Stanislav Aranovskiy
email: stanislav.aranovskiy@centralesupelec.fr.

Denis Efimov

Dmitry Sokolov

Jian Wang

Igor Ryadchikov

Alexey Bobtsov

State estimation for a locally unobservable parameter-varying system: one gradient-based and one switched solutions

Keywords: parameter-varying systems, switched observer, nonlinear observer, switched Lyapunov function

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

State estimation for a locally unobservable parameter-varying system: one gradient-based and one switched solutions Stanislav Aranovskiy * Denis Efimov * * Dmitry Sokolov * * * Jian Wang * * * * Igor Ryadchikov † Alexey Bobtsov ‡

INTRODUCTION

State estimation for parameter-varying and time-varying systems is a long-standing problem arising in numerous applications. Whereas a solution based on Kalman filtering has been proposed by H. [START_REF] Cox | On the estimation of state variables and parameters for noisy dynamic systems[END_REF], see Cox (1964), nowadays this problem still attracts researchers' attention; particularly, it is due to its importance to the state estimation of nonlinear systems, see [START_REF] Bastin | Stable adaptive observers for nonlinear time-varying systems[END_REF] and [START_REF] Nijmeijer | New directions in nonlinear observer design[END_REF].

Many results are available under the assumption that the system to be estimated is always observable, e.g., the adaptive estimation for a parametrically uncertain system proposed by [START_REF] Zhang | Adaptive observer for multiple-inputmultiple-output (MIMO) linear time-varying systems[END_REF], the set-membership approach by [START_REF] Jaulin | Nonlinear bounded-error state estimation of continuous-time systems[END_REF] and interval observers by [START_REF] Efimov | Interval state observer for nonlinear time varying systems[END_REF], and the finite-/fixed-time estimation studied by [START_REF] Rios | Time-Varying Parameter Identification Algorithms: Finite and Fixed-Time Convergence[END_REF]. However, this assumption is violated in some applications: for example, Liu et al. (2014) have shown that a class of power systems is not observable for some combinations of input signals. Nevertheless, if uniform complete observability may be assumed (see [START_REF] Rugh | Linear system theory[END_REF]; [START_REF] Jazwinski | Adaptive-gain second-order sliding mode observer design for switching power converters[END_REF]; [START_REF] Besançon | Nonlinear Observers and Applications[END_REF]), then observers with time-varying or parameter-varying gains can be designed. Particularly, these gains can be chosen based on the observability Gramian of the system ensuring exponential convergence, see [START_REF] Rugh | Linear system theory[END_REF]. The drawback of this approach is that the gains have to be computed online as a solution of a nonlinear matrix differential equation that can be undesirable for embedded real-time applications.

One class of parameter-varying observers, where the gains can be precomputed off-line, is the switched observers.

The switched observers are typically applied for state estimation of systems operating in a finite number of modes, where the commutation among these modes obeys a supervisory signal. The conventional approach to the switched observer design is the common Lyapunov function as described, for example, by [START_REF] Alessandri | Switching observers for continuous-time and discrete-time linear systems[END_REF]. However, the existence of a common Lyapunov function is a restrictive assumption that does not hold if some of the operation modes are not observable. To relax this assumption, one can utilize the idea of the average dwell time (ADT), where the stability of the switched system under the ADT assumption has been studied by [START_REF] Hespanha | Stability of switched systems with average dwell-time[END_REF]. Examples of switched observers with the ADT assumptions can be found, e.g., in the works by [START_REF] Nouailletas | LMI design of a switched observer with model uncertainty: Application to a hysteresis mechanical system[END_REF] and [START_REF] Pettersson | Designing Switched Observers for Switched Systems Using Multiple Lyapunov Functions and Dwell-Time Switching[END_REF]. In addition, [START_REF] Tanwani | Observability for Switched Linear Systems: Characterization and Observer Design[END_REF] have formulated necessary and sufficient conditions for observability of switched systems with unobservable modes.

Novelty and Contribution. In this paper, we consider the state estimation problem for a linear parameter-varying system that is unobservable for certain values of the varying parameters. Our research is motivated by the recent work of [START_REF] Ryadchikov | Differentiator-based velocity observer with sensor bias estimation : an inverted pendulum case study[END_REF] where it has been shown that a simple mechanical system becomes unobservable in some operating modes if the position sensor has a constant bias. Drawing on that result as a motivating example, we propose two novel observers that do not require to compute the matrix gains online. The first solution is to construct (with a nonlinear procedure) a novel instrumental output that can be then used to design a gradient-based observer. The convergence of this observer can be shown under the persistency of excitation assumption for the system trajectories. The second solution is the switched observer, where the operation mode is changed when the system looses observability. It is worth noting that the considered system is not switched, and the operation modes are introduced instrumentally allowing us for a switched observer design. The gains of the switched observer can be obtained off-line as a solution to an LMI problem. We also derive the dwell-time condition to be imposed on the switching signal to establish the exponential convergence.

The rest of the paper is organized as follows. In Section 2 we present an illustrative example to motivate the considered problem and show that some straightforward solutions do not apply. The formal problem statement is given in Section 3. The nonlinear gradient-based observer is proposed in Section 4, and the switched observer is presented in Section 5. Section 6 provides illustrative simulations, and the conclusion of the paper is given in Section 7. Finally, an auxiliary proof can be found in the Appendix.

Used notations. For integers n, m, we define I n as the n × n unit matrix, and 0 n×m and 1 n×m as n × m matrices of zeros and ones, respectively.

MOTIVATING EXAMPLE

This section presents a motivating example, which, as it has been shown by [START_REF] Ryadchikov | Differentiator-based velocity observer with sensor bias estimation : an inverted pendulum case study[END_REF], arises in robotics applications when a position sensor has a constant bias. Consider the linear parameter-varying (LPV) single-input single-output (SISO) system ẋ(t

) = A(z(t))x(t) + β(y(t), u(t)), y(t) = Cx(t), (1)
where 1 x ∈ R 3 is the state, u ∈ R and y ∈ R are the known input and output signals, respectively, the function

β : R × R → R 3 is known, A(z) := 0 1 0 0 0 cos(z) 0 0 0 , C := [1 0 1] , (2)
and z is a known piecewise continuous time-varying signal. We also assume that the input signal u is such that 1 When clear from the context, in the sequel the argument of time is omitted.

trajectories of the system (1) are bounded. The goal is to design an observer of the unmeasurable state x.

It is worth noting that the observability matrix of the system (1) is given by

O = 1 0 1 0 1 0 0 0 cos(z)
, and the system is not observable for cos(z) = 0. This obviously implies that some constrains should be imposed for the signal z, e.g., the states x 1 and x 3 cannot be reconstructed for z(t) ≡ ± π 2 . To simplify the motivating example, assume that z ∈ Q z := z | cos 2 (z) ≥ δ 2 z for some δ z ∈ (0, 1), i.e., the system is always observable but cos(z) may change its sign at some isolated instants of time.

One classic solution for linear time-varying systems is to construct a linear time-varying observer in the form

ẋ = A(z)x + β(y, u) -HC (C x -y) , (3)
where the time-varying symmetric gain matrix

H(t) ∈ R 3×3 is the solution of the matrix differential equation Ḣ = HA (z) + A(z)H -HC CH + Q
for some H(0) = H 0 > 0, and Q > 0 is the design parameter. It is known (see, e.g., [START_REF] Rugh | Linear system theory[END_REF]; Rueda-Escobedo et al. ( 2019)) that the observer (3) ensures exponential convergence if the system is uniformly observable, that is there exist T O , δ 1 , and δ 2 , all positive, such that for all t

δ 1 I 3 ≤ t+T O t Φ (τ, t)C CΦ(τ, t)dτ ≤ δ 2 I 3 ,
where Φ(•, •) is the state-transition matrix of the system (1). The uniform observability can be connected with the assumption that the system (1) does not stuck in the domain where cos(z) ≈ 0. However, the implementation of the observer (3) in embedded systems has certain drawbacks since computation of the gain matrix H(t) requires to solve online 6 differential equations with quadratic terms that may be sensitive to numerical methods. Thus, in what follows we aim to designs that are less demanding for online computations than the observer (3), e.g., by the means of off-line gains precalculation.

Let us now show that some straightforward constantgain and parameter-varying solutions do not apply to this problem. To this end, consider the observer

ẋ = A(z)x + β(y, u) -L(z)(C x -y), (4)
where the parameter-varying gain vector L(z) is to be defined, and x is the estimate of x. Define the estimation error as e := x -x, then ė = (A(z) -L(z)C) e. A simple solution would be to find a constant vector L stabilizing the system for all values of z ∈ Q z , e.g., with a common Lyapunov function. However, it can be shown that such a solution does not exist; hence, we have to calculate a vector function L(z). Methods of design of LPV observers typically consider the quadratic Lyapunov function V = e P e, where the matrix P can be constant or parameter-varying, P = P (z). The main drawback of the (continuous in z) parameter-varying matrix P (z) is that the time derivative of the Lyapunov function will depend on the time derivative of the signal z implying some probably restrictive assumptions on its continuity and differentiability. Concerning the constant matrix P , it can be shown that the system (1) does not admit such a solution, or more precisely, there do not exist a parametervarying gain vector L(z) and a positive-definite constant matrix P , such that for all z ∈ Q z the linear matrix inequality (LMI) (A(z) -L(z)C) P + P (A(z) -L(z)C) < 0 holds. The proof of this claim is straightforward and we omit it for brevity.

Motivated by the discussed shortcomings, we consider designs of two possible observers. First, we propose a nonlinear gradient-like observer that is based on a novel instrumental output of the system and requires the persistency of excitation of the system's trajectory. Next, we design a switched observer, where we consider the Lyapunov function V = e P (z)e with P (z) being piecewise constant. Thus, we do not impose assumption on the time derivative of z but study possible jumps of the value of the Lyapunov function when switches occur and cos(z) changes the sign.

In next section, we formulate the problem statement, for which the system (1) is a particular case.

PROBLEM STATEMENT

Consider an LPV SISO system in the following form ẋ = A(q)x + β(u, y, t), y = Cx, (5) where x ∈ R 3 is the state, u ∈ R is the known input, y ∈ R is the measured output signal, A(q) := 0 q 1 0 0 0 q 2 0 0 0

, C := [1 0 0] ,
with q = (q 1 , q 2 ), q i ∈ [-1, 1], i ∈ ī := {1, 2}, being a time-varying parametric variable available for measurements. The function β is known. The goal is to design an observer for x.

The main issue for solution of this problem is that the system looses its observability for zero values of varying parameters q. Indeed, the observability matrix of the system is

O = 1 0 0 0 q 1 0 0 0 q 1 q 2 .
As it can be seen from the observability matrix, the system is observable when all elements of q are not zero, the rank of the observability matrix equals 2 and the state x 3 is not observable when q 1 = 0 and q 2 = 0, and for q 1 = 0 only the state x 1 is observable. Thus, a hypothesis has to be introduced that the system does not spend much time in the mode where an element of q equals zero. This hypothesis will be formulated as a sufficient condition for the signal q ensuring observer convergence.

