Luciano Goncalves Moreira

Boering Leonardo

João Manoel Groff

Da Gomes
email: jmgomes@ece.ufrgs.br

Silva

L G Moreira

L B Groff

J M Gomes Da Silva

Event-Triggered State-Feedback Control for Continuous-Time Plants Subject to Input Saturation

Keywords: Event-trigger, Saturation, Networked Control Systems, Linear Plants

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Introduction

Nowadays many control systems are implemented over general purpose shared networks to take advantage of the effectiveness in terms of cost and flexibility that such networks offer. In this configuration, the communication between sensors, controllers and actuators is done using the network and occurs at discrete time instants. This scenario describes what is commonly known as networked control systems (NCS) (???). This field has attracted attention since the 1990's decade, as shown in ? and the references cited therein. Issues such as communication bandwidth and energy consumption arise from this distributed paradigm (??) and have been addressed with different techniques, among them, event-triggered control.

In an event-triggered control system, data is sampled and transmitted only when a trigger condition is violated. In this context, it is of primary importance to develop systematic methods to tune the trigger aiming at reducing the number of data transmissions while ensuring the stability of the closed-loop system. In ?, a threshold on the relative state measurement error is used in the trigger condition and it is shown that the resulting inter-event times are lower-bounded. More recently, other issues have been addressed in the literature: the problem of designing an event-trigger strategy for state-feedback controlled single-input-single-output linear plants using a state observer is addressed in ? for continuous-time systems and in ? for discrete-time systems. The co-design between the trigger function and controller gains for LTI plants is addressed in ?. In ?, a design method for event-triggered PI controllers that uses a quadratic criterion for the triggering function and for the tuning of the controller parameters is presented.

On the other hand, control input saturation is a phenomenon present in almost all control systems (?). In the context of event-triggered control, algorithms based on linear-quadratic criteria to design periodic, event-triggered and self-triggered controllers are proposed in ?, taking explicitly into account the actuator saturation. In ?, an event-triggered controller is designed without considering the saturation and examples of how the saturation impacts the performance of the system are given. In ?, stability issues under control saturation are addressed considering PI and generic dynamic output feedback controllers. For discrete-time systems, ? addresses saturating state-feedback control laws.

The present paper addresses the design of an eventtriggered state-feedback strategy for continuous-time linear plants subject to input saturation, considering that the plant and the controller are in different nodes and that the control signal is kept constant between two successive sampling instants. Differently from ?, instead of using hybrid systems theory, a direct Lyapunovbased approach is applied to derive the stability conditions. In contrast to ?, which focus on estimating the region of attraction of the origin for a given control law and given trigger parameters, we address the problem of designing these parameters in order to ensure that a given region of interest is included in the region of attraction of the closed-loop system under the eventtrigger strategy. Moreover, the co-design of the trigger parameters and of the control law is also addressed. The trigger function considered in this paper takes into account a weighted deviation between the last sampled state and the current one. Stability conditions in the form of linear matrix inequalities (LMIs) are derived and since the parameters of the trigger function appear explicitly in these conditions, convex optimization problems aiming at reducing the sampling activity are proposed as means to select these parameters. Numerical examples illustrate the application of the methods.

The paper is organized as follows: Section 2 presents basic definitions and formally states the problem we address. Section 3 recalls stability conditions for continuous-time (non-triggered) LTI systems subject to input saturation. Section 4 presents our event-trigger strategy and the stability conditions for the closed-loop system in regional and global contexts, considering that the controller gains were previously designed. Convex optimizations problems suitable for selecting the trigger parameters aiming at reducing the sampling activity are presented in Section 4.1. Section 5 extends the results from previous section showing how to design the trigger parameters and the controller gains simultaneously (co-design). Section 6 illustrates the application of the method.

Notation

• R represents the set of real numbers.

• The saturation function is defined as

sat(v)      -u 0 if v < -u 0 v if -u 0 ≤ v ≤ u 0 u 0 if v > u 0
if v is a vector, the definition holds element-wise. • A denotes the transpose of matrix A.

• He{A} A + A . sat ẋ = Ax + Bu ν(t) u(t) x(t)
evt-trigger

(Trigger times t k ) sampler ZOH K x(t k ) node 1 node 2 Fig. 1 Closed-loop system diagram
• The symbol * stands for symmetric blocks within a matrix. • tr(A) denotes the trace of matrix A.

• A (j) and x (j) denote the j-th row of matrix A and vector x, respectively. • diag(X, Y ) denotes the block-diagonal matrix composed by the blocks X and Y . • x y with x, y ∈ R n means that x (j) ≤ y (j) , ∀j = 1 . . . n. • E(P ) = {x ∈ R q ; x P x ≤ 1} is the ellipsoid defined by a matrix P ∈ R q×q , P = P > 0.

• || • || denotes the Euclidean norm. • Co{•} denotes a convex hull.

Problem Statement

Consider a continuous-time linear plant defined by the following equation:

ẋ(t) = Ax(t) + Bu(t) (1)
where x ∈ R n is the state vector; u ∈ R m is the input, limited in amplitude such that -u 0 u u 0 , with u 0 0 ∈ R m ; A and B are real-valued constant matrices of appropriate dimensions. We assume that the plant is observable and controllable.

We consider a networked control implementation where plant and controller are in separate nodes and are connected through a general purpose network forming the closed-loop system depicted in Figure 1.

At instants t k , k = 0, 1, 2, . . . determined by the event-trigger generator, a sample of the plant state is sent to the controller node. Between two trigger instants, the controller input is held at the constant sampled value by means of zero-order holders. We assume t 0 = 0.

