Zeashan Hameed Khan
email: zeashan-hameed.khan@gipsa-lab.inpg.fr

Jean Marc Thiriet
email: jean-marc.thiriet@gipsa-lab.inpg.fr

Denis Genon-Catalot
email: denis.genon-catalot@iut-valence.fr

Wireless Network architecture for Diagnosis and Monitoring Applications

Keywords: Wireless network, Condition monitoring, Distributed diagnosis, Windmill farms I

This paper describes a distributed wireless network architecture for remote diagnosis and monitoring. A windmill farm is considered as a target application, which is further grouped into small clusters of windmills communicating with each other for the purpose of distributed diagnosis. Wireless links are commonly used for long-range communication in SCADA systems. In order to extend wireless networks for remote diagnosis applications, strict requirements on availability, robustness, reliability and performance have to be satisfied in order to meet industrial standards. This paper is intended to present a wireless network architecture for tele-operation of large industrial applications.

INTRODUCTION

Wireless networks are getting popular due to their flexibility of design, reduced costs, better power management, ease in maintenance, and easy deployment in remote and hardto-reach areas where conventional wired infrastructure is difficult to install. They have been successfully deployed in many industrial applications such as maintenance, monitoring, control, security, etc [START_REF] Doe | Industrial Wireless Technology for the 21st Century"-Report[END_REF]. Thus, the dependence of control and diagnosis of industrial systems on physically connected communication infrastructure has been relaxed by the developing wireless technologies, ranging from wireless sensor networks to broadband networks [START_REF] Doe | Industrial Wireless Technology for the 21st Century"-Report[END_REF], [START_REF] Särkimäki | Applicability of ZigBee technology to electric motor rotor measurements[END_REF].

In this paper, we consider the utilization of wireless networks for data transfer needs of condition monitoring, maintenance and remote diagnosis. Network architecture is proposed for condition dependent monitoring and fault diagnosis of large scale distant windmill installation. The focus is to address the issues faced by wireless communication systems to deliver diagnosis data on different supervision levels. Thus it is a network/diagnosis co-design problem. The level of intelligence associated with each diagnosis segment will result in overall distributed diagnosis scheme. In addition, the operational challenges are exacerbated when different RF links have to be used to satisfy the requirements of bandwidth, payload, delay, jitter, range, noise immunity, interference rejection and cost for communication [START_REF] Labiod | Wi-Fi, Bluetooth, ZigBee and WiMAX[END_REF], [START_REF] Rackley | Wireless Networking Technology: From Principles to Successful Implementation[END_REF].

II. CONTEXT

Diagnosis and Monitoring of industrial systems inherently requires a distributed approach, which offer some widely recognized advantages like the possibility of enhanced reliability due to component redundancy, improved performance by exploiting concurrency and better resource management and utilization through sharing. A local performance measure thus plays an important role in distributed hierarchy. The goal of communication, at different levels, for monitoring and supervision of windmill farms, can be achieved with low cost wireless network architecture, taking into account the constraints of noise, interference, bandwidth limitations and security [START_REF] Särkimäki | Wireless technologies in condition monitoring and remote diagnosis of electric drives; requirements and applications[END_REF], [START_REF] Salvadori | Monitoring and Diagnosis in Industrial Systems Using Wireless Sensor Networks[END_REF]. A compromise between requirements and limitations will result in a promising solution of the stated problem.

III. PROPOSITION OF A GENERAL ARCHITECTURE

The network based communication architecture can be visualized at various levels, each of which having their own constraints (end to end QoS, time delays, distance, network traffic, packet bursts, multi path effects, EMI etc). The terminology used to represent a cluster of windmills installed within a closer distance is termed as a 'Farm'. While a group of windmill farms is jointly labeled as a 'Park'.

A. Windmill Based Network (WBN)

The WBN is composed of sensors connected to each subsystem. The data acquisition of various components of a windmill is performed by a local supervisor node. The windmill based network is essentially a low power wireless mesh network. Each sub component of windmill e.g. gear box, generator, nacelle etc have sensors attached to them, which send data to the local supervisor node through a wireless network. The parameters of interest include vibration data of gear box and generator, health indicator signals of electrical subsystems, generated voltage, status of AC/DC/AC block, Gear box oil temperature etc.

B. Farm Based Network (FBN)

This network will be responsible for the communication within the various windmills of a farm for control and

C. Inter-Farm (Park) Based Network (PBN)

The communication task between various distributed windmill farms of a region is taken care by an inter-farm network. Fig. 1 shows a PBN composed of many FBNs. One of the tasks for this level of communication can be as simple as to exchange diagnosis information to ensure interrupt free voltage stabilization to the grid with optimized use of windmills with changing electrical loads.

