D Sbarbaro

S Tarbouriech

J M Gomes Da Silva

An event-triggered observer based control strategy for SISO systems

This work addresses the problem of devising a simple event-triggering strategy for observable and controllable SISO systems. The solution considers an observer based controller and an eventtriggering strategy taking into account solely local variables. Sufficient conditions to ensure the global stability of the event-triggering strategy are derived in terms of linear matrix inequalities (LMIs). Two simulation examples illustrate the application of the proposed method to both stable and unstable systems.

I. Introduction

Nowdays the implementation of modern control system requires the use of digital communication networks using both wired and wireless technologies. In this context aperiodic event-and self-triggering strategies have been proposed to deal with issues such as: communication, energy and computation constraints [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF]. In particular, in many distributed applications the point of measurement is geographically separated from the location of the control processing. The sensor information is therefore sent by a wireless network, where the energy consumption can be a critical issue, since these devices are in general feed by batteries. Indeed, there are peaks of energy consumption for transmission/reception of data. Hence, reduce the sampling activity, i.e., the instants where the measurement information is transmitted is of great importance. On the other hand, in classical networks, it can be of interest to reduce the number of messages sent through the network, alleviating in this way the traffic and problems with delays and package losses.

Self-triggering strategies choose the sampling instant based on the available measurements and on predictions of the plant response. On the other hand, event-triggering controllers consider only the current measurements in order to define the sampling instant. Self-triggering strate-gies for observer based controllers have been proposed in [START_REF] Almeida | Observer based self-triggered control of linear plants with unknown disturbances[END_REF] based on a cascade interconnection of a discrete-time observer and a controller designed for state feedback. This approach has also been extended to deal with interconnected systems [START_REF]Observer based self-triggered control of an acyclic interconection linear plants[END_REF].

For event-triggered strategies, [START_REF] Lehmann | Event-based output feedback control[END_REF] proposes the use of a state observer in the event generator and an upper bound on the estimation error for designing an event triggering mechanism to guarantee a stable closed-loop system. A more general approach is proposed by [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF], where three architectures for dynamic output feedback controllers are presented. The event-triggering conditions depend only on the norms of the local variables and are obtained using Lyapunov arguments. Dynamic output feedback controllers are also addressed in [START_REF] Chen | Stability of event-triggered outputfeedback control systems[END_REF], where the asymptotic stability of the resulting closed-loop system is guaranteed by a condition in terms of an LMI (Linear Matrix Inequality) and in [START_REF]Observer based event-triggered control for certain and uncertain linear systems[END_REF] these results are extended to deal with uncertain systems.

In this paper, we consider a Lyapunov approach, similarly to [START_REF] Seuret | Event-triggered sampling algorithms based on a Lyapunov function[END_REF] and [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF], to deal with the design of the eventtriggereing strategy for observer based state feedback controllers. In this context, as the plant evolves continuously, whereas the control signal is updated depending on discrete-time events, event-triggered control systems can be cast as hybrid or impulsive systems. Nevertheless, instead of considering the classical hybrid framework to study mixed continuous and discrete dynamics as defined in [START_REF] Goebel | Hybrid Dynamical Systems. Modeling, Stability and Robustness[END_REF], we use an alternative direction as proposed in [START_REF] Gomes Da | Eventtriggered PI control design[END_REF]. The approach we follow presupposes that a state feedback and an observer are designed, considering the separation principle. Then, an event-triggering strategy is proposed based on the decrease of a Lyapunov function. A key feature of the approach is that it only requires available measurable information. Moreover, the stability of the closed-loop sampled data system under the eventtriggering sampling strategy is formally proven based on the Lyapunov theory.

This paper is organized as follows: Section 2 states the problem to be addressed. Section 3 presents the observer based state feedback controller design in a continuoustime context. Section 4 is then dedicated to the eventtriggering strategy. Section 5 illustrates the implementation of this strategy to control both stable and unstable systems. Finally, Section 6 provides some final remarks and hints for future works.

Notation. For any matrix A. A denotes the transpose. He{A} = A+A . For two symmetric matrices of the same dimensions, A and B, A > B means that A -B is positive definite. I and 0 denote respectively the identity and the null matrix of appropriate dimensions.

II. Problem Statement

Consider the following continuous-time linear plant:

ẋp (t) = Ax p (t) + Bu(t) y p (t) = Cx p (t) (1)
where x p ∈ n , u ∈ , y p ∈ are the plant state, input and output, respectively. Matrices A, B and C are constant matrices of appropriate dimensions. In addition, the pair (A, B) is controllable and the pair (A, C) is observable.

Let us consider an observer based state feedback controller to drive the output to a given nonzero constant set-point r:

ẋo (t) = Ax o (t) + Bu(t) + K o (y(t) -y o (t)) y o (t) = Cx o (t), u(t) = K c x o (t) + K r r.
(2) where x o ∈ n and y o ∈ are the state and output of the observer, respectively. Furthermore, K r ∈ is a feedforward gain, K o ∈ n and K c ∈ n are the observer and controller gains respectively. Design conditions regarding the three gains will be given in the next section.

In this paper, we are interested in looking at the eventtriggered implementation of the controller represented by [START_REF] Almeida | Observer based self-triggered control of linear plants with unknown disturbances[END_REF]. In particular, we consider a sensor node with the capability to estimate the state of the system and a controller based on the estimated state. The sensor and controller are in different nodes of the network as depicted in Figure 1. The block ST represents the eventtriggered sampling strategy. The estimated state is sampled in discrete instants of time and the control action is supposed to be constant between two subsequent sampling instants t k and t k+1 . Note however, that differently from classical digital control approaches, the sampling interval [t k , t k+1 ) is not constant. The system dynamics in the interval [t k , t k+1 ) can therefore be described as follows:

ẋp (t) = Ax p (t) + Bu(t k ) ẋo (t) = Ax o (t) + Bu(t k ) + K o C x(t) u(t k ) = K c x o (t k ) + K r r, ∀t ∈ [t k , t k+1 ) (3)
where x(t) = x p (t)x o (t) ∈ n represents the state estimation error. The main aim of this work is to devise an event-triggered strategy to sample and to update the control signal applied to the plant based solely on available signals, that is, using only the signals u(t), x o (t) and y p (t).

With this aim, two steps should be followed. Firstly, the design of the observer and a state feedback control law should be carried out considering a continuous-time behavior of the closed-loop system and the achievement of some performance requirements. From this step a Lyapunov function, which certificates the asymptotic stability of the closed-loop system, is determined. The second step is the definition of the event-triggered strategy, which will be based on the Lyapunov function determined in the first step.

These steps are described in the following sections.

III. Controller Design

In this section we consider the continuous-time system described by ( 1)-( 2). The control design is carried out according to the separation principle. The observer gain K o is designed to render A + K o C Hurwitz. Thus the equilibrium point xeq = 0 of the state estimation error dynamic:

ẋ(t) = (A + K o C)x(t) (4)
is globally asymptotically stable; i.e. lim t→∞ x(t) = 0, and the estimation output error ẽ(t) = C x(t) (5) also converges to zero; i.e. lim t→∞ ẽ(t) = 0. The dynamic of the observer state with the state feedback controller u(t) = K c x o (t) + K r r is given by:

ẋo (t) = (A + BK c )x o (t) + BK r r + K o ẽ(t) (6)
For a given constant reference signal r, the equilibrium point (x p,eq , x o,eq ) of ( 4) and ( 6) satisfies

x o,eq = x p,eq (A + BK c )x o,eq + BK r r = 0

u eq = K c x o,eq + K r r (7)
This equilibrium point is then uniquely determined by r provided that (A + BK c ) is Hurwitz as follows:

x o,eq = -(A + BK c ) -1 BK r r. ( 8
)
The feedforward gain K r can be designed as K r = -(C(A + BK c ) -1 B) -1 so that y p,eq = r. The inverse of C(A+BK c ) -1 B is well defined if there are no zeros at the origin for the closed-loop system. The dynamics of the error between the observer state x o (t) and its equilibrium point value, xo (t) = x o (t)x o,eq , is given by

ẋo (t) = (A + BK c )x o (t) + K o ẽ(t) (9)
where ẽ(t) can be interpreted as an input. The controller gain K c is designed so that A + BK c is Hurwitz and therefore one can render (9) input-to-state stable with respect to ẽ. In other words, if ẽ is bounded then the state, xo , will also be bounded [START_REF] Sontag | Mathematical Control Theory -Deterministic finite dimensional systems[END_REF]. The following theorem characterizes the stability conditions and provides a Lyapunov function for the composite system, obtained from the equations ( 4), ( 5) and ( 9), in terms of a single LMI.

Theorem 1:

Consider given matrices K o , K c , Q c = Q c > 0, Q o = Q o > 0, if there exist matrices P o = P o > 0, P c = P c > 0 such that Ω < 0, where Ω = He{ P c (A + BK c ) + Qc 2 P c K o C 0 P o (A + K o C) + Qo 2 }, (10)
then the equilibrium point of the system composed by ( 4), ( 5) and ( 9) is asymptotically stable.

Proof: Consider the following candidate Lyapunov function:

V = x o P c xo + x P o x (11)
with P c = P c > 0 and P o = P o > 0. Then, the timederivative of ( 11) along the trajectories given by ( 4), ( 5) and ( 9) is

V = x o He{P c (A + BK c )}x o + x He{P o (A + K o C)}x +x o P c K o C x + x C K o P c xo (12
) Note that Ω defined in [START_REF] Goebel | Hybrid Dynamical Systems. Modeling, Stability and Robustness[END_REF] allows to write:

xo x Ω xo x = V + x o Q c xo + x Q o x. ( 13
)
Hence if Ω < 0 one gets

V < -(x o Q c xo + x Q o x) < 0. ( 14
)
Thus V is a Lyapunov function for the system described by ( 4), ( 5) and ( 9). Hence, one can conclude that the equilibrium point (x p,eq , x o,eq ) of the system composed by ( 4), ( 5) and ( 9) is asymptotically stable.

