S Tarbouriech

A Seuret
email: aseuret@laas.fr

J M Gomes Da Silva

D Sbarbaro
email: dsbarbar@udec.cl

Observer-based event-triggered control co-design for linear systems *

Keywords: Event-triggered observer, stability, co-design, constant reference tracking, LMI

The paper deals with an observer-based event-triggered control strategy for linear systems using only local (that is available) variables. Sufficient conditions based on linear matrix inequalities (LMI) associated to convex optimization problems are proposed to ensure the asymptotic stability of the closed loop and the output convergence to a constant reference in both emulation and co-design contexts. Indeed, the proposed approach allows either to design the event-triggering rules or co-design the event-triggering rule along with the controller gain.

Introduction

Nowadays the implementation of modern control systems is performed through 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 limited communication capabilities, energy and computation constraints. 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 through 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, to reduce the sampling activity, i.e., the instants where the measurement information is transmitted is of great importance. On the other hand, in classical wired networks, it can be of interest to reduce the number of messages sent through the network, alleviating in this way the traffic and problems regarding delays and package losses. For more details, the reader may refer, for instance, to [START_REF] Postoyan | Triggering mechanism using freely selected sensors for linear timeinvariant systems[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Heemels | An introduction to event-triggered and selftriggered control[END_REF] and references therein.

Self-triggered strategies pre-define the sampling instants based on the available measurements and on predictions of the plant response. On the other hand, event-triggered controllers consider only the current measurements in order to define the next sampling instant. Self-triggering strategies 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] Almeida | 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 node and an upper bound on the estimation error for designing an event-triggered mechanism to guarantee asymptotic stability of the closed-loop system. A more general approach is proposed in [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] Ch | Stability of event-triggered output-feedback 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] Ch | Observer based event-triggered control for certain and uncertain linear systems[END_REF] these results are extended to deal with uncertain systems. However, none of these works have addressed the problem of tuning key parameters of the event generator to minimize the number of triggering events. In the event-triggered control framework, one can consider two approaches. In the first one, which corresponds to an emulation problem, the controller is given a priori (see, for example, [START_REF] Heemels | An introduction to event-triggered and selftriggered control[END_REF], [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF], [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback laws[END_REF] and references therein). The second approach, which is called a co-design problem, performs the design of both the controller and the event-triggering rule, simultaneously, and is addressed in a few papers. The control parameters design is then carried out by using a fully continuous-time approach [START_REF] Aström | Event based control[END_REF][START_REF] Seuret | Event-triggered sampling algorithms based on a Lyapunov function[END_REF][START_REF] Seuret | Event-triggered sampling for linear closed-loop systems with plant input saturation[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] or a fully discrete-time approach [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF]. Of course, the attainable performances are affected by the choice of the continuous-time or the discrete-time synthesis approach, as for example discussed in [START_REF] Abdelrahim | Co-design of output feedback laws and eventtriggering conditions for linear systems[END_REF][START_REF] Antunes | Dynamic programming formulation of periodic event-triggered control: Performance guarantees and co-design[END_REF] where some optimality criteria are included in the synthesis phase.

