Anatolii Khalin

Denis Efimov

Rosane Ushirobira

On robust observer design for a class of time-varying continuous-and discrete-time Persidskii systems

This paper considers the state estimation problem for a class of non-autonomous nonlinear systems. We propose conditions on the existence and stability of a nonlinear observer based on the invariant manifold approach in both continuousand discrete-time scenarios. The requirements are formulated using Linear Matrix Equalities (LME) and Inequalities (LMI). We present two possible applications of the result, a reducedorder observer (e.g., an observer for unmeasured states) and regression in linear and nonlinear, continuous-and discretetime settings. With nonlinear regression being a sophisticated case, the parameter estimation problem for a particular output equation (when the fusion of linear and nonlinear sensors is weighted) is investigated. Two nonlinear examples demonstrating the efficiency of results are provided.

I. INTRODUCTION

Designing observers is one of the central problems in modern control theory and dynamical systems analysis. Once presented for linear setting [START_REF] Luenberger | Observing the state of a linear system[END_REF], the problem of observer design was vastly applied and studied for nonlinear cases (e.g., [START_REF] Khalil | Nonlinear systems[END_REF] [7], [START_REF] Besanc ¸on | Nonlinear observers and applications[END_REF], [START_REF] Thau | Observing the state of non-linear dynamic systems[END_REF]) and became a popular research subject in the field. Reducing the order of the observer is a relevant sub-problem, the main idea being to separate the dynamics of measured states from unmeasured ones and disregard the estimate of known variables. This approach greatly simplifies the systems' analysis, modeling, and real-life applications. First stated in [START_REF] Luenberger | Observing the state of a linear system[END_REF], such an idea presented for the linear case had many continuations in nonlinear analysis, some of them using linearization techniques [START_REF] Sundarapandian | Reduced order observer design for nonlinear systems[END_REF], others based on solutions of partial differential equations (PDEs). Reducedorder observers were also studied for discrete-time systems both in linear [START_REF] Leondes | Reduced-order observers for linear discretetime systems[END_REF] and nonlinear settings [START_REF] Boutayeb | A reduced-order observer for nonlinear discrete-time systems[END_REF], [START_REF] Sundarapandian | Reduced order observer design for discrete-time nonlinear systems[END_REF].

The idea presented in this paper follows the so-called invariant manifold approach proposed in [START_REF] Karagiannis | Nonlinear observer design using invariant manifolds and applications[END_REF], [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear systems[END_REF], where it was specifically applied to reduced-order observers for a general class of nonlinear time-varying systems. It provides conditions for the existence of a solution, described by a nonlinear invertible function (chosen as a solution of the corresponding partial differential equation (PDE)), embedded into an invariant manifold containing both the observer and system dynamics. Recently, such a technique was used primarily in adaptive control and estimation and named the immersion and invariance approach [START_REF] Astolfi | Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems[END_REF]. The application of the invariant manifold method is non-trivial. It might lead to some difficulties related to the requirement of the 1 A. Khalin, R. Ushirobira and D. Efimov are with Inria, Univ. Lille, CNRS, UMR 9189 -CRIStAL, F-59000 Lille, France {Anatolii.Khalin, Denis.Efimov, Rosane.Ushirobira}@inria.fr existence of an invertible solution of a PDE, complicating its implementation (the choice of a solution is case-dependent).

In this note, we study a rather general class of nonlinear systems called Persidskii [START_REF] Persidskij | Problem of absolute stability[END_REF], including Lur'e systems, used in many areas, such as mechanical and electrical engineering, biology, genetics, and networks. Examples of such systems can be found, for instance, in neural networks [START_REF] Forgione | Continuous-time system identification with neural networks: Model structures and fitting criteria[END_REF], [START_REF] Kambhampati | Approximation of non-autonomous dynamic systems by continuous time recurrent neural networks[END_REF], electrical circuits [START_REF] Chua | Chua's circuit: an overview ten years later[END_REF], mechanical robotic systems [START_REF] Spong | Modeling and control of elastic joint robots[END_REF], and bioreactors [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. The chosen system class is of particular interest regarding the invariant manifold approach because a set of LMEs can replace the conventional PDE of the invariant manifold method, and a set of LMIs can provide conditions for the system's stability. As we show later in the paper, this procedure for the considered class of systems is more constructive in practice and easy to implement. This paper also proves that our result can be applied to the reduced-order state observation problem and fits a conventional linear regression solution, a nonlinear one in some instances, displaying a broader field of implementation and problem-solving for a class of Persidskii systems. A preliminary version of this work can be found in [START_REF] Khalin | On observer design for a class of time-varying persidskii systems based on the invariant manifold approach[END_REF]. Although this paper is based on this preliminary version, there are numerous advances in the present work as we consider a wider class of systems, with disturbances, and perform a deeper analysis of nonlinear regression.

The paper is structured as follows. The notation and preliminaries are presented in Section II, and the problem statement is in Section III. The main result on conditions of the observer existence and convergence is provided in Section IV for two cases: continuous-and discrete-time. Some applications, including reduced-order observers and parameter estimation in regression analysis (both in linear and nonlinear settings and in discrete time), are given in sections V and VI. Two nonlinear examples in Section VII show the efficiency of the presented estimators. A conclusion is given in Section VIII.

II. PRELIMINARIES A. Notation

• The sets of real and natural numbers are denoted by R and N, respectively. Set R + := {𝑡 ∈ R | 𝑡 ≥ 0}. We denote by R 𝑛 the set of real vectors of dimension 𝑛 and by R 𝑛×𝑚 the set of real matrices of dimension 𝑛 × 𝑚.

→ R 𝑛 such that ∥𝑢∥ ∞ := ∥𝑢∥ [0,+∞) < +∞.
• For a sequence 𝑢 = (𝑢 𝑘 ) 𝑘 ∈N ⊂ R 𝑛 , define the norm |𝑢| ∞ = sup 𝑘 ∈N ∥𝑢 𝑘 ∥, and denote by 𝐿 𝑛 ∞ the set of such sequences with finite norm |𝑢| ∞ < +∞.

• Let 𝜆 min ( 𝐴) and 𝜆 max ( 𝐴) denote the minimal and the maximal eigenvalues of a symmetric matrix 𝐴, respectively. Denote by 𝐼 𝑛 the 𝑛 × 𝑛 identity matrix. We refer to [START_REF] Khalil | Nonlinear systems[END_REF] for definitions of input-to-state stability.

B. Input excitation

Definition 1. A function 𝜙 : R + → R 𝑛 is persistently exciting (or 𝜙 is PE), if there exist 𝑇, 𝜇 > 0 such that for all 𝑡 ∈ R + ,

∫ 𝑡+𝑇 𝑡 𝜙(𝑠)𝜙 ⊤ (𝑠)𝑑𝑠 > 𝜇𝐼 𝑛 , where 𝜙 ⊤ denotes the function R + → R 1×𝑛 , 𝑡 ↦ → 𝜙(𝑡) ⊤ .

C. Invariant manifold approach

Consider a time-varying nonlinear system in general form:

𝑦(𝑡) = 𝑓 1 (𝑦(𝑡), 𝑥(𝑡), 𝑡), ∀𝑡 ∈ R + 𝑥(𝑡) = 𝑓 2 (𝑦(𝑡), 𝑥(𝑡), 𝑡), (1)
where 𝑦(𝑡) ∈ R 𝑚 is the measured part of the state and 𝑥(𝑡) ∈ R 𝑛 is the unmeasured part and the functions 𝑓 1 and 𝑓 2 are assumed to guarantee the forward completeness of (1). Definition 2. The dynamical system

𝜉 (𝑡) = 𝛼(𝑦(𝑡), 𝜉 (𝑡), 𝑡), ∀𝑡 ∈ R + (2)
where 𝜉 (𝑡) ∈ R 𝑝 (𝑝 ≥ 𝑛), is called an observer for the system (1) if there exist mappings

𝛽 : R 𝑚 × R 𝑝 × R → R 𝑝 and 𝜙 𝑦,𝑡 : R 𝑛 → R 𝑝
with 𝜙 𝑦,𝑡 left-invertible and such that the manifold

M = {(𝑦(𝑡), 𝑥(𝑡), 𝜉 (𝑡), 𝑡) ∈ R 𝑚 × R 𝑛 × R 𝑝 × R | 𝛽(𝑦(𝑡), 𝜉 (𝑡), 𝑡) = 𝜙 𝑦,𝑡 (𝑥(𝑡)) (3)
has the following properties:

• all trajectories of the extended system (1), (2) starting on the manifold M remain there for all future times, i.e., M is forward invariant; • all trajectories of the extended system (1), [START_REF] Besanc ¸on | Nonlinear observers and applications[END_REF] starting in a neighborhood of M asymptotically converge to M.

To find the mappings 𝛽, 𝛼, and 𝜙 𝑦,𝑡 , consider the estimation error: 𝑧(𝑡) = 𝛽(𝑦(𝑡), 𝜉 (𝑡), 𝑡) -𝜙 𝑦,𝑡 (𝑥(𝑡)), the dynamics of which are given by

𝑧 = 𝜕 𝛽 𝜕𝑦 𝑓 1 + 𝜕 𝛽 𝜕𝜉 𝛼 + 𝜕 𝛽 𝜕𝑡 - 𝜕𝜙 𝑦,𝑡 𝜕𝑦 𝑓 1 - 𝜕𝜙 𝑦,𝑡 𝜕𝑥 𝑓 2 -
𝜕𝜙 𝑦,𝑡 𝜕𝑡 and must admit the origin to be (locally) asymptotically stable. Existence conditions for these mappings formulated in terms of PDE, and a more detailed description of the approach can be found in [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear systems[END_REF].

III. PROBLEM STATEMENT

To broaden our result's applicability, we consider in this paper a discrete-time setting and a continuous one.

A. Continuous-time

Consider a time-varying nonlinear system in the Persidskii form: 𝑥(𝑡) = 𝐴 0 (𝑡)𝑥(𝑡) + 𝐴 1 (𝑡) 𝑓 (𝐻 (𝑡)𝑥(𝑡)) + 𝑄(𝑡)𝑢(𝑡) + 𝑑 (𝑡), 𝑦(𝑡) = 𝐷 0 (𝑡)𝑥(𝑡) + 𝐷 1 (𝑡) 𝑓 (𝐻 (𝑡)𝑥(𝑡)) + 𝑣(𝑡), [START_REF] Boutayeb | A reduced-order observer for nonlinear discrete-time systems[END_REF] where 𝑥 : R + → R 𝑛 is the state function and 𝑦 : R + → R 𝑝 is the output function; 𝑢 : R + → R 𝑚 is an essentially bounded external input function;

𝑓 : R 𝑟 → R ℓ is a nonlinear continuous function, 𝐴 0 (𝑡) ∈ R 𝑛×𝑛 , 𝐴 1 (𝑡) ∈ R 𝑛×ℓ , 𝐻 (𝑡) ∈ R 𝑟 ×𝑛 , 𝑄(𝑡) ∈ R 𝑛×𝑚 , 𝐷 0 (𝑡) ∈ R 𝑝×𝑛 , 𝐷 1 (𝑡) ∈ R 𝑝×ℓ are known time-varying matrices; 𝑑 ∈ L 𝑛
∞ and 𝑣 ∈ L 𝑞 ∞ are unknown essentially bounded disturbances. We assume that 𝑓 allows the forward existence and uniqueness of a solution of the system (4).

Let us build an observer for system (4) of the form:

𝜔(𝑡) = 𝑆 0 (𝑡)𝜔(𝑡) + 𝑆 1 (𝑡) 𝑓 (𝐽 (𝑡)𝜔(𝑡)) + 𝐵(𝑡)𝑦(𝑡) + 𝑂 (𝑡)𝑢(𝑡), (5)
where 𝜔 : R + → R 𝑞 is the state function and 𝑆 0 (𝑡) ∈ R 𝑞×𝑞 , 𝑆 1 (𝑡) ∈ R 𝑞×ℓ , 𝐽 (𝑡) ∈ R 𝑟 ×𝑞 , 𝐵(𝑡) ∈ R 𝑞× 𝑝 , 𝑂 (𝑡) ∈ R 𝑞× 𝑝 are time-varying matrices to be chosen in a way allowing the estimation (of a part) of the state of (5) (see Definition 2).