D. Distant Supervision Network (DSN)

This level of communication network will involve long distance communication between windmill parks that are distributed usually far away (especially for the case of offshore windmills) and the supervisor at the diagnosis centre.

IV. NETWORK BASED DISTRIBUTED DIAGNOSIS

The distributed diagnosis concept of windmill farms is based on the network resources available at different levels. Typically, faults in windmill subsystems develop slowly e.g. contamination of oil in gear box, rise in temperature of generator windings, excessive nacelle vibration due to structural faults etc. So the measurement data from sensors related to condition monitoring may not be time critical. The measurement data can be processed and analyzed at different levels of the network. The amount of data to be transferred can be reduced sufficiently by emphasizing local analysis.

The measurement parameters traditionally used in vibration measurements are displacement (µm), velocity (mm/s) and acceleration (m/s 2 ). ISO 10816-3 classifies vibration velocity amplitudes in the frequency range of 10 to 1000 Hz as a function of turbine output [7]. Impact-like phenomena, such as bearing and gear faults, friction and insufficient lubrication, can be studied by means of acceleration measurements more effectively than using velocity measurements. For wind turbines, gear manufacturers have defined typical frequency range of 10 Hz to 100 Hz. The frequency information of vibrations is continuously monitored at a higher rate and a local check is maintained on permissible amplitude of the vibration.

In addition, sensor faults of individual windmills can be detected and a collaborative monitoring system at the farm based network (FBN) will diagnose and improve farm based electricity generation. For example, in case of anemometer fault at one windmill, farm supervisor node can estimate and still maneuver the faulty windmill for optimum wind direction based on data received from other anemometers of the farm windmills. In this work, an approach for distributed hierarchic network architecture is proposed for the applications of tele-diagnosis and distant monitoring of a distributed windmill energy generation system. Network simulation with different heterogeneous architectures and protocols is being performed to visualize the constraints, taking into account geographic terrain for long distance communication. In addition, diagnosis tasks at each level of communication need to be allocated for distributed monitoring. This concept can be extended to energy generation from heterogeneous sources (e.g. Wind, Solar, Fuel cells, Hydroelectric etc.) and their collaborative monitoring by utilizing heterogeneous wireless networks [START_REF] Jia | Study on Network Selection for Next-Generation Heterogeneous Wireless Networks[END_REF].

  This work is sponsored by HEC-SFERE grant under OSS-2006 Program diagnosis purpose. The exchange of data is between individual windmills as well as with the farm supervisor node, which is responsible for the data management and diagnosis decision at the farm level. The information of interest may include the wind direction relative to windmill farm, voltage generated by each individual turbine and condition monitoring parameters to decide and optimize farm productivity.

Figure. 1 .

 1 Figure. 1. General hierarchical Network architecture V. CONCLUSION


Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

New Finite-Time and Fast Converging Observers with a Single Delay

Keywords: Reduced order observer, finite time, discrete measurements

We provide new reduced order observer designs for a key class of nonlinear dynamics. When continuous output measurements are available, we prove that our observers converge in a fixed finite time in the absence of perturbations, and we prove a robustness result under uncertainties in the output measurements and in the dynamics, which bounds the observation error in terms of bounds on the uncertainties. The observers contain a dynamic extension with only one pointwise delay, and they use the observability Gramian to eliminate an invertibility condition that was present in earlier finite time observer designs. We also provide analogs for cases where the measurements are only available at discrete times, where we prove exponential input-to-state stability. We illustrate the advantages of our new observers using a DC motor dynamics.

I. INTRODUCTION

Finite and fixed time observers present an obvious advantage by providing estimates of the state variables of systems in finite time [START_REF] Lopez-Ramirez | Finitetime and fixed-time observer design: Implicit Lyapunov function approach[END_REF]. Fixed time observers are special cases of finite time observers where the finite convergence time is independent of the initial state. Several types of fixed time observers are available. Some use discontinuous dynamic extensions, such as [START_REF] Raff | An observer that converges in finite time due to measurement-based state updates[END_REF]. Others use time-varying high gains [START_REF] Holloway | Prescribed-time observers for linear systems in observer canonical form[END_REF]. Others use delays to solve output feedback stabilization problems or homogeneity conditions [START_REF] Silm | A note on distributed finite-time observers[END_REF].

In earlier works, e.g., [START_REF] Engel | A continuous time observer which converges in finite time[END_REF], [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], and [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF], fixed time observers are designed using dynamic extensions and a delay τ . The designs rely on the invertibility of a matrix which can be problematic because it is not invertible for all τ 's and because, when it exists, the inverse can contain big terms when the delays are close to values where it is not invertible. We refer to such delays as artificial delays, because although they are not present in the given dynamics, they occur in the observers. The work [START_REF] Mazenc | Asymptotic stabilization of linear time-varying systems with input delays via delayed static output feedback[END_REF] provides an exact calculation of state variables using a formula with several delays and the inversion of a matrix, which can also be problematic because it may be noninvertible for some delay values.