NONLINEAR OBSERVER

In this section we present a nonlinear observer that is based on construction of an instrumental output signal and a gradient-based estimate update that is widely used in adaptive control. To this end, we define a new instrumental output y † (t) as described in the following proposition. Proposition 1. Consider system (5). Define the signals φ 1 (t), φ 2 (t), and ξ(t) as solutions of φ1 (t) = -aφ 1 (t) + aq 1 (t),

φ2 (t) = -aφ 2 (t) + φ 2 (t)q 2 (t), ξ(t) = -aξ(t) + a 2 y(t) + ah(t), (6)
with zero initial conditions, where a > 0 is the tuning parameter and

h(t) := β 1 (t) - 1 a φ 1 (t)β 2 (t) + 1 a φ 2 (t)β 3 (t),
where β i (t) is the ith element of β(u(t), y(t), t). Define the signal y † (t) := (a + 1)y(t) -ξ(t).

(7) Then it holds y † (t) = w (t)x(t) + ε(t), (8) where w(t) := [1 φ 1 (t) -φ 2 (t)] (9) and ε(t) is a (generic) exponentially decaying term.

Proof. The proof is based on iterative use of the Swapping Lemma, see Ioannou and Sun (1996). Let p = d dt be the differential operator, then from (6) we get

φ 1 (t) = a p + a q 1 (t), φ 2 (t) = 1 p + a [φ 1 (t)q 2 (t)] .
For brevity, here and below we omit exponentially decaying terms caused by initial conditions. From (5),

ap p + a y(t) = a p + a [q 1 (t)x 2 (t)] + a p + a β 1 (t). (10)
Applying the Swapping Lemma,

a p + a [q 1 (t)x 2 (t)] = x 2 (t)φ 1 (t) - 1 p + a [φ 1 (t) ẋ2 (t)] = x 2 (t)φ 1 (t) - 1 p + a [φ 1 (t)q 2 (t)x 3 (t)] - 1 p + a [φ 1 (t)β 2 (t)] . (11)
Applying the Swapping Lemma again,

1 p + a [φ 1 (t)q 2 (t)x 3 (t)] = x 3 (t)φ 2 (t) - 1 p + a [φ 2 (t)β 3 (t)] . (12)
Substituting ( 11) and ( 12) in (10), we obtain

ap p + a y(t) - a p + a h(t) = φ 1 (t)x 2 (t) -φ 2 (t)x 3 (t).
Finally, it is straightforward to verify that

ap p + a y(t) - a p + a h(t) = ay(t) -ξ(t),
where ξ(t) obeys ( 6), that completes the proof. 2

To design the observer, we have to assume that the trajectory q(t) sufficiently excites the system and does not stay at zero. More formally, it can be formulates as the following assumption. Assumption 2. The trajectory q(t) is such that the vector signal w(t) defined in ( 9) is persistently excited, i.e., there exist T w and δ w such that for all t ≥ 0 it holds t+Tw t w(s)w (s)ds ≥ δ w I 3 .

The nonlinear observer and its applicability are stated in the following proposition. Proposition 3. Consider the system (5) with the new instrumental output y † (t) defined in (7). Consider the nonlinear observer ẋ(t) =A(q(t))x(t) + β(u, y, t)

+ Γw(t) y † (t) -w (t)x(t) , (13)
where Γ > 0 is the tuning parameter. If the trajectory q(t) satisfies Assumption 2, then there exists Γ 0 > 0 such that |x(t) -x(t)| → 0 for all Γ > Γ 0 .

Proof. Define the estimation error e(t) := x(t) -x(t).

Recalling ( 8) and neglecting the exponentially decaying term ε(t), the error dynamics is given by ė = A(q)e -Γww e. Define the Lyapunov function candidate V (e) := 1 2 e Γ -1 e. Then V = e Γ -1 A sym (q) -ww e,

where A sym (q) := 1 2 A(q) + A (q) is the symmetric part of A(q). Since the matrix A(q) is bounded and w(t) satisfies Assumption 2 for some T w and δ w , there exist Γ 0 > 0 sufficiently large such that for any Γ > Γ 0 there is T ≥ T w and δ ∈ (0, δ w ] such that t+T t Γ -1 A sym (q(s)) -w(s)w (s) ds ≥ δI 3 for all t ≥ 0. Thus, the same arguments as used for the proof of asymptotic stability of persistently exited systems can be applied, see Ioannou Sun (1996). 2

The observer (13) provides exponential convergence under the PE assumption on w(t), which is related to the excitation of the trajectory q(t), and cannot be applied if, for example, q(t) converges to a constant vector. Note that analysis and verification of the convergence conditions remain a challenging problem. In the following Section, we present a switched observer that does not impose the PE assumption.

SWITCHED OBSERVER

Observer design

To present the switched observer, note that the vector q belongs to a square in R 2 that has (±1, ±1) as its vertices. Let us enumerate these vertices in any particular order, v k , k ∈ k := {1, 2, 3, 4}. Let us say that q ∈ Q k , k ∈ k, if q belongs to a smaller square that has the vertices at the origin and v k . For example, one possible enumeration is

v 1 := (1, 1), v 2 := (1, -1), v 3 := (-1, -1), v 4 := (-1, 1), Q 1 := { (q 1 , q 2 ) | 1 ≥ q 1 ≥ 0, 1 ≥ q 2 ≥ 0 }, Q 2 := { (q 1 , q 2 ) | 1 ≥ q 1 ≥ 0, -1 ≤ q 2 < 0 }, Q 3 := { (q 1 , q 2 ) | -1 ≤ q 1 < 0, -1 ≤ q 2 < 0 }, Q 4 := { (q 1 , q 2 ) | -1 ≤ q 1 < 0, 1 ≥ q 2 ≥ 0 }. (14)
Define the sign function as

sgn (q i ) := 1 if q i ≥ 0, -1 if q i < 0,
and the set of matrices

A k :=   0 sgn v k 1 0 0 0 sgn v k 2 0 0 0   .
In other words, the matrix A k has the same structure as the matrix A(q) where the parameters q 1 , q 2 are replaced by their extremes in the set Q k , k ∈ k, and the matrices A k form a convex polytope where the matrix function A(q) is embedded. Note that all pairs (C, A k ) are observable. For the previously considered example, for the sets ( 14) we have

A 1 = 0 1 0 0 0 1 0 0 0 , A 2 = 0 1 0 0 0 -1 0 0 0 , A 3 = 0 -1 0 0 0 -1 0 0 0 , A 4 = 0 -1 0 0 0 1 0 0 0 . ( 15
)
Define the function

f s : R 2 × R 3 → R 2 , f s (q, x) = (q 1 -sgn (q 1 )) x 2 (q 2 -sgn (q 2 )) x 3 ,
and consider the switched observer ẋ = A s (q)x + β(u, y, t) + Bf s (q, x)

-L s (q) (C x -y) , (16)
where A s (q) is the switched matrix,

A s (q) = A k if q ∈ Q k , B := 1 0 0 1 0 0 , (17)
and L s (q) is the switched gain defined as L s (q) := L k when q ∈ Q k , k ∈ k, and the gains L k are computed as

L k = O -1 k O 1 L 1 ,
where O k is the observability matrix of the pair (C, A k ) and L 1 is to be defined.

In what follows, we show that if a certain LMI is feasible, then there exists L 1 such that under some assumptions on the signal q the observer (16) ensures exponential convergence of the estimate x to the state vector x of the system (5).

Convergence of the observer

To present the convergence result, we define the function γ : R → R as γ(q i ) := |q i | (2 -|q i |) . Note that γ(0) = 0, γ(1) = 1, and γ(q i ) is monotonically increasing as |q i | varies from 0 to 1.

Next choose δ q ∈ (0, 1) and compute δ γ := γ(δ q ), 0 < δ γ < 1. Suppose that for some δ γ there exist matrices P 1 = P 1 > 0 and L 1 being a solution of the matrix inequality M 1 ≤ 0, where

M 1 := F 1 + (1 -δ γ ) I 3 P 1 B B P 1 -I 2 ,
and

F 1 := (A 1 -L 1 C) P 1 + P 1 (A 1 -L 1 C) .
(18) Let λ M and λ m be the maximum and the minimum eigenvalues of P 1 , respectively. Then define

µ := λ M λ m -1, (19)
and η(q) := min (γ(q 1 ), γ(q 2 ))

λ M , η 0 := δ γ λ M . ( 20
)
Now we are in the position to impose an assumption on the signal q implying that the trajectory should not cross the borders between the sets Q too often and should not remain for a long time in small vicinities of these borders having |q i | ≤ δ q for some i yielding min (γ(q 1 ), γ(q 2 )) ≤ δ γ . More formally, this assumption can be formulated as follows.

Assumption 4. For the trajectory q(t) there exist T q > 0 and κ > 0 such that for all t 0 it holds:

• during the time interval [t 0 , t 0 + T q ] the trajectory q(t) crosses the borders between the sets Q not more then n Q ≥ 0 times; • during the time interval [t 0 , t 0 + T q ] the trajectory q(t) satisfies

t0+Tq t0 η(q(τ ))dτ ≥ n Q ln (1 + µ) + (η 0 + κ) T q .
Finally, applicability of the proposed switched observer is summarized in the following Theorem. Theorem 5. Consider the system (5). Choose δ γ ∈ (0, 1) such that there exist P 1 > 0 and L 1 satisfying the matrix inequality 

 (A 1 -L 1 C) P 1 + P 1 (A 1 -L 1 C) P 1 B + (1 -δ γ ) I 3 B P 1 -I 2   ≤ 0, (21)
where A 1 is defined in (15), and B is defined in (17).

Then if the trajectory q(t) satisfies Assumption 4 for η(q) and η 0 defined in (20) and µ defined in (19), then the observer (16) ensures exponential convergence of the estimate x(t) to the state vector x(t) of the system (5).

The proof of Theorem 5 is omitted due to the lack of space. Remark 6. Using standard methods for matrix inequalities, see [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], and defining H 1 := L 1 P 1 , the matrix inequality (21) can be rewritten as the LMI  

A 1 P 1 + P 1 A 1 -C H 1 -H 1 C P 1 B + (1 -δ γ ) I 3 B P 1 -I 2   ≤ 0,
which can be efficiently solved for P 1 and H 1 , see [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]. Then L 1 = P -1 1 H 1 .

ILLUSTRATIVE EXAMPLE

In this section, we present an illustrative example of the system (5), where

β(u, y, t) := - 9 4 1 y(t) - 0 0 1 u(t)
and u(t) = 0.15 + sin(2πt). The trajectory q = (q 1 , q 2 ) is constructed as q 1 (t) := cos(t), q 2 (t) := cos(t + π 4 ), and the vertices v k and sets Q k , k ∈ k, are chosen as ( 14). The trajectories x(t) of the system are depicted in Figure 1, where x(0) = [1 2 -1] .