In this work, inspired by [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF] and [START_REF] Seuret | Event-triggered sampling algorithms based on a Lyapunov function[END_REF], the design of the event-triggering strategy for observer-based state feedback is proposed based on the decrease of a Lyapunov function. In the context of event-triggered control, the plant evolves in continuous time, whereas the control signal is updated depending on discretetime events, the resulting closed-loop system 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] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. The paper then deals with an observer-based event-triggered control strategy for linear systems using only local (that is available) variables. Let us emphasize that the design of event-triggered controllers based on measured signals is a challenging problem (see, for example, [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback laws[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L ∞gain and improved and decentralised event-triggering[END_REF] and references therein). Sufficient conditions based on LMI associated with convex optimization problems are proposed to ensure the asymptotic stability of the closed loop and the output convergence to a constant reference in both emulation (see, for example, [START_REF] Postoyan | Event-triggered and self-triggered stabilization of distributed networked control systems[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Wang | Event-triggering in distributed networked control systems[END_REF][START_REF] Heemels | An introduction to event-triggered and selftriggered control[END_REF] and references therein) and co-design contexts (see, for example, [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF] and references therein). Indeed, the proposed approach allows either to design the event-triggering rules or co-design the eventtriggering rule with the controller gain. Moreover, the results we propose in the paper are complementary to those previously cited in the sense that we pursue an event-triggering strategy for observer-based controllers allowing to track a constant reference. The stability of the closed-loop sampled-data system under the event-triggering strategy is formally proven based on the Lyapunov theory. Furthermore, following the idea presented in [START_REF] Mazo | An ISS self-triggered implementation of linear controllers[END_REF], Zeno behaviors are avoided thanks to a minimum dwell-time, which is explicitly forced as a design parameter of the LMI conditions. The paper can then be considered as a comprehensive version of [START_REF] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF], where the event-triggering strategy was based on a simple and rough algorithm, without a guarantee of the absence of Zeno phenomenon. In addition, the LMI conditions are proposed both in emulation and co-design contexts and are expressed in a very simple and compact form, differently from those of [START_REF] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF]. The originality of the paper relies on the design of the event-triggered control based on two conditions: one to ensure the continuous-time stability conditions and the second one to adjust the co-design among all possible solutions of the first condition thanks to a tunable parameter. The problem of tuning the control strategy is also addressed from a simple optimization criterion, which cope with the implicit objective to reduce the number of updates by playing on the optimization of event-triggered rule and on the parameter related to the expected average sampling rate of the event-triggered implementation.

The paper is organized as follows. Section 2 presents the system under consideration, the sampled-data control implementation and then the event-triggered problem we intend to solve. In Section 3, upon giving basic ingredients on which the approach is based, the event-triggered strategy is proposed in a context of emulation. Then, Section 4 is dedicated to the co-design of event-triggered control, that is to the design of both the controller gain and the event-triggering rule. Section 5 proposes an optimization method to compute the parameters of the event-triggering rule and the controller gain (in the co-design approach). In Section 6, several simulations illustrate the application of the proposed methodology and the influence of different parameters such as the dwell-time and the observer gain. Section 7 ends the paper with concluding remarks and potential future works. Notation. For any matrix A, A denotes its transpose and He{A} = A + A . For two symmetric matrices of the same dimensions, A and B, A > B means that A -B is symmetric positive definite. I and 0 stand respectively for the identity and the null matrix of appropriate dimensions. Let us point out the main definitions of states used throughout the paper: x p , x o , x, (x p,eq , x o,eq , u eq ), and ε eq denote respectively the plant state, the observer state, the state estimation error, the equilibrium point for the plant state, the observer state and the input, and the difference between the observer state and its equilibrium point.

Problem Statement

System data

Consider the following continuous-time linear plant:

ẋp (t) = A p x p (t) + B p u(t), y p (t) = C p x p (t), (1)
where x p (t) ∈ R n , u(t) ∈ R m , y p (t) ∈ R p are the state, the input and the output of the plant, respectively. The matrices A p , B p and C p are constant and of appropriate dimensions. In addition, the pairs (A p , B p ) and (C p , A p ) are controllable and observable. Moreover, system (1) satisfies the following assumption.

Assumption 1 m ≥ p and rank(

A p B p C p 0 ) = n + p.
Assumption 1 means that the plant has no transmission zeros at zero [START_REF] Young | An approach to the linear multivariable servomechanism problem[END_REF].

An observer-based state feedback controller to drive the output to a given nonzero constant set-point r is defined by:

   ẋo (t) = A p x o (t) + B p u(t) -K o (y p (t) -y o (t)), y o (t) = C p x o (t), u(t) = K c x o (t) + K r r, (2)
where x o (t) ∈ R n , y o (t) ∈ R p and r ∈ R p are the state and the output of the observer and the constant reference signal, respectively. Furthermore, K r ∈ R m×p is a feedforward gain, K o ∈ R n×p and K c ∈ R m×n are the observer and controller gains, respectively. By considering the continuous-time system described by ( 2), the control design is carried out according to the separation principle. The observer gain K o is then designed to make A p + K o C p Hurwitz and the state estimation error dynamics is given by:

ė(t) = (A p + K o C p )e(t), (3)
where the state estimation error, e(t) = x p (t)-x o (t) ∈ R n , is globally asymptotically stable, i.e. lim t→∞ e(t) = 0. Consequently, the estimation output error e y (t) = C p e(t) also asymptotically converges to zero, i.e. lim t→∞ e y (t) = 0. On the other hand, the dynamic of the observer with the state feedback controller u(t) = K c x o (t) + K r r is given by:

ẋo (t) = (A p + B p K c )x o (t) + B p K r r -K o e y (t). (4)
For a given constant reference signal r, the equilibrium point (x p,eq , x o,eq ) of ( 3) and ( 4) satisfies

x o,eq = x p,eq , (A p + B p K c )x o,eq + B p K r r = 0, C p x o,eq = r.