To overcome these shortcomings, we revisit the problem of estimating the state variables of a system in finite time using an artificial delay. For a family of unperturbed systems that are affine in the unmeasured state, we propose a new family of observers that converge in fixed time when continuous output measurements are available. The observers only estimate unmeasured variables, and so are reduced order. A key aspect of the observer design we propose is that it relies on the introduction of only one pointwise delay, which can be arbitrarily chosen. The delay is the fixed convergence time. Our fundamental tool is the observability Gramian. We also establish a robustness result for the observers with respect to additive disturbances on the output measurement and on the dynamics. We then provide an analog for cases where the measurements are only available at discrete instants, using the key ideas of the pioneering paper [START_REF] Karafyllis | From continuous-time design to sampleddata design of observers[END_REF]. In this case, the exponential convergence rate is proportional to the logarithm of the size of the largest sampling interval.

We use standard notation, which we simplify when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. The standard Euclidean 2-norm, and its induced matrix norm, are denoted by and| • | S denotes the essential supremum over any set S. We use I to denote an identity matrix of arbitrary dimension.

| • |, | • | ∞ is the usual L ∞ sup norm,

II. STUDIED SYSTEM

We consider the class of continous-time systems

χ(t) = M χ(t) + Ψ(N χ(t), t) + δ 1 (t) Y (t) = N χ(t) + δ 2 (t) (1)
where χ is valued in R n , the output Y is valued in R q , the time dependence in Ψ can represent the effects of a control, and the locally essentially bounded measurable functions δ 1 and δ 2 represent disturbances. We assume that (1) is forward complete, and that Ψ is locally Lipschitz. We also assume that the pair (M, N ) is observable and that N has full rank.

From [6, pp. 304-306], we can deduce that there is a linear change of coordinates which yields the system

       ξ1 (t) = A 1 ξ 1 (t) + F 1 (Y (t) -3 (t), t) + 1 (t) ξ2 (t) = A 2 ξ 1 (t) -kξ 2 (t) + F 2 (Y (t) -3 (t), t) + 2 (t) Y (t) = ξ 2 (t) + 3 (t) (2)
which is affine in the unmeasured variable ξ 1 , where ξ 1 is valued in R n-q , ξ 2 is valued in R q , A 1 ∈ R (n-q)×(n-q) , A 2 ∈ R q×(n-q) , the pair (A 1 , A 2 ) is observable, and k > 0 is a constant such that A 1 + kI is invertible. Then F 1 and F 2 are locally Lipschitz, and the measurable locally essentially bounded functions i represent disturbances. Although the -kξ 2 (t) term can be incorporated into the function F 2 in (2), we keep it separate to facilitate the analysis that follows, and we write ξ 2 (t) as Y (t)-3 (t) in the F i 's in [START_REF] Engel | A continuous time observer which converges in finite time[END_REF] to facilitate our study of the key special case where 3 is the zero function.

Changing the tuning parameter k can be done by changing F 2 . One can always choose it such that A 1 +kI is invertible, by taking k larger than the spectral radius of A 1 .

III. CONTINUOUS MEASUREMENT CASES A. Assumptions and Statement of Theorem

In this section, we construct an observer for the system (2) which will provide our fixed time convergence, under the following assumption: Assumption 1: Either (i) there are two constants K 1 ≥ 0 and K 2 ≥ 0 such that

|F i (a, t) -F i (b, t)| ≤ K i |a -b| for i = 1, 2 (3)
for all t ≥ 0 and a and b in R q or (ii) 3 (t) = 0 for all t ≥ 0.

Let us introduce any positive constant τ and the function λ : R → R q×(n-q) defined by

λ(r) = A 2 (A 1 + kI) -1 I -e (A1+kI)r (4)
which is well-defined because we choose k > 0 such that the matrix A 1 + kI is invertible. We also use the matrix

S = 0 -τ λ(m) λ(m)dm ∈ R (n-q)×(n-q) . (5)
In the appendix below, we prove that since the pair

(A 1 , A 2 )
is observable, S is invertible. Then we define the matrices

N = 0 -τ λ(m) dm ∈ R (n-q)×q , R = S -1 N ∈ R (n-q)×q
, and

H = ((A 1 + kI) -1 ) A 2 ∈ R (n-q)×q (6)
and we introduce the dynamic extension

         ˙ξ 1 (t) = A 1 ξ1 (t) + F 1 (Y (t), t) ˙ξ 2 (t) = A 2 ξ1 (t) -k ξ2 (t) + F 2 (Y (t), t) ψ1 (t) = -kψ 1 (t) + H[Y (t) -ξ2 (t)] ψ2 (t) = -(A 1 + 2kI)ψ 2 (t) + H[Y (t) -ξ2 (t)] ( 7
)
where ξ1 is valued in R n-q , ξ2 is valued in R q , and ψ 1 and ψ 2 are valued in R n-q . Finally, in terms of the functions