Therefore, the closed-loop system reads:

ẋ(t) = Ax(t) + Bu(t) u(t) = sat(ν(t k )) ∀t ∈ [t k , t k+1 ) (2)
where ν(•) ∈ R m is the output of the controller. In this paper we consider a state-feedback control law:

ν(t k ) = Kx(t k ) (3)
where K ∈ R m×n is the gain matrix. The equilibrium point is the origin.

Considering the system formed by ( 2) and (3), we aim at designing an event-triggered controller, that is, the controller gain and a trigger strategy that defines the trigger instants, to reduce the number of events. In the context of a networked implementation as described in Figure 1, this corresponds to reduce the number of messages exchanged between the nodes, so that we can save communication bandwidth and energy. Thus, we can state the problem we want to address as:

Problem 1 Given a continuous-time stabilizing statefeedback controller for (1), devise an event-triggered implementation of this control law that guarantees the regional (or global, when possible) asymptotic stability of the origin of the closed-loop system (2)-(3) for initial conditions in a given set X 0 ⊂ R n containing the origin, while reducing the number of data transmission events between the sensor/plant node and the controller node. This is known as emulation design problem.

Problem 2 Devise controller gains and an eventtriggered strategy that guarantee the regional (or global, when possible) asymptotic stability of the origin of the closed-loop system (2)-(3) for initial conditions in a given set X 0 ⊂ R n containing the origin, while reducing the number of data transmission events between the sensor/plant node and the controller node. This is referred to as co-design problem.

Non-triggered system dynamics

In this section we recall some stability conditions for the system composed by the direct connection between (1) and a continuous-time implementation of (3), i. e. ν(t) = Kx(t), through a saturating actuator, which leads to the system:

ẋ(t) = Ax(t) + B sat(Kx(t)) (4)
Given a gain vector K, (4) is regionally stable in a set X 0 if there exist a diagonal matrix S > 0 ∈ R m×m , matrices W = W > 0 ∈ R n×n , Z ∈ R m×n satisfying the following relations (see ? for details):

He{(A + BK) W } B S -Z SB -Z -2 S < 0 W W K (j) -Z (j) * u 2 0(j) ≥ 0 j = 1 . . . m (5) and X 0 ⊂ E( W -1 ). In this case, V (x) = x W -1 x is such that V (x) < 0, ∀x ∈ E( W -1 ), i. e., E( W -1
) is a contractive invariant set and a region of asymptotic stability (RAS) of the origin for system (4). A RAS of an equilibrium point is a subset of its region of attraction which has a analytical representation and can be used as an estimate for the region of attraction ?.

Event-trigger strategy for given controller gains

In this section, we assume that the gain matrix K has been designed such that ( 4) is regionally stable in the target set X 0 . This is guaranteed if K is such that the linear matrix inequalities in (5) are satisfied and X 0 ⊂ E( W -1 ). Then we propose an event-triggered strategy and provide stability conditions for the closedloop system under this strategy by means of a quadratic Lyapunov function.

The event trigger strategy consists basically in defining what we call a trigger condition. This condition is therefore continuously monitored. If, at instant t, the trigger condition is verified, then the control signal is updated ad we have t k+1 = t.

Defining δ(t) = x(t k )-x(t), i.e., δ(t) is a measure of the difference between the last sampled state to compute (3) and the current continuous state, in ? the proposed trigger condition consists in verifying if ||δ(t)|| = σ 0 ||x(t)||. In other words, while ||δ(t)|| < σ 0 ||x(t)||, i.e., the relative difference between the last sampled state and the current one is lesser than σ 0 , the control signal is kept unchanged. In particular, it is shown that this trigger criterion ensures that the closed-loop (2)-( 3) is asymptotically stable and that a minimum inter-event time exist (i. e., Zeno solutions are avoided) provided σ 0 is selected from an interval of values determined by the system dynamics.

In this paper, we consider the generalized trigger condition δ (t)Q δ δ(t) > x (t)Q x x(t), where Q δ and Q x are symmetric positive definite matrices of dimension n. In this case, while δ Q δ δ ≤ x Q x x, the control applied to the plant is kept constant. This generalization adds degrees of freedom and can allow a larger reduction in the sampling activity. The term δ (t)Q δ δ(t) -x (t)Q x x(t) is a relative measure of the deviation between the last sampled state and the current one, with Q δ and Q x acting as weights. The relation between these matrices plays a role similar to σ 0 in ?: the "larger" Q x and the "smaller" Q δ the more we let the current state deviate from the last sampled one and the less sampling is expected. Hence, we can rewrite our event-trigger strategy in the following more convenient form:

Algorithm 1 Sampling criterion if f (δ(t), x(t)) δ (t)Q δ δ(t) -x (t)Q x x(t) > 0 then t k = t; u(t) = Kx(t k ); else u(t) = u(t k ); end if
Now we provide conditions that ensure the asymptotic stability of the origin of ( 2)-( 3) when the eventtrigger strategy Algorithm 1 is used.