For the nonlinear observer (13) we choose a = 1 and Γ = diag (1, 8, 10). For the chosen q 1 (t), q 2 (t), and a it is straightforward to verify that (in steady state) the signals φ 1 (t) and φ 2 (t) defined in (6) satisfy

φ 1 (t) = 2 -1 2 cos(t + π 4 ),
φ 2 = A 2 sin(2t + ψ 2 ) for some A 2 > 0 and ψ 2 ∈ [0, 2π); thus w(t) defined in (9) satisfies Assumption 2 for T w = 2π.

For the switched observer ( 16) we choose δ γ = 0.1 and compute a feasible solution to the matrix inequality ( 21

The estimation errors e(t) for both observers are depicted in Figure 2 and illustrate convergence of the state estimation errors. The Lyapunov function curve for the switched observer is shown in Figure 3, where the level jumps can be observed at the switch instances, i.e., where q 1 (t) or q 2 (t) change their signs.

CONCLUSION

We have considered the problem of state estimation for a parameter-varying system that is not observable for some values of the time-varying parameters. To this end, two observers have been proposed. The nonlinear observer ( 13) is based on construction of a new instrumental output that allows to apply a gradient-like estimate update and yield exponential convergence under the persistency of excitation condition. The switched observer ( 16) is a linear observer with switched gains and ensure exponential convergence under the dwell-time conditions. Applicability of the observers is illustrated with numerical simulations.

For further researches, we intend to extend the presented observers to a more general case of systems with dimension greater than 3 and also consider other classes of observers, such as observers with finite/fixed-time convergence.

Fig. 1 .

 1 Fig. 1. The states x(t) in the illustrative example.

  ) yielding L 1 = [73 442 68] .

Fig. 2 .

 2 Fig. 2. Estimation errors e(t) in the illustrative example.

Fig. 3 .

 3 Fig. 3. The Lyapunov function V (t) for the switched observer in the illustrative example.

This work was supported by 111 project No. D17019, China, by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and by Government of Russian Federation (Grant 08-08). S. Aranovskiy was also supported by Rennes Metropole (France), project AIS-19C0326.


B Borschinger

C Bartoli

C Chandeysson

C Guilbaud

L Parisi

J F Bourgeay

E Buisson

C E Morris
email: cindy.morris@paca.inra.fr

A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups

Keywords: Plant pathogen, microbial ecology, pathobiome, multiplex PCR, orchards

Aims:

The aim of this study was to develop a rapid PCR-based method for the specific detection of individual phylogroups of the Pseudomonas syringae complex.

Methods and Results:

Seven primer pairs were developed by analyzing whole genomes of 54 P. syringae strains. The specificity and sensitivity of these primer pairs were assessed on 236 strains from a large and comprehensive P. syringae collection. The method was also validated by

Introduction

In plant pathology, competitive or synergistic interactions among microorganisms have been disregarded in disease etiology. The emerging pathobiome concept pointed out that etiological agents can work in concert with indigenous microflora to induce disease [START_REF] Vayssier-Taussat | Shifting the paradigm from pathogens to pathobiome: new concepts in the light of meta-omics[END_REF]. The impact on disease of infra-specific diversity of microorganisms in individual hosts has been ignored.

In the case of plant pathogenic bacteria, analyses of the structure of the pathogenic populations generally concern the metapopulation/population at large in a region [START_REF] Scortichini | The population structure of some plant pathogenic bacteria: an ecological and adaptive perspective[END_REF], but rarely addresses the structure of pathogen populations in individual plants. Phenomena of gain of fitness, as well as synergistic interactions, are not uncommon within strains of different genetic lines belonging to the same bacterial species and could enhance diseases [START_REF] Kamoun | A plant-inducible gene of Xanthomonas campestris pv. campestris encodes an exocellular component required for growth in the host and hypersensitivity on nonhosts[END_REF][START_REF] Hirano | Role of the Hrp type III protein secretion system in growth of Pseudomonas syringae pv. syringae B728a on host plants in the field[END_REF][START_REF] Scortichini | The population structure of some plant pathogenic bacteria: an ecological and adaptive perspective[END_REF][START_REF] Lamichhane | Synergisms between microbial pathogens in plant disease complexes: a growing trend[END_REF]. On the other hand, competition between strains of the same ecological niche could attenuate disease symptoms [START_REF] Kerr | Biological control of crown gall through bacteriocin production[END_REF][START_REF] Moss | Biological control of bacterial spot of tomato using hrp mutants of Xanthomonas campestris pv. vesicatoria[END_REF][START_REF] Kawaguchi | Biological control of bacterial spot on peach by nonpathogenic Xanthomonas campestris strains AZ98101 and AZ98106[END_REF]. In spite of these observations, conventional diagnosis of plant diseases has Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

(3), 714-723. , DOI : 10.1111/jam.13017

Accepted Article Accepted Article

This article is protected by copyright. All rights reserved.

focused on the characterization of the dominant organisms in infected tissue. To address the issue of microbial complexes responsible for disease occurrence, detection methods that facilitate the assessment of the population structure in individual plants are needed. In addition, metagenomics approaches are in part unable to characterize the structure of different microbial communities at species level [START_REF] Handelsman | Metagenomics: application of genomics to uncultured microorganisms[END_REF][START_REF] Walker | Phylogeny, culturing, and metagenomics of the human gut microbiota[END_REF][START_REF] Barret | Emergence shapes the structure of the seed microbiota[END_REF]. In light of these facts, we have developed a set of PCRs to rapidly identify the phylogenetic structure of the plant pathogenic bacterium Pseudomonas syringae [START_REF] Berge | A user's guide to a data base of the diversity of Pseudomonas syringae and its application to classifying strains in this phylogenetic complex[END_REF]. This method requires strain isolation, thereby allowing subsequent evaluation of the pathogenicity of the isolated strains.

P. syringae is a common environmental bacterium that is ubiquitous in both aquatic and agricultural habitats where it can have both saprophytic and pathogenic behavior. P. syringae is also an important plant pathogen causing diseases such as kiwifruit canker reported in all kiwi-producing regions:

Australasia, Europe and South America [START_REF] Hirano | Population biology and epidemiology of Pseudomonas syringae[END_REF][START_REF] Gaignard | Pseudomonas syringae, bactérie épiphyte, glaçogène et pathogène[END_REF][START_REF] Vanneste | Recent progress on detecting, understanding and controlling Pseudomonas syringae pv. actinidiae: a short review[END_REF][START_REF] Donati | New insights on the bacterial canker of kiwifruit (Pseudomonas syringae pv. actinidiae)[END_REF]. Since the beginning of the 20 th century, 55 cases of bacterial diseases in 25 countries and on 25 species of trees of agronomic importance were declared as being caused by this bacterium [START_REF] Lamichhane | Disease and frost damage of woody plants caused by Pseudomonas syringae: seeing the forest for the trees[END_REF]. Likewise, in this same period P. syringae has been identified as the cause of nearly 70 new disease outbreaks affecting over 40 annual crop species (Lamichhane et al. 2015).

P. syringae is now recognized as a complex of at least 13 phylogroups [START_REF] Berge | A user's guide to a data base of the diversity of Pseudomonas syringae and its application to classifying strains in this phylogenetic complex[END_REF]. Each phylogroup displays unique phenotypes and it can be distinguished according to the sequence of one (for example citrate synthase) or a few (Multilocus Sequence Typing) housekeeping genes (Berge et al. 2014). Studies on the non-agricultural habitats (water, litter, wild plants, etc) showed that multiple P. syringae phylogroups can co-exist in the same sample [START_REF] Morris | Surprising niche for the plant pathogen Pseudomonas syringae[END_REF][START_REF] Monteil | Soil water flow is a source of the plant pathogen Pseudomonas syringae in subalpine headwaters[END_REF].

However, there is a paucity of information about such co-existence in crops and about the possible role of co-existing strains in disease occurrence. In light of the current well-defined P. syringae classification and the numerous whole genome sequences that are available [START_REF] Berge | A user's guide to a data base of the diversity of Pseudomonas syringae and its application to classifying strains in this phylogenetic complex[END_REF], we have sought to develop PCR primers that target as many of the phylogroups as possible to facilitate analysis of the P. syringae populations associated with diseased plants as well as with a range of environmental substrates. These molecular markers are more encompassing than those that have been (3), 714-723. , DOI : 10.1111/jam.13017

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developed to detect genetic lines that have pathovar names such as P. syringae pv. phaseolicola (Mosqueda-Cano and Herrera-Estrella 1997), P. syringae pv. tomato [START_REF] Zaccardelli | Identification and in planta detection of Pseudomonas syringae pv. tomato using PCR amplification of hrpZ Pst[END_REF], P.

syringae pv. aesculi [START_REF] Green | Infection of horse chestnut (Aesculus hippocastanum) by Pseudomonas syringae pv. aesculi and its detection by quantitative real-time PCR[END_REF]) and P. syringae pv. actinidiae [START_REF] Sawada | Comparative analysis of Pseudomonas syringae pv. actinidiae and pv. phaseolicola based on phaseolotoxin-resistant ornithine carbamoyltransferase gene (argK) and 16S-23S rRNA intergenic spacer sequences[END_REF][START_REF] Sawada | Phylogenetic analysis of Pseudomonas syringae pathovars suggests the horizontal gene transfer of argK and the evolutionary stability of hrp gene cluster[END_REF][START_REF] Koh | DNA markers for identification of Pseudomonas syringae pv. actinidiae[END_REF][START_REF] Rees-George | Detection of Pseudomonas syringae pv. actinidiae using polymerase chain reaction (PCR) primers based on the 16S-23S rDNA intertranscribed spacer region and comparison with PCR primers based on other gene regions[END_REF][START_REF] Gallelli | Detection of Pseudomonas syringae pv. actinidiae, causal agent of bacterial canker of kiwifruit, from symptomless fruits and twigs, and from pollen[END_REF][START_REF] Balestra | A multiplex PCR assay for detection of Pseudomonas syringae pv. actinidiae and differentiation of populations with different geographic origin[END_REF][START_REF] Gallelli | Real-time and qualitative PCR for detecting Pseudomonas syringae pv. actinidiae isolates causing recent outbreaks of kiwifruit bacterial canker[END_REF][START_REF] Loreti | The Italian inter-laboratory study on the detection of Pseudomonas syringae pv. actinide[END_REF]. Pathovar-specific markers could nevertheless be used in conjunction with phylogroup markers and also with the "universal" marker of all members of the P.

syringae complex (Guilbaud et al., in press). This latter primer pair, called "Psy", allows for the identification of strains in the P. syringae complex with over 97% accuracy. This allows a real gain in time when identifying strains of P. syringae. However , this latter method does not identify the phylogroup affiliation of P. syringae.