It is important to note that the Lyapunov function V given in [START_REF] Gomes Da | Eventtriggered PI control design[END_REF], cannot be directly used to design eventtriggering strategies, since the state estimation error is not available. In order to find useful conditions, that can be directly used in the design of implementable eventtriggering strategies, we explore the fact that the closedloop system, described by equations ( 4), ( 5) and ( 9), has a cascade structure. Thus, the following theorem provides a simplified stability condition along with a Lyapunov function.

Theorem 2: Consider the system composed by (4), ( 5) and ( 9) with fixed matrices K o and K c , and matrices

Q o = Q o > 0, Q c = Q c > 0, Q e = Q e
> 0, if there exist P c = P c > 0 and P o = P o > 0 satisfying the following conditions

He{P o (A + K o C)} + Q oe < 0 (15) He{P c (A + BK c )} + Q c P c K o K o P c -Q e < 0 (16)
with

Q oe = Q o + C Q e C
, then the equilibrium point; i.e. the origin, of the composite system is asymptotically stable.

Proof: Let V 1 = x P o x a Lyapunov candidate function for system (4). Its time-derivative along the trajectories of ( 4) is given by:

V1 = x He{P o (A + K o C)}x. ( 17
)
By using condition (15) one gets

V1 + x Q oe x < 0. ( 18
)
Thus V 1 is a Lyapunov function for system (4). Let V 2 = x o P c xo a quadratic function associated to system [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF]. Its time-derivative along the trajectories of ( 9) is given by:

V2 = x o He{P c (A + BK c )}x o + x o P c K o ẽ + (19) ẽ K o P c xo .
By using condition (16) it follows that

V2 + x o Q c xo -ẽ Q e ẽ < 0. ( 20
)
Let us now consider the composite quadratic function 18) and (20), V can be bounded as follows

V = V 1 + V 2 . Using (
V = V1 + V2 < ẽ Q e ẽ -x Q oe x -x o Q c xo . (21)
Thus, by replacing

Q oe = Q o + C Q e C, one obtains V < -(x o Q c xo + x Q o x) < 0. ( 22
)
Hence the quadratic function V = V 1 + V 2 is a Lyapunov function of the composite system given by ( 4), ( 5) and ( 9). Then, one can conclude that the equilibrium point; i.e. the origin, of the composite system (4), ( 5) and ( 9) is asymptotically stable.

IV. Event-triggered strategy

In this section we propose an event-triggered implementation of the observer based state feedback controller issued from Theorem 2.

Considering variables xo , u and ẽ, the observer equation in [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF] , for t ∈ [t k , t k+1 ), can be rewritten as follows:

ẋo (t) = Ax o (t) + B ūs (t) + K o ẽ(t) (23)
with ūs (t) = u(t k )-u eq , where from [START_REF]Observer based event-triggered control for certain and uncertain linear systems[END_REF] recall that u eq = K c x o,eq + K r r. Define now the vector

ξ(t) = [x o (t) ūs (t) ẽ(t)] ∈ n+2 (24)
and the matrix

M =   P c A + A P c + Q c P c B P c K o B P c 0 0 K o P c 0 -Q e   . (25)
The event-triggered sampling strategy can therefore be described by using the following algorithm to define the sampling event:

Algorithm 1: if ξ (t)Mξ(t) > 0 then sample, i.e:

t k+1 = t k ← k + 1 ūs (t) = u(t) -u eq otherwise: ūs (t) = u(t k ) -u eq
The following theorem relies upon Theorem 2 to ensure the asymptotic stability of the closed-loop sampled-data system considering the event-strategy proposed in Algorithm 1.

Theorem 3: Consider the system composed by ( 4), ( 5) and ( 23), matrices P o and P c satisfying conditions of Theorem 2 and the event-triggered sampling strategy given in Algorithm 1, where M is defined as in (25). Then it follows lim t→∞ xo (t) = 0 and lim t→∞ x(t) = 0, or equivalently, the closed-loop system is asymptotically stable.

Proof:

Let V (x o , x) = V 1 (x)+V 2 (x o ) be a candidate Lyapunov function, where V 2 (x o ) = x o P c xo , V 1 (x) = x P o x,
and matrices P o and P c satisfying (15) and ( 16) respectively. Consider the following notation:

• Vs = V1 (x) + V2s (x o , ūs , ẽ) for denoting the timederivative of V along the trajectories of system (4), ( 5), (23).

• V = V1 (x) + V2 (x o , ẽ) for denoting the timederivative of V along the trajectories of system (4), ( 5), [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF]. By using (15) we have

V1 (x) < -x Q oe x, ∀t. (26)
where

Q oe = Q o + C Q e C > 0.
From the definition of matrix M and vector ξ, it follows:

ξ(t) Mξ(t) = V2s (x o , ūs , ẽ) + φ(x o , ẽ) (27)
where

φ(x o , ẽ) = x o Q c xo -ẽ Q e ẽ. If ξ(t) Mξ(t) 0, ∀t ∈ [t k-1 , t k ), then one gets V2s (x o , ūs , ẽ) -φ(x o , ẽ), ∀t ∈ [t k-1 , t k ) (28)
and therefore from ( 26) and (28)

Vs < -x o Q c xo -x Q o x < 0 ∀t ∈ [t k-1 , t k ). (29)
Let us consider now the sampling instant t = t k . At this instant, the control and the sensor values are instantaneously updated and it follows that:

ūs (t k ) = u(t k ) -u eq = K c xo (t k ) Ax o (t k )+B ūs (t k )+K o ẽ(t k ) = (A+BK c )x o (t k )+K o ẽ(t k ) and V2s (x o (t k ), ūs (t k ), ẽ(t k )) = V2 (x o (t k ), ẽ(t k )). (30)
Suppose now that ( 16) is verified and from the proof of Theorem 2, it follows that:

V2s (x o (t k ), ūs (t k ), ẽ(t k )) + φ(x o (t k ), ẽ(t k )) < 0. (31)
Thus, we conclude that

ξ(t k ) Mξ(t k ) < 0 (32)
which means that the interval between two sampling instants is not empty. From this fact and from ( 4) and ( 9), it follows that the state converges asymptotically to the origin under the sampling strategy. Note that the considered Lyapunov function is continuous at the sampling instants, so that the indicated properties are enough to conclude on asymptotic stability.

V. Numerical examples Two simple simulation examples illustrate the strategy proposed in this work.

Example 1: The first example considers an open-loop stable system described by the following state space matrices:

A = 0 1 -2 -4 , B = 0 1 , C = 1 0 . (33)
The observer and controller parameters are defined as follows:

K o = -1 1 , K c = -1 2 , K r = 3. ( 34
) Given matrices Q c and Q e Q c = 1 0 0 1 , Q e = 10, (35)
The simulation considers different initial conditions for the observer and the system and at time t = 10[s] an unit step change in the reference. The behavior of the controller is summarized in figure 2; where the sampling events are represented by a boolean variable s e ; i.e. s e = 1 if a sampling event occurs. As seen in figure 2 the strategy samples only when is necessary; i.e. once the system reaches the steady state no sampling is required. Figure 3 shows the evolution of the Lyapunov function and the switching condition. Notice that at every sampling event, the ξ(t) Mξ(t) is forced to be negative.

Example 2: The second example considers an unstable system having a pole in the right hand side of the complex plane. This type of structure arises for instance in modeling levitation systems. The parameters of the state space model are:

A = 0 1 4 0 , B = 1 0 , C = 1 0 . ( 37
)
The observer and controller parameters are defined by the following values

K o = -3.5 7 , K c = -6 3 , K r = 2. (38) Given matrices Q c = 1 0 0 1 , Q e = 1000 (
As in the previous example, the simulation starts with a reference equal to zero and different initial conditions for the system and the observer, at time t = 10[s] an unit step change in the reference is simulated. As seen in figure 4, compared to the stable case of the previous example, the controller is able to stabilize the system by generating more sampling events, since in between sampling events the system is in fact in open-loop. Figure 5 shows the evolution of the Lyapunov function and the switching condition.

VI. Final Remarks

In this paper a systematic method for designing eventtriggering strategies for observer-based controllers has been presented. The design of the observer and the controller parameters follows the separation principle. The event-triggering design method is based on Lyapunov and ISS arguments providing simple triggering strategies that only use information of available signals. Hence, since a Lyapunov-based approach is considered, the stability under the event-triggered sampling strategy is formally guaranteed. The proposed strategy has been illustrated by considering both stable an unstable systems, showing that the stability of the closed-loop system is ensured. Since the strategy is formulated using state space models, its extension to multi-input multi-ouput systems is straightforward for zero reference signals. For non zero references, however, the system must be square and the closed-loop system must not have transmission zeros at the origin in order to compute the feedforward gain K r . The resulting event strategy will sample all the input and output signals at once.

In addition, the current work lets several directions of research open. Among them, it should be interesting to use additional constraints in order to enrich the stability conditions to decrease their conservatism, as, for example, some constraint about the sampling error. Furthermore, it should be challenging to consider the magnitude limitation of the actuator or of the sensor explicitly in the event-triggering method with the objective to characterize the domain of attraction around the equilibrium point (see [START_REF] Seuret | Eventtriggered sampling for linear closed-loop systems with plant input saturation[END_REF] and [START_REF] Kiener | Actuator saturation and anti-windup compensation in event-triggered control[END_REF] for some interesting directions on this).

Fig. 1 .

 1 Fig. 1. Observer based controller.

  the following values of matrices P o and P c are obtained by solving the LMIs (16) and (15):

Fig. 3 .

 3 Fig. 3. Lyapunov function V (t) and switching condition.

  0

Fig. 4 .

 4 Fig.[START_REF] Lehmann | Event-based output feedback control[END_REF]. Output of the continuous plant yp(t) (blue) and the estimated one yo(t) (red), control signal u(t), control error r(t) -yp(t) and sampling events.

Fig. 5 .

