Xavier Leoncini

Anatoly Neishtadt

Alexei Vasiliev

Directed transport in a spatially periodic potential under periodic non-biased forcing

Keywords: numbers: 05, 45, -a, 05, 60, Cd

Transport of a particle in a spatially periodic harmonic potential under the influence of a slowly time-dependent unbiased periodic external force is studied. The equations of motion are the same as in the problem of a slowly forced nonlinear pendulum. Using methods of the adiabatic perturbation theory we show that for a periodic external force of general kind the system demonstrates directed (ratchet) transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: the case of the external force of small amplitude, and the case of the external force with amplitude of order one. The obtained formulas can also be used in case of a non-harmonic periodic potential.

I. INTRODUCTION

In recent years, studies of transport phenomena in nonlinear systems have been attracting a growing interest. In particular, a large and constantly growing number of papers are devoted to dynamics in systems which allow for directed [on average] motion under unbiased external forces and are referred to as ratchet systems. Their intensive study was motivated by problems of motion of Brownian particles in spatially periodic potentials, unidirectional transport of molecular motors in biological systems, and recognition of "ratchet effects" in quantum physics (see review [1] and references therein). Generally speaking, ratchet phenomena occur due to lack of symmetry in the spatially periodic potential and/or the external forcing. It is interesting, however, to understand microscopic mechanisms leading to these phenomena. A possible approach is to neglect dissipation and noise terms and arrive at a Hamiltonian system with deterministic forcing. Thus, one can make use of results obtained and methods developed in the theory of Hamiltonian chaos. Many papers studying chaotic transport in such Hamiltonian ratchets appeared in the last years (see, e.g. [2,3,4,5,6,7,8]). In particular, in [6] the ratchet current is estimated in the case when there are stability islands in the chaotic domain in the phase space of the system. The borders of such islands are "sticky" [9] and this stickiness together with desymmetrization of the islands are responsible for occurrence of the ratchet transport.

We consider the problem of motion of a particle in a periodic harmonic potential U(q) = ω 2 0 cos q, where q is the coordinate and ω 0 = const, under the influence of unbiased time periodic external forcing. The equations are the same as in the paradigmatic model of a nonlinear pendulum under the action of external torque with zero time average. We study the case when the external forcing is a time-periodic function of large period of order ε -1 , 0 < ε ≪ 1, and use results and methods of the adiabatic perturbation theory. If ε is small enough, there are no stability islands in the domain of chaotic dynamics (see [10]). Thus, the mechanism of ratchet transport in this system differs from one suggested in [6].

The main objective of this paper is to find a formula for the average velocity V q = q of a particle in the chaotic domain on very large time intervals. Chaos in the system is a result of multiple separatrix crossings due to slow variation of the external forcing. At each crossing the adiabatic invariant ("action" of the system) undergoes a quasi-random jump (see [11,12]). We show that these jumps result in effective mixing and uniform distribution of the action along a trajectory in the chaotic domain. On the other hand, direction and value of velocity depends on the immediate value of the action. Thus, to find the average velocity of transport on time intervals of order or larger than the mixing time, we find formulas for displacement in q at a given value of the action and then integrate them over the interval of values of the action corresponding to the chaotic domain. We demonstrate that for an external force of general kind (i.e. with zero time average but lowered time symmetry, cf. [5]), there is directed transport in the system and obtain an analytic formula for the average velocity V q of this transport.

In Section 2, we obtain the main equations in the case of external force of small amplitude and describe the diffusion of adiabatic invariant due to multiple separatrix crossings. This diffusion makes dynamics chaotic in a large domain of width of order 1. In Section 3, we derive the formula for the average velocity of transport on very long time intervals (of order of typical diffusion time and larger) and check it numerically. In Section 4, we consider the case when the external forcing is not small, of order 1. Width of the chaotic domain in this situation is large, of order ε -1 . In this case chaos arises as a result of scattering of the adiabatic invariant on the resonance. Formula for the average velocity of transport in this case turns out to be much simpler than in the case of external forcing of small amplitude.

II. MAIN EQUATIONS. DIFFUSION OF THE ADIABATIC INVARIANT

We start with a basic equation describing a nonlinear pendulum under the influence of an external force f (t): q + ω 2 0 sin q = f (t).

We assume that f (t) is small in amplitude periodic function of time of large period: f (t) ≡ εf (τ ) = εf (τ + 2π), where τ = εt. Moreover, we assume that f(t) is a function with zero time average. This system is Hamiltonian, with time-dependent Hamiltonian function

H = p 2 2 -ω 2 0 cos q -εf (εt)q. ( 2
)
We make a canonical transformation of variables (p, q) → (p, q) using generating function W = (p-F (εt))q, where F (τ ) =f (τ )dτ . Thus, F (τ ) is a periodic function defined up to an additive constant, which we are free to choose. To make the following presentation more clear, we choose this constant in such a way that the minimal value of F is F min > 4ω 0 /π.

Note that q ≡ q. After this transformation, Hamiltonian of the system acquires the form (bars over q are omitted):

H = (p -F (τ )) 2 2 -ω 2 0 cos q. ( 3
)
Phase portrait of the system at a frozen value of τ (we call it the unperturbed system) is shown in Fig. 1. The separatrix divides the phase space into the domains of direct rotations (above the upper branch of the separatrix), oscillations (between the separatrix branches), and reverse rotations (below the lower branch of the separatrix). Introduce the "action"

I associated with a phase trajectory of the unperturbed system on this portrait. In the domains of rotation, I equals an area between the trajectory, the lines q = -π, q = π, and the axis p = 0, divided by 2π; in the domain of oscillations, this is an area surrounded by the trajectory divided by 2π. It is known that I is an adiabatic invariant of (3): far from the separatrix its value is preserved along a phase trajectory with the accuracy of order ε on long time intervals (see, e.g., [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF]). Location of the separatrix on the (q, p)-plane depends on the value of F (τ ). As τ slowly varies, the separatrix slowly moves up and down, and phase points cross the separatrix and switch its regime of motion from direct rotations to reverse rotations and vice versa.

Recall known results on variation of the adiabatic invariant when a phase point crosses the separatrix. The area surrounded by the separatrix is constant, and hence, capture into the domain of oscillations is impossible in the first approximation (in the exact system, only a small measure of initial conditions correspond to phase trajectories that spend significant time in this domain; thus their influence on the transport is small). To be definite, consider the situation when the separatrix on the phase portrait slowly moves down. Thus, phase points cross the separatrix and change the regime of motion from reverse rotation to direct rotation. Let the action before the separatrix crossing at a distance of order 1 from the separatrix be I = I -and let the action after the crossing (also at a distance of order 1 from the separatrix) be I = I + . In the first approximation, we have

I + = I -+ 8ω 0 /π, i.e.
the action increases by the value of the area enclosed by the separatrix divided by 2π (see, e.g., [14,15]). We shall call this change in the action a "geometric jump". If the separatrix contour slowly moves up, and a phase point goes from the regime of direct rotation to the regime of reverse rotation, the corresponding value of the action decreases by the same value 8ω 0 /π. Thus, in this approximation, the picture of motion looks as follows. While a phase point is in the domain of reverse rotation, the value of I along its trajectory stays constant:

I = I -.
After transition to the domain of direct rotation, this value changes by the value of the geometric jump. The transition itself in this approximation occurs instantaneously.

After the next separatrix crossing, the adiabatic invariant changes again by the value of the geometric jump, with the opposite sign, and returns to its initial value I -. We call this approximation adiabatic.

In the next approximation, the value of action at the separatrix crossing undergoes a small additional jump. Consider for definiteness the case when the separatrix contour on the phase portrait moves down, and I -and I + are measured when it is in its uppermost and lowermost positions, accordingly. Results of [11,12] imply the following formula for the jump in the adiabatic invariant:

2π(I + -I -) = 16ω 0 + 2a(1 -ξ)εΘln(εΘ) + aεΘln 2π(1 -ξ) Γ 2 (ξ) -2bεΘ(1 -ξ),
where

a = ω -1 0 , b = ω -1 0 ln(32ω 2 0 ), Θ = 2πF ′ (τ * ).
Here F ′ is the τ -derivative of F , τ * is the value of τ at the separatrix crossing found in the adiabatic approximation, Γ(•) is the gamma-function. Value ξ is a so-called pseudo-phase of the separatrix crossing; it strongly depends on the initial conditions and can be considered as a random variable uniformly distributed on interval (0, 1) (see, e.g., [12]). Thus, value of the jump in the adiabatic invariant at the separatrix crossings has a quasi-random component of order εlnε.

