for all 𝜏 ∈ I, and more precisely, there exists 𝑄 𝜂 ∈ S 𝑛 𝑛𝑜 >0 such that

𝐽 ⊤ 𝜂 (𝜏)𝑄 𝜂 𝐽 𝜂 (𝜏) -𝑄 𝜂 < 0, ∀𝜏 ∈ I. ( 40
)
Using Lemma 2, we proceed to prove the GES of the error ( z𝑜 , η) with respect to the value ( z𝑜 , η) (𝑡 𝑗 𝑚 , 𝑗 𝑚 ). Then the GES with respect to ( z𝑜 , η) (0, 0) is proven using Lemma 3. Last, because the transformations ( 24) and (38) are linear with 𝜏 ↦ → 𝑒 𝐹 𝜏 bounded with a strictly positive lower bound on the compact set [0, 𝜏 𝑀 ], we obtain

(35) observing that ( z𝑜 , η) = V 𝑜 V 𝑛𝑜 -𝐾 𝑛𝑜 V 𝑜 𝑒 -𝐹 𝜏 (𝑥 -x). □
Remark 5 Applying Schur's lemma and then the elimination lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] to [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF]

If such a 𝑄 𝑛𝑜 is obtained, the gains 𝐿 𝑑,𝑛𝑜 (•) and 𝐾 𝑛𝑜 are then found by using [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF] with 𝑄 𝑛𝑜 known. If I is infinite, then there is an infinite number of LMIs to solve. Actually, it is worth noting that the exponential term 𝑒 𝐹 𝜏 contained in all the 𝜏-dependent matrices in (41) can be expanded using residue matrices [START_REF] Ferrante | State Estimation of Linear Systems in the Presence of Sporadic Measurements[END_REF], as

𝑒 𝐹 𝜏 = 𝜎 𝑟 ∑ 𝑖=1 𝑚 𝑟 𝑖 ∑ 𝑗=1 𝑅 𝑖 𝑗 𝑒 𝜆 𝑖 𝜏 𝜏 𝑗-1 ( 𝑗 -1)! + 𝜎 𝑐 ∑ 𝑖=1 𝑚 𝑐 𝑖 ∑ 𝑗=1 2𝑒 ℜ(𝜆 𝑖 ) 𝜏 (ℜ(𝑅 𝑖 𝑗 ) cos(ℑ(𝜆 𝑖 )𝜏) -ℑ(𝑅 𝑖 𝑗 ) sin(ℑ(𝜆 𝑖 )𝜏)) 𝜏 𝑗-1 ( 𝑗 -1)! , (42)
where 𝜎 𝑟 and 𝜎 𝑐 are the numbers of distinct real eigenvalues and complex conjugate eigenvalue pairs; 𝑚 𝑟 𝑖 and 𝑚 𝑐 𝑖 are the multiplicity of the real eigenvalue 𝜆 𝑖 and of the complex conjugate eigenvalue pair 𝜆 𝑖 , 𝜆 * 𝑖 in the minimal polynomial of 𝐹; 𝑅 𝑖 𝑗 ∈ R 𝑛 𝑥 ×𝑛 𝑥 are matrices corresponding to the residues associated to the partial fraction expansion of (𝑠𝐼-𝐹) -1 . This in turn allows 𝑒 𝐹 𝜏 to be written as a finite sum of matrices affine in 𝑁 scalar functions 𝛽 𝑖 𝑗 = 𝑒 𝜆 𝑖 𝜏 𝜏 𝑗-1 , 𝛾 𝑖 𝑗 = 𝑒 ℜ(𝜆 𝑖 ) 𝜏 cos(ℑ(𝜆 𝑖 )𝜏)𝜏 𝑗-1 , and 𝛾 * 𝑖 𝑗 = 𝑒 ℜ(𝜆 𝑖 ) 𝜏 sin(ℑ(𝜆 𝑖 )𝜏)𝜏 𝑗-1 . It then implies that (41) can be solved in a polytopic approach, i.e., the LMIs are satisfied for all 𝜏 ∈ I compact if they are satisfied at the finite number 2 𝑁 of vertices of the polytope formed by these scalar functions when 𝜏 varies in I. Alternatively, the LMIs can be solved in a grid-based approach followed by post-analysis of the solution's stability as in [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF], possibly with a theoretical proof extended from [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF].

LMI-based observer design in the 𝒙-coordinates

In this section, we show that the observer can equivalently be implemented directly in the original 𝑥-coordinates, with dynamics given by

                       x = 𝐹 x + 𝑢 𝑐 + P𝐻 ⊤ 𝑐 𝑅 -1 (𝜏) (𝑦 𝑐 -𝐻 𝑐 x) P = 𝜆P + 𝐹P + P𝐹 ⊤ -P𝐻 ⊤ 𝑐 𝑅 -1 (𝜏)𝐻 𝑐 P 𝜏 = 1 x+ = 𝐽 x + 𝑢 𝑑 + D 𝑛𝑜 𝐿 𝑑,𝑛𝑜 (𝜏) (𝑦 𝑑 -𝐻 𝑑 x) P + = (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 )𝑃 -1 0 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 ) ⊤ 𝜏 + = 0, (43)
with jumps still triggered at the same time as H . The GES of the error is proven in Theorem 4.

Theorem 4

Under Assumptions 1, 2, and 3, given any 𝑃 0 ∈ S 𝑛 𝑜 >0 , there exists 𝜆 ★ > 0 such that for any 𝜆 > 𝜆 ★ , there exist 𝜌 1 > 0 and 𝜆 1 > 0 such that for any solution 𝑥 ∈ S H (X 0 , U) and for any solution ( x, P, 𝜏) to observer (43) with P (0, 0) = (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 )𝑃 -1 0 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 ) ⊤ , 𝜏(0, 0) = 0, and jumps triggered at the same time as in 𝑥, ( x, P, 𝜏) is complete and we have

|𝑥(𝑡, 𝑗) -x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(0, 0) -x(0, 0)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀(𝑡, 𝑗) ∈ dom 𝑥. ( 44
)
Proof Pick a solution 𝑥 ∈ S H (X 0 , U) and a solution ( ζ, P, 𝜏) to observer (43) with P (0, 0) = (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 )𝑃 -1 0 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 ) ⊤ , and 𝜏(0, 0) = 0, with jumps triggered at the same time as in 𝑥. Consider ( ẑ, 𝑃, 𝜏) solution to observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF], with ẑ(0, 0) = V x(0, 0), 𝑃(0, 0) = 𝑃 0 , and 𝜏(0, 0) = 0, with jumps triggered at the same time as in 𝑥. First, notice from its dynamics that 𝜏 = 𝜏. Then, applying Theorem 3, we get

|𝑥(𝑡, 𝑗) -x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(0, 0) -x(0, 0)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀(𝑡, 𝑗) ∈ dom 𝑥,
where x = 𝑒 𝐹 𝜏 D ẑ. The proof consists in showing that x = x, thus obtaining (44).

Observe that

x = 𝐹 x + 𝑢 𝑐 + 𝐿 𝑐 (𝑃, 𝜏) (𝑦 𝑐 -𝐻 𝑐 x), (45a)
during flows and

x+ = 𝐽 x + 𝑢 𝑑 + 𝐿 𝑑 (𝜏) (𝑦 𝑑 -𝐻 𝑑 x), (45b)
at jumps where

𝐿 𝑐 (𝑃, 𝜏) = 𝑒 𝐹 𝜏 D 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) 𝐾 𝑛𝑜 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) = 𝑒 𝐹 𝜏 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 )𝑃 -1 D ⊤ 𝑜 𝑒 𝐹 ⊤ 𝜏 𝐻 ⊤ 𝑐 𝑅 -1 (𝜏), (46a)
and

𝐿 𝑑 (𝜏) = D 0 𝐿 𝑑,𝑛𝑜 (𝜏) = D 𝑛𝑜 𝐿 𝑑,𝑛𝑜 (𝜏). (46b)
From ( 23), 𝐿 𝑐 (𝑃, 𝜏) can be re-written as 𝐿 𝑐 (𝑃, 𝜏) = P𝐻 ⊤ 𝑐 𝑅 -1 (𝜏) where

P = 𝑒 𝐹 𝜏 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 )𝑃 -1 (D 𝑜 + D 𝑛𝑜 𝐾 𝑛𝑜 ) ⊤ 𝑒 𝐹 ⊤ 𝜏 . ( 47
)
Calculating P while noting that the time derivative of 𝑃 -1 is -𝑃 -1 𝑃𝑃 -1 , we obtain the same flow/jump dynamics as P in observer (43). Besides, P (0, 0) = P (0, 0), so P = P thanks to the uniqueness of solutions. We deduce that x follows the same dynamics as x, and since

x(0, 0) = D ′ ẑ(0, 0) = x(0, 0), we have x = x, which concludes the proof. □

It is interesting to see that the observability provided at jumps by the fictitious output in the non-observable subspace D 𝑛𝑜 is stored into P at jumps. This allows the use of 𝑦 𝑐 to correct the estimate in the non-observable subspace during flows, while the Riccati dynamics of P instead excites only the observable directions provided by [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF]. In terms of implementation, observer (43) is of a larger dimension than that in observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF] but it allows us to avoid the online inversion of the change of variables. Actually, observer (43) has the same dimension and the same flow dynamics as the Kalman-like observer proposed in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]. The difference lies in (i) the jump dynamics, which here contains a priori gains 𝐾 𝑛𝑜 and 𝐿 𝑑,𝑛𝑜 instead of dynamic gains computed online via P, and (ii) the quadratic detectability assumption (3) which replaces the UCO in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF].

Remark 6 Denote 𝐿 𝑑 (𝜏) = D 𝑛𝑜 𝐿 𝑑,𝑛𝑜 (𝜏) and observe that 𝐿 𝑑,𝑛𝑜 (𝜏) = V 𝑛𝑜 𝐿 𝑑 (𝜏). Similarly, denote 𝐾 ′ 𝑛𝑜 = D 𝑛𝑜 𝐾 𝑛𝑜 and observe that 𝐾 𝑛𝑜 = V 𝑛𝑜 𝐾 ′ 𝑛𝑜 . The conditions in Assumption 3 for the design of 𝐿 𝑑,𝑛𝑜 (•) and 𝐾 𝑛𝑜 are equivalent to solving for 𝐿 𝑑 (•) and 𝐾 ′ 𝑛𝑜 directly in the 𝑥-coordinates and for all 𝜏 ∈ I,

★ ⊤ 𝑄 𝑛𝑜 V 𝑛𝑜 (𝐽 -𝐿 𝑑 (𝜏)𝐻 𝑑 -𝐾 ′ 𝑛𝑜 V 𝑜 𝐽)𝑒 𝐹 𝜏 D 𝑛𝑜 -𝑄 𝑛𝑜 < 0, (48a) V 𝑜 𝐿 𝑑 (𝜏) = 0, (48b) V 𝑜 𝐾 ′ 𝑛𝑜 = 0. (48c)
Actually, these are projections of more conservative LMIs (where variables have the full dimensions corresponding to the plant) onto the observable subspaces.