 5 Fig. 5. Lyapunov function V (t) and switching condition.

  Fig.2. Output of the continuous plant yp(t) (blue) and the estimated one yo(t) (red), control signal u(t), control error r(t) -yp(t) and sampling events.

	y p (t),y o (t)	1 2
			0 0	2	4	6	8	10	12	14	16	18	20
								t[s]
			5
	u(t)		0
		0 -5	2	4	6	8	10	12	14	16	18	20
								t[s]
	r(t)-y p (t)		0 5
		0 -5	2	4	6	8	10	12	14	16	18	20
								t[s]
		1.5
	e		1
	s	0.5
			0 0	2	4	6	8	10	12	14	16	18	20
								t[s]

This work was supported by ANR project LimICoS contract number 12 BS03 005 01, HYCON2 Network of Excellence grant agreement 257462, ADNEC project STIC-AmSud grant number 13STIC-03 from CAPES (Brazil), Conycit (Chile) and CNRS (France). D. Sbarbaro is with the Department of Electrical Engineer-He is also supported by CNPq (Brazil) grants 30.6210/2009-6 (PQ) and 480638/2012-8 (Universal).


A H K Palmeira

J M Gomes Da Silva

S Tarbouriech

I M F Ghiggi
email: ilcaghiggi@hotmail.com

Alessandra Helena
email: alessandrakimurap@gmail.com

Kimura Palmeira

João Manoel Gomes
email: jmgomes@ece.ufrgs.br

Da Silva

Stability Analysis of Sampled-data Control Systems under Magnitude and Rate Saturating Actuators *

This paper addresses the problem of stability analysis of sampled-data control of linear systems in the presence of input magnitude and rate saturation. A position-type feedback modeling for the actuator is considered. Based on the use of a discrete-time quadratic Lyapunov function, a loopedfunctional and generalized sector relations (to cope with nested saturation functions), LMI conditions are derived to assess local (regional) and global stability of the sampled-data closed-loop systems under aperiodic sampling strategies.

I. Introduction

Actuator saturation is an ubiquitous feature in control systems. Due to physical or security limitations, the actuators are constrained to provide signals with both magnitude and rate (the derivative) inside certain intervals. The effective magnitude or rate saturation of the control signal is source of instability and performance degradation. Motivated by that, a large amount of works have been devoted to the study of the impact of control saturation on the behavior of closed-loop systems. Most of these works deal only with the problem of magnitude saturation (see for instance [START_REF] Hu | Control systems with actuator saturation: analysis and design[END_REF], [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], [START_REF] Zaccarian | Modern anti-windup synthesis[END_REF] and references therein). On the other hand, the rate of the control signal is also critical and deserves to be taken into account explicitly in many applications. This is the case in the aeronautic field, where the rate limitations in hydraulic actuators can cause the so-called pilot-inducedoscillations (PIO), which can lead to an effective disaster [START_REF] Miller | Maneuvering flight control with actuator constraints[END_REF], [START_REF] Rundqwist | Phase compensation of rate limiters in jas 39 gripen[END_REF]. Other examples are electromechanical actuators (such as valve positioners) and pneumatic actuators, which clearly present dynamic limitations.

Regarding the problem of stability analysis and stabilization of systems presenting magnitude (position) and rate saturation, many works can be cited. Basically, two approaches are followed. The first one considers the design of the controller (possibly nonlinear) in order to generate a signal respecting the magnitude and rate bounds of the actuator [START_REF] Kapila | Fixed structue controller design with amplitude and rate nonlinearities[END_REF], [START_REF] Kapila | LMI-based control of linear systems with actuator amplitude and rate nonlinearities[END_REF], [START_REF] Gomes Da | Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints[END_REF]. The second approach considers a first-order model to represent the dynamics of the actuator, the so-called position-type feedback model. In this case, the saturation in rate appears as the saturation of the actuator state and when the time-constant of the first-order model tends to zero the "ideal" rate limiter is recovered [START_REF] Stoorvogel | Output regulation for linear systems to amplitude plus rate saturating actuators[END_REF]. Considering this approach we can cite works dealing with global [START_REF] Stoorvogel | Output regulation for linear systems to amplitude plus rate saturating actuators[END_REF], semi-global [START_REF] Lin | Semi-global stabilization of linear systems with position and rate-limited actuators[END_REF] and local (regional) stability [START_REF] Gomes Da | Local stabilization of linear systems under amplitude and rate saturation actuators[END_REF]. Moreover, considering simultaneously rate and magnitude saturation leads to a system presenting two nested saturations. General results concerning systems with nested saturations can therefore be applied to consider the global or the regional stability or stabilization problems (see for instance [START_REF] Bateman | An analysis and design method for linear systems under nested saturation[END_REF], [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF], [START_REF] Zhou | An improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation[END_REF]). It should be noted that the aforementioned references consider both the plant and the actuator model in continuous-time. Some counterpart results regarding discrete-time models can be found, for instance, in [START_REF] Gomes Da | Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints[END_REF], [START_REF] Bateman | An analysis and design method for discrete-time linear systems under nested saturation[END_REF].

On the other hand, regarding the case of sampled-data control in the presence of magnitude saturating actuators we can cite for instance [START_REF] Fridman | Robust sampleddata stabilization of linear systems: an input delay approach[END_REF] and [START_REF] Seuret | Taking into acccount period variations and actuator saturation in sampled-data systems[END_REF]. However to the knowledge of the authors, there is a lack of works formally dealing with the problem of sampled-data control in the presence of rate saturation. In particular, it should be noticed that since the position-type feedback modeling is nonlinear (it presents a state saturation), the classical discretization strategies considering periodic sampling and the use of zero-order holders cannot be applied. Furthermore, the use of discretized models can be highly imprecise in this case. Another important aspect, which is quite related to the new paradigm of networked control [START_REF] Zhang | Stability of networked control systems[END_REF], regards the possibility of aperiodic sampling.

This work aims somehow at filling this gap by providing a method to analyze the stability of sampleddata control-loops in the presence of magnitude and rate saturating actuators. We consider the position-type feedback modeling for the actuator and the possibility of having aperiodic sampling. In this context, conditions based on the use of a discrete-time Lyapunov function and a looped-functional [START_REF] Seuret | A novel stability anlysis of linear systems under asynchronous samplings[END_REF] are derived to assess the stability of the closed-loop sampled-data system. The nested saturation nonlinearities resulting from the actuator modeling are taking into account by the application of generalized sector conditions as in [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF]. The proposed conditions are in a quasi-LMI form (that is, if a scalar is fixed, LMI conditions are recovered) and can be easily incorporated in convex optimization problems aiming at maximizing estimates of the region of attraction of the origin or the maximum inter-sampling time for which the stability is ensured regionally or, when possible, globally. Notation. In this paper, the sets N, R n , R n×n and K denote respectively the set of positive scalars, ndimensional vectors, n × n matrices of real elements and the set of continuous functions from an interval [0, T k ] to R n . The superscript ′ T ′ stands for matrix transposition. A (i) denote the ith row of matrix A, A (i, j) is its element in position (i, j), He{A} = A + A T . The symbols I and 0 represent the identity and zero matrices of appropriate dimensions. The notation A > (or <)0 means that A is a symmetric and positive (or negative) definite matrix. The symbol ⋆ stands for symmetric blocks in a matrix. || • || denotes the Euclidean norm.

II. Problem Statement

The plant to be controlled is a linear continuous-time system described in state space as follows:

ẋp (t) = Ax p (t) + Bν(t), (1)
where x p ∈ R n p and ν ∈ R m represent the vectors of plant state and input, respectively, and A and B are constant matrices of appropriate dimensions. In order to model the actuator constraints in position and rate, for each actuator we consider a first-order model [START_REF] Gomes Da | Local stabilization of linear systems under amplitude and rate saturation actuators[END_REF] given by:

ẋa(i) (t) = sat r(i) (-λ (i) x a(i) (t) + λ (i) sat p(i) (u (i) (t))), (2)
for i = 1, ..., m. u ∈ R m and x a ∈ R m represent the input and the actuator state, respectively. The saturation functions are defined as follows

sat p(i) (ω 1(i) ) = sign(ω 1(i) )min{ ω 1(i) , ūp(i) }, sat r(i) (ω 2(i) ) = sign(ω 2(i) )min{ ω 2(i) , ūr(i) },
where ūp(i) and ūr(i) are the symmetrical position and rate bounds, respectively. We consider that the control law is a sampled-data state feedback. The state is sampled at instant t = t k and the value of the control is kept constant, by means of zero-order hold (ZOH), until the next sampling time, t = t k+1 . In other words, u(t) is given as follows:

u(t) = u(t k ) = K p K a x p (t k ) x a (t k ) , t k ≤ t < t k+1 , (3)
where t k is the sampling instant, and k ∈ N is an increasing sequence of positive scalars. We suppose that the difference between two successive sampling instants, defined by

T k = t k+1 -t k , satisfies 0 ≤ τ 1 ≤ T k ≤ τ 2 .
Thus, the closed-loop system, issued from the connection between (1), ( 2) and (3), with ν(t) = x a (t), can be described by:

ẋp (t) = Ax p (t) + Bx a (t), ẋa (t) = sat r (-Λx a (t) + Λsat p (K p x p (t k ) + K a x a (t k ))), for t k ≤ t < t k+1 . (4) where Λ ∈ R m×m is a diagonal matrix with Λ (i,i) = λ (i) .
The model described by ( 4) can be rewritten using the generic model of systems with nested saturations [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF], as follows:

ẋ(t) = A 2 x(t) + B 2 sat 2 (A 1 x(t) + B 1 sat 1 (u(t))), u(t) = u(t k ) = Cx(t k ), t k ≤ t < t k+1 , (5)
with the plant and actuator states represented by the augmented state vector

x(t) = x p (t) T x a (t) T T , x ∈ R n , sat 1 (•) = sat p (•), with the bound ū1 = ūp , sat 2 (•) = sat r (•)
, with ū2 = ūr , and

A 2 = A B 0 0 , B 2 = 0 I , C = K p K a , A 1 = 0 -Λ , B 1 = Λ. ( 6
) Define now χ k (τ) = x(t k + τ), τ ∈ [0, T k ].
Then, the closed-loop dynamics in the interval [0, T k ] can be represented by:

χk (τ) = A 2 χ k (τ) + B 2 sat 2 (A 1 χ k (τ) + B 1 sat 1 (Cχ k (0))). (7)
The set of all initial conditions (χ 0 (0) ∈ R n ), such that, the corresponding trajectories of the system (7) converge asymptotically to the origin corresponds to the so-called region of attraction of the origin (R a ).Due to the difficulty to determine analytically R a , a problem of interest is to compute an estimate (D) of this region, such that, D ⊂ R a ⊆ R n , taking into account both the actuator constraints and the fact that the interval between two successive sampling instants can vary.