Accumulation of small quasi-random jumps due to multiple separatrix crossings produces diffusion of adiabatic invariant (see, e.g., [12]). On a period of F (τ ) (after two separatrix crossings) the action changes by a value of order εlnε. Hence, after N ∼ ε -2 (lnε) -2 separatrix crossings the adiabatic invariant varies by a value of order one. As a result, in time of order t dif f ∼ ε -3 (lnε) -2 the value of adiabatic invariant is distributed in all the range of values corresponding to the domain where phase points cross the separatrix on the phase plane; its distribution is close to the uniform one. We have checked this fact numerically for a sample function F (τ ) at various parameter values. Poincaré sections and distribution histograms of I in all the cases look similar; see an example in Fig. 2. All the points are mapped onto the interval q ∈ (-π, π). F (τ ) = A 1 + 2 exp -α(sin τ ) 2 with A = 10, α = 16, ε = 0.005, ω 0 = 1. The empty region in the chaotic sea corresponds to phase points eternally locked in the domain of oscillations; they never enter the chaotic domain and do not participate in the transport. Right panel: Histogram of I on the segment (I min , I max -8ω 0 /π)) along the same phase trajectory.

III. AVERAGE VELOCITY OF THE TRANSPORT

Our aim is to find a formula for average velocity V q along a phase trajectory on time intervals of order t dif f or larger. We first only take into consideration the geometric jumps, and afterwards, to obtain the final result, we take into account the mixing due to small quasi-random jumps. To simplify the consideration, assume that function F (τ ) has one local minimum F min and one local maximum F max on the interval (0, 2π). The main results are valid without this assumption.

Introduce Ĩ, defined in the domains of rotation as follows: it equals the area bordered by the trajectory, the line p = F (τ ), and the lines q = -π, q = π, divided by 2π. Thus, Ĩ = |F (τ ) -I|. Frequency of motion in the domains of rotation is ω( Ĩ), where ω( Ĩ) at Ĩ > 4ω 0 /π is the frequency of rotation of a standard nonlinear pendulum with Hamiltonian H 0 = p 2 /2ω 2 0 cos q, expressed in terms of its action variable Ĩ. We do not need an explicit expression for function ω( Ĩ). From Hamiltonian (3) we find q = p -F (τ ). Consider a phase trajectory of the system frozen at τ = τ in a domain of rotation. Let the value of action on this trajectory be I = I 0 . Then the value of q averaged over a period T of rotation equals

T 0 | q| dt/T = 2π/T = ω(|F (τ) -I 0 |).
Now consider a long phase trajectory in the case of slowly varying τ . Let on the interval (τ 1 , τ 2 ) a phase point of (3) be below the separatrix contour. In the adiabatic approximation, the value I 0 of the adiabatic invariant along its trajectory is preserved on this interval. Hence,

at τ ∈ (τ 1 , τ 2 ) we have 2πF (τ ) -2πI 0 ≥ 8ω 0 , (4)
and the equality here takes place at τ = τ 1 and τ = τ 2 . In the process of motion on this time interval, q changes (in the main approximation) by a value

∆q -(I 0 ) = - 1 ε τ 2 τ 1 ω(F (τ ) -I 0 )dτ. ( 5
)
On the interval (τ 2 , τ 1 + 2π) the phase trajectory is above the separatrix contour, and the value of the adiabatic invariant equals Î0 = I 0 + 8ω 0 /π due to the geometric jump. On this interval we have

2πF (τ ) -2πI 0 ≤ 8ω 0 . (6)
In the process of motion on this time interval, q changes by a value ∆q

+ (I 0 ) = 1 ε τ 1 +2π τ 2 ω(|F (τ ) -Î0 |)dτ. ( 7
)
Total displacement in q on the interval (τ 1 , τ 1 + 2π) equals ∆q(I 0 ) = ∆q -(I 0 ) + ∆q + (I 0 ), and the average velocity on this interval is ε∆q(I 0 )/(2π).

Consider now the motion on a long enough time period ∆t ∼ t dif f . Due to the diffusion in the adiabatic invariant described above, in this time period values of I 0 , defined as a value of I when the phase point is below the separatrix contour, cover the interval (I min , I max -8ω 0 /π).

Here I min = F min -4ω 0 /π and I max = F max + 4ω 0 /π. Assuming that the distribution of I on this interval is uniform, to find the average velocity, we integrate ε∆q(I 0 )/(2π) over this interval. Integrating (5) over I 0 and changing the order of integration we find

Imax-8ω 0 /π I min ∆q -dI 0 =- 1 ε 2π 0 dτ F (τ )-4ω 0 /π I min ω(F (τ ) -I 0 )dI 0 = - 1 ε 2π 0 dτ F (τ )-I min 4ω 0 /π ω(η)dη.
Now we take into account the equality ω( Ĩ) = ∂H 0 ( Ĩ)/∂ Ĩ (recall that H 0 ( Ĩ) is the Hamiltonian of a nonlinear pendulum as a function of its action variable) and obtain

- 2π 0 dτ F (τ )-I min 4ω 0 /π ω(η)dη = - 2π 0 (H 0 (F (τ )-I min )-H s 0 ) dτ, ( 8
)
where H s 0 is the value of H 0 on the separatrix. Similarly, integrating ( 7) we obtain

Imax-8ω 0 /π I min ∆q + dI 0 = 1 ε 2π 0 (H 0 (I max -F (τ ))-H s 0 ) dτ. (9)
Adding ( 8) to ( 9) and dividing by 2π(F max -F min )/ε we find the expression for the average velocity V q of transport on long time intervals:

V q = 1 2π(F max -F min ) × 2π 0 (H 0 (I max -F (τ ))-H 0 (F (τ )-I min )) dτ. (10)
In (10), H 0 (I) can be found as the inverse function to Ĩ(h), which defines action as a function of energy in domains of rotation of a nonlinear pendulum. For the latter function, the following formula holds (see, e.g., [START_REF] Sagdeev | Nonlinear Physics: from Pendulum to Turbulence and Chaos[END_REF]):

Ĩ(h) = 4 π ω 0 κ E (1/κ) , κ ≥ 1, (11)
where

κ 2 = (1 + h/ω 2 0 )/2, E(•)
is the complete elliptic integral of the second kind. If function F (τ ) has several local extremes on the interval (0, 2π), F min and F max in (10) are the smallest and largest values of F respectively.

It can be seen from ( 10) that for function F (τ ) of general type V q is not zero, and hence there is the directed transport in the system. We checked this formula numerically for a sample function F (τ ) = A (1 + 2 exp [-α(sin τ ) 2 ]) , α > 0 at various values of parameters ε and α. Typical plots of q against time t are shown in Fig. 3. The results of numerical checks of formula (10) are represented in Table I. To find numerical values of V q presented in the table, we integrated the system with Hamiltonian (3) on a long time interval ∆t = 2π • 10 6 /ε with a constant time step of π/100 (5th order symplectic scheme [START_REF] Mclachlan | [END_REF]). Use of a symplectic scheme for long time simulations of Hamiltonian systems is necessary in order to ensure that creeping numerical error do not end up washing off the invariant tori bounding the chaotic domain. The table demonstrates satisfactory agreement between the formula and the numerics.

Finally, we note that formula (10) can be used also in the case of arbitrary (non-harmonic)

spatially-periodic time independent potential in place of the term -ω 2 0 cos q in (2), (3). Of course, in this case function H 0 is different from the Hamiltonian of the nonlinear pendulum, but it always can be found, at least numerically.