Example 4

The spiking behavior of a cortical neuron may be modeled with state

𝜂 = (𝜂 1 , 𝜂 2 ) ∈ R 2 as            𝜂 = 0.04𝜂 2 1 + 5𝜂 1 + 140 -𝜂 2 + 𝐼 ext 𝑎(𝑏𝜂 1 -𝜂 2 ) when 𝜂 1 ≤ 𝑣 𝑚 𝜂 + = 𝑐 𝜂 2 + 𝑑 when 𝜂 1 = 𝑣 𝑚 (49)
where 𝜂 1 is the membrane potential, 𝜂 2 is the recovery variable, and 𝐼 ext represents the (constant) synaptic current or injected DC current [START_REF] Izhikevich | Simple Model of Spiking Neurons[END_REF]. We pick here the parameters as 𝐼 ext = 10, 𝑎 = 0.02, 𝑏 = 0.2, 𝑐 = -55, 𝑑 = 4, and 𝑣 𝑚 = 30 (all in appropriate units), thus characterizing the neuron type and its firing pattern [START_REF] Izhikevich | Simple Model of Spiking Neurons[END_REF]. The solutions of this system are known to have a dwell time with flow lengths remaining in a compact set I = [𝜏 𝑚 , 𝜏 𝑀 ] where 𝜏 𝑚 > 0, and the jump times can be detected from the discontinuities of the measured output 𝑦 𝑐 = 𝜂 1 . Since 𝑦 𝑐 = 𝜂 1 is known during flows, we treat 0.04𝜂 2 1 + 140 + 𝐼 ext as a known term that can be compensated using output injection with 𝑢 𝑐 = (0.04𝑦 2 𝑐 +140+𝐼 ext , 0). On the other hand, in the jump map of (49), we assume 𝑐 and 𝑑 are unknown and include them in the state to be estimated along with (𝜂 1 , 𝜂 2 ). We show here that, using the decomposition in this paper, we can design a flow-based observer using the knowledge of 𝑦 𝑐 = 𝜂 1 during flows only, although the flow dynamics are not observable. In other words, we take 𝑦 𝑑 = 0, which comes back to not using any output injection at jumps. We re-model system (49) extended with (𝑐, 𝑑) into the form (1) with 𝑥 = (𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 ) = (𝜂 1 , 𝜂 2 , 𝑐, 𝑑) ∈ R 4 and the matrices

𝐹 = 5 -1 0 0 𝑎𝑏 -𝑎 0 0 0 0 0 0 0 0 0 0 , 𝐻 𝑐 = 1 0 0 0 , 𝐽 = 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 , 𝐻 𝑑 = 0 0 0 0 . (50)
We see that 𝑥 1 and 𝑥 2 are instantaneously observable from 𝑦 𝑐 and following [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF], where 𝜇 𝑖 , 𝑖 = 1, 2, . . . , 8 are known exponential functions of 𝜏. We see that for any 𝜏 ∈ I, 𝑧 𝑛𝑜 cannot be seen from 𝑦 𝑑 because 𝐻 𝑑,𝑛𝑜 (𝜏) = 0, but it can be accessed through 𝑧 𝑜 via 𝐽 𝑜𝑛𝑜 (𝜏) (hidden dynamics). Solving [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF], we obtain 𝐾 𝑛𝑜 = 1 0 0 1 with any 𝑄 𝑛𝑜 ∈ S 1 >0 and any 𝐿 𝑑,𝑛𝑜 (•). Let us then take 𝐿 𝑑,𝑛𝑜 = 0. In this application, we see that (𝑥 3 , 𝑥 4 ) is estimated thanks to its interaction with 𝑧 𝑜 at jumps, the latter being estimated during flows, namely we exploit the hidden observability analyzed in Theorem 1. Then, an LMI-based observer as in observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF] or (43), with the mentioned gains 𝐾 𝑛𝑜 and 𝐿 𝑑,𝑛𝑜 , a weighting matrix taken as 𝑅 = Id, and a large enough 𝜆, can be designed for this system.

we get 𝑧 𝑜 = 𝜇 1 (𝜏) 𝜇 2 (𝜏) 0 0 𝜇 3 (𝜏) 𝜇 4 (𝜏) 0 0 𝑥 ∈ R 2 , 𝑧 𝑛𝑜 = 0 0 1 0 0 0 0 1 𝑥 = (𝑥 3 , 𝑥 4 ) ∈ R 2 ,

KKL-based observer design from discrete uniform backward distinguishability in observability decomposition

The idea of this section is to replace the LMI-based design of the observer for the 𝑧 𝑛𝑜 part with a systematic KKL-based one. To do that, following [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF], we exploit a discrete-time KKL observer design for the discrete-time linear time-varying system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF], for which observability is assumed as suggested by Theorem 2. The main reasons for relying on this discrete-time design, as opposed to a discrete-time Kalman(-like) design, for instance, are two-fold:

• Compared to a Kalman design [START_REF] Jr | Conditions for Asymptotic Stability of the Discrete Minimum-variance Linear Estimator[END_REF], it admits a strict Input-to-State Stable (ISS) Lyapunov function, allowing for an interconnection with the high-gain flow-based observer of 𝑧 𝑜 ; • Compared to a Kalman-like design [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF], its gain with respect to the fictitious output in system ( 32) is constant, allowing us to re-produce in the analysis a similar change of variables as (38) in the LMI-based design of Theorem 3.

For this method, we make the following assumption.

Assumption 4 Given a subset I ⊂ [0, +∞), the matrix 𝐽 𝑛𝑜 (𝜏) is uniformly invertible for all 𝜏 ∈ I, i.e., there exists 𝑠 𝐽 > 0 such that ∥𝐽 -1 𝑛𝑜 (𝜏) ∥ ≤ 𝑠 𝐽 for all 𝜏 ∈ I.

Remark 7 Contrary to discrete-time systems, the jump map of a hybrid system has little chance of being invertible since it is not a discretization of some continuoustime dynamics. However, here thanks to the transformation [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF], the flow dynamics are somehow merged with the jumps and thus it is reasonable to expect 𝐽 𝑛𝑜 (𝜏) to be invertible for all 𝜏 ∈ I. On the other hand, the ability to deal with the non-invertibility of the dynamics has been studied in [START_REF] Moore | Coping with Singular Transition Matrices in Estimation and Control Stability Theory[END_REF] (in the linear context). With hybrid systems, it can even be coped with using the non-uniqueness of system representation (see for instance [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]Example 2]). Note that while simulations suggest that the invertibility of the jump dynamics is typically not necessary in the Kalman-like design of [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], and may only be for theoretical analysis, it is needed here to implement observer (60) below, namely to compute 𝑇 + correctly. [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] Consider the discrete-time system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N. Following the KKL spirit, we look for a transformation (𝑇 𝑘 ) 𝑘 ∈N such that in the new coordinates 𝜂 𝑘 := 𝑇 𝑘 𝑧 𝑛𝑜,𝑘 , system (32) follows the dynamics

Discrete-time KKL observer design for system

𝜂 𝑘+1 = 𝛾 𝐴𝜂 𝑘 + 𝐵𝑦 𝑘 , (51)
where 𝐴 ∈ R 𝑛 𝜂 ×𝑛 𝜂 is Schur, 𝐵 ∈ R 𝑛 𝜂 ×𝑛 𝑑,ext where 𝑛 𝑑,ext := 𝑛 𝑦,𝑑 + 𝑛 𝑜 such that the discrete-time pair ( 𝐴, 𝐵) is controllable, and 𝛾 ∈ (0, 1] is a design parameter. It then follows that the transformation (𝑇 𝑘 ) 𝑘 ∈N must be such that for every (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N,

𝑇 𝑘+1 𝐽 𝑛𝑜 (𝜏 𝑘 ) = 𝛾 𝐴𝑇 𝑘 + 𝐵𝐻 𝑑,ext (𝜏 𝑘 ), (52)
with 𝐻 𝑑,ext (•) defined in system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] in Theorem 1. The interest of this form is that an observer for system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF] in the 𝜂-coordinates is a simple filter of the output

η𝑘+1 = 𝛾 𝐴 η𝑘 + 𝐵𝑦 𝑘 , (53)
making the error η𝑘 = 𝜂 𝑘 -η𝑘 verify

η𝑘+1 = 𝛾 𝐴 η𝑘 , (54)
and thus is exponentially stable. Then, if (𝑇 𝑘 ) 𝑘 ∈N is uniformly left-invertible, the estimate defined by ẑ𝑛𝑜,𝑘 = 𝑇 * 𝑘 η, where (𝑇 * 𝑘 ) 𝑘 ∈N is a bounded sequence of left inverses of (𝑇 𝑘 ) 𝑘 ∈N verifying 𝑇 * 𝑘 𝑇 𝑘 = Id for all 𝑘 ≥ 𝑘 ★ for some 𝑘 ★ ∈ N, is such that for any (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N, there exist 𝑐 1 > 0 and 𝑐 2 ∈ (0, 1) such that for any initial conditions 𝑧 𝑛𝑜,0 and ẑ𝑛𝑜,0 and for all 𝑘 ∈ N,

|𝑧 𝑛𝑜,𝑘 -ẑ𝑛𝑜,𝑘 | ≤ 𝑐 1 𝑐 𝑘 2 |𝑧 𝑛𝑜,0 -ẑ𝑛𝑜,0 |. (55)
From [27, Corollary 1], we know that this is possible under the uniform backward distinguishability of system (32) as defined next.

Definition 5 (Uniform backward distinguishability of system (32)) Given (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N, system (32) is uniformly backward distinguishable if for each 𝑖 ∈ {1, 2, . . . , 𝑛 𝑑,ext } its output dimension, there exists 𝑚 𝑖 ∈ N >0 such that there exists 𝛼 𝑚 > 0 such that for all 𝑘 ≥ max 𝑖 𝑚 𝑖 , the backward distinguishability matrix sequence (O 𝑏𝑤 𝑘 ) 𝑘 ∈N defined as

O 𝑏𝑤 𝑘 = (O 𝑏𝑤 1,𝑘 , O 𝑏𝑤 2,𝑘 , . . . , O 𝑏𝑤 𝑛 𝑑,ext ,𝑘 ) ∈ R ( 𝑛 𝑑,ext 𝑖=1 𝑚 𝑖 )×𝑛 𝑛𝑜 , (56a)
where

O 𝑏𝑤 𝑖,𝑘 := 𝐻 𝑑,ext,𝑖 (𝜏 𝑘-1 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-1 ) 𝐻 𝑑,ext,𝑖 (𝜏 𝑘-2 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-2 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-1 ) . . . 𝐻 𝑑,ext,𝑖 (𝜏 𝑘-(𝑚 𝑖 -1) )𝐽 -1 𝑛𝑜 (𝜏 𝑘-(𝑚 𝑖 -1) ) . . . 𝐽 -1 𝑛𝑜 (𝜏 𝑘-1 ) 𝐻 𝑑,ext,𝑖 (𝜏 𝑘-𝑚 𝑖 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-𝑚 𝑖 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-(𝑚 𝑖 -1) ) . . . 𝐽 -1 𝑛𝑜 (𝜏 𝑘-1 ) , (56b)
where 𝐻 𝑑,ext,𝑖 (•) denotes the 𝑖 th row of the extended output matrix 𝐻 𝑑,ext (•) of system [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF], has full rank and satisfies

O 𝑏𝑤⊤ 𝑘 O 𝑏𝑤 𝑘 ≥ 𝛼 𝑚 Id > 0.
Note that the forward version (O 𝑓 𝑤 𝑘 ) 𝑘 ∈N of (O 𝑏𝑤 𝑘 ) 𝑘 ∈N , which is much easier to compute, can be considered as in [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF]Remark 3], under the additional assumptions that the 𝑚 𝑖 are the same for 𝑖 ∈ {1, 2, . . . , 𝑛 𝑑,ext } and 𝐽 𝑛𝑜 is uniformly invertible, namely there exists 𝑐 𝐽 > 0 such that (𝐽 -1 𝑛𝑜 (𝜏 𝑘 )) ⊤ 𝐽 -1 𝑛𝑜 (𝜏 𝑘 ) ≥ 𝑐 𝐽 Id for all 𝑘 ∈ N. Theorem 5, which is a particular case of [27, Theorems 2 and 3], states the existence and uniform left-invertibility of (𝑇 𝑘 ) 𝑘 ∈N solving (52).