(

The two last equations of ( 5) can be rewritten as follows

A p B p C p 0 I 0 K c K r x o,eq r = 0 r . (6)
Thanks to Assumption 1, the pseudo-inverse

A p B p C p 0 # exists such that A p B p C p 0 A p B p C p 0 #
= I n+p and from equation ( 6) one gets

K r = -K c I A p B p C p 0 # 0 I . (7)
Such an expression of the feedforward control gain K r is valid for any controller gain K c . Let us define the error dynamics between the observer state x o (t) and its equilibrium point, ε eq (t) = x o (t)x o,eq , which satisfies εeq

(t) = (A p + B p K c )ε eq (t) -K o e y (t). (8)
where e y (t) can be interpreted as an input to this system. Assuming that the controller gain K c is designed such that (A p + B p K c ) is Hurwitz, system ( 8) is input-to-state stable with respect to e y and the unforced linear system (i.e. with e y = 0) is asymptotically stable. Since ( 4) is also supposed to be asymptotically stable, it follows that lim t→∞ e y (t) = 0, which implies that lim t→∞ ε eq (t) = 0. We can therefore conclude that if (A p + B p K c ) and (A p + K o C p ) are Hurwitz one has lim t→∞ x o (t) = lim t→∞ x(t) = x eq , i.e. the output y p converges asymptotically to the desired reference r.

While the problem of continuous-time or periodic sampled-data implementation of such a class of controller has been widely studied in the literature, our objective is to address the problem of an aperiodic sampled-data implementation of such an observer based-controller based on an event-triggered control strategy.

Sampled-data implementation of the control input

In this paper, we consider that the control input, u, is not assumed to be continuously implemented but is updated at certain instants {t k } k∈N , which form a sequence of strictly increasing positive scalar and which is defined in the sequel. We consider that the control action is held constant between two sampling instants (t k and t k+1 ) through a zero order hold. Note, however, that differently from classical digital control approaches, the sampling interval t k+1t k is not assumed to be constant. In such a situation, the closed-loop system can be represented by

   ẋp (t) = A p x p (t) + B p u(t k ), ẋo (t) = A p x o (t) + B p u(t k ) -K o e y (t), u(t k ) = K c x o (t k ) + K r r, ∀t ∈ [t k , t k+1 ), (9)
where we recall that e y (t) = C p e(t) ∈ R p and e(t) = x p (t)x o (t) ∈ R n . Note that the error dynamics e between the plant and he observer states is still governed by (3), the error dynamics ε eq between the observer state x o and its equilibrium x o,eq is affected by the sampled-data implementation of the control input u. Thus, the closed-loop system can be re-written as:

εeq (t) = (A p + B p K c )ε eq (t) + B p δ(t) -K o e y (t), ė(t) = (A p + K o C p )e(t), (10)
where we use the same formulation as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] to define δ(t)

δ(t) = K c (ε eq (t k ) -ε eq (t)).
The variable δ(t) can be seen as a measure of the difference between the continuous-time and the sampled control input. Note that δ(t) depends only on the observer variables and is therefore available at the controller node. The sensor and controller are supposed to be in different nodes of the network as depicted in Figure 1, the SW block representing the event-triggered sampling strategy.