IV. THE CASE OF NOT SMALL EXTERNAL FORCING

In this section we study the case when the external forcing is not small. In this case, amplitude of function f(t) in ( 1) is a value of order one: f (t) ≡ f (τ ) = f (τ + 2π). The equations of motion are:

q = p, ṗ = -ω 2 0 sin q + f (τ ), τ = ε. (12)
One can see from the second equation, that magnitude of momentum p can reach values of order ε -1 . Making the canonical transformation with generating function W = (pε -1 F (εt))q, where F (τ ) is defined in Section II, we obtain the Hamiltonian:

H = (p -ε -1 F (τ )) 2 2 -ω 2 0 cos q. (13)
Introduce p = εp and rescaled time t = ε -1 t. We denote the derivative w.r.t. t with prime and thus obtain:

q ′ = p -F (τ ), p′ = -ε 2 ω 2 0 sin q, τ ′ = ε 2 . ( 14
)
This is a system in a typical form for application of the averaging method. We average over fast variable q and obtain the averaged system p′ = 0,

τ ′ = ε 2 . ( 15
)
The averaged system describes the dynamics adequately everywhere in the phase space except for a small neighborhood of the resonance at p -F (τ ) = 0, where the "fast" variable q is not fast. When a phase trajectory of the averaged system crosses the resonance, value of the adiabatic invariant (in this case it coincides with p) undergoes a quasi-random jump of typical order √ ε 2 = ε (see, e.g., [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF]18]). Thus, the situation in this case is similar to one studied in Section 2, but the typical value of a jump is of order ε, and, accordingly, the typical diffusion time is t dif f ∼ ε -3 . Phase trajectories of the averaged system that cross the resonance correspond to values of p belonging to the interval (F min , F max ). Therefore the chaotic domain of the exact system is, in the main approximation, a strip F min ≤ p ≤ F max .

Captures into the resonance followed by escapes from the resonance (see [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF]18]) are also possible in this system. However, probability of capture is small, of order ε, and hence impact of these phenomena on the transport is small.

In Fig. 4 we represent a sample of Poincaré section of a long phase trajectory of ( 13) and the corresponding histogram of p for this trajectory. The plots show that the distribution of p is close to the uniform one.

To find V q , we have to integrate this expression over p from F min to F max (i.e., over the chaotic domain) and divide the result by (F max -F min ) and by the length of the period of the external forcing 2π/ε. Thus we obtain

V q = ε 2π(F max -F min ) Fmax F min ∆q dp. ( 17
)
Substituting ∆q from ( 16) and integrating, one straightforwardly obtains V q = 1 4πε 2π 0 (F max + F min -2F (τ )) dτ. (18) directed transport on long time intervals. Direction and average velocity of the transport in the chaotic domain are independent of initial conditions and determined by properties of the external force. We studied two different cases: the case of small amplitude of the external force and the case, when this amplitude is a value of order one. We have obtained an approximate formula for average velocity of the transport and checked it numerically.

The final formulas (10) and ( 19) are valid for any smooth periodic potential (not necessarily harmonic one).

FIG. 1 :

 1 FIG. 1:Phase trajectories of the system (3) with sample function F (τ ) =A 1 + 2 exp -α(sin τ ) 2 , A = 10, α = 8, τ = π/2, ω 0 = 1. The bold line is the separatrix.

FIG. 2 :

 2 FIG. 2: Left panel: Poincaré section at τ = 0 mod 2π of a long phase trajectory (5 • 10 4 dots) .

α = 4 α = 8 α = 16 εFIG. 3 :

 4163 FIG.3: Left panel: q against t for ten different initial conditions (comparatively short time interval), α = 4, ε = 0.05. Central panel: q against the number of periods of the external force for a sample trajectory (10 4 periods), α = 16, ε = 0.05. Right panel: q against the number of periods of the external force for the same trajectory (10 5 periods), α = 16, ε = 0.05. Parameter A = 10 in all cases.

FIG. 4 :

 4 FIG.4: Left panel: Poincaré section at τ = 0 mod 2π of a long phase trajectory (5 • 10 4 dots) of system[START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF]. All the points are mapped onto the interval q ∈ (-π, π). F (τ ) = A 1 + 2 exp -α(sin τ ) 2 with A = 0.5, α = 4, ε = 0.01, ω 0 = 1. Right panel: Histogram of p along the same phase trajectory.

Acknowledgements

The work was partially supported by the RFBR grants 06-01-00117 and NSh 691.2008.1.

A.V. thanks Centre de Physique Théorique in Luminy and Ricardo Lima for hospitality in the fall of 2007 and numerous discussions.

Another possibility to find V q is to use already obtained formula (10). This way leads to the same result. When using (10), one should keep in mind that in the considered case H 0 in this formula is the Hamiltonian of the averaged system, i.e. H 0 = I 2 /2, I max = F max , I min = F min , and that according to [START_REF] Arnold | Mathematical aspects of classical and celestial mechanics[END_REF] we should put ε -1 F instead of F everywhere in the formula.

Thus, ( 10) is much simplified, and we again arrive at formula (18). Factor ε -1 is due to the fact that a phase point spends significant time moving at large velocities corresponding to

Note that formula (18) can be rewritten in a more elegant form as:

where the angle brackets denote time average. The results of numerical checks of the formula are represented in Table II. Like in the previous Section, we integrated the system with Hamiltonian (13) on a long time interval ∆t = 2π • 10 6 /ε.

Remarkably, formula (19) is valid for arbitrary smooth periodic potential (not necessarily harmonic) in place of the term -ω 2 0 cos q in (2), (3). The potential may also depend periodically on time with the same period as that of the external force.

V. SUMMARY

To summarize, we have considered the phenomenon of the directed transport in a spatially periodic harmonic potential adiabatically influenced by a periodic in time unbiased external force. We have shown that for the external force of a general kind the system exhibits


Carla De Souza

Valter J S Leite
email: valter@ieee.org

Sophie Tarbouriech

Eugênio B Castelan
email: eugenio.castelan@ufsc.br

Event-triggered Policy for Dynamic Output Stabilization of Discrete-time LPV Systems under Input Constraints ⋆

This paper concerns the event-triggered dynamic output-feedback control of discrete-time linear parameter-varying (LPV) systems subject to saturating actuators. Two independent event-triggering schemes are introduced to determine whether the current signals should be transmitted a) from the sensor to the controller and b) from the controller to the actuator. As a result, the communication resources can be significantly saved. Both the emulation-based problem and the co-design problem are addressed. Sufficient conditions based on linear matrix inequalities (LMIs) are derived to ensure the regional asymptotic stability of the origin for the closed-loop system. A convex optimization procedure is proposed to determine the controller matrices and the event-triggering parameters aiming at reducing the number of updates on the independent channels sensor-to-controller and controller-toactuator. At last, numerical examples are employed to testify to the validity of the proposed methods.

Introduction

In the last two decades, an extensive research effort has been devoted to network control systems (NCS). These systems are characterized by the interconnection among control systems devices through a communication network. Several advantages arise from its use, such as lower costs, ease of deployment and maintenance, flexibility (see, for example, [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] and references therein). However, the communication resources of the network are often limited, which motivate new challenges in the control of systems. In this context, event-triggered control (ETC) emerged as an alternative to traditional sampled-data control [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. The idea of the ETC is to execute control tasks after the occurrence of an event, generated by a specified event-triggering mechanism (ETM), rather than the elapse of a certain period of time, as in most traditional digital control setups. In such a way, ETC is able to significantly reduce the number of communications between process and controller, and controller and actuator, while maintaining a satisfactory closed-loop performance. The existing approaches for the design of event-triggered controllers can be issued either from emulation-based approach [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF][START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF] or from co-design approach [START_REF] Peng | Event-triggered communication and  ∞ control co-design for networked control systems[END_REF][START_REF] Tarbouriech | Observer-based event-triggered control co-design for linear systems[END_REF][START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for the  2 -stabilization of linear systems[END_REF]. In the emulation-based framework, only the controller or the ETC is designed while the other part is given, whereas in the co-design framework both are simultaneously designed. Although, several event-triggering schemes and control strate-gies have been proposed in literature (see, for example, [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF][START_REF] Yue | A delay system method for designing event-triggered controllers of networked control systems[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF][START_REF] Zhang | Event-triggered dynamic output feedback control for networked control systems[END_REF][START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF][START_REF] Tanwani | Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization[END_REF][START_REF] Zhang | Input-to-state stabilization of nonlinear discrete-time systems with event-triggered controllers[END_REF][START_REF] Liu | Robust event-triggered control for networked control systems[END_REF][START_REF] Ge | Dynamic event-triggered distributed coordination control and its applications: A survey of trends and techniques[END_REF]), only a few explore the characteristics of the discrete-time linear parameter-varying (LPV) systems subject to input saturation.