Theorem 5 Consider 𝑛 𝜂 ∈ N >0 , 𝛾 > 0, and a pair ( 𝐴, 𝐵) ∈ R 𝑛 𝜂 ×𝑛 𝜂 × R 𝑛 𝜂 ×𝑛 𝑑,ext . Under Assumption 4, for any (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N and any 𝑇 0 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 , the transformation (𝑇 𝑘 ) 𝑘 ∈N initialized as 𝑇 0 and satisfying (52) uniquely exists under the following closed form

𝑇 𝑘 = (𝛾 𝐴) 𝑘 𝑇 0 𝑘-1 𝑝=0 𝐽 -1 𝑛𝑜 (𝜏 𝑝 ) + 𝑘-1 ∑ 𝑝=0 (𝛾 𝐴) 𝑘-𝑝-1 𝐵𝐻 𝑑,ext (𝜏 𝑝 ) 𝑘-1 𝑟= 𝑝 𝐽 -1 𝑛𝑜 (𝜏 𝑟 ). (57)
Moreover, for any (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N such that system (32) is uniformly backward distinguishable for some 𝑚 𝑖 ∈ N >0 , 𝑖 ∈ {1, 2, . . . , 𝑛 𝑑,ext } and any controllable pairs ( Ã𝑖 , B𝑖 ) ∈ R 𝑚 𝑖 ×𝑚 𝑖 × R 𝑚 𝑖 with Ã𝑖 Schur, 𝑖 ∈ {1, 2, . . . , 𝑛 𝑑,ext }, there exists 0 < 𝛾 ★ ≤ 1 such that for any 0 < 𝛾 < 𝛾 ★ , there exist 𝑘 ★ ∈ N >0 , 𝑐 𝑇 > 0, and 𝑐 𝑇 > 0 such that (𝑇 𝑘 ) 𝑘 ∈N in (57) with

𝐴 = diag( Ã1 , Ã2 , . . . , Ã𝑛 𝑑,ext ) ∈ R 𝑛 𝜂 ×𝑛 𝜂 , (58a)
𝐵 = diag( B1 , B2 , . . . , B𝑛 𝑑,ext ) ∈R 𝑛𝜂 ×𝑛 𝑑,ext , (58b)
where

𝑛 𝜂 := 𝑛 𝑑,ext 𝑖=1 𝑚 𝑖 , verifies 𝑇 ⊤ 𝑘 𝑇 𝑘 ≥ 𝑐 𝑇 Id for all 𝑘 ≥ 𝑘 ★ and ∥𝑇 𝑘 ∥ ≤ 𝑐 𝑇 for all 𝑘 ∈ N.
In other words, for 𝛾 sufficiently small, (𝑇 𝑘 ) 𝑘 ∈N is uniformly left-invertible and upper-bounded after 𝑘 ★ . Note that the dependence of 𝛾 ★ , 𝑐 𝑇 , 𝑐 𝑇 , and 𝑘 ★ on (𝜏 𝑘 ) 𝑘 ∈N and 𝑇 0 is only through 𝛼 𝑚 and the 𝑚 𝑖 coming from the uniform backward distinguishability and the upper bounds of 𝑇 0 and 𝐽 -1 𝑛𝑜 (•). Note also that the (non-uniform) injectivity of 𝑇 can be obtained from non-uniform distinguishability conditions, as seen in [27, Example 1], which may suffice in some cases to ensure the convergence of the KKL observer.

Proof These results are the particular case of [

∑ 𝑖=1 𝑘-1 ∑ 𝑝=𝑘-𝑚 𝑖 ★ ⊤ 𝐻 𝑑,ext,𝑖 (𝜏 𝑝 )𝐽 -1 𝑛𝑜 (𝜏 𝑝 ) . . . 𝐽 -1 𝑛𝑜 (𝜏 𝑘-2 )𝐽 -1 𝑛𝑜 (𝜏 𝑘-1 ) ≥ 𝑐 𝑜 Id > 0. ( 59
)
as in Section 3.1, those dynamics are picked so that 𝑇 coincides with the (𝑇 𝑘 ) 𝑘 ∈N in Section 4.1 at jumps and so that the corresponding discrete-time KKL error dynamics (54) appear after a certain change of coordinates, modulo some errors on 𝑧 𝑜 (see (66) below). The difficulty comes here from the fact that the discrete-time output 𝑦 𝑘 in the discrete-time KKL dynamics (51) is not fully available at jumps since it contains the fictitious output.

Assumption 5 Given 𝑗 𝑚 defined in Assumption 2, there exist 𝑚 𝑖 ∈ N >0 for each 𝑖 = 1, 2, . . . , 𝑛 𝑑,ext and 𝛼 𝑚 > 0 such that for every complete solution 𝑥 ∈ S H (X 0 , U), the sequence of flow lengths (𝜏 𝑗 ) 𝑗 ∈N, 𝑗 ≥ 𝑗 𝑚 where 𝜏 𝑗 = 𝑡 𝑗+1 -𝑡 𝑗 is such that system (32), scheduled with that (𝜏 𝑗 ) 𝑗 ∈N, 𝑗 ≥ 𝑗 𝑚 , is uniformly backward distinguishable with the parameters 𝑚 𝑖 and 𝛼 𝑚 following Definition 5.

Theorem 6 Suppose Assumptions 1, 2, 4, and 5 hold. Define 𝑛 𝜂 :=

𝑛 𝑑,ext
𝑖=1 𝑚 𝑖 and consider for each 𝑖 ∈ {1, 2, . . . , 𝑛 𝑑,ext } a controllable pair ( Ã𝑖 , B𝑖 ) ∈ R 𝑚 𝑖 ×𝑚 𝑖 × R 𝑚 𝑖 with Ã𝑖 Schur. Define then

𝐴 = diag( Ã1 , Ã2 , . . . , Ã𝑛 𝑑,ext ) ∈ R 𝑛 𝜂 ×𝑛 𝜂 , (61a)
𝐵 𝑑,𝑛𝑜 = diag( B1 , B2 , . . . , B𝑛 𝑦,𝑑 ) ∈ R 𝑛 𝜂 ×𝑛 𝑦,𝑑 , (61b)
𝐵 𝑜𝑛𝑜 = diag( B𝑛 𝑦,𝑑 +1 , B𝑛 𝑦,𝑑 +2 , . . . , B𝑛 𝑑,ext ) ∈ R 𝑛 𝜂 ×𝑛 𝑜 . ( 61c
)
Given any 𝜆 1 > 0, any 𝑃 0 ∈ S 𝑛 𝑜 >0 , and any 𝑇 0 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 , there exists 0 < 𝛾 ★ ≤ 1 such that there exists 𝜆 ★ > 0 such that for any 0 < 𝛾 < 𝛾 ★ and any 𝜆 > 𝜆 ★ , there exist j ∈ N >0 , saturation levels 𝑠 𝑇 > 0, 𝑠 𝐽 > 0, and scalar 𝜌 1 > 0 such that for any solution 𝑥 ∈ S H (X 0 , U) and any solution ( ẑ𝑜 , η, 𝑃, 𝑇, 𝜏) to observer (60) with 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, the chosen ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), sat 𝑠 𝑇 at level 𝑠 𝑇 , sat 𝑠 𝐽 at level 𝑠 𝐽 , and jumps triggered at the same time as in 𝑥, ( ẑ𝑜 , η, 𝑃, 𝑇, 𝜏) is complete and we have |𝑥(𝑡, 𝑗)x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(𝑡 j , j)x(𝑡 j , j)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀(𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ j, (62) with x obtained by x = 𝑒 𝐹 𝜏 D ẑ with D defined in [START_REF] Tanwani | Observability for Switched Linear Systems: Characterization and Observer Design[END_REF].

The parameter j is related to 𝑗 𝑚 in Assumption 4 and to the number of jumps needed to get the uniform left-injectivity of (𝑇 𝑘 ) 𝑘 ∈N in Theorem 5.

Proof First, according to Assumption 2, the flow lengths of solutions in S H (X 0 , U) are in the compact set [0, 𝜏 𝑀 ], so there exists 𝑐 𝑇 ,𝑚 > 0 such that for every solution 𝑥 ∈ S H (X 0 , U) and any solution ( ẑ𝑜 , η, 𝑃, 𝑇, 𝜏) to observer (60) with 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, and any 𝛾 ∈ (0, 1], with jumps triggered at the same time as in 𝑥, we have ∥𝑇 (𝑡, 𝑗) ∥ ≤ 𝑐 𝑇 ,𝑚 for all (𝑡, 𝑗) ∈ dom 𝑥 such that 𝑗 ≤ 𝑗 𝑚 . Then, from Assumptions 2, 4, and 5, and according to Theorem 5 starting from jump 𝑗 𝑚 , there exists 0 < 𝛾 ★ 0 ≤ 1 such that for all 0 < 𝛾 < 𝛾 ★ 0 , there exist 𝑗 ★ ∈ N >0 , 𝑐 𝑇 > 0, and 𝑐 𝑇 > 0 such that for every solution 𝑥 ∈ S H (X 0 , U), the solution (𝑇 𝑗 ) 𝑗 ∈N, 𝑗 ≥ 𝑗 𝑚 to (52) with 𝜏 𝑗 = 𝑡 𝑗+1 -𝑡 𝑗 , initialized at any 𝑇 𝑗 𝑚 verifying ∥𝑇 𝑗 𝑚 ∥ ≤ 𝑐 𝑇 ,𝑚 , is uniformly left-invertible for all 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ and uniformly bounded for all 𝑗 ≥ 𝑗 𝑚 , i.e.,

𝑇 ⊤ 𝑗 𝑇 𝑗 ≥ 𝑐 𝑇 Id, ∀ 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ , ∥𝑇 𝑗 ∥ ≤ 𝑐 𝑇 , ∀ 𝑗 ≥ 𝑗 𝑚 . It follows that (𝑇 † 𝑗 ) ⊤ 𝑇 † 𝑗 ≤ 1 𝑐 𝑇
Id, for all 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ , and there exists a saturation level 𝑠 𝑇 > 0 such that sat 𝑠 𝑇 (𝑇 † 𝑗 ) = 𝑇 † 𝑗 for all 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ . Pick 0 < 𝛾 < 𝛾 ★ 0 and consider a solution 𝑥 ∈ S H (X 0 , U) and a solution ( ẑ𝑜 , η, 𝑃, 𝑇, 𝜏) to observer (60) with 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, the chosen ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), the saturation level 𝑠 𝑇 , and jumps triggered at the same time as in 𝑥. Following [START_REF] Sferlazza | Time-Varying Sampled-Data Observer With Asynchronous Measurements[END_REF] where 𝐽 𝑜 (𝜏) = 𝐽 𝑜 (𝜏) + 𝐽 𝑜𝑛𝑜 𝑇 † 𝐵 𝑜𝑛𝑜 , with 𝑇 + seen as a uniformly bounded input, and with the flow and jump sets