B. Discrete-time

Now consider a time-varying nonlinear discrete-time system in the same form:

𝑥 𝑘+1 = 𝐴 0,𝑘 𝑥 𝑘 + 𝐴 1,𝑘 𝑓 (𝐻 𝑘 𝑥 𝑘 ) + 𝑄 𝑘 𝑢 𝑘 + 𝑑 𝑘 , 𝑦 𝑘 = 𝐷 0,𝑘 𝑥 𝑘 + 𝐷 1,𝑘 𝑓 (𝐻 𝑘 𝑥 𝑘 ) + 𝑣 𝑘 , (6)
where 𝑥 𝑘 ∈ R 𝑛 , 𝑢 𝑘 ∈ R 𝑚 , 𝑦 𝑘 ∈ R 𝑝 are the state, bounded external input and the output vector at time instant 𝑘 ∈ N, respectively; 𝑓 :

R 𝑟 → R ℓ is a nonlinear function, 𝐴 0,𝑘 ∈ R 𝑛×𝑛 , 𝐴 1,𝑘 ∈ R 𝑛×ℓ , 𝐻 𝑘 ∈ R 𝑟 ×𝑛 , 𝑄 𝑘 ∈ R 𝑛×𝑚 , 𝐷 0,𝑘 ∈ R 𝑝×𝑛 , 𝐷 1,𝑘 ∈ R 𝑝×ℓ are known time-varying matrices at time instant 𝑘; 𝑑 ∈ 𝐿 𝑛
∞ and 𝑣 ∈ 𝐿 𝑞 ∞ are unknown bounded disturbances. Let us build an observer for system [START_REF] Chua | Chua's circuit: an overview ten years later[END_REF]:

𝜔 𝑘+1 = 𝑆 0,𝑘 𝜔 𝑘 + 𝑆 1,𝑘 𝑓 (𝐽 𝑘 𝜔 𝑘 ) + 𝐵 𝑘 𝑦 𝑘 + 𝑂 𝑘 𝑢 𝑘 , (7)
where 𝜔 𝑘 ∈ R 𝑞 is the state-vector and

𝑆 0,𝑘 ∈ R 𝑞×𝑞 , 𝑆 1,𝑘 ∈ R 𝑞×ℓ , 𝐽 𝑘 ∈ R 𝑟 ×𝑞 , 𝐵 𝑘 ∈ R 𝑞× 𝑝 , 𝑂 𝑘 ∈ R 𝑞× 𝑝
are time-varying matrices to be chosen. This paper aims to establish conditions for using ( 5) and ( 7) as (reduced-order) observers for (4) and [START_REF] Chua | Chua's circuit: an overview ten years later[END_REF]. Conditions for the existence of a static relationship between solutions of ( 4), [START_REF] Chua | Chua's circuit: an overview ten years later[END_REF], and ( 5), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF] have to be established (an analog of (3)). Also, convergence conditions must be set. Next, the goal is to apply the proposed observer to different estimation problems: reduced-order state observer design and parameter estimation in linear and nonlinear settings.

IV. MAIN RESULT

To this end, the invariant manifold method [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear systems[END_REF] will be adapted to the presented problem statement to obtain a simple linear interconnection between the solutions of ( 4), [START_REF] Chair | Optimal data fusion in multiple sensor detection systems[END_REF], and ( 6), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF]. However, instead of using the direct Lyapunov method for stability analysis of the error 𝑧, which would be analogous to the result in [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear systems[END_REF], we separately prove two properties of the manifold from Definition 2.

First, the existence conditions of a static relationship between solutions of the systems (4), [START_REF] Chair | Optimal data fusion in multiple sensor detection systems[END_REF], and systems ( 6), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF] should be established, meaning that the manifold connecting 𝑥, 𝜔, and 𝑦 is invariant. Next, to prove the attractiveness, we use the concept of globally convergent or incrementally input-to-state stable systems [22], i.e., that if the observer state 𝜔 is initiated with different initial conditions (and with disturbances), then it asymptotically converges to the same solution (in its neighborhood). Thus, the found invariant solution is a steady-state one. This two-step approach allows us to obtain a constructive design for observer matrices.

A. Steady-state estimation 1) Continuous-time:

Proposition 1. Assume that there exist time-varying matrices Π(𝑡) ∈ R 𝑞×𝑛 , Υ(𝑡) ∈ R 𝑞× 𝑝 and also matrices 𝑆 0 (𝑡), 𝑆 1 (𝑡), 𝐵(𝑡), 𝑂 (𝑡), that satisfy the following linear equalities ∀𝑡 ∈ R +1 :

Υ𝐷 1 = 0, 𝐽 (Π + Υ𝐷 0 ) = 𝐻, (8)
and

𝑆 0 (Π + Υ𝐷 0 ) + 𝐵 -Υ 𝐷 0 -(Π + Υ𝐷 0 ) 𝐴 0 -Π -Υ 𝐷 0 = 0, 𝐵𝐷 1 -(Π + Υ𝐷 0 ) 𝐴 1 + 𝑆 1 = 0, (9)
(Π + Υ𝐷 0 ) 𝑄 = 𝑂.
Then, for 𝑑 = 0 and 𝑣 = 0:

𝜔(𝑡) = Π(𝑡)𝑥(𝑡) + Υ(𝑡)𝑦(𝑡), 𝑡 ∈ R + ( 10
)
is a solution of the system (4), ( 5), for any 𝑥(0) ∈ R 𝑞 and 𝜔(0) = Π(0)𝑥(0) + Υ(0)𝑦(0).

Proof. Taking the derivative of (10) and using equations ( 4), ( 5), since Υ𝐷 1 = 0, we have:

𝑆 0 (Π + Υ𝐷 0 )𝑥 + 𝑆 1 𝑓 (𝐽 (Π𝑥 + Υ𝑦)) + 𝐵 (𝐷 0 𝑥 + 𝐷 1 𝑓 (𝐻𝑥)) + 𝑂𝑢 = (Π + Υ𝐷 0 ) ( 𝐴 0 𝑥 + 𝐴 1 𝑓 (𝐻𝑥) + 𝑄𝑢) + Π𝑥 + Υ𝑦 + Υ 𝐷 0 𝑥 + 𝐷 1 𝑓 (𝐻𝑥) .
Also Υ𝐷 1 = -Υ 𝐷 1 , so substituting (8) leads to the relation

𝑆 0 (Π + Υ𝐷 0 ) + (𝐵 -Υ)𝐷 0 -(Π + Υ𝐷 0 ) 𝐴 0 -Π -Υ 𝐷 0 𝑥 = (-𝐵𝐷 1 + (Π + Υ𝐷 0 ) 𝐴 1 -𝑆 1 ) 𝑓 (𝐻𝑥) + ((Π + Υ𝐷 0 )𝑄 -𝑂) 𝑢,
which is satisfied thanks to (9) in a disturbance-free setting. □ Remark 1. It is worth mentioning that the conditions of Proposition 1 restrict the class of systems to which the results can be applied. Note that there are additional free variables in ( 8) and ( 9), i.e., matrices for the observer (5) to be set according to the design problem (see sections V-VII).

We do not claim that a valid choice of matrices is always possible since it depends on the system's observability (on the problem's dimensions and the properties of matrices describing the dynamics of ( 4)). Nevertheless, other matrices in ( 8) and ( 9) can also be tuned if required (Υ and Π); hence the restrictions imposed in Propositions 1 are not significant.

2) Discrete-time:

Proposition 2. Assume that for each 𝑘 ∈ N there exist Π 𝑘 ∈ R 𝑞×𝑛 , Υ 𝑘 ∈ R 𝑞× 𝑝 and also matrices 𝑆 0,𝑘 , 𝑆 1,𝑘 , 𝐵 𝑘 , 𝑂 𝑘 satisfying for each 𝑘 ∈ N:

Υ 𝑘 𝐷 1,𝑘 = 0, 𝐽 𝑘 Π 𝑘 + Υ 𝑘 𝐷 0,𝑘 = 𝐻 𝑘 , (11)
and

𝑆 0,𝑘 Π 𝑘 + Υ 𝑘 𝐷 0,𝑘 + 𝐵 𝑘 𝐷 0,𝑘 -Π 𝑘+1 + Υ 𝑘+1 𝐷 0,𝑘+1 𝐴 0,𝑘 = 0, 𝐵 𝑘 𝐷 1,𝑘 -Π 𝑘+1 + Υ 𝑘+1 𝐷 0,𝑘+1 𝐴 1,𝑘 + 𝑆 1,𝑘 = 0, (12)
Π 𝑘+1 + Υ 𝑘+1 𝐷 0,𝑘+1 𝑄 𝑘 = 𝑂 𝑘 .
Then, for 𝑑 = 0 and 𝑣 = 0:

𝜔 𝑘 = Π 𝑘 𝑥 𝑘 + Υ 𝑘 𝑦 𝑘 , 𝑘 ∈ N (13)
is a solution of the system (6), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF], for any 𝑥 0 ∈ R 𝑞 and

𝜔 0 = Π 0 𝑥 0 + Υ 0 𝑦 0 .
The proof of Proposition 2 is analogous to the continuous case, where instead of taking the derivative of (10) we take the next time instant for [START_REF] Kambhampati | Approximation of non-autonomous dynamic systems by continuous time recurrent neural networks[END_REF]. Note that Remark 1 is applicable to Proposition 2 for (11) and ( 12) accordingly.

Observers ( 5), ( 7) of ( 4), ( 6) have the same shape of nonlinearity, so under suitable interconnections among the matrices in Propositions 1 and 2, the obtained relation between solutions is linear, reducing the complexity of analysis significantly and opening space for many applications.

B. Convergence to the invariant solution

In Propositions 1 and 2, only the existence of relations [START_REF] Sandy | Confusion: Sensor fusion for complex robotic systems using nonlinear optimization[END_REF], ( 13) is proven. It should be established further whether this relation is attracting for (4), ( 6) and ( 5), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF] or not.

1) Continuous-time: Assume that the conditions of the Proposition 1 are verified: Assumption 1. There exist Π and Υ such that Υ𝐷 1 = 0 and the equalities (8) and (9) are satisfied for (4), [START_REF] Chair | Optimal data fusion in multiple sensor detection systems[END_REF].

Next, consider the following dynamical system with a copy dynamics of (5):

ω(𝑡) = 𝑆 0 (𝑡) ω(𝑡) + 𝑆 1 (𝑡) 𝑓 (𝐽 (𝑡) ω(𝑡)) + 𝐵(𝑡)𝑦(𝑡) + 𝑂 (𝑡)𝑢(𝑡) + 𝛿(𝑡),
where 𝛿 ∈ L 𝑞 ∞ represents an exogenous perturbation (the influence of 𝑑 and 𝑣). Let us introduce the error 𝑒 := 𝜔ω between two solutions of (5), initiated for different initial conditions, with the same inputs (𝑦 and 𝑢). Then we have the following dynamics:

𝑒(𝑡) = 𝑆 0 (𝑡)𝑒(𝑡) + 𝑆 1 (𝑡) ( 𝑓 (𝐽 (𝑡)𝜔(𝑡)) -𝑓 (𝐽 (𝑡) ω(𝑡))) -𝛿(𝑡). (14)
Now we can state a theorem about the convergence and robust stability of the observer.

Theorem 1. Let Assumption 1 be satisfied. Assume there exist 𝐹 (𝑡) ∈ R 𝑞×𝑘 and 𝑊 (𝑡) = 𝑊 (𝑡) ⊤ ∈ R 𝑞×𝑞 such that

𝑒 ⊤ 𝐹 ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω)) ≤ 𝑒 ⊤ 𝑊 𝑒,
for all 𝜔, ω ∈ R 𝑞 , and that there exist 𝑃(𝑡) = 𝑃(𝑡) ⊤ ∈ R 𝑞×𝑞 , Ξ(𝑡) = Ξ(𝑡) ⊤ ∈ R 𝑞×𝑞 such that the inequalities

𝛼 1 𝐼 𝑛 ≤ 𝑃, 𝛼 1 𝐼 𝑞 ≤ Ξ 𝑃 ≤ 𝛼 2 𝐼 𝑛 , 𝑃𝑆 1 = 𝐹, 𝑃 + 𝑆 ⊤ 0 𝑃 + 𝑃𝑆 0 + 2𝑊 + Ξ + 𝛾𝑃 < 0 have a solution for some 0 < 𝛼 1 < 𝛼 2 < +∞ and 𝛾 > 0.
Then, the system (14) is input-to-state stable (ISS) and ( 5) is globally convergent.