The framework of the NCS considered in this paper is illustrated in Figure 1, where the process to be controlled is a LPV system with a saturating actuator.

To reduce the number of the sensor and the controller data transmissions in the network while preserving the stability and some certain control performance, two event generators are introduced, one in the channel between the sensor and the controller and another in the channel between the controller and the actuator. Event-triggering conditions are embedded in the event generators to determine whether the current signals (output and input) should be transmitted through the networks at every instant. As a result, the usage of the communication resources can be significantly saved.

In order to study the event-triggering scheme depicted in Figure 1, both the plant and the controller are supposed to be LPV. In the literature, some results regarding the eventtriggered control for LPV systems without saturating inputs can be found. [START_REF] Li | Event-triggered control for discrete-time uncer-tain linear parameter-varying systems[END_REF] addresses the co-design problem of event generator and state-feedback controller for discrete-time LPV systems where only estimated parameters satisfying certain uncertainty level is known. An event-triggered  ∞ control for discrete-time LPV systems is proposed in [START_REF] Li | Co-design of event-triggered  ∞ control for discrete-time linear parameter-varying systems with networkinduced delays[END_REF] by jointly designing a mixed ETM and a state feedback controller. [START_REF] Braga | Discretization and event triggered digital output feedback control of LPV systems[END_REF] investigates the problem of discretization and digital state/output feedback control design for continuous-time LPV systems subject to a time-varying networked-induced delay. [START_REF] Golabi | Event-triggered control for discrete-time linear parametervarying systems[END_REF][START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF] address an event-based reference tracking control for discrete-time LPV systems by simultaneously designing event-triggering conditions and a state feedback controller. A co-design condition in a sense of input-to-state practically stable (ISpS) of a mixed ETM and a static outputfeedback controller is established in [START_REF] Xie | Output-based event-triggered control for networked control systems: tradeoffs between resource utilisation and robustness[END_REF] for stabilization of discrete-time LPV systems.

Similarly, some results considering the event-triggering control for systems subject to saturating inputs but outside the LPV paradigm have been published. [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] proposes a procedure to design a static state feedback that maximizes an estimate of the domain of attraction of saturated discretetime system for a given triggering function. A cone complementary linearization algorithm is proposed in [START_REF] Zuo | Co-design of eventtriggered control for discrete-time systems with actuator saturation[END_REF] for solving the non-convex optimization problem in order to obtain the co-design of a state-feedback controller with saturation. [START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF][START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] address the co-design problem for an event-triggered controller based on a static state feedback for a discrete-time system subject to actuator saturation. Also, the co-design based on a dynamic state-feedback controller is proposed in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF].

In the current paper, we address the problem depicted in Figure 1 by expanding the results obtained in [START_REF] De Souza | Eventtriggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF] and [START_REF] De Souza | Codesign of an event-triggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF], in which only an ETM affected the output signal. We consider an approach based on Lyapunov theory as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Tarbouriech | Observer-based event-triggered control co-design for linear systems[END_REF][START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF][START_REF] Moreira | PI event-triggered control under saturating actuators[END_REF]. Although the proposed ETM is simpler than that one studied in [START_REF] De Souza | Emulation-based dynamic output-feedback control of saturating discrete-time LPV systems[END_REF], the contribution of the current paper includes the co-design, which expects to lead to lower transmission rates. Thus, for both emulation and co-design cases, convex optimization procedures are formulated to design a) the output and control event generator parameters and b) the parameterdependent dynamic output-feedback controller. In all cases, the objective is to reduce data transmission on the sensorto-controller and controller-to-actuator channels. We ensure the regional asymptotic stability of the closed-loop system and characterize an estimate of the respective basin of attraction of the origin.

This paper is organized as follows. Section 2 introduces the class of systems under consideration, and states the problem to be solved. Some preliminaries results useful in the development of the conditions are given in Section 3. Section 4 presents the conditions of emulation, while Section 5 presents those of the co-design. In Section 6 are proposed optimization procedures with different control objectives. Numerical examples are given in Section 7, which verify the effectiveness of the presented approach. Finally, some concluding remarks end the paper.

Notation: ℝ, and ℝ + represent the set of real and nonnegative real numbers, respectively. The matrix stands for the null matrix of appropriate dimensions and corresponds to the identity matrix with dimensions × . ℝ × is the set of matrices with real entries and dimensions × . = { 1 , 2 } denotes the block-diagonal matrix composed by the blocks 1 and 2 . ( ) indicates the ℎ line of a vector or a matrix . [ , ] denotes the set of integer numbers belonging to the interval from ∈ ℕ to ∈ ℕ, ≥ . The symbol ⋆ represents the symmetric blocks within a matrix, • represents an element that has no influence on development. For ∈ ℝ , ‖ ‖ = √ ⊤ denotes the Euclidean norm and

‖ ‖ 2 is defined by ⊤ with < = ⊤ ∈ ℝ × .

Problem Statement

The plant is described by the following model:

, +1 = ( ) , + ( ) ( ̂ ), = , , (1)
where , ∈ ℝ is the state vector, ̂ ∈ ℝ is the most recently transmitted value of the control input ∈ ℝ , and ∈ ℝ is the measurable output. The symmetric saturation function, ( ̂ ), is given by

( ̂ ( ) ) = ( ̂ ( ) ) min(| ̂ ( ) |, ̄ ( ) ), (2)
with ̄ ( ) > 0, ∈ [1, ], the limit of the saturation. The vector of time-varying parameters , which are assumed measurable and available on-line [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF], lies in the unitary simplex defined by

Θ ≜ ∑ =1 ( ) = 1, ( ) ≥ 0, ∈ [1, ] . ( 3
)
The matrices ( ) ∈ ℝ × and ( ) ∈ ℝ × belong to a polytopic set given by the convex combination of known vertices as follows

( ) ( ) = ∑ =1 ( ) , ∈ Θ. ( 4
)
Remark 1. In this paper, the system (1) can be regarded as an LPV discrete-time model issued from an identification process [START_REF] Verdult | Nonlinear system identification: State-space approach[END_REF] or as a discretized version of the following continuous-time process

̇ ( ) = ̃ ( ( )) ( ) + ̃ ( ( )) ( ( )), ( ) = ̃ ( ). (5)
Note that there are several methods of discretization in the literature. One of them consists of discretizing the LPV system [START_REF] De Souza | Codesign of an event-triggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF] in the same way as for LTI systems, under the assumption of slow parameter variation. In such a case, the matrices ( ), ( ) and are given by the approximations

( ) = ̃ ( ( )) , ( ) = ∫ 0 ̃ ( ( ))
̃ ( ( )), and = ̃ , where denotes the sampling period and = ∈ Θ, ∀ ∈ [ , ( + 1) ) [START_REF] Ramezanifar | Sampled-data filtering for linear parameter varying systems[END_REF][START_REF] Toth | Discretisation of linear parameter-varying state-space representations[END_REF]. In this sense, we assume the plant is controlled in a periodic eventtriggering way [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF][START_REF] Wu | Event-triggered control for discrete-time linear systems subject to bounded disturbance[END_REF]. Also, the measurement is made periodically with the time interval at the sampling instants and the control input is updated employing a zero-orderolder (ZOH). Therefore, the approach proposed can be used to treat both continuous-time and discrete-time systems. It is also worth noting that, using this approach, the challenging problem of the minimum inter-event time for event-triggered output feedback control is overcome, since the inter-event time are at least lower bounded by the sampling period.