Plant Observer K K r y(t) x (t) x (t )

Problem Statement

In this paper, we are interested in the event-triggered implementation of the controller represented by [START_REF] Abdelrahim | Stabilization of nonlinear systems using eventtriggered output feedback laws[END_REF]. This means that an event generator algorithm is included in the controller to decide whether or not the control input has to be updated. The basic idea is therefore to decide when to sample based on the available information. Following the event-triggered control strategy proposed in [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF][START_REF] Tallapragada | On event triggered tracking for nonlinear systems[END_REF], the sampling instants are determined from the following logic:

t k+1 = min{t ≥ t k + T, s.t. f (δ(t), y a (t)) ≥ 0}. ( 11
)
where y a represents the vector of available information to the controller (which corresponds in our case to y a (t) = [ε eq (t) , e y (t) ] and the function f : R m × R (n+p) → R has to be defined efficiently such that the asymptotic stability of the closed-loop system (10) under the event-triggered rule described in ( 11) is ensured.

The logic in [START_REF] Goebel | Hybrid Dynamical Systems. Modeling, Stability and Robustness[END_REF] means that the next sampling time will occur at least T time units ahead the last one. In this case, T is the minimal dwell-time, which will be instrumental to prevent Zeno solutions. Moreover, note that for t ≥ t k + T the control will not be updated until f (δ(t), y a (t)) ≥ 0.

Event-triggered Strategy Design

The main objective 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 available signals u(t), x o (t) and y p (t). In view of [START_REF] Goebel | Hybrid Dynamical Systems. Modeling, Stability and Robustness[END_REF], this corresponds basically to design T and f in order to ensure the asymptotic stability of the sampled-data system (9).

Preliminary result

Let us first present a general formulation, inspired from [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF], [START_REF] Tallapragada | On event triggered tracking for nonlinear systems[END_REF] and on which the main result of the event-triggered strategy developed below is based. Consider a generic linear system

   ẋ(t) = Ax(t) + BKy(t k ), ∀t ∈ [t k , t k+1 ), y(t) = Cx(t), u(t) = Ky(t) (12)
where x ∈ R n A , u ∈ R n B and y ∈ R n C represent the state, the input and the output of system [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] and where the matrices A, B,C and K have appropriate dimensions and are such that A + BKC is Hurwitz. The following theorem constitutes the first contribution of the paper. It states a Lyapunov-based condition to ensure the asymptotic stability of the closed-loop system [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] under an event-triggering strategy and to prevent Zeno phenomena.

Theorem 1 Consider a positive scalar T , a function f : R n B × R n C → R and the triggering rule

t k+1 = min{t ≥ t k + T, s.t. f (K(y(t k ) -y(t)), y(t)) ≥ 0}. ( 13
)
Consider a positive definite function V (x), for which there exist two positive scalars ε 1 and ε 2 such that

ε 1 x 2 ≤ V (x) ≤ ε 2 x 2 . ( 14
)
Assume that the function V (x) satisfies:

V (x(t)) -f (K(y(t k ) -y(t)), y(t)) < 0, ∀t ∈ [t k + T , t k+1 ) ∀k ∈ N, (15)
and

∆V T (x) = V (x(t k + T )) -V (x(t k )) < 0, ∀k ∈ N. (16)
Then, system [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] with the triggering rule (13) is asymptotically stable and the inter-sampling intervals are lower bounded by T .

Proof. Let us split the stability analysis in two intervals, namely [t k , t k + T ] and [t k + T, t k+1 ). From ( 16), the function V (x) satisfies

V (x(t k + T )) < V (x(t k )), ∀k ∈ N. (17)
Consider now the interval [t k + T, t k+1 ). From [START_REF] Heemels | An introduction to event-triggered and selftriggered control[END_REF] and the use of S-procedure, one obtains [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF]. Then, we conclude that V (x(t)) < 0 and therefore

V (x(t)) < V (x(t k + T )), ∀t ∈ [t k + T, t k+1 ), ∀k ∈ N. (18)
Hence, from ( 17) and ( 18), we conclude that V (x(t k+1 )) < V (x(t k )). Moreover, since system ( 12) is linear and since the function V (x) satisfies ( 14), there exists a positive scalar β such that max t∈[t k , t k +T ] V (x(t)) ≤ βV (x(t k )), for all k ∈ N. Hence, associating this property to relations ( 13) and [START_REF] Marchand | Lyapunov event-triggered control: a new event strategy based on the control[END_REF], i.e. to the fact that V (x) < 0 on the interval [t k + T , t k+1 ), prevents the trajectories of the system from blowing up in between every two events. Therefore, asymptotic stability is ensured and a lower bound on the inter-sampling times is given by T by using similar arguments to Lemma 1 in [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF].