Proof. Assumption 1 ensures the existence of an estimate [START_REF] Sandy | Confusion: Sensor fusion for complex robotic systems using nonlinear optimization[END_REF]. So, select a Lyapunov function 𝑉 (𝑒) = 𝑒 ⊤ 𝑃𝑒, with 𝑃 given by the theorem conditions (this Lyapunov function is positive definite and radially unbounded in 𝑒 due to the introduced restriction). Its derivative for [START_REF] Karagiannis | Nonlinear observer design using invariant manifolds and applications[END_REF] takes the form:

𝑉 = 𝑒 ⊤ (𝑆 ⊤ 0 𝑃 + 𝑃𝑆 0 + 𝑃)𝑒 + 2𝑒 ⊤ 𝑃𝑆 1 ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω)) -2𝑒 ⊤ 𝑃𝛿 ≤ 𝑒 ⊤ 𝑆 ⊤ 0 𝑃 + 𝑃𝑆 0 + 𝑃 + 2𝑊 + Ξ + 𝛾𝑃 𝑒 -𝑒 ⊤ Ξ𝑒 + 𝛾 -1 𝛿 ⊤ 𝑃𝛿
, where 𝐹 = 𝑃𝑆 1 and 𝑊, 𝛾 are given in the formulation of the theorem. According to the imposed conditions, the above expression is non-positive and:

∥𝑒(𝑡) ∥ ≤ √ 𝜆 max (𝑃) 𝜆 min (𝑃)
𝑒 -0.5 𝜆 min (Ξ) 𝜆max ( 𝑃) 𝑡 ∥𝑒(0) ∥ + 𝛾 -0.5 ∥𝛿∥ , ∀𝑡 ≥ 0.

□

The conditions presented in Theorem 1 are given for an illustration, and any other conditions for convergence to zero and robust stability of 𝑒 in [START_REF] Karagiannis | Nonlinear observer design using invariant manifolds and applications[END_REF] as in [START_REF] Mei | On convergence conditions for generalized persidskii systems[END_REF] can be used.

2) Discrete-time: Assume that the conditions of the Proposition 2 are verified: Assumption 2. There exist Π 𝑘 and Υ 𝑘 such that Υ 𝑘 𝐷 1,𝑘 = 0 and the equalities (11) and (12) are satisfied for (6), [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF].

Next, consider the following dynamical system with a copy dynamics of ( 7):

ω𝑘+1 = 𝑆 0,𝑘 ω𝑘 + 𝑆 1,𝑘 𝑓 (𝐽 𝑘 ω𝑘 ) + 𝐵 𝑘 𝑦 𝑘 + 𝑂 𝑘 𝑢 𝑘 + 𝛿 𝑘 ,
where 𝛿 ∈ 𝐿 𝑞 ∞ is an auxiliary bounded disturbance as before. Let us introduce the error 𝑒 𝑘 := 𝜔 𝑘 -ω𝑘 between two solutions of [START_REF] Fridman | Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems[END_REF], initiated for different initial conditions, with the same inputs (𝑦 𝑘 and 𝑢 𝑘 ). Then we have the following system:

𝑒 𝑘+1 = 𝑆 0,𝑘 𝑒 𝑘 + 𝑆 1,𝑘 ( 𝑓 (𝐽 𝑘 𝜔 𝑘 ) -𝑓 (𝐽 𝑘 ω𝑘 )) -𝛿 𝑘 . (15)
Now we can state a theorem about the convergence and robust stability of the observer.

Theorem 2. Let Assumption 2 be satisfied. Assume there exists 𝛾 > 0 such that

| 𝑓 (𝐽 𝑘 𝜔 𝑘 ) -𝑓 (𝐽 𝑘 ω𝑘 )| ≤ 𝛾|𝑒 𝑘 |
for all 𝜔 𝑘 , ω𝑘 ∈ R 𝑞 , and that there exist

𝑃 𝑘 = 𝑃 ⊤ 𝑘 ∈ R 𝑞×𝑞 , Ξ = Ξ ⊤ ∈ R 𝑞×𝑞 and Γ = Γ ⊤ ∈ R 𝑞 such that the inequalities Σ 𝑘 𝑆 ⊤ 0,𝑘 𝑃 𝑘+1 𝑆 1,𝑘 -𝑆 ⊤ 0,𝑘 𝑃 𝑘+1 𝑆 ⊤ 1,𝑘 𝑃 𝑘+1 𝑆 0,𝑘 𝑆 ⊤ 1,𝑘 𝑃 𝑘+1 𝑆 1,𝑘 -𝛼𝐼 -𝑆 ⊤ 1,𝑘 𝑃 𝑘+1 -𝑃 𝑘+1 𝑆 0,𝑘 -𝑃 𝑘+1 𝑆 1,𝑘 𝑃 𝑘+1 -Γ ≤ 0, 𝛼 1 𝐼 𝑛 ≤ 𝑃 𝑘 ≤ 𝛼 2 𝐼 𝑛 , 𝛼 1 𝐼 𝑞 ≤ Ξ
where Σ 𝑘 = 𝑆 ⊤ 0,𝑘 𝑃 𝑘+1 𝑆 0,𝑘 -𝑃 𝑘 + Ξ + 𝛼𝛾 2 𝐼, have a solution for some 0 < 𝛼 1 < 𝛼 2 < +∞ and 𝛼 > 0. Then the system (15) is ISS and (7) is globally convergent.

Proof. Assumption 2 ensures the existence of an estimate [START_REF] Kambhampati | Approximation of non-autonomous dynamic systems by continuous time recurrent neural networks[END_REF], so let us select a Lyapunov function 𝑉 𝑘 = 𝑒 ⊤ 𝑘 𝑃 𝑘 𝑒 𝑘 , where 𝑃 𝑘 is given in the conditions of the theorem. Consider the difference 𝑉 𝑘+1 -𝑉 𝑘 and denote Δ 𝑓 𝑘 := 𝑓 (𝐽 𝑘 𝜔 𝑘 ) -𝑓 (𝐽 𝑘 ω𝑘 ) then using the bound defined in the theorem, we have

𝑉 𝑘+1 -𝑉 𝑘 ≤ 𝑒 𝑘 Δ 𝑓 𝑘 𝛿 𝑘 ⊤ Φ 𝑘 𝑒 𝑘 Δ 𝑓 𝑘 𝛿 𝑘 -𝑒 ⊤ 𝑘 Ξ𝑒 𝑘 + 𝛿 ⊤ 𝑘 Γ𝛿 𝑘 .
where Φ 𝑘 denotes the matrix in the theorem's statement.

According to the imposed conditions, Φ 𝑘 ≤ 0 and the error 𝑒 𝑘 is ISS for the input 𝛿 𝑘 . □

For brevity, the formulations of the results in this section deal with global stability properties based on the global characteristics of the nonlinearities. It is possible to get local stability results by restricting the domain of validity of the constraints for 𝑓 , as it will be illustrated in Section VII.

V. REDUCED-ORDER OBSERVER DESIGN

Let us demonstrate how the generic results presented in the previous section can be used to estimate the system's unobserved states.

A. Linear reduced-order observer

The first application of the result presented in the previous section is a reduced-order observer for the linear case, where 𝑆 1 = 0, 𝐷 1 = 0, and all other matrices are known and constant in (4) (the time-invariance has been imposed to simplify the presentation and comparison). Then we have an ordinary LTI system of the form:

𝑥(𝑡) = 𝐴 0 𝑥(𝑡) + 𝑄𝑢(𝑡) + 𝑑 (𝑡), 𝑡 ∈ R + , 𝑦(𝑡) = 𝐷 0 𝑥(𝑡) + 𝑣(𝑡). (16)
As in the classical problem of reduced-order observer design, we can partition our state 𝑥(𝑡) ∈ R 𝑛 with variables 𝑦(𝑡) ∈ R 𝑝 representing a set of directly measured state variables and 𝑤(𝑡) ∈ R 𝑛-𝑝 as a set of unmeasured states:

𝑦(𝑡) 𝑤(𝑡) = 𝐷 0 Π 𝑥(𝑡) + 𝑣(𝑡) 0 . (17)
In practice, 𝑣 is often considered a bounded measurement noise. The task is to build an observer, effectively reducing the dynamic order of (5) from 𝑛 to 𝑛 -𝑝. Therefore, consider the following LTI system (the respective presentation of ( 5)):

𝜔(𝑡) = 𝑆 0 𝜔(𝑡) + 𝐵𝑦(𝑡) + 𝑂𝑢(𝑡), (18)
where 𝜔(𝑡) ∈ R 𝑛-𝑝 is the observer state, and 𝑆 0 ∈ R (𝑛-𝑝) × (𝑛-𝑝) , 𝐵 ∈ R (𝑛-𝑝) × 𝑝 , 𝑂 ∈ R (𝑛-𝑝) ×𝑚 are constant matrices to be determined. Then, Proposition 1 can be applied for the considered case. If the following matrix equalities are verified:

𝑆 0 (Π + Υ𝐷 0 ) + 𝐵𝐷 0 -(Π + Υ𝐷 0 ) 𝐴 0 = 0, (19) (Π + Υ𝐷 0 ) 𝑄 = 𝑂,
then there exists a solution

𝜔(𝑡) = Π𝑥(𝑡) + Υ𝑦(𝑡), ∀𝑡 ∈ R + (20)
for any 𝑥(0) ∈ R 𝑞 and 𝜔(0) = Π𝑥(0) + Υ𝑦(0), connecting ( 16) and [START_REF] Leondes | Reduced-order observers for linear discretetime systems[END_REF]. Applying Assumption 1 to ( 20), [START_REF] Khalil | Nonlinear systems[END_REF], and ( 18), we can use Theorem 1 (we ask for the existence of a positive definite symmetric matrix 𝑃 ∈ R 𝑝× 𝑝 such that 𝑆 ⊤ 0 𝑃 + 𝑃𝑆 0 < 0) to show that ( 18) is globally asymptotically stable (the estimation error is input-to-state stable). Furthermore, a global asymptotic observer for 𝑤, makes it a reduced-order observer for [START_REF] Khalil | Nonlinear systems[END_REF]. This classical result is well-known and was first presented in [START_REF] Luenberger | Observing the state of a linear system[END_REF]. The purpose of considering such a case here is to demonstrate that the traditional result for linear systems can be obtained through the previous idea.

B. Nonlinear reduced-order observer

Consider ( 4) and ( 5) with constant matrices for simplicity, and let 𝐷 1 = 0. We have a nonlinear system:

𝑥(𝑡) = 𝐴 0 𝑥(𝑡) + 𝐴 1 𝑓 (𝐻𝑥(𝑡)) + 𝑄𝑢(𝑡) + 𝑑 (𝑡), 𝑡 ∈ R + , 𝑦(𝑡) = 𝐷 0 𝑥(𝑡) + 𝑣(𝑡), (21)
where 𝑥(𝑡) ∈ R 𝑛 is the full state, 𝑦(𝑡) ∈ R 𝑝 is the measured part of the state and 𝐴 0 , 𝐴 1 , 𝑄, 𝐻 are matrices of corresponding dimensions, 𝑑 ∈ L 𝑛 ∞ is an external disturbance or a modeling error and 𝑣 ∈ L 𝑝 ∞ is a measurement noise. Our goal is to design a reduced-order observer for (21) of a smaller dimension 𝑞 = 𝑛 -𝑝:

𝜔(𝑡) = 𝑆 0 𝜔(𝑡) + 𝑆 1 𝑓 (𝐽𝜔(𝑡)) + 𝐵𝑦(𝑡) + 𝑂𝑢(𝑡), (22)
where 𝜔(𝑡) ∈ R 𝑞 , and constant matrices are of corresponding dimensions. All of them have to be defined using the presented method. Proposition 1 gives us the existence conditions of a solution in a disturbance-free scenario (𝑑 = 0, 𝑣 = 0 and Π is obtained from the representation ( 17)):

𝜔(𝑡) = Π𝑥(𝑡) + Υ𝑦(𝑡) = (Π + Υ𝐷 0 )𝑥(𝑡) = 𝑍𝑥(𝑡), (23)
where 𝑍 ∈ R 𝑞×𝑛 is a matrix connecting solutions of the initial system (21) and the observer (22). Then, ( 8) and ( 9) in this case are as follows:

𝐽 𝑍 = 𝐻, ( 24
)
𝑆 0 𝑍 = 𝑍 𝐴 0 -𝐵𝐷 0 , 𝑆 1 = 𝑍 𝐴 1 , 𝑂 = 𝑍𝑄.
The equations above can be solved with respect to 𝑆 0 , 𝑆 1 , 𝐵, 𝑂 and 𝐽, knowing 𝑍. Theorem 1 can be directly applied with a constant matrix 𝑃 to prove the convergence of the observer (22) to a hyperplane given by [START_REF] Persidskij | Problem of absolute stability[END_REF]. Since the order of ( 22) is lower than the one of ( 21), we can say that ( 22) is an asymptotical reduced-order observer. Remark 2. The same result can be derived for time-varying matrices. The difference will lay only in matrix equations from Proposition 1 and in the form of 𝑆 0 (𝑡) and 𝑆 1 (𝑡).