To stabilize the system (1), we adopt the following parameter-dependent dynamic output-feedback controller:

, +1 = ( ) , + ( ) ̂ -( )Ψ( ̂ ), = ( ) , + ( ) ̂ , (6)
where , ∈ ℝ is the control state, Ψ( ̂ ) ∶ ℝ → ℝ is the dead-zone non-linearity defined by Ψ( ̂ ) = ̂ -( ̂ ), and ̂ is the most recently transmitted value of the output measurement to the controller. The anti-windup action, represented by the matrix ( ) ∈ ℝ × , is added to mitigate the effects caused by the saturating actuators [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF]. Therefore, it acts only when the saturation occurs, i.e., wherever Ψ( ̂ ) ≠ . About the controller matrices, let us consider the following assumption: Assumption 1. The matrices of the controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] are supposed to have the following structure:

( ) ( ) = ∑ =1 ∑ = (1 + ) ( ) ( ) 2 2 , ( ) ( ) = ∑ =1 ( ) , ( ) = ∑ =1 ( )
, with ∈ Θ and = 1 if ≠ and = 0 otherwise.

The controller structure introduced by Assumption 1 has a quadratic dependency on the parameter . Let us stress that any dynamic controller given in a standard polytopic form can be described according to Assumption 1 by using the fact that:

∑ =1 ( ) ∑ =1 ( ) = 0.5 ∑ =1 ∑ = (1 + ) ( ) ( )
, with ∈ Θ and = 1 if ≠ and = 0 otherwise. However, the structure proposed in Assumption 1 is more general than the polytopic one leading to the fact that any dynamic controller satisfying Assumption 1 cannot always put in a polytopic form. Moreover, a similar but simpler development could be performed assuming ( ) and ( ) with polytopic description since matrices and do not depend on , i.e., they would be time-invariant.

The transmitted output ̂ and the transmitted control ̂ are generated by the following two independent eventtriggering conditions, that shares the same clock,

̂ ∶= , ‖ ̂ -1 -‖ 2 Δ > ‖ ‖ 2 , ̂ -1 , otherwise, (7)
and

̂ ∶= , ‖ ̂ -1 -‖ 2 Δ > ‖ ‖ 2 , ̂ -1 , otherwise, (8)
where the symmetric positive definite matrices Δ , ∈ ℝ × and Δ , ∈ ℝ × are triggering parameters to be designed. These matrices act as weights on the terms associated with the triggering conditions. Their choice has a direct impact on the event-triggering policy, and, thus, on the way to reduce the data transmission. Therefore, by means of the event-triggering conditions ( 7) and ( 8), which are verified periodically, it is decided whether or not to transmit new measurements and control signals, respectively, through the network. Note that, unlike [START_REF] De Souza | Emulation-based dynamic output-feedback control of saturating discrete-time LPV systems[END_REF], the proposed mechanisms do not share the same information, which makes them independent of each other.

Due to the control input saturation, the closed loop behaves as a non-linear system, and the global stability is no longer guaranteed. In this case, the region of attraction   in which belongs the augmented state vector = [ ⊤ , ⊤ , ] ⊤ ∈ ℝ 2 , must be considered. As the exact characterization of   is, generally, a hard task, it is important to characterize subsets with well-defined analytical representation, such as ellipsoidal and polyhedral sets. By denoting   the estimated attraction region, then we are interested in computing

  ⊆   .
From this, the problems we intend to solve can be summarized as follows.

Problem 1 (Emulation problem).

Given the dynamic output feedback controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], which regionally stabilizes the LPV system [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for the  2 -stabilization of linear systems[END_REF] with saturating actuators in the absence of communication networks, design the two independent eventtriggering conditions [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] and [START_REF] Ge | Dynamic event-triggered distributed coordination control and its applications: A survey of trends and techniques[END_REF] to reduce the number of data transmissions on the sensor-to-controller and controllerto-actuator channels, respectively, while preserving the stability of the closed-loop system.

Problem 2 (Co-design problem). Given the LPV system (1)

with saturating actuators, co-design the parameter-dependent dynamic output feedback controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] and the two independent event-triggering conditions ( 7) and ( 8), ensuring the regional asymptotic stability of the closed-loop system, while reducing the number of data transmissions on the sensor-tocontroller and controller-to-actuator channels, respectively.

Preliminaries Results

The LPV system (1) under the dynamic controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], can be represented by the following model:

+1 = ( ) -( )Ψ( ) + ( ) , + ( ) , , = ( ) + ( ) , , = ℂ , (9)
where

= ⊤ , ⊤ , ⊤ ∈ ℝ 2 is the augmented state,
, ∈ ℝ is the error between the latest transmission ̂ and the latest sampling , and , ∈ ℝ is the error between the latest transmission ̂ and the latest sampling . The parameter-varying matrices verify from Assumption 1:

( ) ( ) = ∑ =1 ∑ = (1 + ) ( ) ( ) 2 2 , ( ) ( )
( ) ⊤ = ∑ =1 ( ) ⊤ ,
with ∈ Θ and = 1 if ≠ and = 0 otherwise, and are defined by

= + + ( + ) + , = , = , = + , =
, and ℂ = .

Note that if is updated at instant , then from (7) it follows that , = ̂ -= -= , and if is not updated at instant , then from (7) it also follows that

, = ̂ - = ̂ -1 -.
In other words, the following inequality is always satisfied:

‖ , ‖ 2 Δ ≤ ‖ ‖ 2 . ( 10
)
Similarly, if is updated at instant , then from (8) one gets , = -= , and if is not updated at instant , then from (8) one gets , = ̂ -1 -. Consequently, the following condition always holds

‖ , ‖ 2 Δ ≤ ‖ ‖ 2 . ( 11
)
To investigate the regional asymptotic stability of the closed-loop system (9), we use the following candidate Lyapunov function

( ) = ⊤ -1 ( ) , ( 12
)
where 12) is a Lyapunov function, then the estimated attraction region is computed through an associated level set   =   (1) = { ∈ ℝ 2 ∶ ( ) ≤ 1}, which can be computed as [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF]Lemma 4]:

( ) = ∑ =1 ( ) , with < = ⊤ ∈ ℝ 2 ×2 and ∈ Θ. If (
  =   (1) = ⋂ ∀ ∈Θ ( ( ) -1 , 1) = ⋂ ∈[1, ] ( -1 , 1), (13)
with

( -1 , 1) = ∈ ℝ 2 ∶ ⊤ -1 ≤ 1 . ( 14
)
In addition, to deal with the saturation, we use the following property directly derived from [28, Lemma 1.6, p. 43].

Lemma 1. Let given by ( 6), ̄ ∈ ℝ + , and a matrix

( ) = ∑ =1 ( ) with ∈ ℝ ×2 for [1, ] and ∈ Θ, such that ( ̄ ) ≜ { ∈ ℝ 2 ∶ | ( ) | ≤ ̄ }.

If

∈ ( ̄ ), then for any diagonal positive definite matrix ∈ ℝ × , the following inequality is verified

Ψ( ̂ ) ⊤ (Ψ( ̂ ) -( ( ) -( )) -( ) , -, ) ≤ .

Emulation-based approach

In this section, we provide a solution to Problem 1. In this case, we assume that the dynamic output-feedback controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], which can regionally stabilize the system (1) in the absence of communication networks, is available and we design the parameters of the event-triggering rules [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] and ( 8) that minimize the update rate on both channels. Theorem 1. Consider the LPV system [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for the  2 -stabilization of linear systems[END_REF] in closed-loop with the dynamic output-feedback controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], where the matrices , , , , and of the controller are given. Suppose that there exist symmetric positive definite matrices [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF] (provided at the top of the next page) and the following LMI condition are feasible,

∈ ℝ 2 ×2 , Δ , ̂ ∈ ℝ × , Δ , ̂ ∈ ℝ × , a positive definite diagonal matrix ∈ ℝ × , matrices ∈ ℝ 2 ×2 and ∈ ℝ ×2 , with ∈ [1, ] and ∈ [ , ], such that
+ ⊤ - ⋆ ( ) ̄ 2 ( ) > , ∈ [1, ], ∈ [1, ]. (16)
Then, the closed-loop system (9) subject to the ETMs ( 7) and [START_REF] Ge | Dynamic event-triggered distributed coordination control and its applications: A survey of trends and techniques[END_REF] with matrices Δ , = ̂ -1 , Δ and = ̂ -1 is regionally asymptotically stable and has a reduced number of data transmissions on the sensor-to-controller and the controller-to-actuator channels. Moreover, the region   , computed in ( 13)- [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], is an estimate of the region of attraction of the origin for the closed-loop system.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 ( + - - ) - -1 2 ( + ) 2 ⋆ ⋆ ⋆ 1 2 1 2 ( + ) 1 2 -1 2 ( + ) ̂ ⋆ ⋆ 1 2 ( + ) 1 2 ( + ) ̂ ⋆ ℂ ̂ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ > , , ∈ [1, ], ∈ [ , ], (15)
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⋆ ⋆ ( ) ( ) ( ) -( ) ( +1 ) ⋆ ⋆ ( ) ( ) ̂ ⋆ ℂ ̂ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ > . ( 19
)
Proof 1. By supposing the feasibility of ( 16), multiply its left-hand side by ( ) and sum it up for