Remark 1

The event-triggered rule (13) allows to avoid Zeno behavior since the inter-sampling times are lower bounded by the positive scalar T .

Note that Theorem 1 does not induce real conservatism in the sense that if the continuous-time condition (15) holds then it is always possible to find a small enough T such that conditions (15)-( 16) are verified at the same time.

Event-triggered strategy

Following the conditions presented in Theorem 1 by considering n A = n, n B = m, n C = n + p and a quadratic function f , a way to design the event-triggered rule using only the available signals is stated below.

Theorem 2 For given controller and observer gains K c and K o and a positive scalar T > 0, suppose that there exist symmetric positive definite matrices P ε , P, P e , Q ε and Q δ of appropriate dimensions such that the following matrix inequalities

Φ 1 =       He{P ε (A p + B p K c )} (A p + B p K c ) P -P ε K o C p + P(A p + K o C p ) P ε B p * He{P e (A p + K o C p ) -P K o C p } P B p * * -Q δ     I 0 0 C p 0 0   * -Q ε     < 0, (19)
are verified with

Λ(T ) = e A p -K o C p 0 A p +K o C p T + T 0 e A p -K o C p 0 A p +K o C p s ds B p K c 0 0 0 .
Then, the event-triggered sampling rule defined by

t k+1 = min t ≥ t k + T, s.t. δ(t) Q δ δ(t) - ε eq (t) e y (t) Q -1 ε ε eq (t) e y (t) ≥ 0 (21)
is such that the origin of system (10) is globally asymptotically stable and, consequently, the output vectors y p and y o of both the plant and the observer converge to the reference signal r. Furthermore, the intersampling times are lower bounded by T .

Proof. Consider the Lyapunov candidate function for system [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF] given by V (ε eq (t), e(t)) = ε eq (t) e(t) P ε P P P e ε eq (t) e(t) ,

where the matrix P ε P P P e is positive definite, which is ensured by the satisfaction of [START_REF] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF]. The goal of this proof is to show that the LMI conditions ( 19) and ( 20) are sufficient conditions for the satisfaction of inequalities ( 15) and ( 16) of Theorem 1 respectively. Consider the time-derivative of V along the trajectories of system [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF], for any t ∈ [t k + T, t k+1 ). The following expression is obtained :

V (ε eq (t), e(t)) -δ(t) Q δ δ(t) + ε eq (t) e y (t) Q -1 ε ε eq (t) e y (t) =   ε eq (t) e(t) δ   M   ε eq (t) e(t) δ   -δ(t) Q δ δ(t) + ε eq (t) e y (t) Q -1 ε ε eq (t)
Φ 0 = M -diag(0, 0, Q δ ) +   I 0 0 C p 0 0   Q -1 ε   I 0 0 C p 0 0   .
Therefore, by applying the Schur complement to Φ 0 one obtains Φ 1 . If the LMI condition [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], i.e. Φ 1 < 0, is satisfied, then the condition (15) of Theorem 1 with f δ(t),

ε eq (t) e y (t) = δ(t) Q δ δ(t) - ε eq (t) e y (t) Q -1 ε ε eq (t) e y (t)
is also satisfied. In order to prove that conditions ( 16) holds, note that solving the linear differential equation [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF]

over the interval [t k , t k + T ] yields ε eq (t k + T ) e(t k + T ) = Λ(T ) ε eq (t k ) e(t k ) . ( 22
)
Hence, condition ( 16) becomes

∆V T (ε eq , e) = V (ε eq (t k + T ), e(t k + T )) -V (ε eq (t k ), e(t k )) = ε eq (t k ) e(t k ) Λ(T ) P ε P P P e Λ(T ) - P ε P P P e ε eq (t k ) e(t k ) .
Applying the Schur complement, it can be seen that the condition Φ 2 < 0 ensures the satisfaction of ( 16). The proof is then concluded by virtue of Theorem 1. We can conclude that the solutions to system [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF] converges asymptotically to 0, meaning that the plant output y p converges asymptotically to the reference r. Moreover, the event-triggered strategy [START_REF] Seuret | Event-triggered sampling algorithms based on a Lyapunov function[END_REF] implicitly ensures that the length between two successive sampling instants is lower bounded by T .