∆ * (p) = e A1p -e -kpI and ∆ * * (p, q) = q p e A1(p-) 1 ( )d (8)
where k is from (2), we let ‡ be the R n-q -valued function

‡ (t) = S -1 t t-τ λ(s -t) H ∆ * (t -s)∆ * * (s, t)ds -S -1 t t-τ λ(s -t) - t s e k(m-t) A 2 ∆ * * (m, s)dm + t s e k( -t) 2 ( )d ds. (9)
In terms of the preceding notation and the functions

i (m) = K i | 3 (m)| + | i (m)| (10)
for i = 1, 2 and the constants

S = |S -1 | and c (τ ) = τ S|A 2 |e |A1|τ +S A 2 (A 1 + kI) -1 e |A1|τ + 1 e |A1|τ , (11)
our first theorem is then as follows: Theorem 1: Let (2) satisfy Assumption 1. Then, with the preceding notation, when 3 is the zero function, we have

ξ 1 (t) = ξ e (t) + ‡ (t) (12)
for all t ≥ τ , where

ξ e (t) = ξ1 (t) + R(ξ 2 (t) -ξ2 (t)) + S -1 e -kτ ψ 1 (t -τ ) -ψ 1 (t) + S -1 ψ 2 (t) -e -(A 1 +2kI)τ ψ 2 (t -τ ) . (13)
Also, if F 1 and F 2 satisfy (3) and 3 = 0, then

ξ 1 (t) = ξ e (t) + (t) (14)
holds for all t ≥ τ where is a function such that

| (t)| ≤ c (τ ) t t-τ |λ(s -t)| t s 1 (m)dmds + S t t-τ |λ(s -t)| t s 2 (m)dmds + S|H| 1+e |A 1 +2kI|τ t t-τ | 3 (s)|ds (15)
for all t ≥ τ , Remark 1: A key feature of the observer ξ e we propose is that it incorporates only one delay τ . This delay can be any positive value because for any τ > 0, S is invertible.

Remark 2: Since

|λ(r)| ≤ |A 2 (A 1 + kI) -1 | 1 + e |A1+kI||r| (16)
for all r ∈ R, we deduce that there are two constants

c ≥ 0 and c ♦ ≥ 0 such that | ‡ (t)| ≤ c |( 1 , 2 )| [t-τ,t] and | (t)| ≤ c ♦ |( 1 , 2 , 3 )| [t-τ,t] for all t ≥ τ .

B. Proof of Theorem 1

We introduce the variables y = Y -ξ2 and

∆ i (t) = F i (Y (t) -3 (t), t) -F i (Y (t), t) and x i (t) = ξ i (t) -ξi (t) for i = 1, 2. (17)
Then simple calculations based on (2) and [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF] give

   ẋ1 (t) = A 1 x 1 (t) + ∆ 1 (t) + 1 (t) ẋ2 (t) = A 2 x 1 (t) -kx 2 (t) + ∆ 2 (t) + 2 (t) y(t) = x 2 (t) + 3 (t). (18)
Here and in the sequel, all equalities and inequalities hold for all t ≥ 0, unless otherwise indicated. By applying variation of parameters to (18), we obtain

x 1 (t) = e A1(t-s) x 1 (s) + t s e A1(t-m) [∆ 1 (m) + 1 (m)]dm (19)
and

x 2 (t) -e -k(t-s) x 2 (s) = ρ 1 (t, s) +A 2 (A 1 + kI) -1 e A1(t-s) -e -k(t-s)I x 1 (s) (20)
for all s ≥ 0 and t ≥ s, where

ρ 1 (t, s) = t s e k( -t) [∆ 2 ( ) + 2 ( )]d + t s e k(m-t) A 2 m s e A1(m-) [∆ 1 ( ) + 1 ( )]d dm, (21)
and where we used the fact that

t s e k(m-t) A 2 e A1(m-s) x 1 (s)dm = e k(s-t) A 2 t s e (A1+kI)(m-s) dmx 1 (s) = e k(s-t) A 2 (A 1 + kI) -1 e (A1+kI)(t-s) -I x 1 (s). (22)
According to (19), we have

x 1 (s) = e A1(s-t) x 1 (t) - t s e A1(s-m) [∆ 1 (m) + 1 (m)]dm, (23)
and our formulas (4) and ( 8) give λ(s -t) = H ∆ * (ts)e A1(s-t) . Hence, we can substitute (23) into (20) to obtain