Theorem 1 If there exist a diagonal positive definite matrix S ∈ R m×m , symmetric positive definite matrices W, Qx , Q δ ∈ R n×n and matrices G 2 , Z 1 ∈ R m×n such that the following LMIs are satisfied:

    He{(A + BK)W } W BK BS -Z 1 * -Qx 0 0 * * -Q δ -G 2 * * * -2S     < 0 (6)      W 0 W K (j) -Z 1(j) W * Q δ K (j) -G 2(j) 0 * * u 2 0(j) 0 * * * Qx      > 0 j = 1 . . . m (7)
then, E(W -1 ) = {x ∈ R n ; x W -1 x ≤ 1} is a RAS for the system (2)-(3) under the sampling strategy given by Algorithm 1, with Q x = Q-1

x . Proof Between two trigger events, that is for t ∈ (t k , t k+1 ), we can re-write (2)-(3) as follows:

ẋ(t) = Ax(t) + B sat Kx(t k ) = = Ax(t) + BKx(t k ) + Bφ Kx(t k ) (8)
where φ(v) sat(v) -v is a dead-zone non-linearity.

Using the relation δ(t) x(t k ) -x(t), we can rewrite (8) as follows:

ẋ(t) = (A + BK)x(t) + BKδ(t) + Bφ Kx(t k ) (9)
Considering a quadratic Lyapunov function candidate V (x(t)) = x (t)P x(t), and dropping the time dependency arguments for clarity, it follows that:

V (x) = x [P (A + BK) + (A + BK) P ]x+ + x P BKδ + x P Bφ(Kx(t k ))+ + δ K B P x + φ (Kx(t k ))B P x (10)
Notice that the argument of φ can be written as

K(x + δ) = K K x δ K a x a (11)
where K a K K and x a x δ .

Hence, provided that x a belongs to the region S 0 = {x a ∈ R 2n ; -u 0 K a x a -G a x a u 0 }, the following generalized sector condition (??) is verified with respect to the deadzone nonlinearity φ:

φ (K a x a )T (φ(K a x a ) + G a x a ) ≤ 0 (12)
with T being a diagonal positive definite matrix and G a = G 1 G 2 a free matrix of appropriate dimensions. Assuming x a ∈ S 0 and applying (12), we can write:

V (x) ≤ V (x) -2φ (K a x a )T φ(K a x a ) + G a x a (13)
Thus, from ( 10) and ( 13), after some algebraic manipulation, we get:

V (x) ≤ x [P (A + BK) + (A + BK) P ]x+ + x P BKδ + x P Bφ+ + δ K B P x + φ B P x- -2φ T φ -2φ T G 1 x- -2φ T G 2 δ (14)
In matrix form:

V (x) ≤   x δ φ   M   x δ φ   (15)
with

M =   He{P (A + BK)} P BK P B -G 1 T * 0 -G 2 T * * -2T   From Algorithm 1, for t ∈ (t k , t k+1 ), it follows that δ Q δ δ -x Q x x ≤ 0. Then, if the following inequality is verified:   He{P (A + BK)} + Q x P BK P B -G 1 T * -Q δ -G 2 T * * -2T   < 0 (16) from (15), we can conclude that V (x) < δ Q δ δ - x Q x x ≤ 0 for t ∈ (t k , t k+1 ), provided x a ∈ S 0 .
Pre-and post-multiplying ( 16) by diag(W, I, S), with W = P -1 , S = T -1 , and considering the variable change Z 1 = G 1 W , ( 16) is equivalent to:

  He{(A + BK)W } + W Q x W BK BS -Z 1 * -Q δ -G 2 * * -2S   < 0 (17)
Applying the Schur's complement to the element He{

(A + BK)W } + W Q x W and defining Qx Q -1 x ,
we retrieve the matrix inequality (6). Therefore, the satisfaction of (6) implies that V (x) < 0 for t ∈ (t k , t k+1 ), provided that x a (t) ∈ S 0 . At the instants t = t k , we have δ(t k ) = 0 and the system reduces to (4). Hence, satisfaction of (6) implies that the first relation of ( 5) holds and that V (x(t k )) < 0, provided x a (t k ) ∈ S 0 . Now we show that (7) guarantees that x a (t) ∈ S 0 , ∀t ≥ 0, provided x(0) ∈ E(W -1 ).

Pre-and post-multiplying (7) by diag(P, I, I, I),

recalling that Z 1 = G 1 W and Qx = Q -1
x , applying Schur's complement twice and recalling that [K -G 1 K -G 2 ] = K a -G a , the following relations are verified for j = 1 . . . m:

P -Q x 0 0 Q δ -(K a(j) -G a(j) ) 1 u 2 0(j) (K a(j) -G a(j) ) > 0 (18)
Pre-and post-multiplying (18) by x δ and x δ respectively, (18) implies that:

x P x + δ Q δ δ -x Q x x - -x a (K a(j) -G a(j) ) 1 u 2 0(j) (K a(j) -G a(j) )x a > 0 (19)
Hence, if t ∈ (t k , t k+1 ), since δ Q δ δ -x Q x x ≤ 0, we conclude that x a (t) ∈ S 0 if x(t) ∈ E(P ). On the other hand, at t = t k , we have δ(t) = 0 and it also follows that x a (t k ) ∈ S 0 if x(t k ) ∈ E(P ). Thus, we can conclude that if x(0) ∈ E(P ), (6) along with (7) effectively ensures that x a (t) ∈ S 0 and V (x) < 0, ∀t ≥ 0, which ensures that E(P ) is an invariant and contractive set with respect to the system (2)-(3), being included in its region of attraction.

Theorem 1 ensures the regional asymptotic stability of the closed-loop system and can be applied to both stable or unstable plants. Nevertheless, if A p is Hurwitz, global stability can be achievable ?. A sufficient condition for the global stability of the closed-loop system under the event-trigger strategy can therefore be stated as a corollary to Theorem 1 as follows.