Here, we propose a set of molecular markers identifying each of 9 major P. syringae phylogroups based on the current phylogenetic classification and the genomes of P. syringae accessible in public databases. We also describe a method to deploy these markers in a multiplex-PCR with the Psy primer set as a universal control for reliable detection and identification of P. syringae phylogroups. An example of the use of this method to characterize communities of P. syringae associated with kiwifruit and apricot plant samples from southeastern France is also reported.

Materials and Methods

Strains and criteria used for primer design

During this study 439 strains were used for 3 different objectives: i) to develop the primer pairs, ii) to validate the designed primers (specificity/sensitivity), and iii) to perform a "blind test" for the multiplex-PCR method on putative P. syringae isolates obtained from kiwifruit and apricot orchards from southeastern France. For primer development, DNA sequences were identified in whole genomes of 54 P. syringae strains (Table S1) available on the Joint Genome Institute (JGI) platform

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(http://jgi.doe.gov/). The distribution of the corresponding strains within the various phylogroups is uneven and reflects the availability of the sequences.

To evaluate the specificity and sensitivity of the primer pairs, 236 strains whose phylogroups were previously determined by citrate synthase (cts) gene sequencing, were used (Table S1). These 236 strains are part of the P. syringae collection of the Plant Pathology research Unit of the INRA PACA Avignon center. This collection is composed by over 7000 P. syringae strains isolated from crops and non-agricultural habitats.

Strains were maintained at -80°C in phosphate buffer containing 20% glycerol and were cultured on King's B medium [START_REF] King | Two simple media for the demonstration of pyocyanin and fluorescin[END_REF]) and incubated at 25°C for 48 hours prior each analysis. The strains used for primer development and for the specificity and sensitivity tests were pure culture strains.

Putative P. syringae isolates, not pure culture strains, were used for the "blind test" described below.

These putative P. syringae isolates came from field sampling aiming to characterize P. syringae populations associated with kiwifruit and apricot orchards in southeastern France (Drôme county).

Overall 174 putative P. syringae isolates were used. These isolates were collected from different substrates including leaves, buds, twigs, ground-cover, soil and irrigation waters (Table S1). For bacterial isolation, leaf, bud, twig and ground cover samples were homogenized with a stomacher in a phosphate buffer solution for 4 minutes. Soil samples were directly suspended with sterilized distilled water. Bacterial isolation from water samples was carried out according to the method previously described [START_REF] Morris | Inferring the evolutionary history of the plant pathogen Pseudomonas syringae from its biogeography in headwaters of rivers in North America, Europe, and New Zealand[END_REF]. For all the obtained suspensions, we performed dilution-plating on a modified King's B medium (KBC) [START_REF] Mohan | An improved agar plating assay for detecting Pseudomonas syringae pv. syringae and P. s. pv. phaseolicola in contaminated bean seed[END_REF]. KBC as used in this study was composed of 9 volumes of King's B medium with a smaller quantity of proteose peptone than the original recipe (1.5g K 2 HPO 4 , 1.5g MgSO 4 7H 2 O, 15g proteose-peptone, 10 ml glycerol and 16g agar in 900 ml distilled water), and 1 volume of boric acid (1.5g H 3 BO 3 in 100 ml distilled water), and contains cycloheximide (200 mg l -1 ) and cephalexin (80 mg l -1 ). The plates were incubated at 25°C for 48-72 hours. Putative P. syringae isolates were identified by the morphology of the colony (size, color, reflectance of light and roughness of the colony border). These isolates were stored in a phosphate buffer. No previous data about their phylogroup affiliation were available for these isolates.

Comment citer ce document : Borschinger, B., Bartoli (3), 714-723. , DOI : 10.1111/jam.13017

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Phylogroup affiliation of these isolates was verified by sequencing the cts gene of the corresponding purified strain of each isolate. The resulting cts sequences were aligned with the cts sequences of the strains from the P. syringae collection described above (1 or 2 representatives per phylogroup) by using clustalW2 software. A phylogenetic tree was constructed with the Neighbor Joining Model and 3000 bootstrap replicates by using MEGA6 software [START_REF] Tamura | MEGA6: Molecular Evolutionary Genetics Analysis Version 6.0[END_REF].

Fifty-four P. syringae whole genomes available on the JGI platform and representative of 10 phylogroups were used to identify genes specific for each phylogroup with the IMG/M system (http://img.jgi.doe.gov). The Abundance Profile Overview tool was used in order to identify unique genes specific for each phylogroup. This tool provides an overview of the relative abundance of protein families (COGs and Pfams) across the 54 selected genomes used in this study. Genes selected for primer designing are listed in Table 1. All gene sequences used in this study are available in Table S2. For Phylogroups 1, 3 and 10 no unique open reading frames or unique DNA sequences were observed. For these phylogroups, DNA regions of some genes (Table 1) were used to design primers based on single nucleotide polymorphisms (SNPs). The SNP containing region was chosen based on a high G/C content. The size of the SNPs on the forward (Fw) and reverse (Rv) primers varied from 4

to 17 bp as reported in Table 1.

PCR conditions and analysis of the amplicons for detection of P. syringae phylogroups All primer pairs were tested individually. After verification of the specificity and the size of the amplicons, primer pairs were combined in multiplex PCRs in order to reduce the number of PCRs required to identify the phylogroups. Reaction mixes for the individual PCR were composed of 5.0 µl of GoTaq green 5X PCR buffer (Promega, France), 1.5 µl of a solution of 25 mmol l -1 MgCl 2 , 0.2 µl of a mix of 25 mmol l -1 dNTPs, 2 µl of a mix of primers consisting of 10 mmol l -1 Fw and Rv primers, 0.13 µl of GoTaq G2 DNA polymerase (Promega, France) with 5 units µl -1 , and 2.0 µl of bacterial suspension at 10 8 cells ml -1 . The final volume of the PCR reaction was 25 µl and milliQ water was

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used to mix dilution. All PCR amplifications were performed for 30 cycles; primer sequences and PCR conditions are listed in Table 2. Reaction mixes for the multiplex PCRs were composed of 6.25 µl of 2X QIAGEN Multiplex PCR Master Mix (Qiagen ® Multiplex PCR kit), 1.25 µl of primer mix with a final concentration of 2 mmol l -1 per primer, 1.25 µl of Q solution, 2.75 µl of milliQ water and 2.0 µl of bacterial suspension at 10 8 cells ml -1 . The final volume reaction for the multiplex PCR was 11.5 µl. Taking into account the amplicon sizes (Table 2), two multiplex PCR reactions were developed: "Multiplex 1" containing the universal primer pair Psy and the primers specific for phylogroup 1 (PG1), phylogroup 3 (PG3) and phylogroup 9 (PG9) and "Multiplex 2" containing the primer pair Psy and those specific for phylogroup 2 (PG2), phylogroup 4 (PG4) and phylogroup 13 (PG13). Amplification was performed for 30 cycles and denaturation and elongation temperatures were 95°C and 72°C, respectively for all reactions. Annealing temperatures are reported in Table 2.

PCR products were migrated at 80V in a 1.5% agarose gel containing ethidium bromide (BET) (Euromedex, EU0070) at a concentration of 0.5 μg ml -1 . Migration times varied depending on the size of PCR amplicons. After migration, PCR products were visualized under ultraviolet light (320 nm).

Sensitivity and performance criteria

To test the sensitivity of the multiplex PCR, different bacterial concentrations were used during PCR amplifications. These concentrations varied from 10 8 to 10 4 bacterial cells ml -1 .

To test the performance of the PCRs we used criteria described in previous studies [START_REF] Eppo | PM 7/98 (1): specific requirements for laboratories preparing accreditation for a plant pest diagnostic activity[END_REF][START_REF] Eppo | PM 7/120 (1) Pseudomonas syringae pv. actinidiae[END_REF][START_REF] Loreti | The Italian inter-laboratory study on the detection of Pseudomonas syringae pv. actinide[END_REF]). These performance criteria are derived from those used in the development of a PCR based detection method for P. syringae pv. actinidiae [START_REF] Loreti | The Italian inter-laboratory study on the detection of Pseudomonas syringae pv. actinide[END_REF]. Briefly, these criteria include i) relative accuracy (A) which corresponds to the similarity of the observed results compared to expected results, ii) diagnostic sensitivity (SE) which is the ability of the method to detect the target strain in a contaminated reference sample, and iii) diagnostic specificity (SP) which corresponds to the capacity of the method to not yield a positive result in an uncontaminated sample reference.

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Results

Eight primer pairs are sufficient to identify nine P. syringae phylogroups

In this work, we developed primer pairs for the identification of 9 P. syringae phylogroups (1, 2, 3, 4, 9, 10 and 13). A primer pair for the detection of the P. syringae phylogroups 7 and 8 was developed and evaluated previously [START_REF] Bartoli | The Pseudomonas viridiflava phylogroups in the P. syringae species complex are characterized by genetic variability and phenotypic plasticity of pathogenicity-related traits[END_REF].

To detect PG1, a primer pair was developed based on the inter-phylogroup variability found in the gyrase B (gyrB) housekeeping gene. The primer pair and the SNPs used to discriminate the PG1 strains among all the rest of the P. syringae phylogroups are reported in Table 1 and2. Except for this primer pair all other primers were designed on non -housekeeping genes. The primer pair for PG1 (Table 2) was designed in the 387-754 bp region of the gyrB gene and amplify a fragment of 321 bp. The corresponding SNP localizations are reported in Table 1.

For PG2, a unique gene candidate (DUF421) was found in the 16 genomes used during this study.

This gene codes for a domain with unknown function, according to the database of conserved protein domain families (Pfam04239). The PG2 primer pair (Table 2) was designed on the 21-404 bp region of the gene, and amplifies a fragment of 341 bp.

For PG3, a gene coding for a DUF1835 domain was found in all the available PG3 genomes and in some strains belonging to the phylogroups 1, 7, 9 and 10. This gene codes for a protein of unknown function and its domain is classed under the number pfam08874. Gene sequences for these four P.

syringae phylogroups displayed a high degree of polymorphism. This polymorphism was used to design a specific primer pair that amplifies only PG3. The localizations of the SNPs used to design the specific PG3 primers are reported in Table 1. PG3 specific primer pair (Table 2) was developed on the 61-937 bp region of the gene and amplifies a fragment of 837 bp.

For PG4 one unique gene was found, coding for a lysine-tRNA ligase (pfam01921). The domain of this protein was only found in this P. syringae phylogroup. The sequences of the primer pair (Table 2) developed on the 605-1489 bp region of the gene amplify a fragment of 845 bp.

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For PG9 a gene involved in bacterial chemotaxis (pfam04509) was found to be specific for all the strains in this phylogroup. The primer pair (Table 2) for this P. syringae phylogroup was developed on the 41-469 bp region of the gene and amplifies a fragment of 389 bp.