λ(s -t)x 1 (t) = x 2 (t) -e -k(t-s) x 2 (s) + ρ 2 (t, s) (24) where ρ 2 (t, s) = -ρ 1 (t, s) +H ∆ * (t -s) t s e A1(s-m) [∆ 1 (m) + 1 (m)]dm. ( 25
)
By left multiplying both sides of (24) by λ(s-t) , we obtain

λ(s -t) λ(s -t)x 1 (t) = λ(s -t) ρ 2 (t, s) +λ(s -t) x 2 (t) -e -k(t-s) λ(s -t) x 2 (s). ( 26
)
By integrating (26) with respect to s over [t-τ, t], we obtain

t t-τ λ(s -t) λ(s -t)dsx 1 (t) = t t-τ λ(s -t) dsx 2 (t) - t t-τ e -k(t-s) λ(s -t) x 2 (s)ds + t t-τ λ(s -t) ρ 2 (t, s)ds for all t ≥ τ. ( 27
)
Hence, our choices in ( 5)-( 6), and the invertibility of S, give

x 1 (t) = Rx 2 (t) + S -1 t t-τ λ(s -t) ρ 2 (t, s)ds -S -1 t t-τ e -k(t-s) λ(s -t) x 2 (s)ds. (28)
Using the formula for λ from (4), we obtain

x 1 (t) = Rx 2 (t) + S -1 t t-τ λ(s -t) ρ 2 (t, s)ds -S -1 t t-τ e -k(t-s) I -e (A 1 +kI)(s-t) Hx 2 (s)ds (29)
with H defined in [START_REF] Luenberger | Introduction to Dynamic Systems[END_REF]. This equality can be rewritten as

x 1 (t) = Rx 2 (t) + S -1 t t-τ λ(s -t) ρ 2 (t, s)ds -S -1 t t-τ e -k(t-s) Hx 2 (s)ds + S -1 t t-τ e -(A 1 +2kI)(t-s) Hx 2 (s)ds. (30)
Since ξ 2 (s) = Y (s) -3 (s), we have Y -ξ2 = x 2 + 3 , so we deduce from ( 7) and (30) that

x 1 (t) = Rx 2 (t) -S -1 ψ 1 (t) -e -kτ ψ 1 (t -τ ) + S -1 ψ 2 (t) -e -(A 1 +2kI)τ ψ 2 (t -τ ) + (t), (31)
where (t) = S -1 t t-τ λ(s -t) ρ 2 (t, s)ds + t t-τ e -k(t-s) H 3 (s)ds - t t-τ e -(A 1 +2kI)(t-s) H 3 (s)ds . (32)
Hence, ( 13) and (17) give ξ 1 = ξ e + for all t ≥ τ . Recalling the formula for ρ 2 in (25) and (17) and our Lipshitz condition (3) on the F i 's, it follows that (32) satisfies

| (t)| ≤ S t t-τ |λ(s -t)|ρ 1 (t, s)|ds +S t t-τ |λ(s -t)||H |J 1 (t, s)ds +S|H| t t-τ e -k(t-s) | 3 (s)|ds + S|H| t t-τ e |A 1 +2kI|(t-s) | 3 (s)|ds (33)
and so also

| (t)| ≤ S t t-τ |λ(s -t)|ρ 1 (t, s)|ds +S t t-τ |λ(s -t)||H |J 2 (t, s)ds +S|H| t t-τ J 3 (t -s)| 3 (s)|ds, where (34) J 1 (t, s) = |∆ * (t -s)| t s e |A1|(m-s) [|∆ 1 (m)| + | 1 (m)|]dm (35)
and the function ∆ * was defined in [START_REF] Mazenc | Continuous discrete sequential observers for time-varying systems under sampling and input delays[END_REF], and where

J 2 (t, s) = e |A1|τ + 1 t s e |A1|(m-s) [K 1 | 3 (m)| + | 1 (m)|]dm (36)
and J 3 (r) = e -kr + e |A 1 +2kI|r , and where S is defined in [START_REF] Mazenc | Asymptotic stabilization of linear time-varying systems with input delays via delayed static output feedback[END_REF]. Also, when 3 = 0, we can use (17) to get ∆ 1 = ∆ 2 = 0, so our formula (9) for ‡ and ( 21) and (25) give = ‡ when 3 = 0. Hence, since the right side of ( 15) is an upper bound for the right side of (34), this allows us to conclude.

IV. DISCRETE MEASUREMENTS CASES A. Assumptions and Statement of Theorem

Although the observer from the preceding section enjoys fixed time convergence and robustness properties, it requires continuous measurements of the output, which might not always be available in practice. Therefore, in this section, we consider the case where the variables are only measured at discrete instants. We propose an observer which combines the one of Theorem 1 and the technique of [START_REF] Karafyllis | From continuous-time design to sampleddata design of observers[END_REF].