From the conditions of Theorem 2, it is important to observe that the lower bound on the inter-sampling times is directly obtained via the satisfaction of (20) without the need of additional calculations as required in [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF]. Furthermore, one also gets the guarantee that P ε P P P e defines a quadratic Lyapunov function for the discrete-time system. Theorem 2 is then the second contribution of the paper.

Note that T appears as a tuning parameter of the event-triggered problem. Contrary to most of the approaches developed in the literature, where the dwell-time T is computed a posteriori. Moreover, for a sufficiently small T there will exist a solution to conditions ( 19)- [START_REF] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF]. However if T is too small the number of control updates (events) tends to be larger as illustrated in Section 6. On the other hand, if T is too large, the conditions may be not feasible. Furthermore, a large T can lead to a significant performance degradation (when compared to the one obtained with a continuous-time implementation). This trade-off should be considered when choosing T .

Theorem 3 Given an observer gain K o and a parameter T > 0, suppose that there exist positive definite symmetric matrices W ε , P e , Q ε and Q δ and a matrix F c of appropriate dimensions such that the following matrix inequalities

Ψ 1 =       He{A p W ε + B p F c )} -K o C p B p * He{P e (A p + K o C p )} 0 * * -Q δ     W ε 0 0 C p 0 0   * -Q ε     < 0, ( 23
)
Ψ 2 =   -W ε 0 0 P e W ε 0 0 P e Λ 1 (T ) + F c 0 Λ 2 (T ) I 0 0 P e * -W ε 0 0 P e   < 0, ( 24
)
are verified with

Λ 1 (T ) = e A p -K o C p 0 A p +K o C p T , Λ 2 (T ) = T 0 e A p -K o C p 0 A p +K o C p s ds B p 0 .
Then, the controller gain K c = F c W -1 ε and the event-triggered sampling rule defined by

t k+1 = min t ≥ t k + T, s.t. δ(t) Q δ δ(t) - ε eq (t) e y (t) Q -1 ε ε eq (t)
e y (t) ≥ 0 [START_REF] Tallapragada | On event triggered tracking for nonlinear systems[END_REF] is such that the origin is asymptotically stable for system [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF] and, consequently, the output vectors of both the plant and the observer converge to the reference signal r. Furthermore, the inter-sampling times are lower bounded by T .

Proof. Based on Theorem 1, by considering P = P = 0, W ε = P -1 ε and F c = K c W ε , then pre-and post-multiplying by diag(W ε ; I; I; I) it follows that condition Φ 1 < 0 is equivalent to the condition Ψ 1 < 0. Furthermore, pre-and post-multiplying Φ 2 by the matrix diag(W ε , I,W ε , I) ensures that Φ 2 < 0 is equivalent to Ψ 2 < 0.

Remark 2 While Theorem 2 only provides the triggering rule for prescribed controller and observer gains, K c and K o and a positive scalar T , Theorem 3 provides both the triggering rule and the controller gain K c for given a given observer gain K o and a given T > 0. Then the conditions of Theorem 3 provide an event-triggered control co-design. Theorem 3 constitutes the third contribution of the paper. The design of the controller K c could be completed by adding some performance constraints as, for example, LQperformance, H 2 , H ∞ performance or by considering some pole placement constraints to satisfy some rate of convergence requirement of the closed-loop system.

Optimization of the event-triggered strategies

It can be checked that the conditions in Theorem 2 are LMIs provided that K c , K o and T are fixed, as classically in an emulation context. In order to choose K c and K o classical design technique, possibly adding performance criteria, can be used. It is important to note that provided that matrices A p + B p K c and A p + K o C p are Hurwitz, there always exists a small enough positive scalar T such that the LMIs of Theorem 2 are feasible. Similarly, conditions of Theorem 3 are linear as soon as K o and T are given. The choice of the observer gain is done such that A p + K o C p is Hurwitz with fast enough eigenvalues. Then, in order to optimize the selection of the event-triggered parameters, we can consider the following convex optimization problems:

• Event-triggered rule design (Theorem 2). For given controller and observer gains K c and K o and a positive scalar T > 0: min