C. Discrete-time reduced-order observer Consider ( 6) and ( 7) and let 𝐷 1,𝑘 = 0, 𝑘 ∈ N. We have a nonlinear discrete-time system:

𝑥 𝑘+1 = 𝐴 0,𝑘 𝑥 𝑘 + 𝐴 1,𝑘 𝑓 (𝐻 𝑘 𝑥 𝑘 ) + 𝑄 𝑘 𝑢 𝑘 + 𝑑 𝑘 , 𝑘 ∈ N, 𝑦 𝑘 = 𝐷 0,𝑘 𝑥 𝑘 + 𝑣 𝑘 , (25)
where 𝑥 𝑘 ∈ R 𝑛 is the full state, 𝑦 𝑘 ∈ R 𝑝 is the measured part of the state and 𝐴 0,𝑘 , 𝐴 1,𝑘 , 𝑄 𝑘 , 𝐻 𝑘 are matrices of corresponding dimensions at time 𝑘, 𝑑 𝑘 is an external disturbance and 𝑣 𝑘 is a measurement noise. Then we build an observer again:

𝜔 𝑘+1 = 𝑆 0,𝑘 𝜔 𝑘 + 𝑆 1,𝑘 𝑓 (𝐽 𝑘 𝜔 𝑘 ) + 𝐵 𝑘 𝑦 𝑘 + 𝑂 𝑘 𝑢 𝑘 , (26)
where 𝜔 𝑘 ∈ R 𝑞 , and all matrices are of corresponding dimensions. Proposition 2 gives us the existence conditions for a solution in a disturbance-free scenario and Π 𝑘 are as follows:

𝜔 𝑘 = Π 𝑘 𝑥 𝑘 + Υ 𝑘 𝑦 𝑘 = (Π 𝑘 + Υ 𝑘 𝐷 0,𝑘 )𝑥 𝑘 = 𝐺 𝑘 𝑥 𝑘 , (27)
where 𝐺 𝑘 ∈ R 𝑞×𝑛 is a matrix at time instant 𝑘, connecting solutions of the initial system (25) and the observer [START_REF] Sundarapandian | Reduced order observer design for nonlinear systems[END_REF]. Then, [START_REF] Forgione | Continuous-time system identification with neural networks: Model structures and fitting criteria[END_REF] and ( 12) are as follows:

𝑆 0,𝑘 𝐺 𝑘 + 𝐵 𝑘 𝐷 0,𝑘 -𝐺 𝑘+1 𝐴 0,𝑘 = 0, (28)
𝐵 𝑘 𝐷 1,𝑘 -𝐺 𝑘+1 𝐴 1,𝑘 + 𝑆 1,𝑘 = 0, 𝐺 𝑘+1 𝑄 𝑘 = 𝑂 𝑘 , 𝐽 𝑘 𝐺 𝑘 = 𝐻 𝑘 .
The equations above can be solved to determine 𝑆 0,𝑘 , 𝑆 1,𝑘 , 𝐵 𝑘 , 𝑂 𝑘 , and 𝐽 𝑘 , knowing 𝐺 𝑘 . Theorem 2 can be directly applied with a matrix 𝑃 𝑘 to prove the convergence of [START_REF] Sundarapandian | Reduced order observer design for nonlinear systems[END_REF].

The results on reduced-order observer design for nonlinear systems in [START_REF] Sundarapandian | Reduced order observer design for nonlinear systems[END_REF] and [START_REF] Sundarapandian | Reduced order observer design for discrete-time nonlinear systems[END_REF] are local and based on linearization analysis, while the general results presented in [START_REF] Karagiannis | Invariant manifold based reduced-order observer design for nonlinear systems[END_REF] cover the considered case, but they are less constructive due to the requirement of the existence of a solution for the PDE.

VI. REGRESSION

Let us consider another possible application of the result presented in Section IV.

A. Linear regression

A well-known and common method in practical estimation problems is the linear parameter regression. The application of our result is less intuitive in this setting than previously. However, it fits the problem statement rather well. To demonstrate it, let us consider a simple linear regression equation:

𝑥(𝑡) = 𝑑 (𝑡), 𝑡 ∈ R + (29) 𝑦(𝑡) = 𝐷 0 (𝑡)𝑥(𝑡) + 𝑣(𝑡), (30)
where 𝑦(𝑡) ∈ R 𝑝 is a measured output, 𝐷 0 (𝑡) ∈ R 𝑝×𝑛 is a regression matrix, 𝑥(𝑡) ∈ R 𝑛 is the vector of unknown parameters to be estimated, which in our case is the statevector and 𝑑 (𝑡) ∈ R 𝑛 is a bounded disturbance signal. Having nonzero disturbance means that the parameter vector 𝑥 may not be necessarily constant but can be affected by external factors and vary in time in a certain region. In the case of linear regression, the disturbance 𝑣 can represent an unmodeled observational or experimental error. To apply Proposition 1, we consider 𝐴 0 ≡ 𝐴 1 ≡ 𝐷 1 ≡ 𝑄 ≡ 0 in (4) and we represent the parameter observer in the required form [START_REF] Chair | Optimal data fusion in multiple sensor detection systems[END_REF] with 𝑆 1 ≡ 𝑂 ≡ 0:

𝜔(𝑡) = 𝑆 0 (𝑡)𝜔(𝑡) + 𝐵(𝑡)𝑦(𝑡), (31)
Then, we can choose Π ≡ 𝐼 𝑛 as the constant identity matrix, Υ ≡ 0, leading to:

𝜔(𝑡) = 𝑥(𝑡), (32)
meaning that (31) is an observer for 𝑥. Applying Proposition 1, we have the equality:

𝑆 0 = -𝐵𝐷 0 . (33)
Since we need to choose the matrices 𝑆 0 and 𝐵, let us assign 𝐵 = Γ𝐷 ⊤ 0 , where Γ is a nonsingular matrix, and can substantiate 𝑆 0 = -Γ𝐷 ⊤ 0 𝐷 0 . Substituting in (31), it results in a conventional gradient estimator:

𝜔(𝑡) = Γ𝐷 ⊤ 0 (𝑡) (𝑦(𝑡) -𝐷 0 (𝑡)𝜔(𝑡)) . (34)
For the asymptotic convergence of this observer, we need a standard additional assumption [START_REF] Bitmead | Persistence of excitation conditions and the convergence of adaptive schemes[END_REF]:

Assumption 3. The matrix 𝐷 0 (𝑡) is PE, for ∀𝑡 ∈ R + .
Remark 3. There exist techniques to relax the PE condition using preprocessing algorithms, such as the Dynamic Regressor Extension and Mixing (DREM) method (see, for example, [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]). To demonstrate our approach in this paper, we prefer to keep Assumption 3 for the sake of generality.

If Assumption 3 is satisfied, then applying Theorem 1 to (32), (30), and (31), we obtain the convergence of the estimator to the state 𝑥 (under Assumption 1, it is expected that there is a matrix P satisfying the conditions of the theorem). This result is well known for parameter estimation in linear regression (see, for instance, [START_REF] Ljung | System identification: theory for the user[END_REF]). Despite the result not being novel, this application demonstrates the broad applicability of the approach presented in Section IV to estimation problems.

B. Nonlinear regression

The estimation problem for nonlinear regression is more complex and might require some additional conditions to guarantee the observer's convergence. Therefore, let us consider a particular case:

𝑥(𝑡) = 𝑑 (𝑡), 𝑡 ∈ R + (35) 𝑦(𝑡) = (1 -𝐷 1 (𝑡))𝐽 (𝑡)𝑥(𝑡) + 𝐷 1 (𝑡) 𝑓 (𝐽 (𝑡)𝑥(𝑡)), (36)
where 𝐷 0 (𝑡) = (1 -𝐷 1 (𝑡))𝐽 (𝑡) and 0 ≤ 𝐷 1 (𝑡) ≤ 1, for all 𝑡 ≥ 0, 𝐴 0 ≡ 𝐴 1 ≡ 𝑄 ≡ 0, 𝑦(𝑡) ∈ R 𝑝 is measured output vector, 𝐷 1 (𝑡) ∈ R 𝑝×ℓ , 𝐽 (𝑡) ∈ R 𝑟 ×𝑛 are regression matrices, 𝑥(𝑡) ∈ R 𝑛 is a vector of unknown parameters and 𝑑 (𝑡) ∈ R 𝑛 is a bounded disturbance signal. Remark 4. A form of regression equation (36) might be found in models where the output is a fusion of data from multiple sensors (i.e., linear and nonlinear in parameters) of different nature, whose combination is used to improve the estimation [START_REF] Sandy | Confusion: Sensor fusion for complex robotic systems using nonlinear optimization[END_REF]. For instance, the possible failure [START_REF] Chair | Optimal data fusion in multiple sensor detection systems[END_REF] or missing data [START_REF] Housfater | Nonlinear fusion of multiple sensors with missing data[END_REF] from some sensors is often considered. In any case, 𝐷 1 is considered a signal that dynamically weights the data from linear and nonlinear sensors. We set Π ≡ 𝐼 𝑛 and Υ ≡ 0, so we are looking for a solution in the form 𝜔 ≡ 𝑥. Using Proposition 1, we obtain the following equalities:

𝐽 = 𝐻, 𝑆 0 + 𝐵(1 -𝐷 1 )𝐽 = 0, 𝑆 1 + 𝐵𝐷 1 = 0.
Let 𝐵 = 𝛾𝐽 ⊤ for some 𝛾 > 0, then, we have the corresponding observer:

𝜔(𝑡) = -𝛾(1 -𝐷 1 (𝑡))𝐽 (𝑡) ⊤ 𝐽 (𝑡)𝜔(𝑡) -𝛾𝐷 1 (𝑡)𝐽 (𝑡) ⊤ 𝑓 (𝐽 (𝑡)𝜔(𝑡)) + 𝛾𝐽 ⊤ 𝑦(𝑡), (37)
Consider a copy dynamics of (37): (39) Theorem 3. Let Assumptions 4 and 1 be satisfied. Assume that

ω(𝑡) = -𝛾(1 -𝐷 1 (𝑡))𝐽 (𝑡) ⊤ 𝐽 (𝑡) ω(𝑡) - 𝛾𝐷
-𝛾𝐷 1 𝑒 ⊤ 𝑃𝐽 ⊤ ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω) -𝐽𝑒) < 𝑒 ⊤ 𝑒 2
, for all 𝜔, ω ∈ R 𝑞 and 𝑒 = 𝜔ω, with 𝑃 defined as above. Then, the system (38) is ISS and (37) is globally convergent.

Proof. Assumption 1 ensures the existence of an invariant solution, so select a Lyapunov function candidate 𝑉 (𝑒) = 𝑒 ⊤ 𝑃𝑒. Then the derivative of 𝑉 along the trajectories of (38) takes the form:

𝑉 = 𝑒 ⊤ 𝑃𝑒 + -𝛾𝐽 ⊤ 𝐽𝑒 -𝛾𝐷 1 𝐽 ⊤ ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω) -𝐽𝑒) ⊤ 𝑃(𝑡)𝑒 + 𝑒 ⊤ 𝑃 -𝛾𝐽 ⊤ 𝐽 (𝑡)𝑒 -𝛾𝐷 1 𝐽 ⊤ ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω) -𝐽𝑒) .
Substituting the expression of the derivative of 𝑃 we have:

𝑉 = -𝑒 ⊤ 𝑒 -2𝛾𝐷 1 (𝑡)𝑒 ⊤ 𝑃𝐽 ⊤ ( 𝑓 (𝐽𝜔) -𝑓 (𝐽 ω) -𝐽𝑒) ,
Using the inequality of the theorem, we obtain 𝑉 < 0 and asymptotic stability of the error 𝑒. □

The current section demonstrates an advantage of the generality of the approach considered in Section IV, which allows the investigation of a rather wide range of linear and nonlinear estimation problems.