For PG10 strains we did not identify a specific candidate gene. Therefore, we selected a gene coding for a DUF2599 domain (pfam10783) present in the genome of the 3 genomes available for the PG10 as well in one strain of the P. syringae PG1. The PG10 specific primer pair (Table 2) was developed on the 435-799 bp region of the gene and amplifies a fragment of 324 bp. Localizations of the SNPs that were used to discriminate between the two phylogroups (PG10 and PG1) are reported in Table 1.

For PG13 several specific genes were found in the one available genome of a strain belonging to this phylogroup. The most accurate and specific primer pair targeted a gene encoding an alphaglucosidase (pfam01055). The primer pair (Table 2) developed on the 599-1220 bp region of the gene amplifies a fragment of 582 bp.

The primer pairs for detection of the nine P. syringae phylogroups are specific with a high level of sensitivity.

Results for the performance criteria demonstrated that the primer pairs we developed here for the detection of the nine P. syringae phylogroups are highly specific and sensitive (Table 3). In fact, we reported that SP is higher than 99% for all the primer pairs, indicating a high specificity level. Only one false positive PCR result was reported with the PG2 and PG13 primer pairs. All PCR results are available in Table S1. A 100% SE was obtained for the phylogroups 2, 3, 9 and 13 primer pairs, indicating a good diagnostic sensitivity. Only 1 false negative PCR result was observed for the PG1 and PG4 primer pairs. The lowest diagnostic sensitivity was observed for the PG10 primer pair, only

Comment citer ce document : Borschinger, B., Bartoli (3), 714-723. , DOI : 10.1111/jam.13017

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Concerning sensitivity, i.e. the amount of bacterial cells ml -1 needed for positive PCR detection, clear bands were obtained in single and multiplex PCR on suspensions with at least 10 6 bacterial cells ml -1 , which corresponds to 2.10 3 cells in the 2µl used for PCR reaction. Weak bands were obtained for bacterial suspensions at 10 5 bacterial cells ml -1 .

A blind test on putative P. syringae isolates from kiwifruit and apricot orchards showed the efficiency of the primer pairs in identifying PGs in natural samples Phylogroup affiliation of 174 putative P. syringae isolates from kiwifruit and apricot orchards was determined in parallel by cts sequencing and with the PCR method. Results based on the primer pair detection method showed the following phylogroup affiliation: i) 31% of the isolates were in P.

syringae PG1, ii) 14% of the isolates clustered with strains of PG2, iii) 8% of the isolates were located in PG7 and 8, iv) 29% and 7% of the isolates clustered in PG13 and with different bacterial species respectively (Figure 1a). Furthermore for 5% of the isolates the phylogroup affiliation could not be determined (Ni) and in 6% of cases a multiple phylogroup identification (Mi) occurred (identification of 2 or more phylogroups for a same isolate) (Figure 1a). Through cts sequencing similar results were observed: i) 34% of the isolates were in the P. syringae PG1, ii) 14% of the isolates clustered with strains of PG2, iii) 1% of the isolates were in PG3, iv) 7% and 1% of the isolates were in PG7 and 8 respectively, v) 2% of the isolates were in PG9, vi) 4% of the isolates were in PG 10, vii) 29% and 7% of the isolates clustered with PG13 and with different bacterial species respectively, and viii) 1%

of the isolates remain without phylogroup identification (Figure 1b). For the isolates whose results did not match between the primer pair detection method and the cts sequencing, the cts sequences revealed that 4%, 2%, 3% and 2% of the strains for PG1, PG2, PG7 and PG13 respectively, were misidentified. None of the 7 isolates of PG10 were detected by the PCR method, they were misidentified as belonging to PG7-8 or to PG1. PCR characterization of 10 isolates revealed that they contained mixtures of 2 or more phylogroups. Purification led to the isolation of the predicted components of the mixture. Among the 9 isolates that could not be identified with PCR, based on the

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cts sequence of the purified strains six represented PG1, two represented PG9 and one could not be identified with the current phylogenetic classification. Twelve isolates did not belong to the species P.

syringae according to the PCR results, and this was confirmed by sequencing the cts gene.

Discussion

The PCR markers developed in this study allow rapid classification and identification of the infraspecific diversity for at least 8 P. syringae phylogroups (PG1, 2, 3, 4, 7, 8, 9 and 13) with a high level of accuracy and for some of the strains in PG10. By coupling phylogroup-specific primer pairs to the P. syringae species-specific primer pair (Guilbaud et al. in press) and to a sensitive selective medium [START_REF] Mohan | An improved agar plating assay for detecting Pseudomonas syringae pv. syringae and P. s. pv. phaseolicola in contaminated bean seed[END_REF], the structure of P. syringae populations associated with plants and other substrates can be quantified. Quantification can be easily performed by counting the number of colonies present on the P. syringae selective medium. From the bacterial isolates obtained by plating, classification of the strains into the corresponding P. syringae phylogroups is then possible by using the primer sets proposed here.

Identifying phylogroups, with a sequencing-free molecular technique, will lead to more efficient sequencing that targets only strains for which phylogroup affiliation is not possible. Targeted sequencing could therefore be focused on the more rare phylogroups such as the phylogroup 5, 6, 11 and 12 as well as any currently unknown phylogroups that occurred in the samples. Understanding phylogroup affiliation of isolates is an important step for the identification of P. syringae lines implicated in a range of diseases. The method proposed here could be then employed for epidemiological studies aiming at characterizing P. syringae diversity associated with diseased plants.

To date, the kiwifruit epidemic is known to be caused by strains of the P. syringae complex that belong to PG1 [START_REF] Bartoli | A framework to gauge the epidemic potential of plant pathogens in environmental reservoirs: the example of kiwifruit canker[END_REF]. By using the primers developed here, one can attempt to detect strains in the P. syringae PG1 associated with disease on kiwifruit plants and then to confirm with more specific primers their identity with the current quasi-clonal epidemic line. In this way association of other P. syringae phylogroups as an important factor for disease occurrence on kiwifruit could be revealed. For example, here we revealed the presence of strains belonging to PG10

Comment citer ce document : Borschinger, B., Bartoli (3), 714-723. , DOI : 10.1111/jam.13017

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and PG13 as epiphytes on ground cover plants and in buds of apricot and kiwifruit orchards.

Previously these phylogroups were only observed in water and other non-plant substrates associated with the water cycle [START_REF] Berge | A user's guide to a data base of the diversity of Pseudomonas syringae and its application to classifying strains in this phylogenetic complex[END_REF]. The potential synergistic or antagonistic roles of these phylogroups in symptom manifestation is unknown and can now be explored.

Furthermore the molecular technique described here leads to considerable time savings in light of the effort required for strain purification and preparation of DNA for sequencing. In fact, when the method is applied to isolates that have not been purified it is possible to identify mixtures of several phylogroups. However, it is not possible to determine the number of isolates of each phylogroup constituting these mixtures nor to identify mixtures of several strains of the same phylogroup.

The cases of misallocation of the phylogroups by PCR can be explained by the extreme genetic variability that exists within the P. syringae complex [START_REF] Berge | A user's guide to a data base of the diversity of Pseudomonas syringae and its application to classifying strains in this phylogenetic complex[END_REF]. Indeed some strains can lose or gain genes as a result of mutation, deletion or insertion. Developing primers to identify species of the same genus or genera sharing the same functions can sometimes be difficult, and differentiating strains within a bacterial species is often even more difficult [START_REF] Romero | Specific detection of Brucella DNA by PCR[END_REF][START_REF] Wang | A universal protocol for PCR detection of 13 species of foodborne pathogens in foods[END_REF][START_REF] Hallin | PCR detection of genes encoding nitrite reductase in denitrifying bacteria[END_REF][START_REF] Kidd | Low rates of Pseudomonas aeruginosa misidentification in isolates from cystic fibrosis patients[END_REF][START_REF] Rees-George | Detection of Pseudomonas syringae pv. actinidiae using polymerase chain reaction (PCR) primers based on the 16S-23S rDNA intertranscribed spacer region and comparison with PCR primers based on other gene regions[END_REF][START_REF] Gallelli | Real-time and qualitative PCR for detecting Pseudomonas syringae pv. actinidiae isolates causing recent outbreaks of kiwifruit bacterial canker[END_REF]). The primer pairs described here should be used in two separate PCR protocols: two multiplexes -one to identify PG1, 3 and 9 and the second for identification of PG2, 4 and 13 -and two independent single PCRs.

The first single PCRs will allow for the identification of strains in PG 7 and 8 with the specific primer pair (shcF) previously described [START_REF] Bartoli | The Pseudomonas viridiflava phylogroups in the P. syringae species complex are characterized by genetic variability and phenotypic plasticity of pathogenicity-related traits[END_REF]. The primer pair shcF could not be incorporated into the multiplex due to incompatibility of the annealing temperatures. The second single PCR will allow the identification of strains in PG10. It was not possible to couple this last primer pair because the size of the amplicon, 324 bp, is similar to amplicons obtained for PG1 (321bp) and PG2 (341 bp).

The method described here will facilitate exploration of population structures of P. syringae

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proposed here will open new perspectives on the importance of assemblages of genotypes of this bacterium in its survival, fitness and pathogenicity.

Comment citer ce document : Borschinger, B., Bartoli This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as an 'Accepted Article ', doi: 10.1111/jam.13017 This article is protected by copyright. All rights reserved.

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This article is protected by copyright. All rights reserved. For some isolates the phylogroup could not be determined (Ni ) or they were assigned to multiple phylogroups because they were composed of several strains (Mi ). Some of the isolates tested didn't belong to the P. syringae species (NP ).

Table S1

List of strains used during this study and related information.

Table S2

List of the gene sequences used during this study.

  Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

  Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

  Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

  assemblages and in particular in samples where different genetic lines co-exist and potentially interact. It will also contribute to the discovery and characterization of unknown P. syringae phylogroups in natural samples. With the increase in the number of available P. syringae genomes, improving molecular markers for P. syringae phylogroup detection is facilitated. The approach Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  , C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120 (3), 714-723. , DOI : 10.1111/jam.13017 ccepted Article

  Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

  PA= positive agreement, or the number of positive results obtained with a sample containing the targeted phylogroup PD = positive deviation (false positive), or the number of positive results obtained with a sample that did not contain the targeted phylogroup ND = negative deviation (false negative), or the number of negative results obtained with a sample that contained the targeted phylogroup NA = negative agreement, or the number of negative results obtained with a sample that did not contain the targeted phylogroup A = 100% * [(PA + NA) / (PA + PD + ND + NA)] SE = 100% [PA / (PA + ND)] SP = 100% * [NA / (NA + PD)] Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120(3), 714-723. , DOI : 10.1111/jam.13017

Figure 1 .

 1 Figure 1. Relative proportion of Pseudomonas syringae phylogroups (PG1 , PG2 , PG3 , PG7 , PG8 , PG7/8 , PG9 , PG10 , PG13 ) in kiwifruit and apricot samples from Southeastern France. Results based on the PCR primers described in this work (A) were compared with those obtained by phylogenetic analysis based on sequences the cts gene (B).For some isolates the phylogroup could not be determined (Ni) or they were assigned to multiple phylogroups because they were composed of several strains (Mi ). Some of the isolates tested didn't belong to the P. syringae species (NP ).