P ε ,P,P e ,Q ε ,Q δ trace(Q δ ) + trace(Q ε ) s.t. (19), (20). (26)
• Event-triggered control co-design (Theorem 3). For given observer gain K o and a positive scalar T > 0: min

W ε ,P e ,Q ε ,Q δ ,F c trace(Q δ ) + trace(Q ε ) s.t. (23), ( 24
), W ε > 0, P e > 0. (27)
The dwell-time T being also a design parameter, whose role is connected to the expected average sampling rate of the event triggered implementation one can seek for minimizing its value through problems ( 26) or ( 27) by iteratively increasing T and testing LMI conditions. Note that convex optimization problems proposed are aimed at reducing as much as possible the occurrences of sampling. This aspect is illustrated in the next section.

Numerical example

The considered example is inspired by the model of a magnetic levitation system as mentioned. These systems can be found in important applications such as precision bearing and transportation systems (see, for example, the Feedback Levitation System (http://www.feedback-instruments.com/)). The system suspends a small metal ball by means of an electromagnetic field generated by an electromagnet. The linearized model for this system can be written as:

m ẍ = k 1 x + k 2 i
where i is the coil current, x is the ball vertical position, m is the ball mass [START_REF] Galvao | A simple technique for identifying a linearized model for a didactic magnetic levitation system[END_REF]. In the paper we have chosen reasonable values at x = 0 for the parameters m, k 1 and k 2 in order to obtain the data A p , B p and C p defining the plant [START_REF] Abdelrahim | Co-design of output feedback laws and eventtriggering conditions for linear systems[END_REF]. as follows:

A p = 0 1 4 0 , B p = 1 0 , C p = 1 0 , r = 5. (28)
Note that the open-loop system is unstable since has a pole in the right hand side of the complex plane. We only address the problem of the design of both the controller gain K c and the event-triggering rules through the matrices Q δ and Q σ , as presented in Theorem 3.

Two observer gains K o are considered:

K o1 = -3.5 -7 , and K o2 = -42 -444 . ( 29
)
These two observer gains have been selected to show the influence of the observer gain on the eventtriggered control. Indeed while the first gain leads to eigenvalues of A p + K o C p equal to -1.5, -2 , the second gain leads to eigenvalues equal to -20, -22.

Then, we apply Theorem 3 and the optimization problem [START_REF] Wang | Event design in event-triggered feedback control systems[END_REF] for several values of design parameter T . Figure 2 depicts six simulations showing, for each of them, the time evolution of the state of the plant and the observer, of a timer τ := tt k representing when the control input is updated and of the control signal obtained considering the initial conditions x p (0) = [-5 0] , x o (0) = [0 0] . We have considered here three values for the design parameter T : T = 0.2, 0.4 and 0.8. It is worth mentioning that the design parameter T corresponds to a guaranteed dwell-time for the eventtriggered control system. Indeed, it can be seen in Figure 2, that the "peaks" of the timer are always greater than the corresponding value of T .

T K o K c Q δ Q -1 ε 0.2 K o1 [-
For both observer gains, the influence of T directly appears in the number of control updates, N u , obtained during the 10s-simulations. Increasing T leads to a notable reduction of N u , for this particular initial condition. Nevertheless, increasing T also affects the performances of the responses. Indeed it can be seen that the simulation obtained with T = 0.2 converges more quickly than for T = 0.4 or 0.8. Moreover for large values of T (here 0.8), the response oscillates around the reference signal r = 5. The same behavior can also be observed for other initial conditions.

We also note that selecting the largest value for T may recover a periodic implementation. Indeed, this is due to the fact that at time t k + T , triggering rule f is already positive, which does not allow enlarging the sampling interval. Therefore, the triggering rule only resumes to a periodic implementation of period T . This remark highlights the tradeoff between the reduction of the number of control updates between periodic and event-triggered implementations.

Comparing now the effects of considering "slow" and "fast" observer gains, Figure 2 also shows that employing "fast" observer gains help reducing the number of control updates N u . This is an expected result since the triggering rule (25) depends explicitly on the estimation error e y .