                               z𝑜 = -𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (
R 𝑛 𝑜 × R 𝑛 𝜂 × R 𝑛 𝑜 ×𝑛 𝑜 × [0, 𝜏 𝑀 ], R 𝑛 𝑜 × R 𝑛 𝜂 × R 𝑛 𝑜 ×𝑛 𝑜 × I. ( 67b
)
Since 𝐴 is Schur, let 𝑄 𝜂 ∈ S 𝑛 𝜂 >0 be a solution to the inequality 𝐴 ⊤ 𝑄 𝜂 𝐴 -𝑄 𝜂 < 0. Using Corollary 1, we prove that there exist 𝜆 ★ > 0 and 0 < 𝛾 ★ ≤ 1 such that we have the arbitrarily fast GES of the error ( z𝑜 , η) with respect to the value ( z𝑜 , η) (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ) when 𝜆 > 𝜆 ★ and 0 < 𝛾 < 𝛾 ★ . Then the GES in the 𝑧-coordinates with respect to ( z𝑜 , z𝑛𝑜 ) (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ) is obtained thanks to the uniform left-invertibility of 𝑇. Last, the arbitrarily fast GES is recovered in the 𝑥coordinates after hybrid time (𝑡 j , j) where j = 𝑗 𝑚 + 𝑗 ★ by seeing that z = V𝑒 -𝐹 𝜏 x with 𝜏 ∈ [0, 𝜏 𝑀 ]. □

Remark 9 Note that it is the rate of convergence that can be arbitrarily fast and not the convergence time since we must anyway wait for 𝑗 𝑚 jumps for the flow lengths to be in I and, more importantly, for the max 𝑖 𝑚 𝑖 jumps giving us uniform backward distinguishability (see Definition 5). Furthermore, speeding up the rate may make 𝑇 poorer conditioned, thus increasing the bound 𝜌 1 , which is the well-known peaking phenomenon. This type of result is typical in high-gain KKL designs (see [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF]). Note though that this arbitrarily fast convergence rate is an advantage compared to the LMI-based design in Section 3 where the rate is fixed once the LMI is solved: Corollary 1 does not apply in that case because the parameters 𝑎 and 𝑄 𝜂 in (84) are not independent, i.e., 𝑄 𝜂 is not such that (84) holds for any 𝑎 > 0.

KKL-based observer design in the (𝒛 𝒐 , 𝒛 𝒏𝒐 )-coordinates

The rectangular shape of 𝑇 in observer (60) makes the dimension of 𝜂 larger than that of 𝑧 𝑛𝑜 , preventing us from easily writing the observer in the 𝑧-coordinates, unlike in Section 3. Following the spirit of [START_REF] Bernard | Expressing an Observer in Preferred Coordinates by Transforming an Injective Immersion into a Surjective Diffeomorphism[END_REF], consider a map Γ : R 𝑛 𝜂 ×𝑛 𝑛𝑜 → R 𝑛 𝜂 ×(𝑛 𝜂 -𝑛 𝑛𝑜 ) such that for any 𝑇 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 , Γ(𝑇) is a full-rank matrix such that its columns are orthogonal to those of 𝑇 and Γ ⊤ (𝑇)Γ(𝑇) ≥ Id. Such a map always exists (see Remark 10 for an explicit construction method). Then, define 𝑇 𝑒 : R 𝑛 𝜂 ×𝑛 𝑛𝑜 → R 𝑛 𝜂 ×𝑛 𝜂 such that with x obtained by x = 𝑒 𝐹 𝜏 D ẑ with D defined in [START_REF] Tanwani | Observability for Switched Linear Systems: Characterization and Observer Design[END_REF].

Proof First, because 𝑇 in observer (69) verifies the same dynamics as in observer (60), for the same constant 𝑐 𝑇 ,𝑚 as in the proof of Theorem 6, for every solution 𝑥 ∈ S H (X 0 , U) and any solution ( ẑ, ω, 𝑃, 𝑇, 𝜏) to observer (69) with 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, and any 𝛾 ∈ (0, 1], with jumps triggered at the same time as in 𝑥, we have ∥𝑇 (𝑡, 𝑗) ∥ ≤ 𝑐 𝑇 ,𝑚 for all (𝑡, 𝑗) ∈ dom 𝑥 such that 𝑗 ≤ 𝑗 𝑚 . Moreover, considering the same 𝛾 ★ , 𝑗 ★ , 𝑐 𝑇 as in the proof of Theorem 6, 𝑇 becomes uniformly left-invertible after hybrid time (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ), i.e.,

𝑇 ⊤ (𝑡, 𝑗)𝑇 (𝑡, 𝑗) ≥ 𝑐 𝑇 Id, ∀(𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ .
It follows that 𝑇 𝑒 (𝑇) also becomes uniformly invertible, since

𝑇 ⊤ 𝑒 (𝑇)𝑇 𝑒 (𝑇) = 𝑇 ⊤ 𝑇 0 0 Γ ⊤ (𝑇)Γ(𝑇) ≥ min{𝑐 𝑇 , 1} Id, ∀(𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ .
After hybrid time (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ), we thus have

((𝑇 𝑒 (𝑇 (𝑡, 𝑗))) † ) ⊤ (𝑇 𝑒 (𝑇 (𝑡, 𝑗))) † ≤ max 1 𝑐 𝑇 , 1 Id, (71)
so that there exists a saturation level 𝑠 𝑇 𝑒 > 0 such that sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 (𝑡, 𝑗))) † ) = (𝑇 𝑒 (𝑇 (𝑡, 𝑗))) † = (𝑇 𝑒 (𝑇 (𝑡, 𝑗))) -1 for all (𝑡, 𝑗) ∈ dom 𝑥 such that 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ . Pick 0 < 𝛾 < 𝛾 ★ and consider a solution 𝑥 ∈ S H (X 0 , U) and a solution ( ẑ, ω, 𝑃, 𝑇, 𝜏) to observer (69) with 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, the chosen ( 𝐴, 𝐵

= (𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) -𝐵 𝑜𝑛𝑜 𝐽 𝑜 (𝜏)) ẑ𝑜 + 𝛾 𝐴(𝑇 ẑ𝑛𝑜 + Γ(𝑇) ω) + 𝐵 𝑑,𝑛𝑜 (𝑦 𝑑 -𝐻 𝑑,𝑜 (𝜏) ẑ𝑜 ) = (𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) + 𝛾 𝐴𝐵 𝑜𝑛𝑜 -𝐵 𝑜𝑛𝑜 𝐽 𝑜 (𝜏)) ẑ𝑜 + 𝛾 𝐴 η + 𝐵 𝑑,𝑛𝑜 (𝑦 𝑑 -𝐻 𝑑,𝑜 (𝜏) ẑ𝑜 ). (75)
Therefore, η has the same dynamics as in the proof of Theorem 6 and so by proceeding similarly, we get the results in the (𝑧 𝑜 , 𝑧 𝑛𝑜 , 𝜔)-coordinates thanks to the uniform invertibility of the transformation. □

KKL-based observer design in the 𝒙-coordinates

In a similar manner as in Section 3.2, the KKL-based observer can equivalently be implemented directly in the original 𝑥-coordinates. Based on the proof of Theorem 4 by noting that 𝑇 = 0 during flows, we can derive the dynamics of the observer in the 𝑥-coordinates as

                               ζ = 𝐹 ′ ζ + (𝑢 𝑐 , 0) + P𝐻 ′⊤ 𝑐 𝑅 -1 (𝜏) (𝑦 𝑐 -𝐻 ′ 𝑐 ζ) P = 𝜆P + 𝐹 ′ P + P𝐹 ′⊤ -P𝐻 ′⊤ 𝑐 𝑅 -1 (𝜏)𝐻 ′ 𝑐 P 𝑇 = 0 𝜏 = 1 ζ+ = 𝐽 ′ ζ + (𝑢 𝑑 , 0) + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † ) (𝐵 𝑑,𝑛𝑜 (𝑦 𝑑 -𝐻 𝑑 x) + 𝛾 𝐴Γ(𝑇) ω) P + = ★ ⊤ 𝑃 -1 0 (D ′ 𝑜 + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇)) † )𝐵 𝑜𝑛𝑜 ) ⊤ 𝑇 + = (𝛾 𝐴𝑇 + 𝐵 𝑑,𝑛𝑜 𝐻 𝑑 𝑒 𝐹 𝜏 D 𝑛𝑜 + 𝐵 𝑜𝑛𝑜 V 𝑜 𝐽𝑒 𝐹 𝜏 D 𝑛𝑜 ) sat 𝑠 𝐽 ((V 𝑛𝑜 𝐽𝑒 𝐹 𝜏 D 𝑛𝑜 ) † ) 𝜏 + = 0, (76)
where ζ = ( x, ω), with jumps still triggered at the same time as H , where

𝐹 ′ = 𝐹 0 0 0 , 𝐻 ′ 𝑐 = 𝐻 𝑐 0 , 𝐽 ′ = 𝐽 0 0 0 , D ′ 𝑜 = D 𝑜 0 , D ′ 𝑛𝑜 = D 𝑛𝑜 0 0 Id .
The fact that observability is pumped from discrete time to continuous time via the interaction of P and 𝑇 at the jump is recovered, while here it is interesting to see that instead of being reset to a constant as in Section 3.2, here P + depends on 𝑇, which adapts to the successive flow lengths.

Theorem 8 Suppose Assumptions 1, 2, 4, and 5 hold. Define 𝑛 𝜂 and ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ) as in Theorem 6. Given any 𝜆 1 > 0, any 𝑃 0 ∈ S 𝑛 𝑜 >0 , and any 𝑇 0 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 , there exists 0 < 𝛾 ★ ≤ 1 such that there exists 𝜆 ★ > 0 such that for any 0 < 𝛾 < 𝛾 ★ and any 𝜆 > 𝜆 ★ , there exist j ∈ N >0 , saturation levels 𝑠 𝑇 𝑒 > 0, 𝑠 𝐽 > 0, and scalar 𝜌 1 > 0 such that for any solution 𝑥 ∈ S H (X 0 , U) and any solution ( ζ, P, 𝑇, 𝜏) to observer (76) with P (0, 0) = ★ ⊤ 𝑃 -1 0 (D ′ 𝑜 + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 0 ) † )𝐵 𝑜𝑛𝑜 ) ⊤ , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, the chosen ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), sat 𝑠 𝑇𝑒 at level 𝑠 𝑇 𝑒 , sat 𝑠 𝐽 at level 𝑠 𝐽 , and jumps triggered at the same time as in 𝑥, ( ζ, P, 𝑇, 𝜏) is complete and we have

|𝑥(𝑡, 𝑗) -x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(𝑡 j , j) -x(𝑡 j , j)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀(𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ j . ( 77
) Proof Define D ′ = D ′ 𝑜 D ′
𝑛𝑜 and its inverse V ′ . Consider 𝜆 ★ , 𝛾 ★ , 𝑗 ★ , 𝑐 𝑇 , and 𝑠 𝑇 𝑒 given by Theorem 7. Pick a solution 𝑥 ∈ S H (X 0 , U) and a solution ( ζ, P, 𝑇, 𝜏) to observer (76) with P (0, 0) = ★ ⊤ 𝑃 -1 0 (D ′ 𝑜 + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 0 ) † )𝐵 𝑜𝑛𝑜 ) ⊤ , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, 0 < 𝛾 < 𝛾 ★ , and 𝜆 > 𝜆 ★ , with jumps triggered at the same time as in 𝑥. Consider ( ẑ, ω, 𝑃, 𝑇, 𝜏) solution to observer (69), with ( ẑ(0, 0), ω(0, 0)) = V ′ ζ (0, 0), 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 (0, 0), 𝜏(0, 0) = 0, and the same parameters, with jumps triggered at the same time as in 𝑥. First, notice from their dynamics that 𝑇 = 𝑇 and 𝜏 = 𝜏. Then, applying Theorem 7, we get

|𝑥(𝑡, 𝑗) -x(𝑡, 𝑗)| ≤ 𝜌 1 |𝑥(𝑡 j , j) -x(𝑡 j , j)|𝑒 -𝜆 1 (𝑡+ 𝑗) , ∀ (
= 𝑒 𝐹 ′ 𝜏 D ′ ( ẑ, ω), so that ζ = 𝐹 ′ ζ + (𝑢 𝑐 , 0) + 𝐿 ′ 𝑐 (𝑃, 𝑇, 𝜏) (𝑦 𝑐 -𝐻 ′ 𝑐 ζ), (78a)
during flows and