The goal of this paper is to assess the local or global asymptotic stability of the origin of the sampled-data system (5) taking into account the actuator constraints and aperiodic sampling intervals T k ∈ [τ 1 , τ 2 ]. Then we focus on the following problems. P1. Given the state feedback gains K p , K a and the bounds τ 1 and τ 2 on T k , maximize an estimate of region R a . P2. Given the state feedback gains K p , K a , a set of admissible initial conditions (X 0 ) and τ 1 , maximize the upper bound on the interval between two successive sampling instants, τ 2 , such that the corresponding trajectories of the system converge asymptotically to the origin, provided that x(0) ∈ X 0 . When the open-loop system is asymptotically stable, we will consider global stability assessment. In the global case, R a corresponds to the whole state space.

III. Preliminaries

Using the notation χ k (τ), the following vector-valued dead-zone functions can be defined from the magnitude and rate saturation:

ψ 1 (χ k ) = ψ k 1 = sat 1 (Cχ k (0)) -Cχ k (0), ψ 2 (χ k ) = ψ k 2 = sat 2 (A 1 χ k (τ) + B 1 sat 1 (Cχ k (0))) -(A 1 χ k (τ) + B 1 (ψ k 1 + Cχ k (0))). (8)
Then, ∀τ ∈ [0, T k ], the closed-loop system (5) can be rewritten, with the matrices defined in [START_REF] Kapila | Fixed structue controller design with amplitude and rate nonlinearities[END_REF], as follows:

χk (τ) = (A 2 + B 2 A 1 )χ k (τ) + B 2 B 1 Cχ k (0) +B 2 B 1 ψ k 1 + B 2 ψ k 2 . (9)
Regarding dead-zone functions, we recall now the following Lemma, that provides a generalized sector condition.

Lemma 1: [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] Define the set

S ( ū) = {ω ∈ R m , υ ∈ R m ; |ω (i) -υ (i) | ≤ ū(i) , for i = 1, ..., m}.
If ω and υ are elements of set S ( ū), then the dead-zone nonlinearity ψ(ω), defined as ψ(ω) = sat(ω) -ω, satisfies ψ(ω) T U(ψ(ω) + υ) ≤ 0, for any diagonal and positive definite matrix U ∈ R m×m .

In order to apply Lemma 1 to the dead-zone functions ψ k 1 and ψ k 2 , consider matrices G 1 , G 21 and G 22 ∈ R m×n and G 23 ∈ R m×m , and define the following sets:

S ( ū1 ) = {χ k (0) ∈ R n ; |(C -G 1 ) (i) χ k (0)| ≤ ū1(i) , for i = 1, ..., m}, (10)
S ( ū2 ) = {(χ k (0), χ k (τ), ψ k 1 ) ∈ R n × R n × R m ; |(A 1 -G 21 ) (i) χ k (τ) + (B 1 C -G 22 ) (i) χ k (0) +(B 1 -G 23 ) (i) ψ k 1 | ≤ ū2(i) , for i = 1, ..., m}.
(11) Then, if χ k (0) ∈ S ( ū1 ), and (χ k (0), χ k (τ), ψ k 1 ) ∈ S ( ū2 ), then the following inequalities are satisfied for any diagonal and positive definite matrices U 1 and U 2 ∈ R m×m :

ψ T k 1 U 1 (ψ k 1 + G 1 χ k (0)) ≤ 0, (12)
ψ T k 2 U 2 (ψ k 2 + G 21 χ k (τ) + G 22 χ k (0) + G 23 ψ k 1 ) ≤ 0. (13
) Based on conditions ( 12) and ( 13), from a discrete-time and positive definite function V (χ k ) and a continuoustime looped-functional W (τ, χ k ), we state now a Theorem that allows to conclude about the regional stability of the closed-loop system (5) in a level set of the function V (χ k ). This result is basically an extension of Theorem 1 in [START_REF] Seuret | Taking into acccount period variations and actuator saturation in sampled-data systems[END_REF], to cope with position and rate saturation.

Theorem 1:

Consider a function V (χ k ) : R n → R + , such that V (χ k ) > 0, ∀χ k = 0, (14)
and a continuous-time functional

V 0 : [0, τ 2 ] × K → R, which satisfies for all χ k ∈ K V 0 (T k , χ k ) = V 0 (0, χ k ) = 0 ∀τ ∈ (0, T k ), V 0 (τ, χ k ) > 0. (15)
Define now W (τ, χ k ) = V (χ k )+V 0 (τ, χ k ) and let Ẇ (τ, χ k ) be the time-derivative of W (τ, χ k ) with respect to τ, along the trajectories of the system (9), for τ ∈ [0, T k ] and 0 < τ 1 ≤ T k ≤ τ 2 . If there exist matrices G 1 , G 21 , G 22 ∈ R m×n , G 23 ∈ R m×m , diagonal and positive definite matrices U 1 and U 2 ∈ R m×m , such that, the following inequalities are satisfied

(C (i) -G 1(i) )χ k (0) 2 ≤ ū2 1(i) V (χ k (0)), (16)
Θ 1(i) Θ 2(i) Θ 3(i)   χ k (τ) χ k (0) ψ k 1   2 ≤ ū2 2(i) V (χ k (τ)), (17)
Ẇ (τ, χ k ) -2ψ T k 1 U 1 [ψ k 1 + G 1 χ k (0)] -2ψ T k 2 U 2 [ψ k 2 +G 21 χ k (τ) + G 22 χ k (0) + G 23 ψ k 1 ] < 0, (18)
for i = 1, ..., m and with

Θ 1 = A 1 -G 21 , Θ 2 = B 1 C -G 22 and Θ 3 = B 1 -G 23 .
Then for any initial condition x(0) = χ 0 (0) in the set

D = {χ 0 ∈ R n ;V (χ 0 ) ≤ 1}, (19)
it follows that:

a) ∆V (χ k ) = V (χ k (T k )) -V (χ k (0)) < 0, b)
the corresponding trajectories of the sampled-data system (5) with T k ∈ [τ 1 , τ 2 ] converge asymptotically to the origin. Proof: See appendix. In the case the open-loop system (1) is asymptotically stable, the following Corollary provides a condition to assess the global asymptotically stability of the origin.

Corollary 1: Consider a function V (χ k ) : R n → R + , such that, V (χ k ) > 0, ∀χ k = 0, and a continuous-time functional

V 0 : [0, τ 2 ] × K → R, which satisfies for all χ k ∈ K, ∀T k ∈ [τ 1 , τ 2 ], V 0 (T k , χ k ) = V 0 (0, χ k ) = 0.
Define now W (τ, χ k ) = V (χ k )+V 0 (τ, χ k ) and let Ẇ (τ, χ k ) be the time-derivative of W (τ, χ k )with respect to τ, along the trajectories of the system (9), for τ ∈ [0, T k ] and 0 < τ 1 ≤ T k ≤ τ 2 . If there exist diagonal and positive definite matrices U 1 and U 2 ∈ R m×m , such that the following inequality is satisfied

Ẇ (τ, χ k ) -2ψ T k 1 U 1 [ψ k 1 + Cχ k (0)] -2ψ T k 2 U 2 [ψ k 2 +A 1 χ k (τ) + B 1 Cχ k (0) + B 1 ψ k 1 ] < 0, (20)
Then, it follows that the origin of the system ( 5) is globally asymptotically stable, for

T k ∈ [τ 1 , τ 2 ].

IV. Asymptotic stability of saturated and sampled-data systems

In this section, LMI conditions for regional and global asymptotic stability analysis of the sampled-data system (5) are derived from Theorem 1 and Corollary 1, considering appropriate choices for V (χ k ) and V 0 (τ, χ k ).

Theorem 2: Consider the sampled-data system (5) with

T k ∈ [τ 1 , τ 2 ], 0 < τ 1 ≤ τ 2 .
If there exist symmetric and positive definite matrices P, S1 and R ∈ R n×n , diagonal and positive definite matrices Ũ1 and Ũ2 ∈ R m×m , a symmetric and positive definite matrix X1 ∈ R (m+n)×(m+n) , matrices Ỹ ∈ R n×n , G1 and G21 ∈ R m×n , Ñ ∈ R (3n+2m)×n and a positive scalar ε, that satisfy the following inequalities, for j = 1, 2 and i = 1, ..., m:

Π1 + τ j Π2 + τ j Π3 < 0, (21)
Π1 -τ j Π3 τ j Ñ ⋆ -τ j R < 0, (22)
P (C Ỹ -G1 ) T (i) ⋆ ū2 1(i) ≥ 0, (23)
P (A 1 Ỹ -G21 ) T (i) ⋆ ū2 2(i) ≥ 0, (24)
with

Π1 = He{M T 1 PM 3 -M T 4 G1 M 2 -M T 5 G21 M 1 -M T 5 B 1 C Ỹ M 2 -M T 5 B 1 Ũ1 M 4 -ÑM 12 +(εM T 1 + M T 3 )((A 2 + B 2 A 1 ) Ỹ M 1 +B 2 B 1 C Ỹ M 2 -Ỹ M 3 + B 2 B 1 Ũ1 M 4 +B 2 Ũ2 M 5 )} -2M T 4 Ũ1 M 4 -2M T 5 Ũ2 M 5 -M T 12 S1 M 12 , Π2 = He{M T 3 S1 M 12 } + M T 3 RM 3 , Π3 = M T 24 X1 M 24 , (25)
and auxiliary matrices of appropriate dimensions

M 1 = [I 0 0 0 0], M 2 = [0 I 0 0 0], M 3 = [0 0 I 0 0], M 4 = [0 0 0 I 0], M 5 = [0 0 0 0 I], M 12 = M 1 -M 2 , M 24 = [M T 2 M T 4 ] T . (26)
Then, for any initial condition x(0) = χ 0 (0) belonging to the set

E (P) = {x ∈ R n ; x T Px ≤ 1}, (27)
where P = Ỹ -T P Ỹ -1 , the corresponding trajectories of the sampled-data system (5) converge asymptotically to the origin.