Corollary 1 If there exist a diagonal positive definite matrix S ∈ R m×m , symmetric positive definite matrices W, Qx , Q δ ∈ R n×n such that the following LMI is satisfied:

    He{(A + BK)W } W BK BS -W K * -Qx 0 0 * * -Q δ -K * * * -2S     < 0 (20)
then, the origin of system (2)-( 3) is globally asymptotically stable under the sampling strategy given by Algorithm 1, with Q x = Q-1

x . Proof It follows the same lines as the proof of Theorem 1 but applying a global generalized sector condition, i. e., using G a = K a , which implies G 1 = G 2 = K. In this case, ( 12) is valid ∀x a ∈ R 2n .

Remark 1 The existence of a lower bound for the interevent times can be proved applying the same arguments used in ? and taking into account that the event-trigger criterion satisfies:

δ(t) 2 ≤ λ max (Q x ) λ min (Q δ ) x(t) 2 (21)
Hence, the possibility of Zeno solutions is eliminated in the proposed strategy.

Optimization problems

Considering a region of admissible initial states X 0 , if X 0 ⊂ E(W -1 ) and conditions of Theorem 1 are satisfied, X 0 is also included in the region of attraction of the origin of the closed-loop system. Moreover, to address Problem 1, we also aim at reducing the number of transmission events. As presented at the beginning of Section 4, the reduction of the transmission activity can be obtained by aiming to find Q x as "large" as possible and Q δ as "small" as possible, while ensuring the stability of the origin of the closed-loop system under the eventtrigger strategy. Considering X 0 specified as an ellipsoid E(P 0 ) = {x ∈ R n ; x P 0 x ≤ 1}, with P 0 = P 0 > 0, and restricting Q x = α -1 I and Q δ = µI, with α, µ ∈ R, this goal can be achieved from the following convex optimization problem: min {W,α,µ,S,G2,Z1}

(α + µ) subject to: (6), (7), W -P -1 0 > 0 (22)
Notice that this optimization problem cannot be normalized to eliminate one of the variables, α or µ, due to the terms BK and K B which do not involve any variable. The last constraint in ( 22) ensures X 0 ⊂ E(W -1 ).

Furthermore, we can add degrees of freedom by letting both Q x and Q δ be generic positive definite matrices. This leads to the following optimization problem: min {W, Qx,Qδ,S,G2,Z1}

tr( Qx + Q δ ) subject to: (6), (7), W -P -1 0 > 0 (23)
In this case, the minimization of the trace of Qx = Q -1

x implies the maximization of the trace of Q x , which is the primary objective.

Co-design

If the controller gain K is a variable, conditions established by Theorem 1 are not LMIs, hence the optimization problems proposed in Section 4.1 cannot be used to obtain the controller and trigger parameters simultaneously, as they become non-convex. Nevertheless, some modifications can be made to linearize the conditions, leading to the following result:

Theorem 2 If there exist a diagonal positive definite matrix S ∈ R m×m , symmetric positive definite matrices W, Qx , Qδ ∈ R n×n and matrices Y, Z 1 , Z 2 ∈ R m×n such that the following LMIs are satisfied:

    He{AW + BY } W BY BS -Z 1 * -Qx 0 0 * * -Qδ -Z 2 * * * -2S     < 0 (24)      W 0 Y (j) -Z 1(j) W * Qδ Y (j) -Z 2(j) 0 * * u 2 0(j) 0 * * * Qx      > 0 j = 1 . . . m (25)
then, E(W -1 ) = {x ∈ R n ; x W -1 x ≤ 1} is a RAS for the system (2)-( 3) under the sampling strategy given by Algorithm 1, with

Q x = Q-1 x , Q δ = W -1 Qδ W -1 and K = Y W -1 .
Proof Pre-and post-multiplying (6) by diag(I, I, W, I),

recalling Z 1 = G 1 W and Qx = Q -1
x and making the variable substitutions Qδ = W Q δ W , Y = KW and Z 2 = G 2 W one obtains that ( 6) and ( 24) are equivalent. Pre-and post-multiplying (7) by diag(I, W, I, I), and making the same variable substitutions one concludes that ( 7) and (25) are equivalent.

Optimization problems for co-design

Theorem 2 cannot be cast directly into an optimization problem like those of Section 4.1 because Q δ does not appear as a decision variable and Qδ = W Q δ W is not affine on W and Q δ , meaning that the optimization objectives would become non-convex. Nevertheless, we can optimize Q δ indirectly by minimizing the trace of Qδ . Thus, we propose the following optimization problem: min {W, Qx, Qδ ,Y,Z,Z2,S} tr( Qx + Qδ ) subject to:

(24), ( 25), W -P -1 0 > 0 (26) Remark 2 When A p is Hurwitz, the same procedures used so far can be applied to develop a similar optimization problem guaranteeing global stability. In this case, additional conditions, e. g. convergence rate conditions, must be included in the optimization problem to turn it well-posed.

Remark 3 The present results (for K given and the codesign) can be straightforwardly extended to the case of an uncertain system described by a polytopic model, i. e. if

A ∈ Co{A 1 , A 2 , . . . , A N } (27) B ∈ Co{B 1 , B 2 , . . . , B M } (28)
and the conditions stated in Theorem 1, in Corolary 1 or in Theorem 2 are verified at the vertices of polytopes ( 27) and ( 28), that is, ∀(A i , B j ), with i = 1, . . . , N and j = 1, . . . , M , by convexity arguments, one can conclude that they are verified for the entire polytope.