ζ+ = 𝐽 ′ ζ + (𝑢 𝑑 , 0) + 𝐿 ′ 𝑑,𝑥 (𝑇) (𝑦 𝑑 -𝐻 𝑑 x) + 𝐿 ′ 𝑑, 𝜔 (𝑇) ω, (78b)
at jumps where

𝐿 ′ 𝑐 (𝑃, 𝑇, 𝜏) = 𝑒 𝐹 ′ 𝜏 D ′ 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇)) † )𝐵 𝑜𝑛𝑜 𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏) = 𝑒 𝐹 ′ 𝜏 (D ′ 𝑜 + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇)) † )𝐵 𝑜𝑛𝑜 )𝑃 -1 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏), (79a)
and

𝐿 ′ 𝑑, 𝑥 (𝑇) = D ′ 0 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † )𝐵 𝑑,𝑛𝑜 = D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † )𝐵 𝑑,𝑛𝑜 , (79b)
𝐿 ′ 𝑑, 𝜔 (𝑇) = D ′ 0 𝛾 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † ) 𝐴Γ(𝑇) = 𝛾D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † ) 𝐴Γ(𝑇). (79c)
From ( 23), we actually have 𝐻 ′ 𝑐 𝑒 𝐹 ′ 𝜏 D ′ 𝑛𝑜 = 0, so 𝐻 ′ 𝑐 𝑒 𝐹 ′ 𝜏 D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇)) † )𝐵 𝑜𝑛𝑜 = 0 and we have 𝐿 ′ 𝑐 (𝑃, 𝑇, 𝜏) = P𝐻 ′⊤ 𝑐 𝑅 -1 (𝜏) where

P = ★ ⊤ 𝑃 -1 (D ′ 𝑜 + D ′ 𝑛𝑜 sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇)) † )𝐵 𝑜𝑛𝑜 ) ⊤ 𝑒 𝐹 ′⊤ 𝜏 . (80)
Calculating P, we obtain the same flow/jump dynamics as P in observer (76). Besides, P (0, 0) = P (0, 0), so P = P thanks to the uniqueness of solutions. We deduce that ζ follows the same dynamics as ζ, and since

ζ (0, 0) = D ′ ( ẑ, ω) (0, 0) = ζ (0, 0),
we have ζ = ζ, which implies x = x and concludes the proof. □

Example 5 Consider the same system as in Example 4. First, with 𝐽 𝑛𝑜 (𝜏) = 1 0 0 1 for all 𝜏 ∈ I, Assumption 4 is satisfied. Since 𝐻 𝑑,𝑛𝑜 (𝜏) = 0 for all 𝜏 ∈ I, we discard the jump output and only consider the fictitious one described by the matrix 𝐽 𝑜𝑛𝑜 (𝜏). We then see that with 𝑚 1 = 𝑚 2 = 1, for all sequences of flow lengths (𝜏 𝑗 ) 𝑗 ∈N ,

O 𝑏𝑤 𝑗 = 𝐽 𝑜𝑛𝑜 (𝜏 𝑗 )𝐽 -1 𝑛𝑜 (𝜏 𝑗 ) = 1 0 0 1 satisfies O 𝑏𝑤⊤ 𝑗 O 𝑏𝑤 𝑗 = 1 0 0 1 > 0 for all 𝑗 ≥ max{𝑚 1 , 𝑚 2 } = 1.
Therefore, Assumption 5 is satisfied and we can thus design a KKL-based observer where 𝑇 is of dimension 2 × 2. Let us take 𝐴 = diag(0.1, 0.2), an empty 𝐵 𝑑,𝑛𝑜 , and 𝐵 𝑜𝑛𝑜 = 1 0 0 1 .

Then, a KKL-based observer as in observer (60), (69), or (76), with the mentioned ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), a weighting matrix taken as 𝑅 = Id, a large enough 𝜆, and a small enough 𝛾, can be designed for this system.

Conclusion

This chapter presents and discusses new results on observer design for general hybrid systems with linear maps and known jump times. After defining and discussing the hybrid (pre-)asymptotic detectability and uniform complete observability conditions, we briefly presented again the Kalman-like observer in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF]. We then propose a decomposition of the state of a hybrid system with linear maps and known jump times into a part that is instantaneously observable during flows and a part that is not. A thorough analysis of the asymptotic detectability of the second part is performed, where we show that this part can actually be detectable from an extended output made of the jump output and a fictitious one thanks to the flow-jump combination.

A high-gain Kalman-like observer with resets at jumps is proposed to estimate the first part, while two different jump-based algorithms are proposed for the second one. Several examples are provided to illustrate the methods. A comparison among the mentioned designs, namely the Kalman-like observer [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF] and the two designs depending on observability decomposition, is presented in Table 1 at the end. While the KKL-based design requires stronger conditions and has a larger dimension than the LMI-based one, it can provide an arbitrarily fast convergence rate of the estimate (achieved after a certain time). Compared to the Kalman-like design in [START_REF] Tran | Kalman-like Observer for Hybrid Systems with Linear Maps and Known Jump Times[END_REF], the LMI-based observer has a smaller dimension, whereas the KKL-based one can be bigger or smaller depending on 𝑛 𝑜 versus 𝑛 𝑛𝑜 . Therefore, for a particular application, if 𝑛 𝑜 is large compared to 𝑛 𝑛𝑜 , going through a decomposition is advantageous dimension-wise. Note also that while the Kalmanlike observer can easily deal with time-varying matrices in the system dynamics and output, the decomposition method must additionally assume the (uniform) existence of the transformation into system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF], for example, to solve the LMI [START_REF] Zhang | On Stability of the Kalman Filter for Discrete Time Output Error Systems[END_REF] for the gains 𝐿 𝑑,𝑛𝑜 (•) and 𝐾 𝑛𝑜 . Furthermore, even though the invertibility of 𝐽 𝑛𝑜 (𝜏) for all 𝜏 ∈ I assumed for the KKL-based method may seem lighter than the invertibility of 𝐽 at the jump times assumed in the Kalman-like approach [28, Assumption 2], it may turn out stronger since 𝐽 † 𝑛𝑜 (𝜏) is used in the KKL implementation (see observer (60)), while 𝐽 -1 is used only for analysis in the Kalman-like design, not in the implementation (see observer [START_REF] Kalman | New Results in Linear Filtering and Prediction Theory[END_REF]).

Future work is to study hybrid systems with nonlinear maps (and known jump times) by finding transformations into coordinates of possibly higher dimensions and Lyapunov-based sufficient conditions to couple different observers [START_REF] Tran | Coupling Flow and Jump Observers for Hybrid Systems with Known Jump Times[END_REF], as well as those with unknown jump times.

where I is a compact subset of [𝜏 𝑚 , 𝜏 𝑀 ] for some positive 𝜏 𝑚 and 𝜏 𝑀 . Then for any 𝑃 0 ∈ S 𝑛 𝑜 >0 , there exists 𝜆 ★ > 0 such that for all 𝜆 > 𝜆 ★ , there exist 𝜌 1 > 0 and 𝜆 1 > 0 such that any solution ( z𝑜 η, 𝑃, 𝜏) to system (81) with 𝑃(0, 0) = 𝑃 0 , 𝜏(0, 0) = 0, and 𝑢 ∈ U, is complete and verifies

where 𝑟 𝑚 > 0 is a lower bound of the continuous map 𝑅 on the compact set [0, 𝜏 𝑀 ] (thanks to Item 2 in Lemma 2), 𝛼 > 0 (independent of 𝜆) is obtained by applying [START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF] with 𝛿 = 𝜏 𝑚 4 , and 𝜆 𝑚 := 𝑟 𝑚 𝛼. On the other hand, from (85), for all 𝜏 ∈ [0, 𝜏 𝑚 ], P(𝜏) ≥ 𝑒 -𝜆𝜏 𝑚 𝑃 0 , so 𝑒 We now show that this quantity can be made negative definite by successively picking the degrees of freedom. For the η part, ∃𝜖 ★ > 0 such that 0 < 𝜖 < 𝜖 ★ =⇒ 𝑎 -(1 -𝑒 -𝜖 𝜏 𝑀 ) > 0, ∃𝑘 ★ > 0 such that 𝑘 > 𝑘 ★ =⇒ 𝑘 (𝑎 -(1 -𝑒 -𝜖 𝜏 𝑀 )) -𝑐 4 > 0, and ∃𝜅 ★ such that 𝜅 > 𝜅 ★ =⇒ 𝑘 (𝑎 -(1 -𝑒 -𝜖 𝜏 𝑀 )) -𝑐 4 -𝑘 2 𝑐 5 𝜅 > 0. Then, for the z𝑜 part, ∃𝜆 ★ > 0 such that 𝜆 > 𝜆 ★ =⇒ 𝑐 1 + 𝑘𝑐 2 + 𝜅𝑐 3 -𝑒 𝜆 4 𝜏 𝑚 𝜆 𝑚 < 0. We deduce that for any 𝜆 > 𝜆 ★ , there exist 𝑎 𝑐 > 0 and 𝑎 𝑑 > 0 such that for all ( z𝑜 , η) ∈ R 𝑛 , 𝑉 ≤ -𝑎 𝑐 𝑉, ∀𝜏 ∈ [0, 𝜏 𝑀 ],

+ -𝑉 ≤ -𝑎 𝑑 𝑉, ∀𝑢 ∈ U, ∀𝜏 ∈ I.

From ( 90) and (94), we conclude according to [13, Definition 7.29 and Theorem 7.30] that the set A = {( z𝑜 , η, 𝜏) ∈ R 𝑛 𝑜 × R 𝑛 𝑛𝑜 × [0, 𝜏 𝑀 ] : z𝑜 = 0, η = 0} is GES for system (86). □

Corollary 1

Let us now consider system (81) with 𝑀 𝜂 (𝜏) replaced by 𝛾𝑀 𝜂 (𝜏) for 𝛾 ∈ (0, 𝛾 ★ 0 ]. Under the same assumptions as in Lemma 2, for any 𝜆 𝑐 > 0 and any 𝑃 0 ∈ S 𝑛 𝑜 >0 , there exists 𝛾 ★ > 0 such that there exists 𝜆 ★ > 0 such that for any 0 < 𝛾 < 𝛾 ★ and any 𝜆 > 𝜆 ★ , there exists 𝜌 𝑐 > 0 such that any solution ( z𝑜 , η, 𝑃, 𝜏) to the new system (81), with 𝑃(0, 0) = 𝑃 0 , 𝜏(0, 0) = 0, and 𝑢 ∈ U, is complete and verifies |( z𝑜 , η) (𝑡, 𝑗)| ≤ 𝜌 𝑐 𝑒 -𝜆 𝑐 (𝑡+ 𝑗) |( z𝑜 , η) (0, 0)|, ∀(𝑡, 𝑗) ∈ dom( z𝑜 , η, 𝑃, 𝜏). (95

)
Proof This is a modification of the proof of Lemma 2. Consider the Lyapunov function in (87). First, let us show with an appropriate choice of 𝜖 that for 𝜆 sufficiently large and 𝛾 sufficiently small, we have for some 𝑎 𝑑 > 0, 𝑉 ≤ -2𝜆 𝑐