Let t j be a sequence such that t 0 = 0 and such that there are two constants T > 0 and T > T such that T ≤ t j+1 -t j ≤ T for all j ≥ 0.

(37)

We continue the notation from Section III except we consider

       ξ1 (t) = A 1 ξ 1 (t) + F 1 (ξ 2 (t), t) + 1 (t) ξ2 (t) = A 2 ξ 1 (t) -kξ 2 (t) + F 2 (ξ 2 (t), t) + 2 (t) Y (t j ) = ξ 2 (t j ) + 3 (t j ) for all j ≥ 0, (38)
under the assumption that F 1 and F 2 satisfy (3) and where k is selected as in Section II. We also use these constants:

ς 1 = S|A2|e |A 1 |τ k 2 0 -τ |λ(s)|(e ks -sk -1)ds ς 2 = S k 0 -τ |λ(s)|(1 -e ks )ds ς 3 = Sτ e |A1|τ |H | 0 -τ |λ(s)|(e |A1|τ + e -ks )ds ς 4 = S|H| 1 k (1 -e -kτ ) + τ e |A 1 +2kI|τ
(39

)
where H is from (6) as before, and τ satisfies the requirements from Section II. We introduce the dynamic extension

                       ˙ξ 1 (t) = A 1 ξ1 (t) + F 1 (ω(t), t) ˙ξ 2 (t) = A 2 ξ1 (t) -k ξ2 (t) + F 2 (ω(t), t) ψ1 (t) = -kψ 1 (t) + H[ω(t) -ξ2 (t)] ψ2 (t) = -(A 1 + 2kI)ψ 2 (t) + H[ω(t) -ξ2 (t)] ω(t) = A 2 ξ e (t) -kω(t) + F 2 (ω(t), t) for all t ∈ [t j , t j+1 ) and j ≥ 0 ω(t j ) = Y (t j ) for all j ≥ 0 (40)
where ξ e is defined as in [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF]. We prove:

Theorem 2: Let the constant T in (37) be such that

T µ < 1, where (41)
µ = |A 2 |q 1 +k+K 2 and q 1 = (ς 1 +ς 3 )K 1 +ς 2 K 2 +ς 4 (42)
using the constants (39). Then we can find positive constants a 1 and a 2 such that all solutions of (38) and (40) satisfy

|ξ 1 (t) -ξ e (t)| ≤ a 1 |ξ 2 -ω| [r-2τ -T ,r] e ln(T µ) τ +T (t-r) +a 2 sup ∈[r-T -2τ,t] [| 1 ( )| + | 2 ( )| + | 3 ( )|] (43)
for all r ≥ 2τ + T and t ≥ r + τ . Remark 3: The inequality ( 43) is of ISS type because

ln(T µ) τ +T < 0, (44)
by ( 41). Moreover, since µ is independent of T , the left side of (44) converges to -∞ as T → 0 + . Therefore, we can have arbitrarily large rates of convergence of the estimation error |ξ 1 (t) -ξ e (t)| to 0 when the i 's are zero, by choosing the sample times t i such that T is small enough.

B. Proof of Theorem 2

Our proof will use the variables

ω(t) = ξ 2 (t) -ω(t), and x i (t) = ξ i (t) -ξi (t) and κ i (t) = F i (ξ 2 (t), t) -F i (ω(t), t) for i = 1, 2. (45)
Then

ω -ξ2 = ξ 2 -ω -ξ2 = x 2 -ω, so we obtain                      ẋ1 (t) = A 1 x 1 (t) + κ 1 (t) + 1 (t) ẋ2 (t) = A 2 x 1 (t) -kx 2 (t) + κ 2 (t) + 2 (t) ψ1 (t) = -kψ 1 (t) + Hx 2 (t) -Hω(t) ψ2 (t) = -(A 1 + 2kI)ψ 2 (t) + Hx 2 (t) -Hω(t) ω(t) = A 2 [ξ 1 (t)-ξ e (t)]-k ω(t)+κ 2 (t)+ 2 (t)
for all t ∈ [t j , t j+1 ) and j ≥ 0 ω(t j ) = -3 (t j ) for all j ≥ 0.