Table 1 .

 1 Genes used for primer development Range of the gene that was used to design the primer pairs. † Phylogroup in which the gene was present as single copy. ‡ The given gene was used to develop specific primers for the phylogroups listed in the table. § SNP localization refers to those genes that were present in more than one phylogroup.

	Strain	Genome Accession Number	Gene ID	Genomic Range (bp) *	Gene length (bp)	Protein Annotation	Presence in phylogroups †	Target Phylogroups ‡	Forward primer (bp) localization	Reverse primer (bp) localization	SNP localisation (bp), Fw primer §	SNP localisation (bp), Rv primer §
	NCPPB 1108	GCA_000177495.1 648421386 1935-4352	2418	DNA gyrase/topoisomerase IV, subunit B	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	1	387-410	732-754	387, 390, 391, 393, 394, 397, 399, 400, 405, 408	732, 733, 735, 738, 741, 744, 747, 750, 751, 753
	CC1458 GCA_000452805.2 2506895564 4542-4985	444	Protein of unknown function. DUF421 domain	2	2	21-43	385-404	na	na
												918, 919, 920, 921,
	NCPPB 3335	GCA_000164015.2 648877554	38103-39338	1236	Protein of unknown function. DUF1835 domain	1, 3, 7, 9, 10	3	61-80	918-937	66, 69, 72, 73, 74, 75, 78, 79	922, 923, 924, 926, 927, 928, 929, 930, 931, 933, 934, 935,
												936
	CC1513 GCA_000452765.2 2506907413	8421-11348	2928	Lysyl-tRNA synthetase, class I	4	4	605-624	1470-1489	na	na
	CC1417 GCA_000452825.2 2506965122 7504-8115	612	Chemotaxis like protein	9	9	41-60	450-469	na	na
	CC1466 GCA_000452785.2 2506901715	17733-18980	1248	Protein of unknown function. DUF2599 domain	1, 10	10	435-458	780-799	435, 443, 447, 448, 456, 458	780, 786, 795, 798
	UB246	GCA_000452865.1 2507270264 7514-9898	2385	Alpha-glucosidase, glycosyl hydrolase family GH31	13	13	599-618	1201-1220	na	na

* Comment citer ce document : Borschinger, B., Bartoli, C., Chandeysson, C., Guilbaud, C., Parisi, L., Bourgeay, J.-F., Buisson, E., Morris, C. E. (Auteur de correspondance) (2016). A set of PCRs for rapid identification and characterization of Pseudomonas syringae phylogroups. Journal of Applied Microbiology, 120

(3), 714-723. , DOI : 10.1111/jam.13017

Table 2 .

 2 Primers and PCR conditions used for identification of Pseudomonas syringae phylogroups The multiplex 1 PCR contains primers for PG1, PG3 and PG9, while multiplex 2 PCR contains primers for PG2, PG4 and PG13. The two multiplex PCR primer sets included the Psy primers for the detection of the whole P. syringae complex. The amplicon sizes for multiplex PCRs are described in the single PCR table section.

	Simplex PCR

*

Table 3 .

 3 Specificity and sensitivity of the primers tested with individual primer pairs

	Target Phylogroups	Number of strains tested	PA PD ND NA	Relative accuracy, A (%)	Diagnostic sensitivity, SE (%)	Diagnostic specificity, SP (%)
	PG 1	131	24 0	1 106	99	96	100
	PG 2	157	38 1	0 119	99	100	99
	PG 3	132	17 0	0 115	100	100	100
	PG 4	167	7	0	1 159	99	87	100
	PG 9	138	14 0	0 124	100	100	100
	PG 10	131	8	0 10 113	92	44	100
	PG 13	191	34 1	0 157	99	100	99

This article is protected by copyright. All rights reserved.

Acknowledgements

This work was partially supported by a grant from the French Ministry of Agriculture (contract n° 21000480, "Amélioration des moyens de lutte contre Pseudomonas syringae pv actinidiae"). The salary of B. Borschinger was provided by a scholarship from the TERSYS research federation.

Conflict of Interest

The authors declare no conflict of interest.


Denis Efimov

Stanislav Aranovskiy

Emilia Fridman

Dmitry Sokolov

Jian Wang

Alexey Bobtsov

Adaptive stabilization by delay with biased measurements

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Adaptive stabilization by delay with biased measurements

Denis Efimov * Stanislav Aranovskiy * * Emilia Fridman * * * Dimitry Sokolov * * * * Jian Wang † Alexey A. Bobtsov ‡

INTRODUCTION

Design of identification algorithms, estimators and regulators for dynamical systems are fundamental and complex problems studied in the control theory. In many cases, due to information transmission in the input-output channels, delays appear in the dynamics of the controlled plant.

Influence of a delay on the system stability is vital in many cases [START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF], and it usually leads to degradation of the performances of regulation or estimation [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]. However, in some cases introduction of a delay may result in an improvement of the system transients (see [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]Shaikhet, 2016, 2017;[START_REF] Efimov | On hyper-exponential output-feedback stabilization of a double integrator by using artificial delay[END_REF] and the references therein). The idea of these papers is that unmeasured components of the state can be calculated using delayed values of the measured variables, which allows a design of observer to be passed by.

The goal of this note is to extend the results obtained in [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]Shaikhet, 2016, 2017) for linear systems to adaptive stabilization of a class of nonlinear systems, which contain a globally Lipschitz nonlinearity, and have a part of the measurements available with a constant bias, which is induced by a sensor error. Since for embedded control and estimation solutions, the amount of computations needed for realization is a critical resource (less important than the used memory in some scenarios), in this note we avoid to design an (reduced order) observer for the state, but introduce delayed measurements in the feedback and adaptation algorithm. The closed-loop system becomes time-delayed, then stability analysis of the regulation error is based on the Lyapunov-Krasovskii functional proposed in [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]Shaikhet, 2016, 2017).

It is important to note that there exist papers devoted to adaptive control of time-delay systems as, for example, [START_REF] Mirkin | Robust adaptiveoutput-feedback tracking for a class of nonlineartime-delayed plants[END_REF][START_REF] Pepe | Adaptive output tracking for a class ofnonlinear time delay systems[END_REF] (the uncertain parameters appear in the state equation only), or papers dealt with adaptive control for systems with (multiplicative) uncertain parameters in the output equation (Zhang and Lin, 2019a,b) (without presence of time delays), but to the best of our knowledge there is no theory on the intersection of these approaches. This work and the considered problem statement is motivated by a pendulum control application, which is finally used for illustration.

The outline of this work is as follows. The preliminaries are given in Section 2. The problem statement and the adaptive control design are presented in sections 3 and 4, respectively. A nonlinear pendulum application is considered in Section 5. The symbol 1, m is used to denote a sequence of integers 1, ..., m. For a symmetric matrix P ∈ R n×n , the minimum and the maximum eigenvalues are denoted as λ min (P ) and λ max (P ), respectively. For a matrix A ∈ R n×m , |A| = λ max (A A) is the induced norm. The identity matrix of dimension n × n is denoted by I n .

Neutral time-delay systems

Consider an autonomous functional differential equation of neutral type with inputs [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

ẋ(t) = f (x t , ẋt , d(t)) (1)
for almost all t ≥ 0, where x(t) ∈ R n and

x t ∈ W 1,∞ [-τ,0]
is the state function,

x t (s) = x(t + s), -τ ≤ s ≤ 0, with ẋt ∈ L n [-τ,0] ; d(t) ∈ R m is the external input, d ∈ L m ∞ ; f : W 1,∞ [-τ,0] × L n [-τ,0] × R m → R n
is a continuous function, that is globally Lipschitz in the second variable with a constant smaller than 1, ensuring forward uniqueness and existence of the system solutions [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. We assume f (0, 0, 0) = 0. For the initial function x 0 ∈ W 1,∞ [-τ,0] and disturbance d ∈ L m ∞ denote a unique solution of the system (1) by x(t, x 0 , d), which is an absolutely continuous function of time defined on some maximal interval [-τ, T ) for T > 0, then x t (x 0 , d) ∈ W 1,∞ [-τ,0] represents the corresponding state function with x t (s, x 0 , d) = x(t + s, x 0 , d) for -τ ≤ s ≤ 0.

Given a locally Lipschitz continuous functional

V : R × W 1,∞ [-τ,0] × L n [-τ,0] → R + define
its derivative in the Driver's form:

D + V (t, φ, d) = lim h→0 + sup 1 h [V (t + h, x h (φ, d), ẋh (φ, d)) -V (t, φ, φ)],
where x h (φ, d) is a solution of the system (1) for φ ∈ W 1,∞ [-τ,0] and d(t) = d for all t ≥ 0 and some d ∈ R m .

ISS of time delay systems

The input-to-state stability (ISS) property is an extension of the conventional stability paradigm to the systems with external inputs [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF][START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF][START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF]. Definition 1. [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF][START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF] The system (1) is called practical ISS, if for all

x 0 ∈ W 1,∞ [-τ,0] and d ∈ L m ∞ there exist q ≥ 0, β ∈ KL and γ ∈ K such that |x(t, x 0 , d)| ≤ β( x 0 W , t) + γ(||d|| ∞ ) + q ∀t ≥ 0. If q = 0 then (1) is called ISS.
For establishment of this stability property, the Lyapunov-Krasovskii theory can be used [START_REF] Pepe | A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems[END_REF][START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF][START_REF] Efimov | On hyper-exponential output-feedback stabilization of a double integrator by using artificial delay[END_REF]. Definition 2. A locally Lipschitz continuous functional V :

R + × W 1,∞ [-τ,0] × L n [-τ,0] → R + (i.e., V (t, φ, φ)) is called simple if D + V (t, φ, d) is independent on φ.
For instance, a locally Lipschitz functional

V : R + × W 1,∞ [-τ,0] → R + is simple, another example of a simple functional is given in Theorem 6 below. Definition 3. A locally Lipschitz continuous functional V : R + × W 1,∞ [-τ,0] × L n [-τ,0] → R + is called practical ISS Lyapunov-Krasovskii functional for the system (1) if it is simple and there exist r ≥ 0, α 1 , α 2 ∈ K ∞ and α, χ ∈ K such that for all t ∈ R + , φ ∈ W 1,∞ [-τ,0] and d ∈ R m : α 1 (|φ(0)|) ≤ V (t, φ, φ) ≤ α 2 ( φ W ), V (t, φ, φ) ≥ max{r, χ(|d|)} =⇒ D + V (t, φ, d) ≤ -α(V (t, φ, φ)).
If r = 0 then V is an ISS Lyapunov-Krasovskii functional. Theorem 4. [START_REF] Fridman | On input-to-state stability of systems with time-delay: A matrix inequalities approach[END_REF] If there exists a (practical) ISS Lyapunov-Krasovskii functional for the system (1), then it is (practical

) ISS with γ = α -1 1 • χ.
Converse results for Theorem 4 can be found in [START_REF] Pepe | Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form[END_REF][START_REF] Efimov | A note on converse Lyapunov-Krasovskii theorems for nonlinear neutral systems in Sobolev spaces[END_REF].