,

  and the form[START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with matrices 𝐽 𝑜 (𝜏) = 0 0 𝜇 5 (𝜏) 𝜇 6 (𝜏), 𝐽 𝑜𝑛𝑜 (𝜏) 𝐻 𝑐,𝑜 (𝜏) = 𝜇 7 (𝜏) 𝜇 8 (𝜏) , 𝐻 𝑑,𝑜 (𝜏) = 0 0 , and 𝐻 𝑑,𝑛𝑜 (𝜏) = 0 0 ,

  𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 η = 0 𝑃 = -𝜆𝑃 + 𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) 𝜏 = 1 z+ 𝑜 = 𝐽 𝑜 (𝜏) z𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏)𝑇 † η η+ = (𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) + 𝛾 𝐴𝐵 𝑜𝑛𝑜 -𝐵 𝑜𝑛𝑜 𝐽 𝑜 (𝜏) -𝐵 𝑑,𝑛𝑜 𝐻 𝑑,𝑛𝑜 (𝜏)) z𝑜 + 𝛾 𝐴 η 𝑃 + = 𝑃 0 𝜏 + = 0, (67a)

  𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ j, where x = 𝑒 𝐹 𝜏 D ẑ. The proof consists in showing that x = x where ζ = ( x, ω), thus obtaining (77). Denote ζ = ( x, ω). We start by observing that ζ

Lemma 2

 2 Assume that: 1. 𝐻 𝑐,𝑜 is continuous on [0, 𝜏 𝑀 ] and such that the pair (0, 𝐻 𝑐,𝑜 (𝜏)) satisfies (27); 2. 𝑅 is continuous on [0, 𝜏 𝑀 ] and 𝑅(𝜏) > 0 for all 𝜏 ∈ [0, 𝜏 𝑀 ]; 3. 𝑀 𝑜 , 𝑀 𝑜 𝜂 , and 𝑀 𝜂𝑜 are bounded on U × I; 4. 𝑀 𝜂 is continuous on I and there exists 𝑄 𝜂 ∈ S 𝑛 𝜂 >0 such that 𝑀 ⊤ 𝜂 (𝜏)𝑄 𝜂 𝑀 𝜂 (𝜏) -𝑄 𝜂 < 0, ∀𝜏 ∈ I. (82)

|

  ( z𝑜 , η) (𝑡, 𝑗)| ≤ 𝜌 1 𝑒 -𝜆 1 (𝑡+ 𝑗) |( z𝑜 , η) (0, 0)|, ∀(𝑡, 𝑗) ∈ dom( z𝑜 , η, 𝑃, 𝜏). (83) Proof First, due to the compactness of I and Item 4 in Lemma 2, there exists 𝑎 > 0 such that for all 𝜏 ∈ I, (82) is strengthened into𝑀 ⊤ 𝜂 (𝜏)𝑄 𝜂 𝑀 𝜂 (𝜏) -𝑄 𝜂 ≤ -𝑎𝑄 𝜂 , ∀𝜏 ∈ I.(84)Since 𝑃(0, 0) = 𝑃 0 , 𝑃 + = 𝑃 0 , and 𝜏(0, 0) = 0, the component (𝑡, 𝑗) ↦ → 𝑃(𝑡, 𝑗) of the solution to system (81) can actually be written as a closed form of the component (𝑡, 𝑗) ↦ → 𝜏(𝑡, 𝑗) by definingP(𝜏) = 𝑒 -𝜆𝜏 𝑃 0 + ∫ 𝜏 0 𝑒 -𝜆( 𝜏-𝑠) 𝐻 ⊤ 𝑐,𝑜 (𝑠)𝑅 -1 (𝑠)𝐻 𝑐,𝑜 (𝑠)𝑑𝑠,(85)namely 𝑃(𝑡, 𝑗) = P(𝜏(𝑡, 𝑗)) for all (𝑡, 𝑗) ∈ dom 𝑥. Note that since 𝑃 0 > 0, P(𝜏) is invertible for all 𝜏 ∈ [0, 𝜏 𝑀 ]. It follows that the solution is complete and ( z𝑜 , η, 𝜏) = -P -1 (𝜏)𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 η = 0 𝜏 = 1z+ 𝑜 = 𝑀 𝑜 (𝑢, 𝜏) z𝑜 + 𝑀 𝑜𝜂 (𝑢, 𝜏) η η+ = 𝑀 𝜂𝑜 (𝑢, 𝜏) z𝑜 + 𝑀 𝜂 (𝜏) η 𝜏 + = 0, (86) with the flow set R 𝑛 𝑜 × R 𝑛 𝜂 × [0, 𝜏 𝑀 ] and the jump set R 𝑛 𝑜 × R 𝑛 𝜂 × I. Consider the Lyapunov function 𝑉 ( z𝑜 , η, 𝜏) = 𝑒 𝜆 2 𝜏 z⊤ 𝑜 P(𝜏) z𝑜 + 𝑘𝑒 -𝜖 𝜏 η⊤ 𝑄 𝜂 η,(87)where 𝑘 > 0 and 𝜖 > 0. We have for all 𝜏 ∈ [𝜏 𝑚 , 𝜏 𝑀 ] ⊇ I, ( 𝜏-𝑠) 𝐻 ⊤ 𝑐,𝑜 (𝑠)𝑅 -1 (𝑠)𝐻 𝑐,𝑜 (𝑠)𝑑𝑠 ≥ 𝑟 𝑚 𝑒 𝑒 𝐹 ⊤ 𝑠 𝐻 ⊤ 𝑐 𝐻 𝑐 𝑒 𝐹𝑠 D 𝑜 𝑑𝑠 ≥ 𝑟 𝑚 𝑒 𝜆 𝑒 𝐹 ⊤ 𝑠 𝐻 ⊤ 𝑐 𝐻 𝑐 𝑒 𝐹𝑠 D 𝑜 𝑑𝑠 ≥ 𝑒 𝜆 4 𝜏 𝑚 𝑟 𝑚 𝛼 Id := 𝑒 𝜆 4 𝜏 𝑚 𝜆 𝑚 Id,

𝜆 2 𝜏𝜆 2 𝑒 𝜆 2 2 ,

 2222 P(𝜏) ≥ 𝑒 -𝜆𝜏 𝑚 𝑃 0 . Besides, as P is continuous on the compact set [0, 𝜏 𝑀 ], there exists 𝑝 𝑀 > 0 such that P(𝜏) ≤ 𝑝 𝑀 Id for all 𝜏 ∈ [0, 𝜏 𝑀 ]. It then follows that there exist 𝜌 > 0 and 𝜌 > 0 defined as𝜌 = min eig 𝑒 -𝜆𝜏 𝑚 𝑃 0 , 𝑒 𝜆 4 𝜏 𝑚 𝜆 𝑚 , eig 𝑘𝑒 -𝜖 𝜏 𝑀 𝑄 𝜂 ,(89a)𝜌 = max 𝑒 𝜆 2 𝜏 𝑀 𝑝 𝑀 , eig 𝑘𝑒 -𝜖 𝜏 𝑚 𝑄 𝜂 ,(89b)such that𝜌|( z𝑜 , η)| 2 ≤ 𝑉 ( z𝑜 , η, 𝜏) ≤ 𝜌|( z𝑜 , η)| 2 , ∀( z𝑜 , η) ∈ R 𝑛 , ∀𝜏 ∈ [0, 𝜏 𝑀 ].(90)During flows, for all ( z𝑜 , η) ∈ R 𝑛 and 𝜏 ∈ [0, 𝜏 𝑀 ], ) -2𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) + P(𝜏) z𝑜 -𝜖 𝑘𝑒 -𝜖 𝜏 η⊤ 𝑄 𝜂 η= 𝑒 𝜆 2 𝜏 z⊤ 𝑜 -𝜆 2 P(𝜏) -𝐻 ⊤ 𝑐,𝑜 (𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) z𝑜 -𝜖 𝑘𝑒 -𝜖 𝜏 η⊤ 𝑄 𝜂 η ≤ -𝜏 z⊤ 𝑜 P(𝜏) z𝑜 -𝜖 𝑘𝑒 -𝜖 𝜏 η⊤ 𝑄 𝜂 η ≤min 𝜆 𝜖 𝑉 .(91)At jumps, for all ( z𝑜 , η) ∈ R 𝑛 , 𝑢 ∈ U, and 𝜏 ∈ I,𝑉 + -𝑉 = ★ ⊤ 𝑃 0 (𝑀 𝑜 (𝑢, 𝜏) z𝑜 + 𝑀 𝑜𝜂 (𝑢, 𝜏) η) -𝑒 𝜆 2 𝜏 z⊤ 𝑜 P(𝜏) z𝑜 + 𝑘 ★ ⊤ 𝑄 𝜂 (𝑀 𝜂𝑜 (𝑢, 𝜏) z𝑜 + 𝑀 𝜂 (𝜏) η) -𝑘𝑒 -𝜖 𝜏 η⊤ 𝑄 𝜂 η. (92)From Young's inequality, (88), (84), and Items 3 and 4 in Lemma 2, there exist non-negative constants 𝑐 𝑖 , 𝑖 = 1, 2, . . . , 5 independent of (𝜆, 𝑘, 𝜖) such that for any 𝜅 > 0, for all ( z𝑜 , η) ∈ R 𝑛 , 𝑢 ∈ U, and 𝜏 ∈ I, 𝑉 + -𝑉 ≤ 𝑐 1 + 𝑘𝑐 2 + 𝜅𝑐 3 -𝑒 𝜆 4 𝜏 𝑚 𝜆 𝑚 z⊤ 𝑜 z𝑜 -𝑘 (𝑎 -(1 -𝑒 -𝜖 𝜏 𝑀 )) -𝑐 4 -𝑘 2 𝑐 5 𝜅 η⊤ 𝑄 𝜂 η. (93)

Lemma 1

 1 System[START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] with known jump times and zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) is preasymptotically detectable if and only if any of its complete solutions (𝑧, 𝜏) with zero inputs (𝑢 𝑐 , 𝑢 𝑑 ) and flow and jump outputs satisfying

  with input (𝜏 𝑗 ) 𝑗 ∈N in I and applying 3., we obtain lim 𝑗→+∞ 𝑗 ∈N 𝑧 𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0. Since 𝑧 𝑛𝑜 is constant during flows, we have lim 𝑡+ 𝑗→+∞

	= 0, implying 2. according to Lemma
	(𝑡 , 𝑗) ∈dom 𝑧
	1.