VII. EXAMPLES Example 1.

Let us consider a two mass-spring system with nonlinear stiffness affected by the white Gaussian noise 𝑣 and disturbance 𝑑, the dynamics of which can be expressed as follows:

                     𝑥 1 = 𝑥 2 , 𝑥 2 = -𝑘 1 (𝑥 1 -𝑥 3 ) -𝑘 2 (𝑥 1 -𝑥 3 ) 3 -𝑎 1 (𝑥 2 -𝑥 4 ) + 𝑑, 𝑥 3 = 𝑥 4 , 𝑥 4 = 𝑘 1 (𝑥 1 -𝑥 3 ) + 𝑘 2 (𝑥 1 -𝑥 3 ) 3 + 𝑎 1 (𝑥 2 -𝑥 4 ) -𝑘 3 𝑥 3 -𝑎 2 𝑥 4 + 𝑢, 𝑦 1 = 𝑥 1 + 𝑣, 𝑦 2 = 𝑥 2 + 𝑣 (40)
with 𝑢(𝑡) = sin(𝑡) and 𝑑 (𝑡) = 0.2 sin(10𝑡), 𝑡 ∈ R + . Presenting the system in the form (4), we have:

𝐴 0 = 0 1 0 0 -𝑘 1 -𝑎 1 𝑘 1 𝑎 1 0 0 0 1 𝑘 1 𝑎 1 -𝑘 1 -𝑘 3 𝑎 1 -𝑎 2 , 𝐴 1 = 0 -𝑘 2 0 𝑘 2 , 𝑓 (𝜇) = 𝜇 3 (𝜇 ∈ R), 𝑄 = 0 0 0 1 ⊤ , 𝐷 0 = 1 0 0 0 0 1 0 0 , 𝐻 = 1 0 -1 0 , 𝐷 1 = 0,
where the states 𝑥 1 and 𝑥 2 correspond to the position and the velocity of the first mass, assumed to be measured. The states 𝑥 3 and 𝑥 4 represent the second mass's position and velocity, which we need to estimate. The signal 𝑢 is the periodic force applied to the second mass to excite the system. The task is to build an observer of the form (22) for the states 𝑥 3 and 𝑥 4 using Proposition 1 (or (26) and Proposition 2). We have the following LMEs: We can assign Π and Υ having full row rank (the matrix Π should have a left inverse with respect to the unmeasured state components of (40), while Υ describes the utilization of the output variables) as, for example:

𝐽 (Π + Υ𝐷 0 ) = 𝐻, (41)
𝑆 0 (Π + Υ𝐷 0 ) = (Π + Υ𝐷 0 ) 𝐴 0 -𝐵𝐷 0 , 𝑆 1 = (Π + Υ𝐷 0 ) 𝐴 1 , 𝑂 = (Π + Υ𝐷 0 )𝑄.
Π = 0 0 1 0 0 0 0 1, Υ = 𝜐 1 0 0 𝜐 2 .
Therefore, we have 16 equations in total, with 16 unknowns. Solving the equations for chosen simulation values of the coefficients (𝑘 1 = 3, 𝑘 2 = 3, 𝑘 3 = 0.6, 𝑎 1 = 0.6, 𝑎 2 = 2), we obtain:

𝑆 0 = 0 1 -3.6 -2.6
, 𝑆 1 = 0 3 , 𝐽 = -1 0 , 𝑂 = 0 1 , 𝐵 = 0 -1 -0.6 0.6

, Υ = -1 0 0 0 .

Then, we have the dynamics of 𝜔, which must satisfy the LMIs in Theorem 1. From the given dynamics, we have the condition:

𝑒 ⊤ 𝐹 ω3 1 -𝜔 3 1 = (𝐹 1 𝑒 1 + 𝐹 2 𝑒 2 ) -𝑒 1 ( ω2 1 + 𝜔 1 ω1 + 𝜔 2 1 ) ≤ 𝑒 ⊤ 𝑊 𝑒,
which has to be satisfied locally in ω, 𝜔 ∈ R 2 by imposing the admissible upper limits on these variables. Assume that ω2 1 + 𝜔 2 1 ≤ 𝜂 for some 𝜂 > 0, then 𝑊 = 0.5𝜂 -𝐹 1 + 1.5𝐹 2 0 0 1.5𝐹 2 .

Consider, for example the following matrices: 𝐹 = 0.25 0 , 𝑊 = -0.1875 0 0 0 .

For 𝛾 = 0.5, 𝜂 = 1.5 the solution of LMIs given by the solver: 𝑃 = 0.267 0.0547 0.0547 0.031 , Ξ = 0.384 -0.006 -0.006 0.027 .

Finally, we build an observer for states 𝑥 3 and 𝑥 4 from the expression 𝜔 = Π𝑥 + Υ𝐷 0 𝑥:

x3 = 𝜔 1 -𝜐 1 𝑥 1 𝑝 1 = 𝜔 1 + 𝑥 1 , x4 = 𝜔 2 -𝜐 4 𝑥 2 𝑝 4 = 𝜔 2 .

VIII. CONCLUSION

The results presented in this paper significantly simplify the design of reduced-order observers for a particular class of systems, avoiding solutions of PDEs that arise in the conventional invariant manifold methodology. The resulting solution is explicit and more constructive than existing results on nonlinear observer design. We have shown two possible applications of our approach in discrete-and continuous-time cases: nonlinear reduced-order observer and nonlinear regression. Two examples, one representing a mechanical system and an academic case for nonlinear regression, demonstrated the applicability of our results in this work.

1 (Assumption 4 .

 14 𝑡)𝐽 (𝑡) ⊤ 𝑓 (𝐽 (𝑡) ω(𝑡)) + 𝐵(𝑡)𝑦(𝑡) + 𝛿(𝑡), then we have error dynamics for 𝑒(𝑡) = 𝜔(𝑡)ω(𝑡): 𝑒(𝑡) = -𝛾(1 -𝐷 1 (𝑡))𝐽 (𝑡) ⊤ 𝐽 (𝑡)𝑒(𝑡) -𝛾𝐷 1 (𝑡)𝐽 (𝑡) ⊤ ( 𝑓 (𝐽 (𝑡)𝜔(𝑡)) -𝑓 (𝐽 (𝑡) ω(𝑡))) -𝛿(𝑡). (38) Remark 5. If 𝐷 1 (𝑡) = 0, ∀𝑡 ∈ R + , then (36) takes the form of linear regression equation (30), where 𝐽 ≡ 𝐷 0 .Theorem 1 can be used for stability analysis of (38), but to get more efficient stability conditions, let us consider its modification. For this purpose, we need an auxiliary hypothesis: The function 𝐽 is PE.Under this condition, consider a time-dependent matrix𝑃(𝑡) = ∫ ∞ 𝑡 Φ(𝑠, 𝑡) ⊤ Φ(𝑠, 𝑡)𝑑𝑠, where Φ(𝑡, 𝑡 0 ) = -𝛾𝐽 ⊤ (𝑡)𝐽 (𝑡)Φ(𝑡, 𝑡 0 ) Φ(𝑡 0 , 𝑡 0 ) = 𝐼.then the following equality is satisfied 𝑃(𝑡) -𝛾𝑃(𝑡)𝐽 (𝑡) ⊤ 𝐽 (𝑡) -𝛾𝐽 ⊤ (𝑡)𝐽 (𝑡)𝑃(𝑡) = -𝐼 𝑛 .

Fig. 1 .

 1 Fig. 1. Example 1. Reduced-order observer for a mass-spring system.

Fig. 2 .

 2 Fig. 2.Example 2. Nonlinear regression for a system given in a form (36).

Fig. 1

 1 Fig. 1 demonstrates the convergence of the observer for unmeasured states, successfully reducing the dimension of the observation problem from 4 to 2. Example 2. Consider the nonlinear regression equation (36) with 𝑑 (𝑡) = 0.1 sin(𝑡) sin(2𝑡) ⊤ , error 𝑣 ≡ 0 and 𝐽 (𝑡) = 1 sin(𝑡) , 𝐷 1 (𝑡) = sin 2 (5𝑡), 𝑓 (𝑡) = arctan(𝑡) for all 𝑡 ∈ R + , and constant parameters 𝑥(0) = 3 5 ⊤ to estimate. Then we build an observer of the form (37), with 𝛾 = 0.75, and the conditions of the Theorem 3 are satisfied. Fig. 2 demonstrates the convergence of the estimates 𝜔 to the time-varying parameter 𝑥.

Throughout the paper, we simplify the notation for variables by writing, for instance, Π in place of Π (𝑡 ) when the time-dependency was once defined, unless the opposite is mentioned.


Sophie Barrau

Anthony Ferri

Antonio Da Costa

Juliette Defebvin

Sébastien Leroy

Rachel Desfeux

Jean-Marc Lefebvre

Nanoscale investigations of and -crystal phases in PVDF based nanocomposites

Keywords: PVDF, -phase, atomic force microscopy, carbon nanotubes, piezoelectric

The impact of carbon nanotubes (CNTs) incorporation into semi-crystalline poly(vinylidene fluoride), PVDF, was investigated at both macro-and nanoscales. A special effort was devoted to probe the local morphology, and mechanical, ferroelectric, piezoelectric and electrical conductivity response by means of atomic force microscopy tools. Incorporation of CNTs mainly induces the development of the polar -phase and as a consequence the coexistence of the -phase with the most stable non-polar -phase is observed. A maximum -phase content is reached at 0.7 wt% CNT loading. The spherulitic morphology of the PVDF -phase is assessed, in conjunction with the lack of any ferroelectric response, while the presence of the polar phase is confirmed, owing to clear piezoresponse signals. Local piezoelectric measurements on -phase domains yield a maximum effective coefficient d33 ~13 pm/V, thus underlining the potential for applications of such functional PVDF-based nanocomposites in advanced piezoelectric devices. An increase in macroscopic conductivity with CNT content is observed, with a percolation threshold achieved for a composition close to 0.7 wt%. Nanoscale investigation of electrical conductivity confirms the presence of some infinite CNT cluster homogeneously distributed over the surface. The macroscopic viscoelastic behavior of the composite reflects the reinforcing effect of CNTs, while the nanomechanical characterization yields a local contact modulus of the -phase domains larger than that of its -phase counterpart, in agreement with the fact that the CNTs act as -phase promoters and subsequently reinforce the -domains.

INTRODUCTION

Polymers with electroactive properties provide numerous opportunities for applications in domains such as biomedicine or energy generation owing to their flexibility, good mechanical acoustic impedance and biocompatibility compared to ceramic materials. Among this class of polymers, poly(vinylidene fluoride) (PVDF) is a semi-crystalline polymer with unique piezo-, pyro-and ferroelectric properties allowing its use for various development in sensors, actuators and energy harvester systems. [START_REF] Ueberschlag | PVDF piezoelectric polymer[END_REF] PVDF displays a complex crystal polymorphism with three main phases of interest ,  and . [START_REF] Lovinger | Poly(Vinylidene Fluoride) In Developments in Crystalline Polymers -1[END_REF] The non-polar -phase (TGTG' conformation) is the most common phase, obtained directly by cooling from the melt at conventional rates. [START_REF] Lovinger | Crystallization and morphology of melt-solidified poly(vinylidene fluoride) Journal of polymer science. Part A-2[END_REF] The electroactive properties are mainly related to the highly polar -phase (TTTT conformation) induced by stretching films initially in -phase or in the presence of fillers. [START_REF] Defebvin | In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching[END_REF] The intermediate polar -phase (TTTGTTTG' conformation) [START_REF] Lovinger | Unit cell of the gamma phase of poly(vinylidene fluoride)[END_REF] focuses attention as a ferroelectric material and because it provides a higher breakdown strength and discharge energy density as compared to the -phase. [START_REF] Li | Electric energy storage properties of poly(vinylidene fluoride)[END_REF] This phase may either be obtained by isothermal crystallization at high temperature, [START_REF] Lovinger | Crystalline transformations in spherulites of poly(vinylidene fluoride)[END_REF] or by solution induced crystallization [START_REF] Prest | The formation of the gamma phase from the alpha and beta polymorphs of polyvinylidene fluoride[END_REF] or as well by filler addition. Indeed fillers such as clays, 9,10 zeolites [START_REF] Lopes | Electrical and thermal behavior of gamma-phase poly(vinylidene fluoride)/NaY zeolite composites[END_REF] or carbon based fillers [START_REF] Li | Role of Ion-Dipole Interactions in Nucleation of Gamma Poly(vinylidene fluoride) in the Presence of Graphene Oxide during Melt Crystallization[END_REF][START_REF] Chae | Rheology, crystallization behavior under shear, and resultant morphology of PVDF/multiwalled carbon nanotube composites[END_REF] may promote the formation of the -phase owing to the existence of favorable interactions between the polymer chains and the filler surface. For instance a 0.2 wt% content of montmorillonite (MMT) clay induces a -phase content of 36%. [START_REF] Sujoy Kumar | Improved breakdown strength and electrical energy storage performance of gammapoly(vinylidene fluoride)/unmodified montmorillonite clay nano-dielectrics[END_REF] Filler addition may also impact material properties. For instance, carbon nanotubes (CNTs) can improved the dielectric and piezoelectric behavior of fluorinated polymer. [START_REF] Levi | Properties of Polyvinylidene Difluoride-Carbon Nanotube Blends[END_REF] However, there still remains a key scientific issue in the understanding of how the behavior of pristine PVDF or PVDF composites at the nanoscale impacts the macroscopic properties. Some studies at local scale have already been reported in the literature on thin films [START_REF] Jia | Enhanced beta-crystalline phase in poly(vinylidene fluoride) films by polydopamine-coated BaTiO3 nanoparticles[END_REF][START_REF] Nunes | Relationship between the microstructure and the microscopic piezoelectric response of the alpha-and beta-phases of poly(vinylidene fluoride)[END_REF][START_REF] Ting | Design and characterization of one-layer PVDF thin film for a 3D force sensor[END_REF][START_REF] Zhiyuan | Mechanisms of polarization switching in graphene oxides and poly(vinylidene fluoride)-graphene oxide films[END_REF] or fibers [START_REF] Baji | Nanoscale investigation of ferroelectric properties in electrospun barium titanate/polyvinylidene fluoride composite fibers using piezoresponse force microscopy[END_REF][START_REF] Liu | Nanoscale investigations on beta-phase orientation, piezoelectric response, and polarization direction of electrospun PVDF nanofibers[END_REF][START_REF] Sharma | Outstanding dielectric constant and piezoelectric coefficient in electrospun nanofiber mats of PVDF containing silver decorated multiwall carbon nanotubes: assessing through piezoresponse force microscopy[END_REF] but have been primarily focused on the -phase. To our knowledge, no deep investigation at the nanoscale on both microstructure and physical properties of the phase, has been reported in the literature.