(46)

We also use the R n-q -valued variables

γ 1 (t) = -S -1 t t-τ λ(s -t) H ∆ * (t -s)J a (t, s)ds -S -1 t t-τ λ(s -t) t s e k(m-t) A 2 J a (s, m)dmds -S -1 t t-τ λ(s -t) t s e k( -t) [κ 2 ( ) + 2 ( )]d ds, (47)
γ 2 (t) = -S -1 t t-τ e -k(t-s) Hω(s)ds + S -1 t t-τ e (A 1 +2kI)(-t+s) Hω(s)ds, (48)
and

x a (t) = Rx 2 (t) -S -1 ψ 1 (t) -e -kτ ψ 1 (t -τ ) + S -1 ψ 2 (t) -e -(A 1 +2kI)τ ψ 2 (t -τ ) , (49)
where

J a (s, m) = m s e A1(m-) [κ 1 ( ) + 1 ( )]d . (50)
The rest of the proof of the theorem consists of two steps. In the first step, we prove that the preceding variables satisfy

ξ 1 (t) = ξ e (t) + γ 1 (t) + γ 2 (t) (51)
for all t ≥ τ . In the second step, we bound γ 1 (t) + γ 2 (t) by the right side of (43) for suitable values of r and t. First Step. Since the (x 1 , x 2 )-dynamics of (46) agree with the first two equations of (18) except with the ∆ i 's replaced by the κ i 's, the same reasoning that led to (30) gives

x 1 (t) = Rx 2 (t) -S -1 t t-τ e -k(t-s) Hx 2 (s)ds +S -1 t
t-τ e -(A 1 +2kI)(t-s) Hx 2 (s)ds + γ 1 (t).

(52) Also, by applying the method of variation of parameters separately to the ψ 1 and ψ 2 dynamics in (46), we obtain t t-τ e -k(t-s) Hx 2 (s)ds = ψ 1 (t) -e -kτ ψ 1 (t -τ ) + t t-τ e -k(t-s) Hω(s)ds

(53) and t t-τ e -(A 1 +2kI)(t-s) Hx 2 (s)ds = ψ 2 (t) -e -(A 1 +2kI)τ ψ 2 (t -τ ) + t t-τ e (A 1 +2kI)(-t+s) Hω(s)ds (54)
for all t ≥ τ . By combining ( 52)-(54), we obtain

x 1 (t) = Rx 2 (t) -S -1 ψ 1 (t) -e -kτ ψ 1 (t -τ ) + t t-τ e -k(t-s) Hω(s)ds + γ 1 (t) + S -1 ψ 2 (t) -e -(A 1 +2kI)τ ψ 2 (t -τ ) + t t-τ e (A 1 +2kI)(-t+s) Hω(s)ds (55)
for all t ≥ τ Hence, our choices (47)-(49), and our choice of x 1 in (45) and our formula [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF] for ξ e , give

x 1 = x a + γ 1 + γ 2 and x 1 -x a = ξ 1 -ξ e . (56)
J 4 (s, m) = m s e |A1|τ [|κ 1 ( )| + | 1 ( )|]d . (58)
Moreover, our choices of the κ i 's in (45) and (3) give |κ 2 ( )| ≤ K 2 | w( )| when s ≤ ≤ t, and so also

J 4 (s, m) ≤ (m -s)e |A1|τ (K 1 | w| [s,m] + | 1 | [s,m] ) (59)
when s ≤ m ≤ t. Therefore, by upper bounding the right side of (57) and then collecting the coefficients of | w| [t-τ,t] and | i | [t-τ,t] for i = 1, 2 in the result, it follows from our choices of the constants ς i in (39) that

|ξ 1 (t) -ξ e (t)| ≤ q 1 | w| [t-τ,t] +q 2 (| 1 | [t-τ,t] +| 2 | [t-τ,t] ) for all t ≥ τ, (60)
where q 1 is from (42) and q 2 = max{ς 1 + ς 3 , ς 2 }.

the output in terms of the state x. Moreover, [7, Section 5.2] produces large coefficients in the final estimation error under discrete time measurements for small τ > 0 values. Hence, we believe that the observer designs from this work offer potential advantages over previously available observers.

B. Illustration of Theorem 2

We next illustrate the sampled output case using the following pendulum dynamics that was studied, e.g., in [START_REF] Dawson | Integrator backstepping control of a brush dc motor turning a robotic load[END_REF]:

   ẋ1 (t) = x 2 (t) ẋ2 (t) = -sin(x 1 (t)), y(t j ) = x 1 (t j ) + 3 (t j ) (72)
with x 1 (t) and x 2 (t) both valued in R. This has the form (38) with

ξ 1 = x 2 , ξ 2 = x 1 , A 1 = 0, F 1 (s, t) = -sin(s), A 2 = 1, F 2 (s, t) = ks, 1 = 2 = 0,

VI. CONCLUSION

We solved significant observer design problems for systems that are linear in the unmeasured variables and with continuous but perturbed measurements, using a novel Gramian approach which eliminates an invertibility condition on the fixed convergence time from prior works such as [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF], while only requiring a single delay in the observer. We also provided an analog for cases where only discrete measurements are available and where we instead get arbitrarily large exponential convergence rates. Our examples illustrated the potential advantages of our new approaches and a tradeoff between sample rates in the measurements and the parameter choices in our observer. We aim to obtain extensions for systems with state delays or measurement delays.