Conclusion

This paper presented a systematic method for designing event-triggering strategies considering observerbased controllers. The event-triggering design method is based on Lyapunov arguments and uses only information on available signals. Hence, since a Lyapunov-based approach is considered, stability under the event-triggered sampling strategy is formally guaranteed. The proposed implementation is parameterized by a dwell-time T , which prevents the Zeno phenomenon occurrence.

Considering an emulation approach (i.e. the state feedback and observer gains is a priori given), design procedure consists in fixing, first, the design parameter T and then solving a convex optimization problem to determine, in a systematic way, the parameters of the event-triggering rule aiming at reducing as much as possible the occurrences of sampling. It is also shown that the conditions can be adapted for a co-design of the estimated state-feedback gain and the event-triggering rules parameters. The work proposed lets the room for future developments. In particular, the co-design not only of the state feedback gain but also of the observer gain is an open problem. One of the difficulties in this case is the non-convexity of condition [START_REF] Tallapragada | Event-triggered dynamic output feedback control of LTI systems[END_REF] if K o is a free variable. One direction to overcome this problem is the application of the results proposed in [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF][START_REF] Briat | Convex conditions for robust stabilization of uncertain switched systems with guaranteed minimum and mode-dependent dwell-time[END_REF] about the equivalence of an inequality such as [START_REF] Sbarbaro | An event-triggered observer based control strategy for siso systems[END_REF] (containing a matrix exponential) to an inequality where the argument of the exponential matrix appears linearly. The drawback in this case is that this equivalent inequality has infinite dimension, since one of the decision variables depends on the time between two impulses (events). At the expense of a substantially numerical complexity increasing, this can be overcome (for instance) by using polynomial matrices and applying Sum of Squares techniques. Note also that to apply this technique, a maximal bound on the dwell-time must be considered. On the other hand, there is another difficulty that prevents so far to obtain convex conditions for the synthesis of K o . It should be observed that due to the the cross product P e K o , it is not possible to make convex [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] if K o and P e are decision variables. An ad hoc way to overcome this would be to apply some iterative algorithm. Another open issue regards the extension of the approach to cope with the perfect (or almost perfect) tracking of non constant signals (such as periodic ones, for instance). This problem seems more involved in the context of event-triggered control, since the reference would also be sampled aperiodically.

Figure 1 :

 1 Figure 1: Observer based controller.

Figure 2 :

 2 Figure 2: Evolution of the plant and observer states with the reference r (top), the timer tt k (middle) and the control input u (bottom) for T = 0.2, 0.4 and 0.8.

Table 1 :

 1 Table showing controller gains and the event-triggered rule parameters Q δ and Q ε for several values of T and of the observer gain K o . Table 1 summarizes the solution K c , Q δ and Q ε obtained for each value of T and for each observer gain considered.

			4.201	0.302	-0.143
	11.9240 -7.0307] 0.6484	0.302	3.519	-0.202
			-0.143 -0.202	4.665
			2.929	0.282	-0.0067
	0.4 K o1 [-6.2664 -3.1687] 1.7767	0.282	3.366	-0.0007
			-0.067 -0.007	0.3720
			2.595	-0.885 -0.032
	0.8 K o1 [-5.4479 -2.7445]	1.623	-0.885	4.052	-0.018
			-0.032 -0.018	6.446
			4.001	0.342	-0.002
	0.2 K o2 [-11.4222 -6.9135] 0.7464	0.342	3.353	-0.006
			-0.002 -0.0006	4.396
			3.337	-0.038 -0.004
	0.4 K o2 [-7.3796 -3.9756] 1.1649	-0.038	3.176	-0.003
			-0.004 -0.003	4.048
			2.413	0.010	-0.002
	0.8 K o2 [-5.0127 -2.5115] 9.3231	0.010	3.081	-0.001
			-0.002 -0.001	3.517

Event-triggered control co-designBased on the stability condition obtained in Theorem 2, the following constructive theorem provides conditions for the co-design of event-triggered control, that is for designing both the controller gain K c and the matrices Q δ and Q ε defining the triggering rule.

This work was supported by ANR projects LimICoS contract number 12 BS03 005 01, SCIDIS, contract number 15-CE23-0014 and STIC-AmSud project "ADNEC". We also like to thank the support of CNPq under grants 30.6210/2009-6 (PQ) and 480638/2012-8 (Universal).