In the present study, the -phase was induced by dispersion of CNTs in the polymer matrix.

The morphology of the co-existing crystal polymorphs was investigated and correlated to the nanoscale ferroelectric, piezoelectric, electrical conductivity and mechanical response by means of advanced tools of atomic force microscopy (AFM). Identification of the two major PVDF crystal phases (namely  and -phases) in the nanocomposites was achieved and the impact of the CNTs on the physical properties was probed at local scale.

EXPERIMENTAL SECTION

Materials. PVDF (Kynar grade 720) was provided by Arkema. Multiwalled CNTs (8.5 nm in diameter and 1.5 μm in length) were obtained from Nanocyl (Nanocyl 7000 series, Belgium).

Elaboration of CNT-PVDF nanocomposites.

Nanocomposites were elaborated following a two-step procedure. CNTs and PVDF were dispersed in N,N-dimethylacetamide (DMAc) (Sigma-Aldrich). The solution was stirred at 40°C for 1h and was subsequently poured into silicone cups and left overnight at ambient temperature and atmospheric pressure (Tamb, Patm).

The residual solvent was removed by placing the nanocomposites in a vacuum oven at 40°C for 10h. Then the "dried" material was melt-compounded using a twin-screw microextruder from DSM Xplore (Geleen, The Netherlands) under the following conditions (210°C, 50 rpm, 5 min under nitrogen atmosphere). The extrudates with a 2 mm diameter were air-cooled at the exit of the die. The samples are denoted NTx where x holds for the CNT content in wt%, namely NT0 for neat PVDF, NT01, NT03, NT07, NT1 and NT12 for composites with 0.1, 0.3, 0.7, 1.0 and 1.2 wt% loading, respectively.

Nanocomposite thin film preparation. Thin films were prepared by spin coating on silicon wafers coated with platinum. The nanocomposite previously elaborated was solubilized in DMAc and the solution was deposited at ambient temperature with a rotational speed of 2000 rpm for 20 s and at an acceleration of 200 m.s -2 .

Characterization. Fourier transform infrared (FTIR) spectra were collected after 16 scans (resolution of 2 cm -1 ) using-ATR (Attenuated Total Reflexion) technique on a Perkin Elmer Spectrum 100 spectrometer. Wide-angle X-ray scattering (WAXS) measurements were performed from a Cu target (λ = 1.54 Å), operated at 20 mA and 40 kV. The sample-to-detector distance was 6 cm. 2D-WAXS patterns were recorded on a CCD camera from Photonic Science Ltd and the radial WAXS intensity profiles were obtained by azimuthal integration using the FIT2D software. Thermal characterization was performed on a DSC Q20 (TA Instruments) calibrated with indium under nitrogen atmosphere. Samples of approximately 10 mg were put in aluminum pans and were analyzed between 20 and 200°C at a heating rate of 20°C.min -1 under nitrogen gas flow. The morphology of nanocomposites was investigated by field emission scanning electron microscopy (FE-SEM) using an Hitachi S4700 apparatus operated at an acceleration voltage of 10 kV. The micrographs were taken from the surface of cryofractured samples previously coated with carbon. Nanoscale measurements were performed by AFM using a commercial microscope (Asylum Research/Oxford Instruments, MFP-3D, USA) working in air at room temperature. Electroactive properties were probed by recording piezoelectric hysteresis loops using dual AC resonance tracking (DART) piezoresponse force microscopy (PFM). [START_REF] Rodriguez | Dual-frequency resonancetracking atomic force microscopy[END_REF] Such experiments were carried out using grounded conductive bottom electrode while Pt/Ir-coated silicon tips and cantilevers with a stiffness of 2.5 N.m -1 (Nanosensors) were used. [START_REF] Balke | Electromechanical Imaging and Spectroscopy of Ferroelectric and Piezoelectric Materials: State of the Art and Prospects for the Future[END_REF] Typical contact-resonance curves obtained during PFM experiments are presented in Figure S1. On macroscale, conductivity measurements were conducted at a frequency of 10-1 Hz by Broadband Dielectric Spectroscopy (Novocontrol BDS 4000 system). Local conductivity (i.e. current detection) was detected by the ORCA conductive-AFM technique. Similar tips as for PFM experiments were used. A bias voltage of 0.5 V was applied between the grounded tip and the conductive bottom electrode during mapping. For the nanoscale I-V spectroscopy, the DC bias was ranging between -10 and +10

V. Dynamic Mechanical Analysis (DMA) was performed on an Ares apparatus from TA Instruments used in torsion mode. Experiments were performed at a frequency of 1 Hz, in the temperature range -30°C < T < 30°C at a heating rate of 5 °C.min -1 . Nanomechanical response was probed by the AM-FM (amplitude modulation -frequency modulation) method from Asylum Research/Oxford Instruments, [START_REF] Garcia | Nanomechanical mapping of soft matter by bimodal force microscopy[END_REF][START_REF] Hurley | Fast, quantitative AFM nanomechanical measurements using AM-FM Viscoelastic Mapping Mode[END_REF][START_REF] Gannepalli | Mapping nanoscale elasticity and dissipation using dual frequency contact resonance AFM[END_REF] using blueDriveTM photothermal excitation on a Cypher S AFM at 2 Hz (further details in Figure S2a). AC160TSA silicon tips (Olympus) with cantilever stiffness of 26 N.m -1 were used for these experiments and the tip/sample interaction was measured in the repulsive regime, which enables to get the contact modulus of the different PVDF phases. A reference polycarbonate sample with known modulus of 2.5 GPa was used to accurately adjust the model parameters such as the tip radius. [START_REF] Garcia | Nanomechanical mapping of soft matter by bimodal force microscopy[END_REF]

RESULTS AND DISCUSSION

Characterization of the polymer crystal structure in nanocomposites combined FTIR, DSC and WAXS. Results are presented in Figure 1a-c. The NT0 FTIR spectrum exhibits the characteristic bands at 766 cm -1 (CF2 bending and skeletal bending), 795 cm -1 (CH2 rocking) and 976 cm -1 (CH out-of-plane deformation) associated to the presence of the PVDF -phase. After CNT incorporation, the FTIR spectra of composites present additional bands at 812 cm -1 (CH2 out-of-plane wag), 833 cm -1 , and 1234 cm -1 (CF out-of-plane deformation) as a signature of the PVDF -phase and a band at 1275 cm - 1 corresponding to the PVDF -phase. [START_REF] Cai | A critical analysis of the alpha, beta and gamma phases in poly(vinylidene fluoride) using FTIR[END_REF] Note the band at 840 cm -1 (CH2 rocking) is attributed to both -and -phases. FTIR data thus assess the coexistence of the three α-, -and -phases in the nanocomposites. It appears that -and -phase development is not linearly dependent on CNT content. Indeed, for both polar crystal phases, band intensities increase with CNT content up to a maximum value reached for NT07 and then decrease at higher CNT content. Assessment of the degree of crystallinity and of the respective fractions of the different crystal polymorphs has been performed through the combined analysis of FTIR and DSC data. According to the method recently developed by Cai et al. [START_REF] Cai | A critical analysis of the alpha, beta and gamma phases in poly(vinylidene fluoride) using FTIR[END_REF] for FTIR data treatment in the particular case of the co-existence of the three crystal phases, the relative fraction of the electroactive -and -phases (FEA) was first calculated. Then quantification of individual -and -phase contributions has been performed by focusing on their specific signatures at 1234 cm -1 and 1275 cm -1 respectively. The relative fractions (F) of electroactive (EA), -, and -phases in the composites are reported in Table 1. The key information issued from the FTIR analysis is that the -phase is a minor component of the polar fraction. Complementary information to estimate the degree of crystallinity and the crystal phase fraction of -and -phases is provided by DSC measurements The DSC thermograms display the melting peak of PVDF -phase at 165°C. A shoulder at 173°C, characteristic of the -phase, [START_REF] Gregorio | Determination of the α, β, and γ crystalline phases of poly(vinylidene fluoride) films prepared at different conditions[END_REF] is also present beyond 0.3 wt% CNT content, and it is more pronounced for NT07. The -phase fraction is considered negligible and the enthalpy of fusion for a 100% -phase PVDF is assumed identical to that of a 100% -phase (H,100 = 93.07 J/g). [START_REF] Prest | The formation of the gamma phase from the alpha and beta polymorphs of polyvinylidene fluoride[END_REF] The results are also reported in Table 1. At first sight WAXS diffractograms are definitely not providing sound information with respect to the relative composition in crystal phases. Indeed, the characteristic peaks of PVDF α-phase are observed at 2 = 17.4°, 18.1°, 19.7° and 26.7°, corresponding to the (100), (020), ( 110) and (021) planes respectively. [START_REF] Gregorio | Effect of crystalline phase, orientation and temperature on the dielectric properties of poly (vinylidene fluoride) (PVDF)[END_REF][START_REF] Davis | Electric-field-induced phase changes in poly(vinylidene fluoride)[END_REF] Regarding the -phase, the main diffraction peak attributed to the (110/200) planes is located at 2 =20.6° and superimposed to those of the α-phase. In the case of the phase the characteristic peaks at 2 = 19.2° and 20.04° corresponding to the (020) and (110/101) planes are as well superimposed to those of the α-phase in the 20° region. A broadening of the latter peak in this particular angular range may however be noticed for NT07 in relation to the coexistence of the α and  phases. The diffraction peak at 2 = 26.7° related to the α-phase nevertheless provides an indirect way to probe the evolution of -phase content. The intensity of the diffraction peak decreases at low CNT content, reaches a minimum for NT07 and then increases again at higher CNT content, thus suggesting that the -phase content is indeed maximum for NT07 as probed by FTIR and DSC.

Table 1. Degree of crystallinity  and relative fraction (F) of electroactive (EA), -, and - A fair agreement is obtained between FTIR and DSC analysis. Incorporation of carbon nanotubes not only induces a change in PVDF crystal phase but seems to impact as well the overall degree of crystallinity in the nanocomposites. There remains however reasonable doubt about the assumption of identical enthalpy of fusion for both -and -phases. In the following, contribution of the -phase fraction will be considered negligible in the nanoscale investigation of properties.

phases of PVDF.  (%) a FEA (%) b F (%) a, b F (%)
To sum up, the structural characterization of nanocomposites reveals that incorporation of carbon nanotubes mainly induces the development of the polar PVDF -phase, and a maximum -phase fraction is observed at 0.7 wt% CNT content. Recently Kumar et al. [START_REF] Kumar | High dielectric permittivity and improved mechanical and thermal properties of poly(vinylidene fluoride) composites with low carbon nanotube content: effect of composite processing on phase behavior and dielectric properties[END_REF] established that the -phase fraction is dependent on CNT functionalization and composite processing, and in all cases, increases with CNT. As stated above, this is not the case in the present study.

Considering that a maximum -phase content is reached for NT07 and then decreases at higher CNT content, there ought to be a limitation to -phase formation beyond this particular composition.