Proof: See appendix. Based on the result stated in Corollary 1, the following Corollary provides a condition to assess the global asymptotically stability of the origin of the sampled-data system [START_REF] Rundqwist | Phase compensation of rate limiters in jas 39 gripen[END_REF].

Corollary 2: Consider the sampled-data system (5) with T k ∈ [τ 1 , τ 2 ], 0 < τ 1 ≤ τ 2 . If there exist symmetric and positive definite matrices P, S1 and R ∈ R n×n , diagonal and positive definite matrices Ũ1 and Ũ2 ∈ R m×m , symmetric and positive definite matrix X1 ∈ R (m+n)×(m+n) , matrices Ñ ∈ R (3n+2m)×n and Ỹ ∈ R n×n and a positive scalar ε, such that, the matrix inequalities (21), ( 22) are satisfied, with G1 = C Ỹ and G21 = A 1 Ỹ , then, the origin is globally asymptotically stable.

V. Numerical and Optimization Issues

Based on the conditions given in Theorem 2 and Corollary 2, we propose now optimization problems to solve Problems P1 and P2 stated in Section II. We first recall the following result.

Lemma 2: [START_REF] Seuret | Taking into acccount period variations and actuator saturation in sampled-data systems[END_REF] Let P be a positive definite matrix and define P = M T PM, with M being a non-singular matrix.

If P 0 I I M + M T -P > 0, (28)
then, P < P 0 .

A. Maximization of the estimate of R a

Note that the set E (P) provided by the satisfaction of the conditions in Theorem 2 is, by definition, included in R a and can be used as an estimate of it. Then, from Problem P1, given T k ∈ [τ 1 , τ 2 ] and considering some size criterion, the goal is to maximize the set E (P). With this aim, we can for instance maximize the minor axis of E (P). This can be accomplished from the following optimization problem min δ subject to (21), ( 22), ( 23 (29)

From Lemma 2 and since P = Ỹ -T P Ỹ -1 , the last inequality in (29) implies that P < δ I. Then, the maximal eigenvalue of P is smaller than δ . Hence, the minimization of δ ensures the maximization of the minor axis of E (P). Note that, for τ 1 and τ 2 given, constraints in (29) are LMIs for a fixed ε. Then the optimal solution of (29) can be obtained by solving LMI-based problems on a grid in ε.

B. Maximization of the sampling interval

From Problem P2, we consider a region of admissible initial conditions for the sampled-data system [START_REF] Rundqwist | Phase compensation of rate limiters in jas 39 gripen[END_REF] given in the following form E (P 0 ) = {x ∈ R n ; x T P 0 x ≤ 1}, with P 0 = P T 0 > 0. (30) Then for τ 1 given, the goal is to find the maximum value of τ 2 , for which the stability is ensured for (31) Note that, from Lemma 2, the last inequality implies that P < P 0 and E (P 0 ) ⊆ E (P). Hence, considering τ 1 given, the optimization problem (31) can be solved as a feasibility LMI problem by iteratively increasing τ 2 and testing the feasibility of LMIs, for a fixed ε.

T k ∈ [τ 1 ,
On the other hand, if the open-loop system is asymptotically stable, we can attempt to find the maximal bound τ 2 on the sampling interval for which the global stability of the sampled-data system (5) is ensured. The following optimization problem can be used in this case max τ 2 subject to (21), (22) (32) with G1 = C Ỹ and G21 = A 1 Ỹ .

VI. Illustrative example

Consider the numerical example borrowed from [START_REF] Gomes Da | Local stabilization of linear systems under amplitude and rate saturation actuators[END_REF], where system (4) is described by the following matrices 1 and2. Figure 1 depicts the signals sat p (u) and ẋa , Figure 2 presents the plant states, the actuator state x a and sat p (u). Table I shows the trade-off between the size of the sampling interval T k = [τ 1 , τ 2 ] and the size of the estimate of the region of attraction E (P) obtained from the optimization problem (29). We consider a fixed value for τ 1 (τ 1 = 0.01) and different values for τ 2 . As τ 2 increases, i.e., larger is the interval [τ 1 , τ 2 ], larger values for δ are obtained, i.e., smaller is E (P). Note that the case of constant sampling periods (τ 1 = τ 2 ) leads to a larger set when compared to the cases of aperiodic sampling.

A = 0 1 10 -0.1 B = 0 1 Λ = 20 C = -

VII. Conclusion

In this paper conditions for the stability analysis of sampled-data linear control systems with actuators presenting both magnitude and rate saturation have been proposed. The control signal has been assumed to be constant between two successive sampling instants and the continuous behavior of the plant and the nonlinear actuator model have been explicitly considered. These conditions being LMIs for a fixed scalar ε, convex optimization problems are proposed to compute an estimate of the region of attraction or, given a set of admissible initial conditions, compute the maximal admissible inter-sampling time for which the convergence of the trajectories to the origin is ensured.

  ), (24), δ I I I Ỹ + Ỹ T -P > 0.

  τ 2 ]. This can be accomplished from the following optimization problem max τ 2 subject to (21), (22), (23), (24), P 0 I I Ỹ + Ỹ T -P > 0.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Signals sat p (u) (in red) and ẋa (in blue), for x(0) = [0.16 -0.74 1.2] T

  problem (29), for ε = 3, we obtain an estimate of the region of the attraction defined with

		 567.3811 176.6072 28.8366 
	P =	 176.6072 54.9722 8.9760	 .	(34)
		28.8366	8.9760	2.1313
	Considering 0.16 -0.74 1.2 the T ∈ E (P) and an aperiodic sampling initial condition x(0) =
	with T k ∈ [0.01, 0.06], simulation results are shown in
	Figures

6.4276 -2 0.4598 . (33) with the bounds of magnitude and rate saturation being ū1 = 1 and ū2 = 10. By considering T k ∈ [0.01, 0.06] and the optimization

TABLE I

 I Values of δ and trace of P, with fixed τ 1 = 0.01 for different values of τ 2 . τ 2 δ × (10 2 ) trace(P) × (10 2 )

	1 × τ 1	4.82	4.81
	2 × τ 1	4.93	4.91
	3 × τ 1	5.06	5.04
	4 × τ 1	5.26	5.22
	5 × τ 1	6.24	5.85
	6 × τ 1	8.04	6.36

*This work was supported by CAPES, CNPQ (grants PQ 306210/2009-6, UNIV. 480638/2012-8), Brazil, and by ADNEC project STIC-AmSud grant number 13STIC-03

Appendix

A. Proof of Theorem 1

Suppose χ 0 (0) ∈ D, then from ( 16) and [START_REF] Seuret | Taking into acccount period variations and actuator saturation in sampled-data systems[END_REF], for i = 1, ..., m, it follows that:

It means that χ 0 (0) ∈ S ( ū1 ) and (χ 0 (0), χ 0 (0), ψ 0 1 ) ∈ S ( ū2 ). Hence, we conclude from [START_REF] Bateman | An analysis and design method for linear systems under nested saturation[END_REF], [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF] and [START_REF] Zhang | Stability of networked control systems[END_REF] that Ẇ (0, χ 0 (0)) < 0. Taking into account that V 0 (τ, χ k ) > 0, for τ ∈ (0, T k ) and V 0 (0, χ k ) = V 0 (T k , χ k ) = 0, it follows that V (χ k ) ≤ W (τ, χ k ). This fact along with Ẇ (0, χ 0 ) < 0, leads to:

Since from [START_REF] Bateman | An analysis and design method for discrete-time linear systems under nested saturation[END_REF], V 0 (0, χ 0 (0)) = 0, then we conclude that V (χ 0 (ρ)) < V (χ 0 (0)), i.e., χ 0 (ρ) ∈ D. Repeating now the reasoning, we conclude that V (χ 0 (τ)) < V (χ 0 (0)), ∀τ ∈ [0, T k ], i.e., (χ 0 (τ), χ 0 (0), ψ 0 1 ) ∈ S ( ū2 ), ∀τ ∈ [0, T k ], and we conclude that

Recalling that

and integrating Ẇ (τ, χ 0 (τ)) over the interval [0, T k ] it follows that

Since χ k+1 (0) = χ k (T k ), this procedure can now be repeated and we conclude that V (χ k ) is decreasing at the sampling instants, i.e, V (χ k ) is a discrete-time Lyapunov function for the system (5).

B. Proof of Theorem 2

Consider a quadratic candidate Lyapunov function V (χ k (τ)) = χ k (τ) T Pχ k (τ), χ k ∈ R n , the matrix P = P T > 0, and a candidate continuoustime functional V 0 (τ, χ k ) defined for all τ ∈ [0, T k ], as follows

with symmetric and positive definite matrices

We show now that the satisfaction of ( 21) and ( 22) implies that Γ(τ, χ k ) < 0. Consider the following vector ξ k (τ)

, and positive definite matrices N ∈ R (3n+2m)×n and R ∈ R n×n . Hence for the integral term of (35), the following inequality holds

(36) On the other hand, from [START_REF] Stoorvogel | Output regulation for linear systems to amplitude plus rate saturating actuators[END_REF], it follows that

, for any matrices Y 1 and Y 2 ∈ R n×n . Hence, using ξ k and the auxiliary matrices from (26) one gets

with

Using (36), (37) and the auxiliary matrices defined in (26), it follows that

with

Consider now the contribution corresponding to terms of the form χT k (τ)(.) χk (τ) in the right hand side of (38), which is given by

For a negative contribution of the term χT k (τ)(.) χk (τ), Y 2 should be a non-singular matrix. Then we can define the matrices

} and the relation Y 1 = εY 2 , for a positive scalar ε. Thus by considering ξk = Ξ -1 ξ k , the inequality (38) is rewritten as

(39) From the auxiliary matrices defined in (26), we have

Considering now the following change of variables

and applying in (39), with Π1 , Π2 and Π3 defined as in (25), it follows that

for j = 1, 2. Note that (21) is directly (41) and, by applying the Schur's complement, ( 22) is equivalent to (42). Thus, we conclude that (21) and ( 22) ensure that Γ(τ, χ k ) < 0, which corresponds to (18) in Theorem 1. Now from Theorem 1, in order to conclude that the trajectories converge to origin asymptotically provided x(0) ∈ E (P), we need to show that relations [START_REF] Fridman | Robust sampleddata stabilization of linear systems: an input delay approach[END_REF] and [START_REF] Seuret | Taking into acccount period variations and actuator saturation in sampled-data systems[END_REF] are satisfied. By right and left-multiplying (23) and ( 24) by diag{ Ỹ -1 , I}, it follows that χ k (0) T {P -

1(i) , and χ k (τ) T {P -1 ū2 2(i)

(A 1(i) -G 21(i) ) T (A 1(i) -G 21(i) )}χ k (τ) ≥ 0, i.e. χ k (τ) T (A 1(i) -G 21(i) ) T (A 1(i) -G 21(i) )χ k (τ) ≤ V (χ k (τ)) ū2 2(i) , where, in particular, we considered G 22 = B 1 C and G 23 = B 1 in the condition [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF]. That concludes the proof.