Numerical example

We illustrate the application of the proposed results considering the following continuous-time unstable plant and u 0 = 5:

ẋ(t) = 0 1 4 0 x + 1 0 sat(u(t)) (29)
We choose K = -9 -5.500 so that the plant without saturation and with a continuous-time controller is stabilized with closed-loop poles in -3 and -6. We consider X 0 as the disk around the origin with radius 1, that is X 0 = {x ∈ R n ; x x ≤ 1}. From this data, the optimization problem (22) gives:

Q x = 0.3256 0 0 0.3256 Q δ = 5.138 0 0 5.138 W -1 = 0.6490 0.2472 0.2472 0.5146
Figure 2 shows results of simulation for x(0) = [ √ 2/2 √ 2/2] (notice that x(0) ∈ X 0 ). Only 26 events occurred in the time interval [0,2]. For comparison purposes, the response considering a non-triggered (i. e., continuous) controller implementation is also depicted in this figure. We can observe that no significant degradation is introduced by the event-triggered implementation. Figure 3 shows the set E(W -1 ) obtained. Note that E(W -1 ) contains X 0 , as expected, and that it does not touch X 0 , which might mean that there is some conservatism induced by the trigger parameters obtained with ( 22). We now use optimization problem (23) to tune the trigger, obtaining: With these values, we obtain a less conservative solution in terms of the triggering activity, as shown in figures 4 and 5. Now only 9 events occurred in the time interval [0,2]. On the other hand, as expected, the RAS obtained is much closer to X 0 than with parameters tunned using (22), since typically there is a trade off between the number of events and the size of the region of attraction. One also observes a slightly larger degradation relative to the continuous-time implementation than that obtained with the previous optimization problem.

Q x = 0.
Table 1 shows a comparison of the number of events and the minimum inter-event times obtained with each of the optimization problems for some initial conditions. It gives evidence that the optimization problem (23) leads to better results in terms of the reduction of the number of events. One can also see that this optimization problem also leaded to a larger minimum interevent time (36 ms, against 25 ms from the other). Note that this table includes initial conditions that are not in X 0 but are in the region of attraction of the closed-loop system. Fig. 5 RAS with event-trigger tunned using (23)

Finally, using the co-design optimization problem (26), we obtain: Figure 6 shows the results of simulation using these parameter values and the same initial condition

Q x = 0.
x(0) = [ √ 2/2 √ 2/2] .
In this case, 14 events occurred in the interval [0, 2] and the response deviates a little more from the continuous-time implementation than the previous results. Figure 7 shows the set E(W -1 ) obtained with the co-design optimization problem. Table 2 shows the number of events and the minimum inter-event times obtained from simulations with various initial conditions. Comparing to the results obtained from optimization problem (26), listed in Table 1, one can see that the co-design leaded to worse numbers of events (an average of 14.2 events against 11 when optimization problem (26) is used), but to better minimum inter-event times (68 ms against 36 ms). Note that includes initial conditions that are not in X 0 but are in the region of attraction of the closed-loop system.

Concluding remarks

In this paper we proposed a methodology to design event-trigger strategies for state-feedback controlled plants subject to input saturation. The method guarantees regional asymptotic stability of the origin for a given set of initial conditions when the plant is unstable. For stable plants, we show how the method can be extended to achieve global stability.

Since the stability conditions are presented in the form of LMIs, convex optimization problems can be proposed to compute the trigger function parameters aiming at a reduction of the number of events (and, therefore, of the data transmissions in a networked implementation) while ensuring the stability of the origin of the closed-loop system. Two different optimization problems are proposed for the case in which the controller gain is defined a priori. Numerical experiments show that the results obtained with the more generic one can lead to a reduction of the number of events.

Stability conditions that allow the simultaneous design of the controller gains and the trigger parameters are also presented. In this case, part of the trig- ger parameters do not appear explicitly in the stability conditions, but since a monotonic function of them do appear, we still can propose a suitable convex optimization problem based on LMI restrictions to tune the controller and the trigger. Numerical experiments show that the co-design method proposed does not always lead to an effective reduction of the number of samples when compared to the case where the controller gain is given; nevertheless, the experiments show that the co-design indeed leads to an effective increase of the minimum inter-event time.

We have also shown that the results can be straightforwardly extended to uncertain systems described by polytopic models. This is possible due to the linear nature of the constraints in the stability theorems.

Future work will address the extension of the proposed approaches to systems with delays to model the communication network and the data computation times.

Fig. 3

 3 Fig.3RAS with event-trigger tunned using(22)

Fig. 2 Fig. 4

 24 Fig.2Simulation results -Comparison between continuous system and event-trigger tunned via(22)

Fig. 6 Fig. 7

 67 Fig.6Simulation results -Comparison between continuous system and event-trigger tunned via (26) -co-design

Table 1

 1 Comparison of optimization problems

		x(0)	Number	Minimum	Number	Minimum
			of		inter-event	of		inter-event
			events	time (ms)	events		time (ms)
			OP (22)	OP (22)	OP (23)	OP (23)
	(-3.00, 3.00)	35	35		12		71
	(-2.00, 1.00)	31	30		14		41
	(-2.00, 2.00)	28	51		6		146
	(-1.00, 0.00)	30	25		15		36
	(-1.00, 1.00)	28	51		11		85
	(-1.00, 2.00)	25	79		8		136
	(0.00, -1.00)	25	63		11		67
	(0.00, 1.00)	25	63		11		67
	(1.00, -2.00)	25	79		8		136
	(1.00, -1.00)	28	51		11		85
	(1.00, 0.00)	30	25		15		36
	(2.00, -2.00)	28	51		6		146
	(2.00, -1.00)	31	30		14		41
	(3.00, -3.00)	35	35		12		71
		2							RAS estimate X
		1
	x 2	0
		-1
		-2
		-2	-1.5	-1	-0.5	0	0.5	1	1.5	2
					x 1