∈ [1, ]. Then, ( ) by ( ) , use the fact that [ ( ) -] ⊤ - × -1 ( )[ ( )-] ≥ or equivalently ⊤ + -( ) ≤
⊤ -1 ( ) , and pre-and post-multiply the resulting inequality by the matrix { -⊤ , 1}, to obtain

-1 ( ) ⋆ ( ) ( ) ̄ 2 ( ) > . ( 17
)
Finally, apply Schur complement and pre-and post-multiply the resulting inequality by ⊤ and , respectively, to obtain

-⊤ ( ) -1 + ⊤ ( ) ⊤ ( ) ( ̄ 2 ( ) ) -1 ( ) ( ) ≤ 0, ( 18
)
which ensures ( ) = ⊤ -1 ( ) ≤ 1, and | ( ) | ≤ ̄ , and consequently,   ⊆ ( ̄ ). Thus, any trajectory of the closed-loop system belonging to   belongs also to ( ̄ ). Therefore, the feasibility of ( 16) implies that the region   is included in ( ̄ ), and consequently, Lemma 1 applies. Moreover, by supposing the feasibility of ( 15), we have from block [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for the  2 -stabilization of linear systems[END_REF][START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for the  2 -stabilization of linear systems[END_REF] that is non-singular. So, first multiply the left-hand side of (15) by +1( ) , ( ) and ( ) , and sum it up for , ∈ [1, ] and ∈ [1, ]. Then, replace ( ) by ( ) and use the fact that [ ( ) -] ⊤ -1 ( )[ ( ) -] ≥ or equivalently ⊤ + -( ) ≤ ⊤ -1 ( ) . Next, pre-and post-multiply the resulting inequality by the matrix { -⊤ , , , -1 , 2 , , } and its transpose, respectively, to obtain the inequality [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF]

(provided at the top of the next page).

After that, apply Schur complement, pre-and post-multiply the resulting inequality by the augmented vector

⊤ = ⊤ ⊤ , ⊤ ,
Ψ( ̂ ) ⊤ and , respectively, and replace [START_REF] Golabi | Event-triggered control for discrete-time linear parametervarying systems[END_REF], to obtain

( ) + ( ) , + ( ) , -( )Ψ( ̂ ) by +1 ac- cording to
⊤ +1 -1 ( +1 ) +1 -⊤ -1 ( ) -2Ψ( ̂ ) ⊤ Ψ( ̂ ) -( ( )-( )) -( ) , -, -, Δ , + -, Δ , + ≤ 0 (20)
Finally, assume that ⊤ +1 -1 ( +1 ) +1 -⊤ -1 ( ) is equivalent to ( +1 ) -( ) = Δ ( ), and denote -1 = , ̂ -1 = , and ̂ -1 = , to get

Δ ( ) < 2Ψ( ̂ ) ⊤ Ψ( ̂ ) -( ( ) -( )) -( ) , -, < ⊤ , Δ , -⊤ + ⊤ , Δ , -⊤ ≤ 0. ( 21
)
Hence, the feasibility of ( 15) ensures the positivity of the function given in [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] and the negativity of Δ ( ). Also, by inequalities [START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF] and [START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF], we have that the event-triggering conditions [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] and ( 8) are always satisfied, respectively.

Therefore, by Lyapunov theory, the regional stability of closed-loop system [START_REF] Golabi | Event-triggered control for discrete-time linear parametervarying systems[END_REF] under the event-triggering mechanisms [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] and ( 8) is ensured whenever the state trajectories evolve inside the estimated attraction region   , computed as in ( 13)- [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], and the sensor-to-controller and controllerto-actuator channels will have a reduced data transmission rate, whenever [START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF] and [START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF] are verified.

Remark 2. Theorem 1 can be adapted to treat particular cases usually found in the literature, in which there is an event generator in only one of the communication channels (see, for example, [START_REF] Zhang | Event-triggered dynamic output feedback control for networked control systems[END_REF][START_REF] De Souza | Eventtriggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF][START_REF] De Souza | Codesign of an event-triggered dynamic output feedback controller for discrete-time LPV systems with constraints[END_REF][START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]). To consider an event generator only in the channel between the sensor and the controller, it is necessary to delete the second and the sixth lines and columns of the LMI [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF]. On the other hand, to admit an event generator only in channel between the controller and the sensor, we have to delete the third and the seventh lines and columns of the LMI [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF].

Co-design approach

In the previous section, the dynamic output-feedback controller is supposed to be known, and capable to regionally stabilize the system (1) without communication networks. Therefore, only the event-triggering mechanisms are designed by Theorem 1. The disadvantage is that the control performance of the closed-loop system may be constrained by the previously selected controller. To overcome such a restriction, a co-design approach of the dynamic controller and the event-triggering mechanisms is proposed in this section, thus providing a solution to Problem 2.

Let us start by introducing some matrices useful to the developments. Thus, inspired by [START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF], we use matrices , , and, ∈ ℝ × to define

= • • , -1 = • • , Φ = , (22)
which yield

Φ = and ̂ = Φ ⊤ Φ = ⊤ ⊤ , (23)
where, by construction, we have

⊤ = ⊤ + ⊤ . ( 24
)
By partitioning matrix = 11 ⋆ 21 22

, one obtains:

̂ = Φ ⊤ Φ = ̂ 11 ⋆ ̂ 21 ̂ 22 , (25)
with

̂ 11 = ⊤ 11 + ⊤ ⊤ 12 + ⊤ 12 + ⊤ 22 , ̂ 21 = ⊤
11 + 12 , and ̂ 22 = 11 . With the aid of matrices in ( 22)-( 25), we can provide a solution to Problem 2 through the next theorem.

Theorem 2. Consider there exist symmetric positive definite matrices

̂ ∈ ℝ 2 ×2 , Δ , ̂ ∈ ℝ × , Δ , ̂ ∈ ℝ × ,
a positive definite diagonal matrix ∈ ℝ × , and matrices , , , ̂ , ̂ , ̂ , ̂ , and ̂ of appropriate dimensions, with ∈ [1, ] and ∈ [ , ], such that (26) (given at the top of this page) and the following LMI conditions are feasible,

̂ + ̂ ⊤ -̂ ⋆ ( ) ̄ 2 ( ) > , ∈ [1, ], ∈ [1, ], (27)
with

Ξ 1 = ( ̂ + ̂ ) ̂ + ̂ , Ξ 2 = ⊤ ( + ) + ̂ + + ( ̂ + ̂ ) ̂ ( + ) + ( ̂ + ̂ ) , Ξ 3 = ⊤ ( + ) + , Ξ 4 = ̂ ̂ + ̂ , Ξ 5 = -( ̂ + ̂ ) -( + ) , and ̂ = ⊤ ⊤ .
Then, by choosing non-singular matrices and such that ( 24) holds, we have that the saturated LPV system (1) under the dynamic output-feedback compensator [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] with matrices defined by

= ⊤ ⊤ ( + ) -1 × ̂ -⊤ ( + ) ̂ ̂ ̂ -1 = ( -1 ) ⊤ ( ̂ -1 -⊤ ), (28)
subject to the ETMs ( 7) and ( 8) with matrices Δ , = ̂ -1 , Δ and = ̂ -1 is regionally asymptotically stable and has a reduced number of data transmissions on the sensor-to-controller and controller-to-actuator channels. Moreover, the region   , computed in ( 13)- [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], is an estimate of the region of attraction of the origin for the closed-loop system.