Vincent Andrieu
email: vincent.andrieu@gmail.com

Bayu Jayawardhana
email: bayujw@ieee.org

Sophie Tarbouriech

Necessary and sufficient condition for local exponential synchronization of nonlinear systems

Based on recent works on transverse exponential stability, some necessary and sufficient conditions for the existence of a (locally) exponential synchronizer are established. We show that the existence of a structured synchronizer is equivalent to the existence of a stabilizer for the individual linearized systems (on the synchronization manifold) by a linear state feedback. This, in turns, is also equivalent to the existence of a symmetric covariant tensor field which satisfies a kind of Lyapunov inequality. Based on this property, we provide the construction of such synchronizer.

I. INTRODUCTION

Controlled synchronization, as a coordinated control problem of a group of autonomous systems, has been regarded as one of important group behaviors. It has found its relevance in many engineering applications, such as, the distributed control of (mobile) robotic systems, the control and reconfiguration of devices in the context of internet-of-things, and the synchronization of autonomous vehicles.

For linear systems, the solvability of this problem and, as well as, the design of controller thereof, have been thoroughly studied in literature. To name a few, we refer to the classical work on the nonlinear Goodwin oscillators [START_REF] Goodwin | Oscillatory behaviour in enzymatic control processes[END_REF], to the synchronization of linear systems in [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF], [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF] and to the recent works in nonlinear systems [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF], [START_REF] De Persis | On the Internal Model Principle in the Coordination of Nonlinear Systems[END_REF], [START_REF] De Persis | On the internal model principle in formation control and in output synchronization of nonlinear systems[END_REF], [START_REF] De Persis | Coordination of passive systems under quantized measurements[END_REF], [START_REF] Scardovi | Synchronization of interconnected systems with applications to biochemical networks: An inputoutput approach[END_REF]. For linear systems, the solvability of synchronization problem reduces to the solvability of stabilization of individual systems by either an output or state feedback. It has recently been established in [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF] that for linear systems, the solvability of the output synchronization problem is equivalent to the existence of an internal model, which is a well-known concept in the output regulation theory.

The generalization to nonlinear systems has appeared recently in literature (see, for example, [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF], [START_REF] De Persis | On the Internal Model Principle in the Coordination of Nonlinear Systems[END_REF], [START_REF] De Persis | On the internal model principle in formation control and in output synchronization of nonlinear systems[END_REF], [START_REF] De Persis | Coordination of passive systems under quantized measurements[END_REF], [START_REF] Scardovi | Synchronization of interconnected systems with applications to biochemical networks: An inputoutput approach[END_REF], [START_REF] Hamadeh | Global state synchronization in networks of cyclic feedback systems[END_REF]). In these works, based on the concept of passivity theory (or, the weakened notions of co-coercive systems), some sufficient conditions are studied that can be used to construct a synchronizer that solves the synchronization problem. For such a class of systems, the synchronizer is constructed based on the relative output/state measurement, as in the linear systems case. In [START_REF] Hamadeh | Contructive synchronization of networked feedback systems[END_REF], small-gain theorem is used to construct a synchronization control law for L 2 -stable systems. If we restrict ourselves to the class of incremental ISS, as discussed in [START_REF] Angeli | Further results on incremental input-to-state stability[END_REF], the synchronizer can again be based on the relative output/state measurement.

In general, with the lack of characterization of controlled synchronization for general nonlinear systems, it is difficult to conclude on the generality of the synchronizer as proposed in the aforementioned works. Using recent results on the transverse exponential contraction, we establish in this paper some necessary and sufficient conditions for the solvability of a (locally) exponential synchronization. It extends the work in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF] where only two interconnected systems are discussed. We show that a very important tool to achieve synchronizing is to construct a symmetric covariant tensor field of order two which Lie derivative has to satisfy some specific properties.

The paper is organized as follows. Section II defines the synchronization problem studied here. Section III is devoted to establish necessary and sufficient conditions to the synchronization problem. Finally Section IV gives the conclusions. Notations. The vector of all ones with a dimension N is denoted by 1 N . We denote the identity matrix of dimension n by Id n .

II. PROBLEM DEFINITION

A. System description and communication topology

In this note, we consider the problem of synchronizing N identical nonlinear systems with N ≥ 2. For every i = 1, . . . , N , the i-th system Σ i is described by

ẋi = f (x i ) + g(x i )u i , i = 1, . . . , N (1)
where x i ∈ R n , u i ∈ R p and the functions f and g are assumed to be C 2 . In this setting, all systems has the same drift vector field f and the same control vector field g : R n → R n×p , but not the same controls in R p . For simplicity of notation, we denote the complete state variables by x = x 1 , . . . , x N which is in R N n . The synchronization manifold D, where the state variables of different systems agree with each other, is defined by

D = {(x 1 , . . . , x N ) ∈ R N n | x 1 = x 2 • • • = x N }.
For every x in R N n , we denote the Euclidean distance to the set D by |x| D .

The communication graph G, which is used for synchronizing the state through distributed control u i , i = 1, . . . N , is assumed to be an undirected graph and is defined by G = (V, E), where V is the set of N nodes (where the i-th node is associated to the system Σ i ) and E ⊂ V × V is a set of M edges that define the pairs of communicating systems. Moreover we assume that the graph G is connected.

Let us, for every edge k in G connecting node i to node j, label one end (e.g., the node i) by a positive sign and the other end (e.g., the node j) by a negative sign. The incidence matrix D that corresponds to G is an N × M matrix such that

d i,k =    +1 if node i is the positive end of edge k -1 if node i is the negative end of edge k 0 otherwise
Using D, the Laplacian matrix L can be given by L = DD whose kernel, by the connectedness of G, is spanned by 1 N . We will need the following lemma on the property of L in our main result.

Lemma 1: Let L = L 11 L 1,2:n L 1,2:n L 2:n,2:n
be a non-zero balanced Laplacian matrix associated to an undirected graph G where L 11 is a scalar. Then, the eigenvalues of the (N -1) × (N -1) matrix L := L 2:n,2:n -1 N -1 L 1,2:n are the same as the non-zero eigenvalues of L with the same multiplicity. Moreover, if the graph is connected then -L is Hurwitz.

The proof of Lemma 1 can be found in Appendix A.

B. Synchronization problem formulation

Using the description of the interconnected systems via G, the state synchronization control problem is defined as follows.

Definition 1: The control laws u i = φ i (x), i = 1 . . . , N solve the local uniform exponential synchronization problem of (1) if the following conditions hold:

1) For all non-communicating pair (i, j) (i.e., (i, j) / ∈ E), ∂φi ∂xj (x) = ∂φj ∂xi (x) = 0; 2) For all x ∈ D, φ(x) = 0 (i.e., φ is zero on D); and 3) The manifold D of the closed-loop system

ẋi = f (x i ) + g(x i )φ i (x), i = 1, . . . , N (2)
is uniformly exponentially stable, i.e., there exist positive constants r, k and λ > 0 such that for all x in

R N n satisfying |x| D < r, |X(x, t)| D ≤ k exp(-λt) |x| D , (3)
where X(x, t) denotes the solution initiated from x, holds for all t in the time domain of existence of solution. When r = ∞, it is called the global uniform exponential synchronization problem.

In this definition, the condition 1) implies that the solution u i is a distributed control law that requires only a local state measurement from its neighbors in the graph G.

An important feature of our study is that we focus on exponential stabilization of the synchronizing manifold. This allows us to rely on the study developed in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF] in which an infinitesimal characterization of exponential stability of a transverse manifold is given. As it will be shown in the following section this allows us to formalize some necessary and sufficient conditions in terms of matrix functions for a synchronizing control law to exist.

III. NECESSARY AND SUFFICIENT CONDITION

A. Necessary conditions

In [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF], a first attempt has been made to give necessary conditions for the existence of an exponentially synchronizing control law for only two agents. In [START_REF] Andrieu | Transverse exponential stability and applications[END_REF], the same problem has been addressed for N agents but without any communication constraints (all agents can communicate with all others). In both cases, it is shown that assuming some bounds on derivatives of the vector fields the following two properties are necessary conditions.

IS Infinitesimal stabilizability. The couple (f, g) is such that the n-dimensional manifold {z = 0} of the transversally linear system

ż = ∂f ∂z (z)z + g(z)ũ (4a) ż = f (z) (4b)
with z in R n and z in R n is stabilizable by a state feedback that is linear in z (i.e., ũ = h(z)z for some function h : R n → R p×n ). CMF Control Matrix Function. For all positive definite matrix Q ∈ R n×n , there exist a continuous function P : R n → R n×n and strictly positive real numbers p and p such that

p Id n ≤ P (z) ≤ p Id n (5)
holds for all z ∈ R n , P has a derivative d f P along f in the following sense

d f P (z) = lim h→0 P (Z(z, h)) -P (z) h (6)
where Z(z, •) is the solution to (4b) with an initial state z in R n so that the inequality

v d f P (z)v + 2v P (z) ∂f ∂z (z)v ≤ -v Qv (7)
holds for all (v, z) in R n × R n satisfying v P (z)g(z) = 0. An important feature of properties IS and CMF comes from the fact that they are properties of each individual agent, independent of the network topology. The first one is a local stabilizability property. The second one establishes that there exists a symmetric covariant tensor field of order two denoted by P whose Lie derivative satisfies a certain inequality in some specific directions. This type of condition can be related to the notion of control Lyapunov function which is a characterization of stabilizability as studied by Artstein in [START_REF] Arststein | Stabilization with relaxed controls[END_REF] or Sontag in [START_REF] Sontag | A Lyapunov-Like characterization of asymptotic controllability[END_REF]. This property can be regarding as an Artstein like condition. The dual of the CMF property has been thoroughly studied in [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a riemannian metric (part i)[END_REF] when dealing with an observer design ([12, Eq. ( 8)], see also [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF] or [START_REF] Andrieu | Observability necessary conditions for the existence of observers[END_REF]).