0

Trajectories

Table 2 ,

 2 as Table1,

Table 2

 2 Number of events and inter-event times -co-design

	x(0)	Number of	Minimum
		events for	inter-event time
		t ∈ [0, 2]	(ms)
	(-3.00, 4.00)	15	68
	(-3.00, 3.00)	13	68
	(-3.00, 5.00)	16	69
	(-2.00, 2.00)	13	68
	(-2.00, 3.00)	16	68
	(-2.00, 5.00)	15	69
	(-1.00, 0.00)	14	68
	(-1.00, 1.00)	13	68
	(-1.00, 2.00)	12	68
	(-1.00, 3.00)	14	68
	(0.00, -2.00)	14	68
	(0.00, -1.00)	15	69
	(0.00, 1.00)	15	69
	(0.00, 2.00)	14	68
	(1.00, -3.00)	14	68
	(1.00, -2.00)	12	68
	(1.00, -1.00)	13	68
	(1.00, 0.00)	14	68
	(2.00, -5.00)	15	69
	(2.00, -3.00)	16	68
	(2.00, -2.00)	13	68
	(3.00, -5.00)	16	69
	(3.00, -4.00)	15	68
	(3.00, -3.00)	13


Pietro Aricò

Maxime Reynal

Jean-Paul Imbert
email: jean-paul.imbert]@enac.fr

Christophe Hurter

Gianluca Borghini

Gianluca Di Flumeri
email: gianluca.diflumeri]@uniroma1.it

Nicolina Sciaraffa
email: nicolina.sciaraffa@uniroma1.it

Antonio Di Florio
email: a.diflorio@hsantalucia.it

Michela Terenzi

Ana Ferreira

J-P Imbert

S Pozzi

E Pavone

M Marucci
email: m.marucci91]@gmail.com

Human-Machine Interaction Assessment by Neurophysiological Measures: A Study on Professional Air Traffic Controllers

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

I. INTRODUCTION

In many operational environments (e.g. aircraft piloting, air-traffic control, industrial process control, robot-assisted surgery) operators have to face with complex systems and machines to accomplish operational activity. Improvements in such technology or even new solutions are often proposed, with the aim to enhance the human-machine interaction (HMI) and consequently increase operator's performance and consequently overall safety. In this context, the most studied user's mental state is the Mental Workload, due to its strong relationship with the user's performance variations [START_REF] Aricò | A passive brain-computer interface application for the mental workload assessment on professional air traffic controllers during realistic air traffic control tasks[END_REF]. The mental workload can be assessed by using different approaches: i) Performance assessment e.g. by using a secondary task, provides an objective but indirect measure of the workload; ii) subjective questionnaires (e.g. NASA-TLX) provide a direct but subjective measure of the perceived workload; iii) neurophysiological measure provide both a direct and objective measure of the experienced workload [START_REF] Borghini | Avionic technology testing by using a cognitive neurometric index: A study with professional helicopter pilots[END_REF]. With respect to the former two techniques, the latter has the advantages to not impact on the main task, since it does not require any input by the user's side, and to be available even online, i.e. during the execution of the task. A lot of works demonstrated as electroencephalography (EEG)based workload measures outperform the other kind of techniques (e.g. ECG, fNIRs) [START_REF] Arico | Passive BCI in Operational Environments: Insights, Recent Advances, and Future Trends[END_REF]- [START_REF] Flumeri | On the Use of Cognitive Neurometric Indexes in Aeronautic and Air Traffic Management Environments[END_REF]. In particular, most of the studies showed that the brain electrical activities mainly involved in the mental workload analysis are the theta and alpha brain rhythms respectively gathered from the Pre-Frontal Cortex (PFC) and the Posterior Parietal Cortex (PPC) regions. Previous studies demonstrated as the EEG theta rhythm over the PFC presents a positive correlation with the mental workload [START_REF] Smith | Neurophysiologic monitoring of mental workload and fatigue during operation of a flight simulator[END_REF], [START_REF] Flumeri | On the Use of Cognitive Neurometric Indexes in Aeronautic and Air Traffic Management Environments[END_REF]. Moreover, published literature stressed the inverse correlation between the EEG power in the alpha frequency band over the PPC and the mental workload [START_REF] Jaušovec | Working memory training: Improving intelligence -Changing brain activity[END_REF]. Depending on such evidences, theta EEG rhythms over frontal sites and alpha EEG rhythms over parietal sites have been used to define an EEG-based workload index, by using the ratio between frontal theta and parietal alpha rhythms. Nowadays, there are not many studies performed in real settings demonstrating the practical advantages of neurophysiological measures for HMI assessment. For example, Borghini et al., [START_REF] Borghini | Avionic technology testing by using a cognitive neurometric index: A study with professional helicopter pilots[END_REF] performed a preliminary study on few helicopter pilots in which it has been investigated a neurophysiological workload measure to compare avionic technologies. The present work targets tower ATCOs working in a small or medium size towers. In this context, sounds emitted by airplanes play a very important role to accurately locate aircrafts even without where it provides important input information. In this context, ATCOs should be able to easily identify within the airfield any kind of stimuli (i.e. aircrafts engine sounds), but nowadays one of the encountered difficulties is to locate such stimuli precisely. In this regard, it has been developed at ENAC an interaction modality (the "Augmented" solution) by which operators could more easily retrieve information from the airfield. This solution has been compared with a "Normal" condition (i.e. without any augmentation) by using performances, subjective and neurophysiological measures.