ROBUST OUTPUT ADAPTIVE REGULATION WITH BIASED MEASUREMENTS

Consider a nonlinear system for the time t ≥ 0:

ẋ1 (t) = x 2 (t), ẋ2 (t) = A 21 x 1 (t) + A 22 x 2 (t) + A 23 x 3 (t) +B 1 (u(t) + Ω(t)θ 2 ) + L 1 φ(x(t)) + d 1 (t), ẋ3 (t) = A 31 x 1 (t) + A 32 x 2 (t) + A 33 x 3 (t) (2) +B 2 (u(t) + Ω(t)θ 2 ) + L 2 φ(x(t)) + d 2 (t), y 1 (t) = x 1 (t) + θ 1 , y 2 (t) = x 3 (t),
where x 1 (t) ∈ R n and x 2 (t) ∈ R n are the position and velocity, respectively,

x 3 (t) ∈ R p is an additional state, x(t) = [x 1 (t) x 2 (t) x 3 (t)] ∈ R 2n+p is the total state vector of (2), the initial conditions x(0) = x 0 ∈ R 2n+p are unknown; u(t) ∈ R m is the control input; y(t) = [y 1 (t) y 2 (t)] ∈ R n+p is the output available for measurements, d(t) = [d 1 (t) d 2 (t)] ∈ R n+p is the disturbance with d ∈ L n+p ∞ ; θ 1 ∈ R n
is the vector of biases in the measurements of the position x 1 (t), θ 2 ∈ R r is the vector of uncertain parameters in the state dynamics, θ = [θ 1 θ 2 ] ∈ R n+r , the regressor Ω : R + → R m×r is a known continuous matrix function; the nonlinearity φ can be partitioned as

φ(x) = φ 1 (x) φ 2 (x 3 ) ,
where φ 1 : R 2n+p → R s1 and φ 2 : R p → R s2 , then we can decompose

L 1 = [L 11 L 12 ], L 2 = [L 21 L 22 ]
, and φ(x) is assumed to be Lipschitz continuous; all the matrices are constant and known having the corresponding dimensions.

Therefore, the considered system is subject to unknown inputs d(t), it contains unknown parameters θ (part of them corrupt the measurements), and the equations of (2) are nonlinear and time-varying due to the presence of φ and Ω, respectively. The goal is to design an (dynamical) output control input u(t) = u(y 1 (t), y 2 (t), Ω(t)) ensuring a practical ISS property of the closed loop system for all x 0 ∈ R 2n+p and all d ∈ L n+p ∞ under the restriction to minimize the computational complexity of the algorithm (in order to be able to use the proposed solution as a component of an embedded system).

We will use the following hypothesis about the properties of (2): Assumption 1. For the regressor function Ω(t) there is a known upper bound Ω ≥ 0 such that sup t≥0 |Ω(t)| ≤ Ω.

The function φ admits a global Lipschitz constant γ ≥ 0:

|φ(x)-φ(x )| ≤ γ|x-x | for all x, x ∈ R 2n+p . The function φ 2 : R p → R s2 is known.
Under the introduced restriction on φ the system (2) has well-defined solutions for all t ≥ 0 for any x 0 ∈ R 2n+p and any d ∈ L n+p ∞ [START_REF] Khalil | Nonlinear Control[END_REF].

MAIN RESULTS

Due to a rather complicated structure of the considered system and introduced uncertainty, clearly, for realization of a robustly stabilizing control it is necessary to use the full state x(t) information. Consequently, it is required to design an estimator for x 1 (t), which is measured with an unknown bias θ 1 , and for its velocity x 2 (t). Facing all uncertain terms presented in (2), i.e. θ 2 and d(t), such an observation problem becomes rather intriguing, and a corresponding observer solving these issues will be also complex and nonlinear. In [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF]Shaikhet, 2016, 2017) an approach is presented for design of a linear delayed output static control for a linear system, which avoids a state observer design by introducing the estimates of x(t) through delayed output y(t) values. Hence, such a method has a low computational capacity (since for delay operation only memory is needed). In this work we will follow the same approach.

Defining x 1 (t -h) = x 1 (0) for t ∈ [0, h], where h > 0 is the delay, the control algorithm proposed in this paper is

u(t) = -(K 1 + K 2 )y 1 (t) + K 2 y 1 (t -h) (3) -K 3 y 2 (t) -K 4 φ 2 (y 2 (t)) + K 1 θ1 (t)
-Ω(t) θ2 (t), where θ1 (t) ∈ R n and θ2 (t) ∈ R r are the estimates of θ 1 and θ 2 , respectively; K i ∈ R m×n for i = 1, 2, K 3 ∈ R m×p and K 4 ∈ R m×s2 are the control gains to be derived.

Similarly, an adaptive law for θ1 (t) can be synthesized:

θ1 (t) = (F 1 + F 2 )y 1 (t) -F 2 y 1 (t -h) (4) +F 3 y 2 (t) + F 4 φ 2 (y 2 (t)) -F 1 θ1 (t),
where

F i ∈ R n×n for i = 1, 2, F 3 ∈ R n×p and F 4 ∈ R n×s2
are the adaptation gains which will be defined later. An adaptive law for θ2 (t) is more sophisticated, and such a choice of the structure will become clear from the stability analysis given next:

θ2 (t) = Ω (t)[(S 1 + S 2 )y 1 (t) -S 2 y 1 (t -h) (5) +S 3 y 2 (t) + S 4 φ 2 (y 2 (t)) -S 1 θ1 (t)] -S 5 θ2 (t),
where

S i ∈ R m×n for i = 1, 2, S 3 ∈ R m×p , S 4 ∈ R m×s2
and S 5 ∈ R r×r are also the adaptation gains.

Remark 5. There is also an algebraic way to solve the problem of estimation of unknown values θ 1 , θ 2 and signals x 1 (t), x 2 (t) (in the framework of indirect adaptive control), which is based on some structural restrictions and auxiliary filtering. Indeed, let p = n and J 1 L 11 = J 2 L 21 for some matrices J 1 and J 2 , then define

ζ(t) = J 1 x 2 (t) - J 2 x 3 (t) with ζ(t) = Y 1 (y 1 (t) -θ 1 ) + Y 2 ẏ1 (t) +Y 3 y 2 (t) + Y 4 (u(t) + Ω(t)θ 2 ) +Y 5 φ 2 (y 2 (t)) + J 1 d 1 (t) -J 2 d 2 (t), where Y 1 = J 1 A 21 -J 2 A 31 , Y 2 = J 1 A 22 -J 2 A 32 , Y 3 = J 1 A 23 -J 2 A 33 , Y 4 = J 1 B 1 -J 2 B 2 , Y 5 = J 1 L 12 -J 2 L 22 .
Let also for brevity d 1 (t) = 0, d 2 (t) = 0, Y 1 = 0. Note that by construction:

ζ(t) = J 1 ÿ1 (t) -J 2 ẏ2 (t),
and equating the expressions for ζ(t) we obtain:

ζ(t) = -Y 1 θ 1 + Y 4 Ω(t)θ 2 ,
which is a linear regressor model with respect to unknown parameters θ 1 and θ 2 that can be used for their identification, where ζ(t) and Ω(t) variables calculated as

Ω(t) = λ 2 (s + λ) 2 Ω(t), ζ(t) = J 1 λ 2 s 2 (s + λ) 2 y 1 (t) -Y 2 λ 2 s (s + λ) 2 y 1 (t) -J 2 λ 2 s (s + λ) 2 y 2 (t) - λ 2 (s + λ) 2 [Y 1 y 1 (t) + Y 3 y 2 (t) +Y 4 u(t) + Y 5 φ 2 (y 2 (t))]
with s being the differentiating operator and λ > 0 is a tuning parameter of the filters. Inversely, if

d 1 (t) = 0, d 2 (t) = 0, Y 1 = 0, θ 2 = 0, then ψ(t) = 1 s + 1 ζ(t) = J 1 s s + 1 y 1 (t) -J 2 1 s + 1 y 2 (t)
is a variable that we can calculate, and

ψ(t) = ζ(t) - 1 s + 1 ζ(t) = ζ(t) - 1 s + 1 [Y 2 ẏ1 (t) + Y 3 y 2 (t) +Y 4 u(t) + Y 5 φ 2 (y 2 (t))].
Hence, for a nonsingular J 1 we obtain:

x 2 (t) = J -1 1 ψ(t), where

ψ(t) = ψ(t) + Y 2 s s + 1 y 1 (t) + 1 s + 1 [Y 3 y 2 (t) +Y 4 u(t) + Y 5 φ 2 (y 2 (t))] + J 2 y 2 (t)
can be calculated using filters. If d 1 (t) = 0 and d 2 (t) = 0, then these approaches lead to a reconstruction of unknown parameters and variables corrupted by noises, and robust estimation tools should be applied. Next, a control design has to be performed. A drawback of such solutions is also their computational complexity comparing to (3), ( 4), ( 5), where just additional adaptation algorithms (observers or filters) are introduced to calculate θ1 (t) and θ2 (t) in order to compensate the influence of θ 1 and θ 2 .

It is worth noting that a complexity of ( 3), ( 4), ( 5) comes from another side. Initially the system ( 2) is delay-free (the state x(t) ∈ R 2n+p ), then (3), ( 4), ( 5) introduces the delay h and transforms the system into the retarded type time-delay dynamics [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF] (the state function is from C 2n+p [-h,0] ), while for the stability analysis we will perform below an additional transformation of the closedloop system to the neutral type (with the state from

W 1,∞ [-h,0]
). Therefore, (3), ( 4), ( 5) needs a rather sophisticated analysis, but it allows a simple realization.

The restrictions on selection of the control and adaptation gains, and the conditions to check, are given in the following theorem. Theorem 6. Let Assumption 1 be satisfied, if for given K i , F i with i = 1, 4 and S i with i = 1, 5 the system of linear matrix inequalities

Q ≤ 0, P = P > 0, α > 0, β > 0, δ > 0, η > 0, (6) M M ≤ ρIs 1 +s 2 , 4 e -h h 2 In ≥ αB B S = G P + qh 2 G Γ ΓA, S2 = -qh 2 G Γ ΓB, S4 = qh 2 G Γ ΓM Q =      Q11 N N N M -αqI3n+p M N M M -βI3n+p 0 0 0 N M M 0 0 0 0 N 0 0 M 0 0 M 0 Q44 0 -S5 0 M -(η -qh 2 )I3n+p 0 -S 5 0 -δIr      , Q11 = P A + A P + qh 2 A Γ ΓA + ρβγ 2 C C + P, Q44 = -2S5 + (qh 2 |B1| 2 Ω2 + )Ir, N = P + qh 2 A Γ Γ, M = qh 2 Γ Γ,
is feasible with respect to P , α, β, δ and η for some q > 0, > 0 and ρ > 0, where

A =   0 In 0 0 A21 -B1K1 A22 -hB1K2 A23 -B1K3 B1K1 A31 -B2K1 A32 -hB2K2 A33 -B2K3 B2K1 F1 hF2 F3 -F1   , B =   0 B1K2 B2K2 -F2   , G =   0 B1 B2 0   , D =   0 In Ip 0   , M =   0 0 L11 L12 -B1K4 L21 L22 -B2K4 0 F4   , C = diag In In Ip 0 , S = S1 hS2 S3 -S1 , S4 = 0 S4 , Γ = 0 In 0 0 .
Then the system (2) with the control (3) and adaptive laws (4), ( 5) is practically ISS.