In this note, we show that properties IS and CMF are still necessary conditions if one considers a network of agents with a communication graph G as given in II-A. Hence, as this is already the case for linear system, we recover the paradigm which establishes that a necessary condition for synchronization is a stabilizability property for each individual agent. However, to obtain this property we need to restrict ourselves to a particular class of synchronizing control laws as given in the following theorem.

Theorem 1: Consider the interconnected systems in (1) with the communication graph G and assume that there exists a control law u = φ(x) where φ(

x) = φ 1 (x) . . . φ N (x)
that solves the local uniform exponential synchronization of (1). Assume that there exists a

C 2 function r : R n → R n such that for all x = 1 N ⊗ z = (z, . . . z) ∈ D, with z ∈ R n we have ∂φ ∂x (x) = r(z) ⊗ L, (8)
where L in R N ×N is the Laplacian matrix associated to the graph. Assume moreover that g is bounded and f , g and the φ i 's have bounded first and second derivatives and the closed-loop system is complete. Then properties IS and CMF hold.

Proof : The proof of IS is decomposed into the following two steps. In the first step, we show the stabilizability property for an N n-dimensional system which is established using the tools as developed in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF]. In the second step, employing the structure of the control law as given in Definition 1, we obtain the desired stabilizability property for the transversally linear system (4a).

Step 1 : This part of the proof follows exactly the same steps as the ones presented in [START_REF] Andrieu | Transverse exponential stability and applications[END_REF]. Let e = e 2 e 3 . . . e N with e i = x i -x 1 , i = 2, . . . N , the closed-loop system (1) with the control law φ is given by

ė = F (e, x 1 ) , ẋ1 = G(e, x 1 ) (9)
with e in R (N -1)n , x 1 in R n and where

F = F 2 F 3 . . . F N (10) F i (e, x 1 ) = f (x 1 + e i ) -f (x 1 ) (11) + g(x 1 + e i ) φi (e, x 1 ) -g(x 1 ) φ1 (x 1 ) , G(e, x 1 ) = f (x 1 ) + g(x 1 ) φ1 (e, x 1 ) , (12)
where we have used the notation φi (e, x 1 ) = φ i (x 1 , x 1 + e 2 , . . . , x 1 + e N ).

Note that we have

|e| 2 = N i=2 |x i -x 1 | 2 , ≤ (N -1)|x| 2 D , (14)
and

|x| 2 D ≤ |e| 2 + (N -1) N i=1 x 1 -x i N 2 ≤ 1 + N -1 N 2 |e| 2 . (15)
Hence, if we denote E(e, x 1 , •) the e components of the solution to (9), then (3) implies for all (e, x 1 ) in

R (N -1)n × R n |E(e, x 1 , t)| ≤ (N -1) 1 + N -1 N 2 k exp(-λt) |e|.
It follows that the manifold e = 0 is locally uniformly (in x 1 ) exponentially stable for [START_REF] Goodwin | Oscillatory behaviour in enzymatic control processes[END_REF]. We conclude with [2, Prop. 1] that the so-called Property UES-TL is satisfied. Moreover, we conclude that there exist strictly positive real numbers k and λ such that any solution ( E( e, z,

•), Z(z, •)) to ˙ e = ∂F ∂ e (0, z) e , ż = G(0, z) (16)
with e in R (N -1)n and z in R n satisfies

| E( e, z, t)| ≤ k exp(-λt)| e| (17)
for all ( e, z, t) in

R (N -1)n × R n × R ≥0 .
By the definition of φ in [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF], it follows that in the manifold where e = 0, we have that φi (0, x 1 ) = 0. This implies that for every i = 1, . . . , N ,

∂F i ∂ e i (0, z) = ∂f ∂z (z) + g(z) ∂ φi ∂ e i (0, z) - ∂ φ1 ∂ e i (0, z) (18)
and for all j = i,

∂F i ∂ e j (0, z) = g(z) ∂ φi ∂ e j (0, z) - ∂ φ1 ∂ e j (0, z) . (19)
Hence, this yields

˙ e i = ∂F i ∂ e (0, z) e = ∂f ∂z (z) e i + g(z) ∂ φi ∂ e (0, z) - ∂ φ1 ∂ e (0, z) e.
Step 2: We will now show that (4a) is stabilizable by a state feedback that is linear in z. In particular, the stabilizing control law for (4a) will be given by

ũ = νr(z) z ( 20
)
where ν is a real non-zero eigenvalue of the matrix L 2:n,2:n -

1 N -1 L 1,2:n =: L, where L = L 11 L 1,2:n L 1,2:n L 2:n,2:n
is the Laplacian of G. By Lemma 1, all eigenvalues of L are the same as the non-zero eigenvalues of L (which are all positive) with the same multiplicity and -L is Hurwitz.

Let v = (v 2 , . . . , v N ) in R N -1 be a left eigenvector of L associated to ν, i.e.
v L = νv By using Π = Id n ⊗v as a mapping from R (N -1)n to R n , we will show that the image of the solution to ( 16) are solution to (4a) with the control law (20). Indeed, by denoting z = Πẽ and by using ( 16), we have

ż = ˙ N i=2 v i e i = ∂f ∂z (z) N i=2 v i e i + g(z) N i=2 v i ∂ φi ∂ e (0, z) - ∂ φ1 ∂ e (0, z) e = ∂f ∂z (z) N i=2 v i e i + g(z) N i=2 v i N j=2 [L ij -L 1j ] r(z) e j = ∂f ∂z (z) N i=2 v i e i + g(z)r(z) v (L 2:n,2:n -1 N -1 L 1,2:n ) e = ∂f ∂z (z) + g(z)νr(z) z.
Denote the solution of the above equation by Z(z, z, t). Since Z(z, z, t) = Π E( e, z, t), it follows from [START_REF] Wieland | An internal model principle is necessary and sufficient for linear output synchronization[END_REF] that

| Z(z, z, t)| ≤ |v| k exp(-λt)| e|.
This proves that the property IS holds.

Proof of CMF : To prove CMF, we use the property IS and [2, Proposition 2]. Indeed, note that if we consider another system with state

(z, z) in R n × R n that is described by ż = F (z, z) , ż = f (z) (21) with F (z, z) = f (z + z) -f (z) + g(z)νr(z) z, ν ∈ R,
it follows from IS and [2, Proposition 2] that there exist a function P : R n → R n such that CMF holds. In particular, we have that, for all

(v, z) in R n × R n , v d f P (z)v +2v P (z) ∂f ∂z (z) + g(z)νr(z) v ≤ -v Qv
which implies that (7) holds when v P (z)g(z) = 0. 2

B. A sufficient condition

The interest that we have in the Property CMF given in Subsection III-A is to use the symmetric covariant tensor P in the design of a local synchronizing control law. Indeed, following one of the main results in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF], we get the following sufficient condition for the solvability of (local) uniform exponential synchronization problem. The first assumption is that, up to a scaling factor, the control vector field g is a gradient field with P as a Riemannian metric.

Theorem 2 (Local sufficient condition): Assume that g is bounded and that f and g have bounded first and second derivatives. Assume that there exists a C 2 function P : R n → R n×n with a bounded derivative that satisfies the following two conditions. 1. There exist a C 2 function U : R n → R which has bounded first and second derivatives, and a C 1 function α : R n → R p which has bounded first and second derivative such that

∂U ∂z (z) = P (z)g(z)α(z) ; (22)
holds for all z in R n ; and 2. There exist a positive definite matrix Q and positive constants p, p and ρ > 0 such that (5) and

v d f P (z)v + 2v P (z) ∂f ∂z (z)v -ρ ∂U ∂z (z)v 2 ≤ -v Qv , (23) holds for all (z, v) in R n × R n .
Then, given a connected graph G with associated Laplacian matrix L = (L ij ), there exists a constant such that the control laws u = φ(x) with φ = φ 1 . . . φ N given by

φ i (x) = -α(x i ) N j=1 L ij U (x j ) (24)
and ≥ solves the local uniform exponential synchronization of (1).

Proof : First of all, note that the control law φ satisfies the condition 1) and 2) in Definition 1. Indeed, for all x = 1 N ⊗ z = (z, . . . , z) in D and for all (i, j)

∂φ i ∂x j (x) = -α(z)L ij ∂U ∂z (z) (25)
Hence, for all x = 1 N ⊗ z in D, we get

∂φ ∂x (x) = -α(z) ∂U ∂z (z) ⊗ L . (26)
It remains to prove that the manifold D of the closed-loop system is locally exponentially stable.

As in the proof of Theorem 1, let us denote e = (e 2 , . . . , e N ) with e i = x 1 -x i . Note that the closed-loop system may be rewritten as in [START_REF] Goodwin | Oscillatory behaviour in enzymatic control processes[END_REF] with the vector fields F and G as defined in ( 10)- [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a riemannian metric (part i)[END_REF] with φ as the control law.

The rest of the proof is to apply [2, Proposition 3]. For this purpose, we need to show that for closed-loop system (10)- [START_REF] Sanfelice | Convergence of nonlinear observers on R n with a riemannian metric (part i)[END_REF] the property ULMTE introduced in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF] is satisfied. More precisely, we need to show that there exists a symmetric covariant 2-tensor field P N : R n → R (N -1)n×(N -1)n which satisfies [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF]Ineq. (9)].