Human-Machine Interaction

II. MATERIAL AND METHODS

A. Experimental Subjects

Ten French professional ATCOs (6 males, Mean Age: ~41±4.6 years) coming from different airports and formations took part to the experiment. All the ATCOs were normal hearing. Their mean experience in hours was 7330 hours (SD = 5349.98). Their mean number of years in Control Tower was 12.8 (SD = 7.24). Informed consent was obtained from each participant after explanation of the study, which conformed to the revised Declaration of Helsinki and was approved by the local institutional ethics committee.

Figure 1. Schematic representation of the two compared interaction modalities (i.e. Normal vs Augmented). In particular, in the Normal one all the sound sources (aircraft) have the same amplification factor. In the Augmented solution, the amplification factor of the sound sources located along the head axis is increased, while the gain of sound sources located outside this axis is decreased.

B. Augmented solution

The augmented solution has been implemented by retrieving the participant head orientation via a Microsoft HoloLens mixed-reality headset (we use here only its inertial measurement unit feature). Hence, we can detect in which location the participant's head is pointing at. This head axis (sagittal) will select sound sources (engine sounds) aligned with its position. All these sound sources are related to aircraft which are spatially placed in the airport vicinity, i.e. in the tower competency. In this way, the sounds that are heard by the participants are spatialized. Figure 1 shows the two experimental conditions. In particular, in the Normal modality all the sound sources have the same amplification factor. In the Augmented solution, the amplification factor of the sound sources located along the head axis is immediately increased, while the gain of sound sources located outside this axis is decreased.

C. Experimental Protocol

The experiment was performed at École Nationale de l'Aviation Civile (ENAC, Toulouse, France). With respect to the aim of comparing the two different interaction modalities (i.e. Normal and Augmented); the experimental hypothesis was that the Augmented solution should be able to enhance ATCO's performance and/or decrease the experienced workload with respect to the Normal solution. To investigate this hypothesis, it has been reproduced a synthetic but realistic Control Tower environment, in particular the Muret airport (France) by using 8 screens and Flight Gear (FG) open flight simulator (Figure 2). The airport airfield has been divided into five distinct "Areas of Interest" (Figure 3). The ATCO has the point of view as the real Control Tower (located in front of the runway) and he had to discriminate the location of aircrafts in that airfield (each area could not contain more than one aircraft at a time) in the two different modalities (i.e. Normal and Augmented). In this regard, the ATCOs had an additional screen (named Input screen, Figure 2) in front of them by which to communicate the location of aircrafts within the five areas. In order to simulate different working scenarios, different trials have been realized as the combination of whether conditions (i.e. sunny or foggy) and difficulty levels (i.e. one or more aircraft at the same time), randomly changed along the experiment. Participants have been trained to use the simulator in both the experimental conditions before the experiment started. Each participant has been asked to perform the experimental task four times for each interaction modality (i.e. Normal and Augmented), in two blocks of two repetitions each (with a resting pause in the middle). The presentation of each modality has been randomized for each subject.

D. Performed Analysis

This section describes the metrics (e.g. behavioural, subjective and neurophysiological indexes) that have been used to compare the effectiveness of the two modalities (Normal vs Augmented). In particular, to quantify performance achieved by the subjects, percentages of correct responses across experimental conditions have been computed and averaged over all the experimental trials. In order to evaluate the mental workload perceived by the ATCOs during the different phases of the experimental protocol, users have been asked to fill the six (Mental demand, Physical demand, Temporal demand, Performance, Effort, Frustration) 100-points range subscales NASA-Task Load indeX (TLX, [START_REF] Hart | Development of NASA-TLX (Task Load Index): Results of Empirical and Theoretical Research[END_REF]). The global workload score from 0 to 100 was obtained for each modality (i.e. Normal vs Augmented) by averaging the individual dimension scale scores. Finally, a neurophysiological mental workload index has been computed from the EEG activity for each subject, as the ratio between frontal theta and parietal alpha frequency bands contributions (WL EEG ). In particular, for each subject, scalp EEG signals have been recorded by the digital monitoring beMicro amplifier (EBNeuro system) with a sampling frequency of 256 (Hz) by 13 Ag/AgCl passive wet electrodes covering the frontal and parietal sites (Fpz, AFz, AF3, AF4, Fz, F3, F4, Pz, P3, POz, PO3, PO4) referenced to both the earlobes and grounded to the left mastoid, according to the 10-20 standard [START_REF] Jurcak | 10/20, 10/10, and 10/5 systems revisited: Their validity as relative head-surface-based positioning systems[END_REF]. In order to compute the WL EEG index the following steps have been performed. First, EEG signals have been bandpass filtered with a fifth-order Butterworth filter and segmented in 2-seconds long epochs, shifted of 125ms [START_REF] Aricò | A passive brain-computer interface application for the mental workload assessment on professional air traffic controllers during realistic air traffic control tasks[END_REF]. Artifacts contributions that could affect the morphology of theta and alpha bands (e.g. eyes blinks and saccades, muscular artifacts, amplifiers saturations) have been removed by following specific procedures available in the EEGLAB toolbox [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF]. All the EEG epochs marked as artifact have been rejected from the EEG dataset with the aim to have an artifact-free EEG signal from which to estimate the brain variations along the different modalities (i.e. Normal vs Augmented). At this point, the Power Spectral Density (PSD) was calculated for each EEG epoch using a Hanning window [START_REF] Aricò | A passive brain-computer interface application for the mental workload assessment on professional air traffic controllers during realistic air traffic control tasks[END_REF] of the same length of the considered epoch (2 seconds). Then, the EEG frequency bands of interest have been defined for each participant by the estimation of the Individual Alpha Frequency (IAF) value [START_REF] Klimesch | EEG alpha and theta oscillations reflect cognitive and memory performance: a review and analysis[END_REF]. In order to have a precise estimation of the alpha peak and, hence of the IAF, each ATCO has been asked to keep the eyes closed for a minute before starting with the experiment. Finally, the theta rhythm (IAF-6 ÷ IAF-2), over the EEG frontal channels (Fpz, AFz, AF3, AF4, Fz, F3 and F4), and the alpha rhythm (IAF-2 ÷ IAF+2), over the EEG parietal channels (Pz, P3, P4, POz, PO3 and PO4) have been divided to compute the WL EEG index. In conclusion, the z-score method [START_REF] Zhang | A Simple Statistical Parameter for Use in Evaluation and Validation of High Throughput Screening Assays[END_REF] has been employed to compute a normalization of WL EEG index distribution.