  with the same 𝜏dependent matrices (𝐺 𝑜 , 𝐺 𝑛𝑜 , 𝐽 𝑜 , 𝐽 𝑜𝑛𝑜 , 𝐽 𝑛𝑜𝑜 , 𝐽 𝑛𝑜 , 𝐻 𝑐,𝑜 , 𝐻 𝑑,𝑜 , 𝐻 𝑑,𝑛𝑜 ) and the same inputs (𝑢 𝑐 , 𝑢 𝑑 ). Since 𝑧 𝑎,𝑜 and 𝑧 𝑏,𝑜 are instantaneously observable during flows according to[START_REF] Tran | Arbitrarily Fast Robust KKL Observer for Nonlinear Timevarying Discrete Systems[END_REF], 𝑦 𝑎,𝑐 = 𝑦 𝑏,𝑐 implies that 𝑧 𝑎,𝑜 = 𝑧 𝑏,𝑜 during each flow interval. Next, as 0 ∉ I, there is no more than one jump at each jump time so that 𝑧 𝑎,𝑜 (𝑡, 𝑗) = 𝑧 𝑏,𝑜 (𝑡, 𝑗) for all (𝑡, 𝑗) ∈ dom 𝑥 𝑎 . Besides, since 𝑢 𝑑 is the same for both solutions, from (26a), we have 𝐽 𝑜𝑛𝑜 (𝜏(𝑡 𝑗 , 𝑗 -1))𝑧 𝑎,𝑛𝑜 (𝑡 𝑗 , 𝑗 -1) = 𝐽 𝑜𝑛𝑜 (𝜏(𝑡 𝑗 , 𝑗 -1))𝑧 𝑏,𝑛𝑜 (𝑡 𝑗 , 𝑗 -1) for all 𝑗 ≥ 1. Therefore, since 𝑧 𝑎,𝑛𝑜 and 𝑧 𝑏,𝑛𝑜 evolve in the same way during flows (with the same 𝑢 𝑐 ) and 𝑦 𝑎,𝑑 (𝑡 𝑗 , 𝑗 -1) = 𝑦 𝑏,𝑑 (𝑡 𝑗 , 𝑗 -1), for all 𝑗 ≥ 1, by defining z𝑛𝑜 = 𝑧 𝑎,𝑛𝑜 -𝑧 𝑏,𝑛𝑜 , we have that z𝑛𝑜 is constant during flows and for all 𝑗 ∈ N,

z𝑛𝑜 (𝑡 𝑗+1 , 𝑗 + 1) = 𝐽 𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗)) z𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐽 𝑛𝑜 (𝜏 𝑗 ) z𝑛𝑜 (𝑡 𝑗 , 𝑗), 𝐻 𝑑,𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗)) z𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐻 𝑑,𝑛𝑜 (𝜏 𝑗 ) z𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0, 𝐽 𝑜𝑛𝑜 (𝜏(𝑡 𝑗+1 , 𝑗)) z𝑛𝑜 (𝑡 𝑗+1 , 𝑗) = 𝐽 𝑜𝑛𝑜 (𝜏 𝑗 ) z𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0.

By considering the sequence ( z𝑛𝑜 (𝑡 𝑗 , 𝑗)) 𝑗 ∈N solution to system

[START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]

with input (𝜏 𝑗 ) 𝑗 ∈N in I and using the asymptotic detectability of system

[START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]

for the particular sequence (𝜏 𝑘 ) 𝑘 ∈N generated by 𝑥 𝑎 and 𝑥 𝑏 , we obtain lim 𝑗→+∞ 𝑗 ∈N z𝑛𝑜 (𝑡 𝑗 , 𝑗) = 0. Since z𝑛𝑜 is constant during flows, we have lim 𝑡+ 𝑗→+∞ (𝑡 , 𝑗) ∈dom 𝑥 𝑎 z𝑛𝑜 (𝑡, 𝑗) = 0, which means that lim 𝑡+ 𝑗→+∞ (𝑡 , 𝑗) ∈dom 𝑥 𝑎

  𝜏)𝑅 -1 (𝜏)𝐻 𝑐,𝑜 (𝜏) 𝜏 = 1 z+ 𝑜 = 𝐽 𝑜 (𝜏) z𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏) z𝑛𝑜 z+

	𝑃 + = 𝑃 0
	𝜏 + = 0,
	(37a)
	with the flow and jump sets

𝑛𝑜 = (𝐽 𝑛𝑜𝑜 (𝜏) -𝐿 𝑑,𝑛𝑜 (𝜏)𝐻 𝑑,𝑜 (𝜏)) z𝑜 + (𝐽 𝑛𝑜 (𝜏) -𝐿 𝑑,𝑛𝑜 (𝜏)𝐻 𝑑,𝑛𝑜 (𝜏)) z𝑛𝑜

  , we see that 𝑄 𝑛𝑜 ∈ S 𝑛 𝑛𝑜 >0 exists if and only if there exists a solution to the LMI

	𝐻 𝑑,𝑛𝑜 (𝜏)	⊥⊤	𝐻 𝑑,𝑛𝑜 (𝜏)	⊥
	𝐽 𝑜𝑛𝑜 (𝜏) 𝑄 𝑛𝑜 𝐽 𝑛𝑜 (𝜏) 𝑄 𝑛𝑜 𝐽 𝑜𝑛𝑜 (𝜏) 𝐽 𝑜𝑛𝑜 (𝜏) 𝐻 𝑑,𝑛𝑜 (𝜏) ⊥	★ 𝑄 𝑛𝑜	> 0,	∀𝜏 ∈ I.

  27, Theorems 2 and 3]. Note that by continuity, 𝜏 ↦ → 𝐽 𝑛𝑜 (𝜏) is uniformly invertible on I because I is compact, and 𝜏 ↦ → 𝐻 𝑑,𝑛𝑜 (𝜏) is uniformly bounded on the compact set [0, 𝜏 𝑀 ]. □ Remark 8 Interestingly, Definition 5, whose nonlinear version is defined in [27, Definition 1], coincides with the uniform complete observability (UCO) condition required by the discrete-time Kalman(-like) filter (see [11, Condition (13)], [34, Assumption 3], or [33, Definition 3]), on the pair 𝐽 𝑛𝑜 (𝜏), 𝐻 𝑑,𝑛𝑜 (𝜏) 𝐽 𝑜𝑛𝑜 (𝜏) , which is the discrete-time version of that in Definition 4 above, i.e., there exist 𝑚 𝑖 ∈ N >0 and 𝑐 𝑜 > 0 such that for all (𝜏 𝑘 ) 𝑘 ∈N with 𝜏 𝑘 ∈ I for all 𝑘 ∈ N, we have for all 𝑘 ≥ max 𝑖 𝑚 𝑖 ,

	𝑛 𝑑,ext

  , define 𝑧(𝑡, 𝑗) := V𝑒 -𝐹 𝜏 (𝑡 , 𝑗) 𝑥(𝑡, 𝑗), ∀(𝑡, 𝑗) ∈ dom 𝑥, and consider the error z = ( z𝑜 , z𝑛𝑜 ) = (𝑧 𝑜 -ẑ𝑜 , 𝑧 𝑛𝑜 -ẑ𝑛𝑜 ). As justified above, ∥𝑇 (𝑡 𝑗 𝑚 , 𝑗 𝑚 ) ∥ ≤ 𝑐 𝑇 ,𝑚 . Since 𝑇 = 0 during flows, the sequence (𝑇 (𝑡 𝑗 , 𝑗)) 𝑗 ∈N, 𝑗 ≥ 𝑗 𝑚 coincides with the sequence (𝑇 𝑗 ) 𝑗 ∈N, 𝑗 ≥ 𝑗 𝑚 solution to (52) with 𝜏 𝑗 = 𝑡 𝑗+1 -𝑡 𝑗 for all 𝑗 ∈ N with 𝑗 ≥ 𝑗 𝑚 . Therefore, 𝑇 ⊤ (𝑡, 𝑗)𝑇 (𝑡, 𝑗) ≥ 𝑐 𝑇 Id, ∀(𝑡, 𝑗) ∈ dom 𝑥, 𝑗 ≥ 𝑗 𝑚 + 𝑗 ★ . (63) We now use Corollary 1 to show exponential convergence of z starting at hybrid time (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ) by putting the error dynamics into the appropriate form. In order to exploit the KKL design, we define 𝜂(𝑡, 𝑗) = 𝑇 (𝑡, 𝑗)𝑧 𝑛𝑜 (𝑡, 𝑗) -𝐵 𝑜𝑛𝑜 𝑧 𝑜 (𝑡, 𝑗), ∀(𝑡, 𝑗) ∈ dom 𝑥. (64) Notice that 𝜂 verifies 𝜂 = (𝑇𝐺 𝑛𝑜 (𝜏) -𝐵 𝑜𝑛𝑜 𝐺 𝑜 (𝜏))𝑢 𝑐 during flows. From Assumptions 2 and 4, after hybrid time (𝑡 𝑗 𝑚 , 𝑗 𝑚 ) we get sat 𝑠 𝐽 (𝐽 † 𝑛𝑜 (𝜏)) = 𝐽 † 𝑛𝑜 (𝜏) and 𝐽 † 𝑛𝑜 (𝜏)𝐽 𝑛𝑜 (𝜏) = Id so that at jumps, 𝜂 + = (𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) + 𝛾 𝐴𝐵 𝑜𝑛𝑜 -𝐵 𝑜𝑛𝑜 𝐽 𝑜 (𝜏))𝑧 𝑜 + 𝛾 𝐴𝜂 + (𝑇 + V 𝑛𝑜 -𝐵 𝑜𝑛𝑜 V 𝑜 )𝑢 𝑑 + 𝐵 𝑑,𝑛𝑜 𝐻 𝑑,𝑛𝑜 (𝜏)𝑧 𝑛𝑜 . (65) Given the dynamics of η in observer (60), the error η := 𝜂η verifies η = 0 during flows and at jumps (after hybrid time (𝑡 𝑗 𝑚 , 𝑗 𝑚 )), η+ = (𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) + 𝛾 𝐴𝐵 𝑜𝑛𝑜 -𝐵 𝑜𝑛𝑜 𝐽 𝑜 (𝜏) -𝐵 𝑑,𝑛𝑜 𝐻 𝑑,𝑛𝑜 (𝜏)) z𝑜 + 𝛾 𝐴 η, (66) which is a contracting dynamics in η. After (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ), we have (i) 𝑇 † 𝑇 = Id so that 𝑧 𝑛𝑜 = 𝑇 † 𝜂 +𝑇 † 𝐵 𝑜𝑛𝑜 𝑧 𝑜 , and (ii) sat 𝑠 𝑇 (𝑇 † ) = 𝑇 † so that ẑ𝑛𝑜 = 𝑇 † η +𝑇 † 𝐵 𝑜𝑛𝑜 ẑ𝑜 . Therefore, after (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ), ( z𝑜 , η, 𝑃, 𝜏) is solution to

  We see that after hybrid time (𝑡 𝑗 𝑚 + 𝑗 ★ , 𝑗 𝑚 + 𝑗 ★ ), η = (𝑇𝐺 𝑛𝑜 (𝜏) -𝐵 𝑜𝑛𝑜 𝐺 𝑜 (𝜏))𝑢 𝑐 during flows and at jumps,η+ = 𝑇 𝑒 (𝑇 + ) ẑ+ 𝑇 + Γ + (𝐽 𝑛𝑜𝑜 (𝜏) ẑ𝑜 + 𝐽 𝑛𝑜 (𝜏) ẑ𝑛𝑜 , 0) + sat 𝑠 𝑇𝑒 ((𝑇 𝑒 (𝑇 + )) † ) (𝛾 𝐴Γ(𝑇) ω + 𝐵 𝑑,𝑛𝑜 (𝑦 𝑑 -𝐻 𝑑,𝑜 (𝜏) ẑ𝑜 -𝐻 𝑑,𝑛𝑜 (𝜏) ẑ𝑛𝑜 )) -𝐵 𝑜𝑛𝑜 (𝐽 𝑜 (𝜏) ẑ𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏) ẑ𝑛𝑜 )= 𝑇 + 𝐽 𝑛𝑜𝑜 (𝜏) ẑ𝑜 + (𝛾 𝐴𝑇 + 𝐵 𝑑,𝑛𝑜 𝐻 𝑑,𝑛𝑜 (𝜏) + 𝐵 𝑜𝑛𝑜 𝐽 𝑜𝑛𝑜 (𝜏)) ẑ𝑛𝑜 + 𝛾 𝐴Γ(𝑇) ω + 𝐵 𝑑,𝑛𝑜 (𝑦 𝑑 -𝐻 𝑑,𝑜 (𝜏) ẑ𝑜 -𝐻 𝑑,𝑛𝑜 (𝜏) ẑ𝑛𝑜 ) -𝐵 𝑜𝑛𝑜 (𝐽 𝑜 (𝜏) ẑ𝑜 + 𝐽 𝑜𝑛𝑜 (𝜏) ẑ𝑛𝑜 )