Proof 2. By supposing the feasibility of [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], from block (1,1), we have that ̂ + ̂ ⊤ > 0, and consequently, ̂ is nonsingular. In view of [START_REF] Ramezanifar | Sampled-data filtering for linear parameter varying systems[END_REF], and are also non-singular, and by rewritten ̂ as

⊤ ⊤ = ⊤ ⊤ - , (29)
we can also verify the non-singularity of ( ⊤ -⊤ ). As a result, it is always possible to choose non-singular matrices and , such that [START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF] is verified. This shows that the gains (28) are well-defined.

Furthermore, consider the matrices ( 22)-( 25) and the following change of variables

̂ ̂ ̂ ̂ = ⊤ ⊤ ( + ) × + ⊤ ( + ) , ̂ = ⊤ + ⊤ , ( 30
)
by pre-and post-multiplying ( 26) by {Φ -⊤ , , , Φ -⊤ , , } and its transpose, respectively, one gets [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF]. Similarly, by pre-and post-multiplying [START_REF] Tanwani | Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization[END_REF] by

{Φ -⊤ , 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ̂ + ̂ ⊤ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ -1 1 2 ( + -Ξ 1 ) - -1 2 ( ̂ + ̂ ) 2 ⋆ ⋆ ⋆ 1 2 Ξ 2 1 2 Ξ 3 1 2 Ξ 4 1 2 Ξ 5 ̂ ⋆ ⋆ 1 2 Ξ 1 1 2 ( ̂ + ̂ ) ̂ ⋆ ̂ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ > , , ∈ [1, ], ∈ [ , ] (26)
and its transpose, respectively, one obtains [START_REF] Li | Co-design of event-triggered  ∞ control for discrete-time linear parameter-varying systems with networkinduced delays[END_REF]. Thus, as in the proof of Theorem 1, these two equivalences allow to conclude the proof.

Remark 3. The design of the dynamic controller ( 7) through [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] in Theorem 2 imposes, for given , and , to compute non-singular matrices and satisfying [START_REF] Scherer | Multiobjective outputfeedback control via LMI optimization[END_REF] or, equivalently, ⊤ = ⊤ -⊤ . However, the choice of these matrices can be performed in different ways, for example, we can set = , for any given scalar , and compute = ( ⊤ -⊤ ) -1 , or even use any matrix decomposition, such as LU and QR factorizations, to determine them. Remark 4. Theorems 1 and 2 can be simplified to deal with LTI and non-saturated systems. In the LTI case, it is required to set = = = 1, which results in fixed matrices. Notice that, the dynamic and input matrices of the controller, and , are retrieved by setting 11 = 0.5 11 and = 0.5 11 , according to Assumption 1, with 11 and 11 calculated as in [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF]. In the non-saturated case, the third line and column of the LMIs ( 15) and ( 26) must be deleted, and the LMIs ( 16) and ( 27) discarded.

Optimization procedures

The main objective here is to reduce the number of data transmissions on the sensor-to-controller and the controllerto-actuator channels. Let us remark that if

̂ -1 - ⊤ ̂ -1 - max ( Δ ) -⊤ min ( ) ≤ 0, ̂ -1 - ⊤ ̂ -1 - max ( Δ ) -⊤ min ( ) ≤ 0,
then, the triggering conditions ( 7) and ( 8) do not hold, avoiding data transmission. Note that, in the worst case, that is, when the conditions become equalities, one gets:

̂ -1 - ⊤ ̂ -1 - max ( Δ ) ⊤ min ( ) ≤ 1 and ̂ -1 - ⊤ ̂ -1 - max ( Δ ) ⊤ min ( ) ≤ 1, (31)
respectively. Since min ( ) -1 = max ( -1 ) and min ( ) -1 = max ( -1 ) , we can rewrite [START_REF] Verdult | Nonlinear system identification: State-space approach[END_REF] as

̂ -1 - ⊤ ̂ -1 - ⊤ ( Δ , -1 ) ≤ 1 and ̂ -1 - ⊤ ̂ -1 - ⊤ ( Δ , -1 ) ≤ 1, (32)
respectively, with ( Δ , -1 ) = max ( Δ ) max ( -1 ) and ( Δ , -1 ) = max ( Δ ) max ( -1 ). Thus, the idea is to minimize ( Δ , ̂ ) and ( Δ , ̂ ), with ̂ = -1 and ̂ = -1 , so that the minimum time required for the expressions on the left hand-side of ( 31) to evolve from 0 to 1 is enlarged. However, ( Δ , ̂ ), and ( Δ , ̂ ) are not convex functions and therefore it can be difficult to optimize them. Nevertheless, one can observe that the eventtriggering functions depend on all the eigenvalues of Δ , ̂ , Δ and ̂ . So, to formulate a convex objective function, we can minimize the sum of all eigenvalues of Δ , ̂ , Δ and ̂ , which leads to the following convex optimization procedure: [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF] and ( 16), or ( 26) and ( 27). [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] Let us point out that the data transmission is indirectly reduced by means of the optimization procedure  1 .

 1 ∶ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ min ( Δ + ̂ ) + ( Δ + ̂ ), subject to ⎧ ⎪ ⎨ ⎪ ⎩ (
Another objective of optimization consists in considering a given region of admissible initial states  0 for which we can reduce the update rate on the sensor-to-controller and the controller-to-sensor channels. In this case, we should ensure that  0 is included in the region of attraction of the closed-loop systems, i.e.  0 ⊆   ⊆   . If  0 is specified as an ellipsoid ( , 1), defined similarly to ( 14), then we have that

⋆ 2 > , or equivalently (34a) ⋆ Φ ⊤ ̂ > , ( 34b
)
with Φ given in [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], for all ∈ [1, ]. However, the LMI (34b) is non-convex due to the presence of the matrix in Φ. To make it convex, we can consider the partitioning

= 11 ⋆ 21 22
and ,0 = , which allows us to dismiss the rows concerning the position of in Φ. With that, the inequality (34b) can be rewritten as

⎡ ⎢ ⎢ ⎣ 11 ⋆ ̂ ⎤ ⎥ ⎥ ⎦ > , ( 35
)
for all ∈ [1, ]. Thus, we have

 2 ∶ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ min ( Δ + ̂ ) + ( Δ + ̂ ), subject to ⎧ ⎪ ⎨ ⎪ ⎩ (15)
, ( 16) and (34a), or ( 26), ( 27) and ( 35), [START_REF] Zhang | Input-to-state stabilization of nonlinear discrete-time systems with event-triggered controllers[END_REF] with ̂ = -1 and ̂ = -1 .

Although both optimization procedures aim at minimizing the data transmission, the optimization procedure  2 differs from the optimization procedure  1 by the inclusion of the restrictions (34a) and [START_REF] Yue | A delay system method for designing event-triggered controllers of networked control systems[END_REF] to take into account a specific region of initial condition. It is important to point out that, the use of the optimization procedure  2 leads to deal with a classical trade-off between the size of the estimate of the basin of attraction and the transmission saving. Indeed, it results that the smaller the estimate of the basin of attraction, the greater the transmission saving.

Simulation results

In this section, some examples addressing both the LPV and the LTI cases, with and without saturating actuators, are explored. First, we present two examples relating to LTI systems and then a third example concerning an LPV system with saturating actuators.

Example 1 Consider the following discretized version, with sampling time = 0.05 seconds, of the system investigated in [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF].

, +1 = 1 0.05 0.1 0.85 , + 0.11 0.11 ̂ , = -1 4 , . (37)
Our objective here is to compare our co-design and emulation proposals with the one in [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF]. The authors in [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF] address the co-design event-triggered dynamic output-feedback control problem for continuous linear time-invariant (LTI) system. The two independent ETMs are based on a condition that depends on the plant output and the controller output taken at different times. The results obtained by [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF] are presented in Table 1, where the initial conditions ,0 = ,0 = 40 -20

⊤ were taken to simulate the closed-loop response of the system. Observe that, in the first channel, [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF] got an average sampling time that corresponds to 3 times the sampling time of the system without ETM; and, in the second channel, the average sampling time found corresponds to 3.6 times the sampling time of the system without ETM. First, to compare the co-design approach, we solve the optimization procedure  1 given in [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] By simulating the closed-loop response of the system for the same initial conditions, we got the average sampling times presented in Table 1. Note that, in the first channel, we obtained an average sampling time that corresponds to almost 5 times the sampling time of the system without ETM, and in the second channel, the average sampling time found corresponds to 7.5 times the sampling time of the system without ETM. Therefore, we increased the average sampling in 53.87% on the sensor-to-controller channel, and 108.33% on controller-to-actuator channel, with respect to [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF].