Assessment of the state of dispersion of the CNTs in the polymer matrix is essential to understand the nanocomposite behavior. Figure 2a shows the SEM micrograph of NT07 as probed on the cryofractured surface of a bulk specimen.  PVDF -phase. In the meantime, some areas display an irregular shape (as indicated by the green cross on Figure 2b) with a topographic contrast similar to those of -spherulites. At this stage, we may postulate that it might refer to the presence of -spherulites located between phase spherulites, thus avoiding -spherulite impingement. The -spherulites are known to exhibit a rather odd bundle type arrangement of crystalline lamellae with "scroll-like" morphology. [START_REF] Lotz | Origine moléculaire de l'enroulement des lamelles cristallines du poly(fluorure de vinylidène), phase gamma[END_REF][START_REF] Vaughan | Etching and morphology of poly(vinylidene fluoride)[END_REF] These nanoscale morphological features are not evidenced on the present figure.

To better distinguish the two PVDF phases, the local electroactive behavior was probed at various locations over the entire surface of the thin film by using the spectroscopic tool of the PFM. The probing tip was used as nanometric top electrode over the surface to locally assess the polarization switching and the electromechanical activity. The piezoresponse loops were achieved in remnant mode, i.e. at zero voltage, to avoid electrostatic interactions. [START_REF] Martin | Ferroelectricity in Si-Doped HfO2 Revealed: A Binary Lead-Free Ferroelectric[END_REF] To illustrate this investigation, an example of scanned surface recorded on larger area (10×10 µm²) than for the previous AFM image (Figure 2b) is depicted on Figure 3a, where the two characteristic morphologies described above are seen. Several locations symbolized by the numbered crosses were analyzed on this scan area, which reflect the electrical response of the whole sample. Both As expected, no ferroelectric signal is measured in the spherulitic zones attributed to the nonpolar PVDF α-phase and identified as red crosses on the AFM image (see Figure 3b). On the contrary, in the areas indicated by the blue crosses numbered from 1 to 7, well-defined ferroelectric loops with two stable states of polarization for phase signal and typical butterflyshape for amplitude response are obtained, as revealed on Figures 3b and3c. This is directly related to the polar nature of the PVDF phase under investigation, i.e. to the ferroelectric phase previously mentioned. As a remark, both coercive voltage and vibration amplitude deduced from the phase and amplitude loops respectively are not similar for the seven recorded signals. This can be explained by the complex nature of the probed volume under the tip, where amorphous phase and CNTs as well as crystalline γ-phase coexist. By considering the piezoactivity at maximum applied voltage for the higher amplitude-voltage butterfly signal (loop #4), the larger local deformation of the film due to the converse piezoelectric effect is 450 pm.

However, this latter value was overestimated owing to the DART PFM method used for the measurement in which the contact resonance of the cantilever enhances the detected PFM signal. Moreover, in the case of a polymer material, relatively high losses are considered resulting in a lower Q quality factor of the cantilever of about 10. [START_REF] Mclachlan | Probing Local and Global Ferroelectric Phase Stability and Polarization Switching in Ordered Macroporous PZT[END_REF][START_REF] Liu | Rapid Nanoimprinting and Excellent Piezoresponse of Polymeric Ferroelectric Nanostructures[END_REF][START_REF] Choi | Enhancement of Local Piezoresponse in Polymer Ferroelectrics via Nanoscale Control of Microstructure[END_REF] Thus, by considering the expression A = d33VacQ, [START_REF] Mclachlan | Probing Local and Global Ferroelectric Phase Stability and Polarization Switching in Ordered Macroporous PZT[END_REF][START_REF] Liu | Rapid Nanoimprinting and Excellent Piezoresponse of Polymeric Ferroelectric Nanostructures[END_REF] where A is the detected PFM amplitude and Vac the driving voltage (3.5 V for our measurements), we can estimate the maximum effective piezoelectric coefficient d33 around 13 pm/V (note that the d33 piezoelectric coefficient of PVDF is negative). This value is lower than -58.5 pm/V reported by Kanik et al. by the similar PFM technique for phase PVDF nanoribbons. [START_REF] Kanik | Spontaneous High Piezoelectricity in Poly(vinylidene fluoride) Nanoribbons Produced by Iterative Thermal Size Reduction Technique[END_REF] However, in the latter case the high effective piezoelectric constant was mainly attributed to the nanoscale confinement, where a larger contribution of the domain wall motion occurs. In the present study, the estimated value is more in agreement with -7 pC/N measured at the macroscopic scale on PVDF -phase based nanocomposites, [START_REF] Lopes | Nucleation of the Electroactive gamma Phase and Enhancement of the Optical Transparency in Low Filler Content Poly(vinylidene)/Clay Nanocomposites[END_REF] the difference could be explained by a nanoscale confinement as mentioned above but with a reduced effect. This is quite significant with respect to the polymeric nature of the electromechanical layer and the absence of polarization step before d33 measurement, opening perspectives for the integration of such nanocomposites in piezoelectric-based applications.

As a result, the presence of both -and -phases is thus confirmed by probing the nanoscale morphology and the PFM ferroelectric and piezoelectric response. AFM data of mixed α-and -phases have already been published by Cebe et al. [START_REF] Ince-Gunduz | Impact of nanosilicates on poly(vinylidene fluoride) crystal polymorphism: Part 1. Melt-crystallization at high supercooling[END_REF] in the case of PVDF nanocomposites incorporating organically modified clay silicates, however not at such nanometric resolution.

Contrary to our observations, these authors report that the -spherulites lie on top of the spherulites. This morphology is explained by clay migration to the surface during the melt-crystallization process, while clays act as a promoter of the PVDF -phase. The 2D nature of the clay platelet and its propensity to lie parallel to the surface is definitely the key factor that yields the difference with our present situation dealing with rod-like nano-reinforcement.

Otherwise, considering that solvent evaporation at ambient temperature is very slow, any memory of the initial mechanical centrifugation of the spin-coating process should be erased prior to full evaporation. No resulting chain orientation is thus expected. For the sake of comparison, the AFM image of NT07 surface of a thick film is shown in Figure S3.

In order to understand the influence of CNTs on the structural evolution of PVDF (presence of a maximum in -phase content), electrical measurements have been performed. The evolution of electrical conductivity as a function of CNT content is reported in Figure 4a. NT0 is an insulating material. Its electrical conductivity has a value of 2×10 -14 S.cm -1 . The incorporation of a low amount of CNTs (NT01) implies the presence of finite size conductive clusters inside the polymer matrix that have no impact on electrical conductivity. Beyond 0.2 wt% CNT loading, the composites display a dramatic increase in dc by 10 orders of magnitude to reach a value of 8×10 -5 S.cm -1 . The percolation threshold, defined at 0.7 wt% of CNTs, indicates the existence of an infinite size conductive cluster within the PVDF matrix. The singularity in the PVDF crystal phase evolution with CNT content thus seems correlated to the electrical percolation threshold. A similar discontinuity of properties at the percolation threshold has already been reported in literature. [START_REF] Barrau | Glass Transition Temperature Depression at the Percolation Threshold in Carbon Nanotube-Epoxy Resin and Polypyrrole-Epoxy Resin Composites[END_REF][START_REF] Carponcin | Discontinuity of physical properties of carbon nanotube/polymer composites at the percolation threshold[END_REF] In the latter case, a glass transition temperature depression was observed. The free volume accessible to macromolecular motions and/or the specific interactions between the conducting fillers and the polymer matrix depending on the conductive network could explain the phenomenon. The existence of this singularity as a function of composition is further supported by the observation of a maximum in crystal growth as shown in Figure S4.

The specific mechanism is however still not fully understood. In the present situation it may be postulated that CNTs promote the formation of PVDF -phase. However CNT aggregation beyond NT07 composition reduces the capacity for CNT to further enhance surface nucleation.

The local conductivity was then probed over the surface of the NT07 sample by the ORCA c-AFM technique. Current map over 10×10 µm² is displayed on Figure 4b where conducting paths (green contrast) are observed, embedded in blue background contrast associated to the absence of signal. Such "hot spots" evidence the electrical conductivity of some areas in the sample, in agreement with the macroscopic measurements previously shown in Figure 4a. Local I-V characteristics were further performed on both insulating (blue) and conductive (green) regions (Figure 4b). As expected, Ohmic behavior is obtained when hot spots are locally probed thus reflecting the occurrence of an electrical percolation path through the composite layer between the top (AFM tip) and bottom (platinum) electrodes, while no current signal is detected for the background region when the bias voltage is varied between -10 V and +10 V (figure plotted between -1 V and +1 V for clarity), confirming the insulating nature of the matrix.

Therefore the conductive properties of NT07 observed at the macroscopic scale actually result from the presence of some infinite cluster (density of 15 in the 1010 µm 2 area) homogeneously distributed. Note that since NT07 is an electrically conductive sample, this precludes the measurement of macroscopic ferroelectric and piezoelectric properties, which further justifies the use of nanoscale probe to gain access to such electrical behavior.

Finally, the mechanical behavior of nanocomposites was analyzed by probing the dynamic viscoelastic response by DMA at a frequency f = 1 Hz. The storage modulus (G') of NT0 and NT07 recorded in the temperature range [-30°C, +30°C], i.e. above the glass transition temperature and below the melting temperature of secondary crystals, [START_REF] Neidhöfer | Structural evolution of PVDF during storage or annealing[END_REF] is displayed in Figure 5a. At T = 20°C, G' has a value of 1 GPa for NT0 as compared to G' of the order of 1.5 GPa for NT07. In the latter case, the viscoelastic response primarily reflects the reinforcing effect of the CNTs even if the sample comprises both α-and -phases. Nanomechanical properties of NT07 thin film were then probed by mapping the AM-FM contact modulus over the surface of the sample. Figure 5b displays the surface morphology of NT07 sample where both α-and -phases have been observed (-phase regions identified by the green crosses). Corresponding modulus mapping is presented in Figure 5c. The contact modulus of the α-phase dominated area (blue

(a)
regions) is of the order of 3.1 GPa, whereas in the -phase (yellow regions) it amounts to 5.9

GPa. These elastic moduli were calculated by using the Hertzian model with punch tip geometry and radius of 8 nm. In this case, the average indentation value calculated during the nanomechanical experiments was found to be included between 0.5 and 0.9 nm for the γ and α phases respectively (see Figure S2b for the indentation map, and Figure S2c for the AM-FM contact modulus contrasts overlaid on the top of the 3D topography). This range of values is in agreement with the one obtained on the reference sample. From these results, since -PVDF is induced by the CNTs it is reasonable to consider that the nanoscale modulus reflects the local mechanical reinforcement by the CNTs. The reinforcing effect of the CNTs is thus confirmed at both macro-and nanoscales.

NT07 composite is of particular interest since it displays a maximum of PVDF -phase content. However looking for macroscopic piezoelectric properties, the composite is electrically conductive and the polar phase fraction is pretty low. Providing a specific nanoscale arrangement of nanotubes, for instance 2D CNT network, could allow the design of a distribution of high density electroactive "wires" in a non polar environment.

CONCLUSIONS

In summary, we report on the macro-and nanoscale investigations of PVDF-based nanocomposites for CNT loadings in the range 0 to 1.2 wt%. CNT incorporation mainly promotes the formation of the polar PVDF -phase and a maximum content is reached at 0.7% CNT content. AFM characterization of the latter sample clearly identifies at high resolution the spherulitic morphology of the PVDF -phase with lamellar thickness around 10 nm. A peculiar morphology is also identified and attributed to the presence of the  crystal phase by means of the PFM technique. The electromechanical behavior was probed in the PVDF -phase regions.

It thus provides a robust answer to nanoscale phase identification and allows to assess the piezoelectric activity in such a confined environment. The maximum effective piezoelectric coefficient was estimated to be d33 ~13 pm/V, which paves the way for application of such PVDF-based composites as sensors, actuators or energy harvester systems. An increase in macroscopic conductivity with CNT content is observed, with a percolation threshold achieved for a composition close to NT07. It is postulated that beyond this composition, CNT aggregation reduces the capacity for -phase formation in relation to the decrease in interfacial area. The current detected locally reveals the presence of some infinite cluster homogeneously distributed over the NT07 surface. At the macroscopic level, the viscoelastic modulus is larger than that of neat PVDF, thus assessing the reinforcing effect of the CNTs. Nanoscale investigations reveal a higher contact modulus in the PVDF -phase regions, as compared to their -phase counterparts. Considering that the -phase is induced by the presence of the CNTs, we may therefore conclude that the latter act both as a -phase nucleant and subsequently provide local mechanical reinforcement of the -phase regions. The direct identification of PVDF α-and phases which was successfully achieved at the nanoscale, allowed a better understanding of impact of the CNTs on the structural and functional properties of the PVDF-based nanocomposites.