	(74)

𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), the saturation level 𝑠 𝑇 𝑒 , and jumps triggered at the same time as in 𝑥. Define for all (𝑡, 𝑗) ∈ dom 𝑥 the change of variables 𝜂(𝑡, 𝑗) = 𝑇 𝑒 (𝑇 (𝑡, 𝑗)) 𝑧 𝑛𝑜 (𝑡, 𝑗) 0 -𝐵 𝑜𝑛𝑜 𝑧 𝑜 (𝑡, 𝑗). (72) We see that for all (𝑡, 𝑗) ∈ dom 𝑥, 𝜂(𝑡, 𝑗) = 𝑇 (𝑡, 𝑗)𝑧 𝑛𝑜 (𝑡, 𝑗) -𝐵 𝑜𝑛𝑜 𝑧 𝑜 (𝑡, 𝑗), (73) and hence it verifies the dynamics of 𝜂 in the proof of Theorem 6. Let us study the dynamics of the image of ( ẑ𝑜 , ẑ𝑛𝑜 , ω) in observer (69) defined as η(𝑡, 𝑗) = 𝑇 𝑒 (𝑇 (𝑡, 𝑗)) ẑ𝑛𝑜 (𝑡, 𝑗) ω(𝑡, 𝑗) -𝐵 𝑜𝑛𝑜 ẑ𝑜 (𝑡, 𝑗). 𝑛𝑜 ω+ -𝐵 𝑜𝑛𝑜 ẑ+ 𝑜 =

  Following the same analysis as in the proof of Lemma 2, we obtain that during flows, 𝑉 ≤min 𝜆 2 , 𝜖 𝑉 for all 𝜏 ∈ [0, 𝜏 𝑀 ], and at jumps thanks to (82), for all 𝑢 ∈ U and 𝜏 ∈ I,𝑉 + -𝑉 ≤ 𝑐 1 + 𝑘𝑐 2 + 𝜅𝑐 3 -𝑒 𝜆 4 𝜏 𝑚 𝜆 𝑚 z⊤ 𝑜 z𝑜 -𝑘 𝑒 -𝜖 𝜏 𝑀 -𝛾 2 -𝑐 4 -𝑘 2 𝑐 5 𝜅η⊤ 𝑄 𝜂 η. (97)

	1	+ 1 𝑉,	𝑉 + < 𝑉 .	(96)
	𝜏 𝑚

Note that 𝐽 is not invertible at jumps, but it can be made invertible by considering an alternative jump map 𝒗 + = 𝒗 + 𝑐𝑦 𝑑,𝒗 -𝑦 𝑑,𝒗 and seeing -𝑦 𝑑,𝒗 as part of 𝑢 𝑑 . Simulations have shown that this invertibility may not be necessary for the Kalman-like observer.

Note that 0 ∉ I and the compactness of I are not needed to prove this direction.

Acknowledgements

We thank Florent Di Meglio and Ricardo Sanfelice for their helpful feedback.

It is thus interesting to compare these two discrete-time observers. In terms of dimensions, the complexity of the Kalman filter is 𝑛 𝑛𝑜 (𝑛 𝑛𝑜 +1) 2 + 𝑛 𝑛𝑜 , while that of the KKL observer is 𝑛 𝑑,ext 𝑖=1 𝑚 𝑖 𝑛 𝑛𝑜 + 𝑛 𝑑,ext 𝑖=1 𝑚 𝑖 . Therefore, the Kalman filter is advantageous in dimension compared to the KKL observer. However, the advantage of the latter (besides being applicable in the nonlinear context) is that there exists a strict ISS Lyapunov function of quadratic form that allows us to prove exponential ISS, unlike the discrete-time Kalman filter [START_REF] Jr | Conditions for Asymptotic Stability of the Discrete Minimum-variance Linear Estimator[END_REF] whose Lyapunov function is not strict. This advantage is then exploited in the next part, where we design a KKLbased observer to estimate the 𝑧 𝑛𝑜 part in the hybrid system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] while the error in 𝑧 𝑜 is seen as a disturbance. Note that a discrete-time Kalman-like observer [START_REF] Ţiclea | Exponential Forgetting Factor Observer in Discrete Time[END_REF] could seem like a possible alternative to the KKL-based one since it also exhibits a strict Lyapunov function. However, the gain multiplied with the fictitious output in the observer must be constant during flows for us to perform the analysis (see 𝐾 𝑛𝑜 in ( 33)), which is not the case in a Kalman-like observer (unless the pair (𝐽 𝑛𝑜 (𝜏), 𝐻 𝑑,𝑛𝑜 (𝜏)) at the jump times is UCO, so without the need for the fictitious output). This is ensured in KKL design since it relies on a transformation into a linear time-invariant form (see below in the proof of Theorem 6).

Next, in Section 4.2, we exploit this section's results for the hybrid system (26). [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] The KKL-based observer we propose for system [START_REF] Tanwani | Error Covariance Bounds for Suboptimal Filters with Lipschitzian Drift and Poisson-sampled Measurements[END_REF] has the form

KKL-based observer design for system

with jumps triggered at the same time as H in the same way as observer [START_REF] Allan | Nonlinear Detectability and Incremental Input/Outputto-State Stability[END_REF],

>0 is a positive definite weighting matrix that is defined and is continuous on [0, 𝜏 𝑀 ] to be chosen only for design purpose, 𝛾 ∈ (0, 1], and ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ) are design parameters to be chosen. Following similar reasoning

which is a square matrix extension of 𝑇, that is invertible whenever 𝑇 is full-rank.

Remark 10 One possible way to construct Γ(𝑇) is to exploit a singular value decomposition of 𝑇. Indeed, given 𝑇 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 with 𝑛 𝜂 ≥ 𝑛 𝑛𝑜 , consider orthonormal matrices 𝑈 (𝑇) ∈ R 𝑛 𝜂 ×𝑛 𝜂 and 𝑉 (𝑇) ∈ R 𝑛 𝑛𝑜 ×𝑛 𝑛𝑜 , as well as a matrix

Let us split 𝑈 (𝑇) as

. By taking Γ(𝑇) = 𝑈 2 (𝑇), we see that Γ(𝑇) is orthogonal to 𝑇 and verifies Γ ⊤ (𝑇)Γ(𝑇) = Id.

An alternative KKL-based observer to observer (60) can then be implemented in the (𝑧 𝑜 , 𝑧 𝑛𝑜 )-coordinates, as

with jumps still triggered at the same time as H and initialized as in Theorem 6 with any ω(0, 0) ∈ R 𝑛 𝜂 -𝑛 𝑛𝑜 . Still in the spirit of [START_REF] Bernard | Expressing an Observer in Preferred Coordinates by Transforming an Injective Immersion into a Surjective Diffeomorphism[END_REF], ω ∈ R 𝑛 𝜂 -𝑛 𝑛𝑜 is an estimate of some fictitious extra state 𝜔 ∈ R 𝑛 𝜂 -𝑛 𝑛𝑜 defined on dom 𝑥 such that 𝜔(𝑡, 𝑗) = 0 for all (𝑡, 𝑗) ∈ dom 𝑥, serving to equalize the dimension in the 𝑧 𝑛𝑜 and 𝜂 coordinates. Note that along the solutions to observer (69), since 𝑇 is constant during flows, 𝑇 𝑒 (𝑇) is also constant during flows and needs to be re-computed only at jumps. Theorem 7 Suppose Assumptions 1, 2, 4, and 5 hold. Define 𝑛 𝜂 and ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ) as in Theorem 6. Given any 𝜆 1 > 0, any 𝑃 0 ∈ S 𝑛 𝑜 >0 , and any 𝑇 0 ∈ R 𝑛 𝜂 ×𝑛 𝑛𝑜 , there exists 0 < 𝛾 ★ ≤ 1 such that there exists 𝜆 ★ > 0 such that for any 0 < 𝛾 < 𝛾 ★ and any 𝜆 > 𝜆 ★ , there exist j ∈ N >0 , saturation levels 𝑠 𝑇 𝑒 > 0, 𝑠 𝐽 > 0, and scalar 𝜌 1 > 0 such that for any solution 𝑥 ∈ S H (X 0 , U) and any solution ( ẑ, ω, 𝑃, 𝑇, 𝜏) to observer (69) with ω(0, 0) ∈ R 𝑛 𝜂 -𝑛 𝑛𝑜 , 𝑃(0, 0) = 𝑃 0 , 𝑇 (0, 0) = 𝑇 0 , 𝜏(0, 0) = 0, the chosen ( 𝐴, 𝐵 𝑑,𝑛𝑜 , 𝐵 𝑜𝑛𝑜 ), sat 𝑠 𝑇𝑒 at level 𝑠 𝑇 𝑒 , sat 𝑠 𝐽 at level 𝑠 𝐽 , and jumps triggered at the same time as in 𝑥, ( ẑ, ω, 𝑃, 𝑇, 𝜏) is complete and we have

6 Appendix: Technical lemmas

Exponential stability of the error dynamics

Consider a hybrid system of the form

where 𝑢 ∈ U is the input, with the flow and jump sets

Let us pick 𝜖 = 2𝜆 𝑐 1 𝜏 𝑚 + 1 and define 𝜆 ★ 0 := 4𝜆 𝑐 1 𝜏 𝑚 + 1 . Then, the first item in (96) holds as soon as 𝜆 ≥ 𝜆 ★ 0 . Now define 𝛾 ★ := min 𝛾 ★ 0 , √ 𝑒 -𝜖 𝜏 𝑀 . For any 0 < 𝛾 < 𝛾 ★ , we have 𝑒 -𝜖 𝜏 𝑀 -𝛾 2 > 𝑒 -𝜖 𝜏 𝑀 -(𝛾 ★ ) 2 > 0, then 𝑘, 𝜅, and 𝜆 are successively picked (based on 𝛾 ★ ) as in the proof of Lemma 2. The final 𝜆 ★ is the larger one between this and 𝜆 ★ 0 . For any 𝜆 > 𝜆 ★ and 0 𝛾 < 𝛾 ★ , we obtain (96). Second, we deduce (95) from (96) and the dwell time condition. From (96), we get

𝑉 (0, 0) for all (𝑡, 𝑗) ∈ dom( z𝑜 , η, 𝑃, 𝜏). Since the flow lengths of system (81) are at least 𝜏 𝑚 > 0 for all 𝑗 ≥ 1, we have 𝑗 ≤ 𝑡 𝜏 𝑚 + 1 so that 𝑡 ≥ 𝑡+ 𝑗-1 1 𝜏𝑚 +1 , for all (𝑡, 𝑗) ∈ dom( z𝑜 , η, 𝑃, 𝜏). Therefore, 𝑉 (𝑡, 𝑗) ≤ 𝑒 2𝜆 𝑐 𝑒 -2𝜆 𝑐 (𝑡+ 𝑗) 𝑉 (0, 0), for all (𝑡, 𝑗) ∈ dom( z𝑜 , η, 𝑃, 𝜏), implying (95). □

Boundedness in finite time

Lemma 3 Consider a hybrid system with state 𝜂 ∈ R 𝑛 𝜂 and input 𝑢 ∈ U ⊂ R 𝑛 𝑢 :

Yes

Yes

Table 1

Comparison of observer designs for hybrid systems with linear maps and known jump times. All the conditions here are sufficient conditions.