APPENDIX: INVERTIBILITY OF THE MATRIX S

Let us prove that the matrix S defined in ( 5) is invertible, which was needed for the observer designs from our theorems. Let V ∈ R n-q be a vector such that SV = 0. Then 0 -τ V λ(m) λ(m)V dm = 0.

(A.1)

As an immediate consequence, we get λ(m)V = 0 for all m ∈ [-τ, 0]. It follows that for all integers j > 0, λ (j) (0)V = 0. Also, simple calculations give λ (j) (0) = -A 2 (A 1 +kI) j-1 . Thus A 2 (A 1 +kI) l V = 0 for all integers l ≥ 0. Using the fact that these equalities are equivalent to

A 2 l j=0 C j l A j 1 V = 0 (A.2)
for suitable nonzero integers C j l , we deduce that A 2 A l 1 V = 0 for all integers l ≥ 0, by induction on l. Since (A 1 , A 2 ) is observable, it follows that V = 0. This allows us to conclude.

  and any positive constant k, by subtracting and adding kx 1 (t) in the x 1 dynamics. Hence, Theorem 2 applies for any positive values k and τ if the sup T of the sampling intervals satisfies T < 1/µ, where µ was defined by (42). For example, by choosing k = 1.2 and τ = 0.6, we get the bound T < 1/µ = 0.0159168. On the other hand, smaller values of T produce faster convergence rates in the exponential input-to-state stability estimate in the theorem; see Remark 3 above. The upper bound 1/µ depends on the choices of k and τ . In Figure 2, we used Mathematica to plot 1/µ on the vertical axis, as a function of k and τ for the preceding values of the other parameters. Our figure shows how smaller k values for given choices

Fig. 1 .

 1 Fig. 1. Bound 1/µ on T for Values k ∈ [1.5, 10.5] and τ ∈ [0.01, 1]. of τ lead to larger values of the upper bound for T . This illustrates the tradeoffs between the choices of τ and k and the sample interval bounds T .

Supported by NSF Grants 1711299 and 2009659 (Malisoff). F. Mazenc is with

By combining (46) and (60), and recalling (3), we get

for all t ∈ [t j , t j+1 ) and all j ≥ 0 when t ≥ τ , where

for all t ∈ [t j , t j+1 ), we deduce that, for all t ≥ T + τ ,

for all t ∈ [t j , t j+1 ) and j ≥ 0 when t ≥ r and r ≥ T + τ , where T = T (|A 2 |q 2 + 1) + 1, the last inequality in (63) used the fact that |ω(t j )| = | 3 (t j )| for all j ≥ 0, and µ is from (42). Using (41), it follows from applying the trajectory based approach from [10, Lemma 1] to the function w

for all t ≥ r with

The theorem now follows by using (64) to upper bound the first right side term in (60).

V. ILLUSTRATIONS A. Illustration of Theorem 1

Consider this model for a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor [START_REF] Dawson | Integrator backstepping control of a brush dc motor turning a robotic load[END_REF]:

and where m is the mass of the link, J is the rotor inertia, L 0 is the length of the link, M 0 is the mass of the load, B 0 is the viscous friction coefficient at the joint, R 0 is the radius of the load, G is the gravitational constant, q(t) is the position of the load (which is the angular motor position), I(t) is the motor armature current, the coefficient K τ characterizes the electromagnetic conversion of armature current to torque, R is the armature resistance, L is the armature inductance, K B is the back-emf coefficient, and V e is the input current voltage. All constants in (66) are positive, and we assume that perturbed values of q are available for measurement.

The preceding model has been studied extensively. For instance, see [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF] for continuous-discrete observers for (66), and [START_REF] Mazenc | Continuous discrete sequential observers for time-varying systems under sampling and input delays[END_REF] for full order observers for (66) with sampling and input delays. However, we believe that the problem we solve in this subsection of building reduced order observers for (66) with arbitrarily small fixed convergence times τ under continuous measurements was open.

By also allowing additive uncertainties in the model (66) and in the measurements, we obtain the dynamics

where Adopting the notation

, and ε 3 (t) = δ 2 (t), we can rewrite the system (67) as

for a constant k > 0 that will be specified.

Then the notation of Sections II-III produces the choices

Then, Theorem 1 provides the exact value

for all t ≥ τ for a function satisfying (15). The preceding observer contrasts significantly with the fixed time observer for (67) that was presented in [7, Section 5.2], whose fixed convergence time τ is required to be such that e -Hτ -e -τ A is invertible where H = A + L A C for a suitable matrix L A , and where C is from the representation y = Cx of