By the assumption on the graph being connected and together with Lemma 1, we have that the matrix 1) be a positive definite matrix solution to the Lyapunov equation

A = -(L 2:n,2:n -1 N -1 L 1,2:n ) is Hurwitz. Let S in R (N -1)×(N -
SA + A S ≤ -νS (27)
where ν is a positive real number. Consider the function P N : R n → R (N -1)n×(N -1)n defined as P N (z) = S ⊗ P (z)

Note that we have G(0, z) = f (z). Moreover we have

d G(0,z) P N (z) = S ⊗ d f P (z).
Note that with properties (18), ( 19) and ( 26), it follows that

∂F ∂ e (0, z) = Id N -1 ⊗ ∂f ∂z (z) + A ⊗ α(z)g(z) ∂U ∂z (z) (28)
Hence,

d G(0,z) P N (z) + P N (z) ∂F ∂ e (0, z) + ∂F ∂ e (0, z) P N (z) = S ⊗ d f P (z) + P (z) ∂f ∂z (z) + ∂f ∂z (z) P (z) + (SA + A S) ⊗ ∂U ∂z (z) ∂U ∂z (z)
With ( 27) and ( 23) this implies that

d f P N (z) + P N (z) ∂F ∂ e (0, z) + ∂F ∂ e (0, z) P N (z) ≤ S ⊗ -Q + (ρ -ν) ∂U ∂z (z) ∂U ∂z (z)
Hence, by choosing > ρ ν , [2, Inequality (9)] holds. The last part of the proof is to make sure that the vector field F has bounded first and second derivatives and that the vector field G has bounded first derivative. Note that by employing the bounds on the functions P , f , g, α and their derivatives, the result follows immediately from [2, Proposition 3]. Hence e = 0 is (locally) exponentially stable manifold for system (10)-( 12) in closed loop with the control (24). With inequalities ( 14) and [START_REF] Scardovi | Synchronization in networks of identical linear systems[END_REF], it implies that inequality (3) holds for r sufficiently small. 2

Remark 1: Note that in [START_REF] Andrieu | Transverse exponential stability and applications[END_REF] with an extra assumption related to the metric (the level sets of U are totally geodesic set with respect to the Riemanian metric obtained from P ), it is shown that global synchronization may be achieved when all agents are connected to all other. It is still an open question to know if global synchronization may be achieved in the communication topology constraint context.

C. Illustrative example

As an illustrative example, consider the case in which the vector fields f and g are given by

ż = f (z) + g(z)u with f (z) = -z 1 + sin(z 2 ) cos(z 1 ) + z 2 0 , g(z) = 0 2 + sin(z 1 )
We will now show that the above system satisfy the hypotheses in Theorem 2. Consider a constant matrix P given by

P = 2 1 1 2
It can be checked that by taking

U (z) = z 1 + 2z 2 ,
Eq. ( 22) holds with α = 1 2+sin(z1) . Using the above U , it follows that

v ∂U ∂z (z) = 0 ⇔ v 1 + 2v 2 = 0 in which case, by taking v = -2 1 s where s ∈ R, Eq. (23) becomes -2 1 P ∂f ∂z (z) -2 1 = -3 -2 ∂f 1 ∂z 1 + ∂f 1 ∂z 2 = -3[-2(-1 + sin(z 2 ) sin(z 1 )) -cos(z 1 ) cos(z 2 ) + 1] ≤ -3.
Hence, (23) holds by taking ρ sufficiently large.

The function ∂f ∂z (z) being periodic in z a1 and z a2 we can assume that z a1 and z a2 are in compact subset. This implies employing some variation on Finsler lemma that there exists ρ and Q such that inequality (23) holds. Hence from Theorem 2, the control law given in (24) solves the local exponential synchronization problem for the N identical systems that exchange information via any undirected communication graph G which is connected.

IV. CONCLUSION

In this paper, based on recent results in [START_REF] Andrieu | On transverse exponential stability and its use in incremental stability, observer and synchronization[END_REF], we have presented necessary and sufficient condition for the solvability of local exponential synchronization of N identical affine nonlinear systems through a distributed control law. In particular, we have shown that the necessary condition is linked to the infinitesimal stabilizability of the individual system and is independent of the network topology. The existence of a symmetric covariant tensor of order two, as a result of the infinitesimal stabilizability, has allowed us to design a distributed synchronizing control law. Note that ṽ is non zero since v is not colinear to 1 N . By a routine algebraic computation, it follows that this vector satisfies This shows that ṽ is an eigenvector with the same non-zero eigenvalue of L. It proves the first claim of the lemma. Note that the multiplicity of the eigenvalue ν is the same for both matrices. Also, if the graph is connected, then the 0 eigenvalue of the Laplacian matrix L is of multiplicity 1 and the other eigenvalues are positive and distinct. Hence the matrix -L is Hurwitz.

2

APPENDIX A. Proof of Lemma 1

 1 From the property of Laplacian matrix, the eigenvalues of L are real and satisfy 0 = λ 1 ≤ λ 2 ≤ . . . ≤ λ N . Let us take the non-zero eigenvalue ν > 0 of L and its corresponding eigenvector v in R N . Note that we can decomposev = v a v b , L = L 11 L 1,2:n L 1,2:n L 2:n,2:n with v a and L 11 in R. It follows that L 11 v a + L 1,2:n v b = νv a (29) L 1,2:n v a + L 2:n,2:n v b = νv b .(30)Moreover, since 1 N is an eigenvector associated to the eigenvalue 0,L 11 + L 1,2:n 1 N -1 = 0 (31) L 1,2:n + L 2:n,2:n 1 N -1 = 0(32)Consider now a vector in R N -1 defined byṽ = v b -1 N -1 v a

[L 2

 2 :n,2:n -1 N -1 L 1,2:n ]ṽ = L 2:n,2:n v b -1 N -1 L 1,2:n v b +[1 N -1 L 1,2:n 1 N -1 -L 2:n,2:n 1 N -1 ]v a = νv b -L 1,2:n v a -1 N -1 L 1,2:n v b + [-1 N -1 L 11 + L 1,2:n ]v a = ν ṽ.

This work was supported by the ANR Project LIMICOS contract number 12BS0300501


B Zhang

B Gupta

B Ducharne

G Sebald

T Uchimoto

Dynamic magnetic scalar hysteresis lump model, based on Jiles-Atherton quasistatic hysteresis model extended with dynamic fractional derivatives

Keywords: Hysteresis, ferromagnetic material, Jiles-Atherton model, fractional derivatives

Accurate and simple magnetic material law is necessary to correctly model the complete electromagnetic systems. In this article, a new formulation based on the scalar quasi-static hysteresis Jiles-Atherton model extended to dynamic behavior using fractional derivative dynamic contribution is proposed. The fractional contribution is solved using convolution which highly reduces the numerical issues. The order of the fractional derivation provides a new degree of freedom and allows to obtain correct simulation results on a very large frequency bandwidth. By using such a formulation, highly space and time consuming space discretization techniques (finite differences, finite elements) are avoided while keeping the global accurate simulation results.

MAIN TEXT

The development of new electromagnetic designs, such as the improvement of already existing ones require precise simulation tools. These numerical tools can also be used for the estimation of ferromagnetic losses. In the micromagnetic non-destructive testing field, they are used for the understanding and the interpretation of non-destructive control techniques such as eddy current testing, magnetic incremental permeability or magnetic Barkhausen noise [START_REF] Yusa | Profile reconstruction of simulated natural cracks from eddy current signals[END_REF] [START_REF] Gupta | A space discretized ferromagnetic model for non-destructive eddy current evaluation[END_REF].

Previous scientific progress in the electromagnetic simulation domain mainly focuses on coupling space discretization techniques (Finite Elements Method (FEM), Finite Differences Method (FDM)) to accurate scalar or vector, dynamic or static hysteresis material law [3][4].

For this material law, the best results come from the extension of a vector quasi-static hysteresis contribution (Preisach model [START_REF] Preisach | Über die magnetische Nachwirkung[END_REF], Jiles-Atherton model [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF]) to a dynamic behavior as it is defined by the separation losses in the Bertotti's theory [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF]. Unfortunately, the usual simultaneous resolution of space discretization techniques and hysteresis models can only be performed by time and high memory space consuming iterative techniques. One of them is the so-called fixed point scheme. If the non-linearity levels are weak, this technique gives accurate results whereasfor higher levels (strong hysteresis, saturation) numerical errors are always observed. In this article, an original alternative to space discretization techniques is proposed.

In this approach, the space discretization is left behind and the focus is on numerically reproducing the time variations of lump quantities (The cross-section average induction field B, and the surface tangent field H). A few months ago, by extending the Preisach model quasistatic contribution to fractional derivation operators, the authors succeeded in obtaining very accurate simulation results on a very large frequency bandwidth [START_REF] Zhang | Preisach's model extended with Dynamic fractional derivation contribution[END_REF]. Unfortunately, the issue here, comes from the Preisach model congruent property, i.e. the similarity of the minor cycles obtained for a given H variations at different levels of B. This property becomes real inconvenient, when we try to plot the butterfly loop | |( ) required by the magnetic incremental permeability technique [START_REF] Yashan | Measurements and semi-analytical modeling of incremental permeability using eddy current coil in the presence of magnetic hysteresis[END_REF]. Indeed, using Preisach model | |( ) is limited to just one curve and exhibits no hysteresis. To improve this, a switch from Preisach's model to Jiles-Atherton's model has been finalized. In this article, we explain with details, how we succeed in simulating the butterfly loop and the B(H) curve on a large frequency bandwidth by extending scalar Jiles-Atherton's quasi-static contribution to dynamic behavior thanks to fractional derivation operators. Fractional derivative is introduced in the lumped quasi-static hysteresis model through a dynamic contribution. Eq. 1. And fig. 1 gives a quick overview of the model equation, n is the fractional order and J.A the Jiles-Atherton model. ( ) . ( ( )) (400 Hz). In the final version of the article, a large number of comparisons simulation/measure will be provided, the good results obtained for the butterfly loop | |( ) will also be illustrated.

n dyn n d B t H t J A B t dt ρ - = - (1)

Fig. 1 .

 1 Fig. 1. Block scheme for the Jiles-Atherton extended dynamic model.

Fig. 2

 2 Fig. 2 gives a first illustration for the model accuracy by comparing simulation and

Fig. 2 .

 2 Fig. 2. Comparison of simulation/measurement for Iron/Silicon non-oriented grains under 400Hz sinus excitation field H.