All proofs are omitted due to space limitations.

The conditions of the theorem connect the control parameters to be tuned (the gains K i , F i , S i and the admissible delay h), the auxiliary constants (q > 0, > 0, ρ > 0) and the variables of linear matrix inequalities (Q ≤ 0, P > 0, α > 0, β > 0, δ > 0 and η > 0), which are obtained applying numerical solvers to (6). It is worth noting that the inequality Q ≤ 0 in ( 6) is always satisfied for a sufficiently small value of h. Remark 7. The presentation above is given by factorizing the LMIs in the briefest way, which, however, may be more conservative. For example, taking

ξ(t) = z (t) R (t) φ (Cz(t)) θ 2 (t) d (t) θ 2
we obtain that if for given K i , F i with i = 1, 4 and S i with i = 1, 5 the system of linear matrix inequalities

Q ≤ 0, P = P > 0, α > 0, β > 0, δ > 0, η > 0, (7)
S = G P + qh 2 G Γ ΓA, S 2 = -qh 2 G Γ ΓB, S 4 = qh 2 G Γ ΓM Q =      Q 11 N B N M B N B M B -4q e -h h 2 In B M M M N M M B M M M -βI s 1 +s 2 0 0 0 D N D M B D M M 0 0 0 0 N D 0 0 B M D 0 0 M M D 0 Q 44 0 -S 5 0 D M D -(η -qh 2 )I n+p 0 -S 5 0 -δIr     , Q 11 = P A + A P + qh 2 A Γ ΓA + βγ 2 C C + P,
is feasible with respect to P , α, β, δ and η for some q > 0 and > 0 (where the meaning of other variables is the same as in the formulation of Theorem 6), then the system (2) with the control (3) and adaptive laws (4), ( 5) is also practically ISS. However, it is worth to stress a more nonlinear nature of (7) comparing with (6).

Introducing additional mild restrictions we can reformulate the conditions of Theorem 6 by considering the control and adaptation gains K i , F i , S i as solutions of LMIs: Corollary 8. Let Assumption 1 be satisfied, if the system of linear matrix inequalities

Q ≤ 0, P -1 = P - ≥ 0, α > 0, β > 0, δ > 0, η > 0, (8)
ρI s 1 +s 2 M M I 3n+p ≥ 0,

I 3n+p P -1 C CP -1 1 ργ 2 In ≥ 0, 2P -1 - α 4 e h ∆ -1 U G -W I GU -IW I 3n+p ≥ 0, ∆ = c In 0 0 0 0 c -1 In 0 0 0 0 Ip 0 0 0 0 In , Σ = G + qh 2 B 1 (ΓA 0 P -1 -B 1 U ), Σ 2 = -qh 2 B 1 B 1 U 2 , 0 S 4 = qh 2 B 1 L 11 L 12 -B 1 K 4 Q =        Q11 I 3n+p I 3n+p 0 I 3n+p 0 Q17 I 3n+p -αqI 3n+p 0 0 0 0 -Γ I 3n+p 0 -βI 3n+p 0 0 0 -Γ 0 0 0 Q 44 0 -S 5 0 I 3n+p 0 0 0 Q 55 0 -Γ 0 0 0 -S 5 0 -δIr 0 Q 17 -Γ -Γ 0 -Γ 0 - 1 qh 2 In        , Q11 = A 0 P -1 + P -1 A 0 -GU -U G + IW +W I + βI 3n+p + P -1 , Q17 = -P -1 A 0 Γ + U B 1 , Q 44 = -2S 5 + (qh 2 |B 1 | 2 Ω2 + )Ir , Q 55 = -(η -qh 2 )I 3n+p ,
is feasible with respect to P -1 , U, W, Σ, K 4 , F 4 , S 4 , S 5 , α, β, η and δ for some given q > 0, c > 0, > 0, µ > 0 and ρ > 0, where

A 0 =    0 I n 0 0 A 21 A 22 A 23 0 A 31 A 32 A 33 0 0 0 0 0    , I =    0 0 0 I n    ,
and the matrices B, G, D, M, C and Γ are defined in Theorem 6. Then for

[ K 1 hK 2 K 3 K 5 ] = UP, [ F 1 hF 2 F 3 F 5 ] = WP,
[ S 1 hS 2 S 3 S 6 ] = ΣP, the system (2) with the control and adaptive laws

u(t) = -(K 1 + K 2 )y 1 (t) + K 2 y 1 (t -h) (9) -K 3 y 2 (t) -K 4 φ 2 (y 2 (t)) -K 5 θ1 (t) -Ω(t) θ2 (t), θ1 (t) = (F 1 + F 2 )y 1 (t) -F 2 y 1 (t -h) (10) +F 3 y 2 (t) + F 4 φ 2 (y 2 (t)) + F 5 θ1 (t), θ2 (t) = Ω (t)[(S 1 + S 2 )y 1 (t) -S 2 y 1 (t -h) (11) +S 3 y 2 (t) + S 4 φ 2 (y 2 (t)) + S 6 θ1 (t)]
-S 5 θ2 (t), is practically ISS.

The last important observation is that for the algorithms (3), ( 4), (5) the variables K , F and S have an additional linear constraint (i.e., K 1 = -K 5 , F 1 = -F 5 and S 1 = -S 6 ), which is hard to formulate in terms of an LMI since K = UP , W = FP and S = ΣP , then a possible solution is to assume that the variable θ1 (t) in the algorithms (9), (10), (11) enters with an independent gain.

Let us consider some results of application of the proposed robust adaptive output control.

SIMULATION RESULTS

To illustrate the proposed approach, we consider an inverted pendulum stabilization problem. We use the dy- namic model and mechanical parameters of the setup previously reported in [START_REF] Aranovskiy | Differentiator-based velocity observer with sensor bias estimation: an inverted pendulum case study[END_REF]. The notation is shown in Fig. 1.

Define the state variable vector x := θ θ θr , where θ is the angle between the pendulum and the vertical, and θ r is the angle of the reaction wheel with respect to the pendulum; note that the reaction wheel velocity θr is assumed to be available due to the equipment specifics. Following [START_REF] Aranovskiy | Differentiator-based velocity observer with sensor bias estimation: an inverted pendulum case study[END_REF], the system dynamics is given by ẋ1

= x 2 , ẋ2 = a 1 sin(x 1 ) -b 1 u, ẋ3 = -a 1 sin(x 1 ) + b 2 u, (12)
where a 1 = 50, b 1 = 1.4, b 2 = 31, and u is the motor current considered as the control input. The measured signals are y 1 = x 1 + θ 1 and y 2 = x 3 , where θ 1 is the unknown constant bias of the pendulum position sensor.

It is straightforward to verify that the model ( 12) can be written in the form (2) with Ω(t) ≡ 0. The goal is to drive the system to the origin and to estimate the bias θ 1 .

In [START_REF] Aranovskiy | Differentiator-based velocity observer with sensor bias estimation: an inverted pendulum case study[END_REF], the goal has been achieved constructing a nonlinear velocity observer for the state x 2 combined with the state-feedback control law; however, only local convergence has been shown. In this section, we apply the control law (3), (4). It can be verified that the LMI ( 7) is feasible with h = 0.006 and K 1 = -68, K 2 = -1764, K 3 = -0.12, F 1 = 1, F 3 = 6 • 10 -4 , and K 4 = F 2 = F 4 = 0, thus the conditions of Remark 7 are met.

For simulations, we choose the bias as θ 1 = -10 • ≈ -0.1745. The initial conditions are chosen such that y 1 (0) = 0, i.e., x 1 (0) = -θ 1 , and x 2 (0) = 0, x 3 (0) = 0.

The simulation results of the pendulum stabilization and the θ 1 estimation with the control law (3), (4) are shown in Fig. 2 for the estimated pendulum position y 1 -θ1 and the bias estimation θ1 . The reaction wheel velocity y 2 is depicted in Fig. 3. The simulation results illustrate the the proposed delay-based control law stabilizes the system and allows for the bias estimation.

CONCLUSIONS

Considering a Lipschitz nonlinear system, whose model contains external perturbation and uncertain parameters, while the measurements are available with a constant bias, the problem of robust output adaptive stabilization has been solved. Due to a severe uncertainty of the plant, the state reconstruction has been avoided by introducing artificial delays of the output in the feedback and adaptation algorithms. Applying the Lyapunov-Krasovskii approach, the conditions of practical ISS have been established, which are based on linear matrix inequalities. The efficacy of the proposed approach is demonstrated in simulations for an inertia wheel nonlinear pendulum. Extension of the proposed method to a more general class of systems is a direction of future research.

  by R the set of real numbers and R + = {s ∈ R : s ≥ 0}. For a Lebesgue measurable function of time d : [a, b] → R m , where -∞ ≤ a < b ≤ +∞, define the norm d [a,b) = ess sup t∈[a,b) |d(t)|, where | • | is the standard Euclidean norm in R m , then ||d|| ∞ = d [0,+∞) and the space of d with d [a,b) < +∞ (||d|| ∞ < +∞) we further denote as L m [a,b] (L m ∞ ). Denote by C n [a,b] , a, b ∈ R the Banach space of continuous functions φ : [a, b] → R n with the uniform norm φ [a,b] = sup a≤s≤b |φ(s)|; and by W 1,∞ [a,b] the Sobolev space of absolutely continuous functions φ : [a, b] → R n with the norm φ W = φ [a,b) + || φ|| [a,b) < +∞, where φ(s) = ∂φ(s) ∂s is a Lebesgue measurable essentially bounded function, i.e. φ ∈ L n [a,b] . A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ(0) = 0; it belongs to class K ∞ if it is also radially unbounded. A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is decreasing to zero for any fixed r > 0.

Fig. 1 .

 1 Fig. 1. The inverted pendulum notation, see Aranovskiy et al. (2019).

  The bias estimate θ1 and the true bias value θ 1 .

Fig. 2 .Fig. 3 .

 23 Fig. 2. Stabilization of the inverted pendulum.

  This work was partially supported by 111 project No. D17019 (China), by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898 and by Israel Science Foundation (grant No 673/19).