III. RESULTS

A. Behavioral data

A paired student's t-test (α=0.05) has been performed between the two modalities in terms of achieved performance. Statistical analysis showed a significantly higher performance (p=0.001) achieved during the Augmented modality with respect to the Normal condition (Figure 4).

B. Subjective data

A paired student's t-test (α=0.05) has been performed between the NASA-TLX scores related to the two experimental conditions. NASA-TLX analysis showed a decreasing of perceived workload of ~5% during the Augmented modality, with respect to the Normal one. Anyhow, this difference was not significant (p=0.214).

C. Neurophysiological data

A paired student's t-test (α=0.05) has been performed to investigate differences between modalities. The statistical analysis performed among the experimental conditions (Figure 5), revealed a significant decreasing of the experienced workload during the Augmented solution with respect to the Normal one (p=0.003).

D. Correlation analysis

A Pearson correlation analysis has been performed among performances and WL EEG scores. Results showed a significant negative correlation between the two variables (R=-0.9; p=0.0024).

IV. DISCUSSIONS

Results highlighted a clear advantage of the Augmented interaction modality with respect to the Normal one. More precisely, both performance and neurophysiological workload index analysis have confirmed such behavior. Furthermore, such indexes showed a significant negative correlation, confirming that a higher performance is followed by a lower experienced workload and vice-versa. On the contrary, subjective analysis did not revealed any significant trend, underlining the poor resolution (in terms of sample size) of such technique in comparison to neurophysiological measures (i.e. same trend, but statistically different). In addition, neurophysiological measures allow to measure operator's mental states not only post-task (as for subjective analysis), but even during the task execution since the measure does not require any input from the user side and does not interfere with the task that he is performing. This feature could allow for a better tuning of the technology the operator is interacting with, in order to optimize the humanmachine interaction enhancing the performance of the whole system.

V. CONCLUSION

The proposed study aimed at investigating the suitability of neurophysiological indexes to compare different interaction modalities by measuring the operator's experienced workload. Achieved results suggested that neurophysiological indicators could provide lower intrasubjects variability with respect to subjective methods (i.e. NASA-TLX), since the same trend resulted not significant for subjective measure, and significant for neurophysiological one. This study represents an example of a new perspective in using neurophysiological measures, in which the objective is to assess the quality of human-machine interaction by the comparison of different technological solutions, with the final aim of enhancing user's performance and increasing safety.

Figure 2 .

 2 Figure 2. Overview of the experimental setup

Figure 3 .

 3 Figure 3. A satellite view of Muret airport, with the five "Areas of Interest" used for this experiment.

Figure 4 .

 4 Figure 4. Performance and standard deviation values achieved by the users across the Normal and Augmented modalities for each repetition. The t-test highlighted a significant increment in performance during the Augmented modality.

Figure 5 .

 5 Figure 5. WLEEG scores and standard deviation related to the workload experienced by the ATCOs, for each modality and repetition. The statistical analysis highlighted a significant decrement in WLEEG scores during the Augmented modality.

Assessment by Neurophysiological Measures: a Study on Professional Air Traffic Controllers

	looking at them. Sound is also crucial with low visibility
	conditions
	Pietro Aricò, Maxime Reynal, Jean-Paul Imbert, Christophe Hurter, Gianluca Borghini, Gianluca Di
	Flumeri, Student Member, IEEE, Nicolina Sciaraffa, Student Member, IEEE, Antonio Di Florio,
	Michela Terenzi, Ana Ferreira, Simone Pozzi, Viviana Betti, Matteo Marucci, Enea Pavone,
	Alexandru C. Telea, Fabio Babiloni, Member, IEEE

ACKNOWLEDGMENT

This work is co-financed by the European Commission by Horizon2020 projects Sesar-06-2015 the embodied reMOte Tower, "MOTO", GA n. 699379; Sesar-01-2015 Human Performance neurometricS Toolbox foR highly automatEd Systems deSign, "STRESS", GA n. 699381; H2020-MG-2016 simulator of behavioural aspects for safer transport, "SimuSafe", GA n. 723386; Horizon2020 project HCO-06-2015, Multidisciplinary tools for improving the efficacy of public prevention measures against smoking, "SmokeFreeBrain", GA n. 681120.