Figure 1 .

 1 Figure 1. Structural characterization of the CNT-PVDF composites. (a) FTIR spectra, (b)

Figure 2 .

 2 Figure 2. Surface morphology of the NT07 composite film. (a) SEM image of cryofractured

  phase and amplitude PFM signals obtained are shown on Figure3b and 3c, respectively.

Figure 3 .

 3 Figure 3. Local electroactive behavior of the NT07 composite. (a) AFM surface morphology,

Figure 4 .

 4 Figure 4. Electrical conductivity properties of the CNT-PVDF composite films. (a) Evolution

Figure 5 .

 5 Figure 5. Mechanical behavior of the NT07 composite films. (a) Viscoelastic behavior

ACKNOWLEDGMENT This work was partly supported through grant ANR-16-CE08-0025, ANR NanoPiC from French National Research Agency. The FE-SEM equipment has been implemented at the electron microscope facility (Institut Chevreul) thanks to the financial support of Région Hautsde-France and European FEDER. The authors are grateful to Région Hauts-de-France and University of Lille for granting a PhD fellowship to J. Defebvin. Région Hauts-de-France and Fonds Européen de Développement Régional (FEDER) are also acknowledged for funding the MFP-3D microscope under Program "Chemistry and Materials for a Sustainable Growth". The authors thank R. Barbattini from Asylum Research/Oxford Instruments for the AM-FM experiments.

ASSOCIATED CONTENT

Supporting information

Additional data including contact-resonance curves during DART PFM experiments (Figure S1), AM-FM method and contact modulus (Figure S2), AFM image of thick NT07 (Figure S3) and isothermal crystallization of nanocomposites (Figure S4).

AUTHOR INFORMATION

Corresponding Author

*Sophie Barrau, E-mail : sophie.barrau@univ-lille1.fr *Anthony Ferri, E-mail : anthony.ferri@univ-artois.fr


domain: spi info

text:
The Internet of Things (IoT) lies at the core of the unprecedented connected era we are currently speeding towards. Both Academia and the Industry strive to push the limits of technology fueled by vertical applications such as: factory 4.0, eHealth, human-centric and tactile IoT, wearable IoT, smart city/building, digital twin, etc., which will shape our future society and significantly impact both economic and societal aspects. Signal Processing (SP) plays an important role in expanding the number of IoT technologies and capitalizing on their applications. This is due to the sophisticated processing of signals and data gathered and shared by connected things.

In this context, the unique challenges from the SP point of view are: (i) system design taking into account the applications' practical limitations and requirements, (ii) gather collective intelligence from heterogeneous and inexpensive IoT sensors/devices (e.g. to perform an inference task in decentralized fashion) (iii) minimize device cost and energy consumption while allowing perpetual monitoring/sensing (iv) grant adaptivity to cope with device mobility use cases, (v) leverage fine-grained "programming" of wireless environments to cope with massive October 2022 DRAFT IoT scenarios. These challenges give rise to the need for developing pervasive, inexpensive, fast, and low-power algorithms (including those performing learning and data fusion), as well as novel theoretical tools, evaluation benchmarks, and methodologies to characterize the performance of the proposed algorithms. The wide diversity of these challenges pushes for a more collaborative effort from engineers and scientists with complementary and interdisciplinary expertise going beyond the standard SP techniques.

Accordingly, the main goal of this Special Issue is to address these challenges and present In "How Far Are Wireless Networks from Being Truly Deterministic?", various emerging applications sustained by IoT technology and machine-type communications are discussed. Specifically, the authors focus on applications that require an extremely low latency and a very small jitter such as: industrial sensing and controlling, remote surgery, and automatic driving. For such applications, delivering information in a deterministic fashion is crucial and one of the the biggest challenges in modern wire-line and wireless communications. In this paper, a review of currently available wire-line deterministic networks is presented and the main challenges behind wireless deterministic networks are discussed. Several promising techniques to enable deterministic communications in wireless networks are then proposed. By elaborating on the physical layer coding/modulation schemes and the channel-access / packet-scheduling at the media access control (MAC) layer, a significant contribution is made towards the realization of wireless deterministic communications in the near future.

In authors focus on federated learning (FL) as a powerful paradigm for distributed learning thanks to its privacy-preserving capabilities. With the use of FL, a network of edge devices can make intelligent decisions without exposing their data to others. Despite its success, the traditional FL is not well suited to many practical applications such as those involving the IoT or cyber-physical systems (CPS), where data access can be intermittent, and edge devices are semi-independent with device-specific dynamic behavior characteristics. The so-called semi-independent devices need to make decisions based on their own data and device characteristics, often independent of other devices and of the information obtained from other devices. Additionally, as new information becomes available, traditional FL must repeat the entire learning process and may not be able to provide timely and tailored solutions to clients. At the opposite, personalized online FL retains the collaborative and privacy-preserving abilities while learning in real time from intermittent data. It further enables devices to learn models customized to the device and the specific tasks it performs. In light of these reasons, personalized Online-FL is ideal for applications where the learning relies on heterogeneous data streams, and local optimization is beneficial. In this work, the authors highlight this new learning paradigm, present some potential applications, and discuss the major challenges in developing successful personalized Online-FL.

In "Impact of Embedded Deep Learning Optimizations for Inference in Wireless IoT Use

Cases", the authors investigate the inference problem via deep learning when the data is processed locally at the edge and on constrained embedded devices. The aim of this approach is to eliminate the need for high-throughput links in wireless IoT toward high-end computing platforms (such as edge or cloud servers) for further processing. Although most published oversized models (trained at the cloud) cannot run on typical embedded devices, with optimizations, efficient embedded inference can be achieved while mitigating the need for raw data transmissions and thus preserving privacy. This approach provides several benefits in terms of latency, reliability, privacy and energy consumption. In this paper, the authors overview different optimization strategies to enable embedded deep learning inference on constrained devices and demonstrate via DRAFT October 2022 numerical results the potential benefits in terms of energy consumption vs. accuracy performance.

Finally, several trends in embedded deep learning use cases are discussed and valuable insights between design-and run-time metrics to predict model memory, storage and energy consumption together with model inference time are also provided.

In "Collaborative Inference for AI-Empowered IoT Devices", the authors build on the similar setup of IoT inference at the edge using neural networks, but taking a different perspective and investigating candidate collaboration strategies among several edge devices, which perform local data processing via neural networks. As opposed to the aforementioned paper focused on how to embed neural network processing on a single edge device, this work leverages the devices' communication capabilities to establish collaborative inference among multiple such devices.

The need to operate in different mobility and connectivity constraints is used as a motivating factor to consider multiple schemes, which can be roughly divided into methods where inference is done remotely, i.e., on the cloud, and those that infer at the edge. The authors identify the key characteristics of each strategy in terms of inference accuracy, communication latency, privacy, and connectivity requirements, providing a systematic comparison between existing approaches.

The paper is concluded with future research challenges and opportunities arising from the concept of collaborative inference.

As a first concrete application, in "Stand-Alone, Affordable IoT Satellite Terminals and Their Opportunistic use for Rain Monitoring", the authors investigate IoT communication via satellite, which represents an essential emerging technology for a variety of applications such as agriculture, financial technology, and homeland security in rural or isolated areas. In this paper, the system under study is a two-way low cost, stand-alone IoT satellite terminals communicating over geostationary satellites at Ku-band frequencies, already deployed in isolated areas in Africa and South America. On top of their designated use for specific IoT applications, the authors propose to exploit such terminals for opportunistic rain monitoring, based on the two-way link quality measurements that are being regularly collected for network management.

ACKNOWLEDGMENTS

The Guest Editorial team would like to express our sincere thanks to all the authors, the anonymous reviewers -who have bravely coped with holiday deadlines and the strict SI schedule -the Associate Editor-in-Chief, Dr. Maurice Khabbaz, and the Editor-in-Chief, Dr. Rath

Vannithamby, for all their kind efforts and precious help that led to this exciting Special Issue.

BIOGRAPHIES OF THE GUEST EDITORS

  "Surface-Based Techniques for IoT Networks: Opportunities and Challenges", an overview of recent techniques in the area of surface-based communications is provided by the authors, describing their high potential and advantages for IoT networks. Specifically, the use of reconfigurable intelligent surfaces is argued to bring significant advantages in terms of energy consumption, self-sustainment, programmability, and self-organization of IoT networks, possibly enabling the use of mmWaves for IoT, and the improvement of back-scattering techniques. The emerging paradigms in the area of surface-based communications are described, including holographic surfaces, active surfaces, and plug-and-play surfaces, and fitted into the IoT context. Some numerical examples are shown to substantiate the potential gains that surface-based communications can provide for IoT. Finally, open research directions are identified together with the main research challenges towards the implementation of energy-neutral, self-organizing surface-based IoT environments. The next three papers address important data processing challenges in the IoT context. In "Personalized Online Federated Learning for IoT/CPS: Challenges and Future Directions", the October 2022 DRAFT

E

  . Veronica Belmega (IEEE S'08, M'10, SM'20), co-Lead GE, is a full professor at the Université Gustave Eiffel and LIGM laboratory, Marne-la-Vallée, France, since May 2022. Previously, she was an associate professor (MCF HDR) with ENSEA graduate school (Sep. 2011 -Apr. 2022) and Deputy Director of ETIS laboratory (Jan. 2020 -Apr. 2022), Cergy, France. She received the M.Sc. (engineering) degree from the University Politehnica of Bucharest, Romania, in 2007, and the M.Sc. and Ph.D. degrees both from the University Paris-Sud 11, Orsay, France, in 2007 and 2010. From 2010 to 2011, she was a Post-doctoral researcher in a joint project between Princeton University, N.J., USA and Supélec, France. In 2015-2017, she was a visiting researcher at October 2022 DRAFT

  Guest Editorial: Pervasive, Efficient, and Smart Signal Processing for IoT E. Veronica Belmega, Senior Member, IEEE, Domenico Ciunzo, Senior Member, IEEE, Mérouane Debbah, Fellow, IEEE, Lutz Lampe, Senior Member, IEEE, Dinh-Thuy Phan Huy, Senior Member, IEEE, Tirza Routtenberg, Senior Member, IEEE, and Chau Yuen, Fellow, IEEE,

  The local October 2022 DRAFT measurements are transmitted to the cloud, where their transformation to rain-intensity estimates is carried out. Preliminary results using real measurements show a good alignment with direct rainfall measurements recorded by a rain gauge. Enabling this feature in two-way communication of IoT satellite terminals would be highly beneficial, as it will provide the ability to monitor the rain in areas where designated weather monitoring equipment is sparse, helping to improve local weather-based applications, and to better track the climate change globally. Finally, in "IoT-based Monitoring in Carbon Capture and Storage Systems", another important application is investigated: the task of Carbon capture and storage (CCS), which is critical for climate-change policies and strategies targeting global warming within the Paris Agreement. , this SI contains 8 peer-reviewed articles that address key challenges in the future IoT networks, with a focus on novel signal processing techniques both in terms of the IoT communication and the IoT data processing aspects. This SI is conceived to reach a large number of researchers, scientists, engineers, industry experts and practitioners (even those not familiar with SP) and help them advancing research practices by digesting these concepts. We thank our readers and we hope that each article in this SI provides novel ideas and a deeper understanding of the variety of SP challenges & tools in the deployment of pervasive, smart and efficient IoT systems.

	CONCLUSIONS
	To sum up
	The overarching technological requirements are well described in the strategic plans, yet several
	barriers exist to the technology widespread deployment, including improved cost-effectiveness
	and enhanced process integration. For the safe and reliable operation of large-scale CCS systems,
	the development of effective IoT-based monitoring tools to ensure flow assurance of CO2
	throughout the CCS value chain is crucial. Furthermore, reliable sensor measurements related
	to different transport parameters such as temperature, pressure, and flow across the process are
	essential to develop these methods. However, sensors are prone to errors due to inherent issues
	or environmental conditions which result in performance degradation of the overall monitoring
	system. Developing techniques for detecting anomalies in the measurements, identifying the
	faulty sensors and accommodating them with appropriately estimated data is one of the paramount
	requirements for the reliable operation of the CCS systems. In this context, the last article
	provides an overview of CCS monitoring and control requirements, emphasizing data-fusion
	DRAFT	October 2022

synergies. The state-of-the-art methods for sensor validation are overviewed and a clear roadmap to further deploy metering technologies for industrial needs is proposed.