Then, to compare the emulation-approach, we design the ETMs ( 7) and ( 8) for both the controller obtained in the co-design and the one obtained by [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF] using the optimization procedure  1 given in [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] with conditions of Theorem 1. For the first case, we have found the same results, and for the second, we got the ETM matrices Δ = 1.8588, = 0.2646, Δ = 3.7788 and = 0.5380. By using these matrices to simulate the closed-loop response of the system, we find the average sampling times presented in Table 1. Thus, using our controller, the sampling average in the control ETM improved 50% in relation to the controller of [START_REF] Ma | Co-design of event generator and dynamic output feedback controller for LTI systems[END_REF].

Example 2 Consider the following discretized version, with sampling time = 0.05 seconds, of the system investigated in [START_REF] Liu | Robust event-triggered control for networked control systems[END_REF] , +1 = 1 0.05 -0.25

1 , + 0 0.05 ̂ , = 1 0 , . (38)
Our objective here is to compare our co-design approach with the one in [START_REF] Liu | Robust event-triggered control for networked control systems[END_REF], where a co-design of a dynamic outputfeedback controllers and two independent ETMs for continuous linear time-invariant system with communication delays are proposed. The results obtained by [START_REF] Liu | Robust event-triggered control for networked control systems[END_REF] are showed in Table 2, where the initial states condition ,0 = 1 0.2 ⊤ and ,0 = 1,2 were taken to simulated the closed-loop response. By taking into account the procedures described in Remark 4 to deal with LTI ( = 0) and non-saturated system, we run the optimization procedure  1 given in [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] For the same initial conditions, we simulated the response of the closed-loop system and got the transmission rates showed in Table 2. In this case, we reduced the update rate in 6.15% and 5.23% on the sensor-to-controller and controller-to-actuator channels, respectively, in relation to [START_REF] Liu | Robust event-triggered control for networked control systems[END_REF].

Example 3 Consider the inverted pendulum shown in Figure 2. This system has been extensively investigated in the literature (see, for example, [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF][START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF][START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]), but without taking into account possible variations in the system parameters. Let us then consider such variations by adding the parametervarying to the system model, as follows arbitrary variation was added just to test our approach, and, therefore, has no direct physical meaning with the continuoustime system's variables. Our objective here is to made the co-design of the dynamic controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] and the two ETMs ( 7) and ( 8), for two different given admissible initial conditions region  0 , such that the number of data transmissions in both channels is as low as possible.

For the first case, let us consider a given region of admissible initial conditions  0 = ( , 1) with 11 = {76, 2} derived from the partitioning of . By using the optimization procedure  2 with conditions of Theorem 2, we design simultaneously the dynamic controller [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] and the two independent ETMs [START_REF] Eqtami | Eventtriggered control for discrete-time systems[END_REF] and ( 8) such that the update on the sensorto-controller and the controller-to-actuator channels are minimized. Through this, we got the ETM matrices Δ = 18.8382, = 0.2365, Δ = 3.6281, and = 0.0516 and the following dynamic controller matrices Figure 3 shows the projection (-) and the cut (-), on the plane defined by the plant states, of the   obtained,  0 (-), and also the projections of some convergent (--) and some divergent trajectories (--) starting from the points marked with • and * , respectively. Notice that, as required,   contains  0 , i.e.  0 ⊂   . In particular, for the convergent trajectory (--) starting in the initial condition 0 = -0.1480 -0.4735 0 0 ⊤ marked with '•', we plot in the Figure 4 the states, the control input, the events of the sensor and the controller, and the parameter-varying as a function of the sampling instants. In the inter-events graph, the events that occur asynchronously in the sensor and in the controller are represented by '•' and '•', respectively, and synchronous by '•'. Thus, we can see the asymptotic stability of the system despite the saturation in the first instants of the simulation. For this case, the update rate between the sensor and controller and between the controller and actuator was 50.33% and 43%, respectively, thus, saving a significant amount of samples to be transmitted. However, the inclusion of  0 yielded an ETM behavior that appears to have some periodicity despite the asynchronous updates of sensor and control ETMs. Moreover, the asynchronous ETMs save transmissions because only one ETM is active over the network.

Then, for the second case, we carried out the co-design for a region of admissible initial conditions less stringent, given by  0 = ( , 1) with 11 = {26.60, 0.70} derived from the partitioning of . For this case, we obtained the ETM matrices Δ = 29.5663, = 0.0881, Δ = 5.8971, and = 0.0221 and the dynamic controller matri- Figure 5 presents the projection and the cut, on the plane defined by the plant states, of the   obtained. For the convergent trajectory (--) starting in the initial condition 0 = -0.2117 -0.3245 0 0 ⊤ marked with "•", we simulated the closed-loop response of the system, and the results can be seen in the Figure 6. In this case, the update rates between the sensor and controller and between the controller and actuator found were 67.33% and 59.67%, respectively. Therefore, in relation to the update rate, there was a slight worse performance than the more restrictive (on the plane defined by the plant states)  0 specification. The ETMs seem to present a more pronounced periodic behavior in this case, which may be connected to the higher transmission rate achieved due to the inclusion of a larger region of initial condition considered here (w.r.t. that one in the previous case).

Another effect of including a larger  0 is the reduction on the asynchronous transmission, supporting the hypothesis of the bigger the region of initial conditions, the smaller the transmissions saving. The inverted pendulum is also investigated in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], where the design of event-triggering static and dynamic state stabilizing controllers for discrete-time linear systems with urating actuators is addressed. The number of updates obtained by [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] are showed in Table 3, where the initial conditions ,0 = 0.2 0.8

⊤ and ,0 = 0 0 ⊤ were taken to simulate the closed-loop response of the system. Observe that since [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] does not consider a communication network between the controller and actuator channel, then the system updates the control at all sampling times. As the authors in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] explore the LTI case with saturation, to compare our approach with theirs, we fix = 0, and set = 2 . Also, we consider a region of admissible initial conditions  0 = ( , 1) with 11 = 11.9987 0.2318 0.2318 0.5873 , which is contained in the region of attraction estimated by [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]. Thus, for the co-design, we run the optimization proce- For the same initial condition, we simulate the closedloop response of the system, and found the updates rates showed in Table 3. Although, in the first channel, the update rates obtained by [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] with Theorem 3.1 and Theorem 4.1 are 21.18% and 17.65% smaller than ours, respectively, in the second channel, they are in both cases 331.03% higher than ours.

Conclusion

In this paper, the dynamic event-triggered control problem was investigated for a discrete-time LPV system subject to actuator saturation. The measured output and the control input are transmitted based on two independent eventtriggering schemes. Both emulation-based approach and codesign of the event-generators parameters and the controller matrices were proposed. The convex conditions in form of linear matrix inequalities (LMIs) ensured the regional asymptotic stability of the closed-loop system for every initial condition belonging to the estimated attraction region. Some optimization procedures were also formulated allowing the minimization of the data transmission on the sensor-to-control and control-to-actuator channels. As future work, it would be interesting to study if the two event-triggering mechanisms ( 7) and ( 8) could be dependent (that is, each ETM could use information from the other transmitted signal).

Figure 1 :

 1 Figure 1: Event-triggering closed-loop system.

  with conditions of Theorem 2, and obtain the ETM matrices Δ = 1.5489, = 0.7091, Δ = 1.4102, and = 0.6456 and the following dynamic controller matrices 1654 0.1180 , and = -1.2314.

  , , ̄ = 1 and | | ≤ 1. Note that, this

Figure 2 :

 2 Figure2: An inverted pendulum[START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF]

Figure 3 :Figure 4 :

 34 Figure 3:   and  0 = ( , 1) with 11 = {76, 2}.

Figure 5 :

 5 Figure 5:   and  0 = ( , 1) with 11 = {26.60, 0.70}.

Figure 6 :

 6 Figure 6: The closed-loop response of system (39) - 0 = ( , 1) with 11 = {26.60, 0.70}.

Table 1

 1 Comparison of the average sampling time -Example[START_REF] Zhang | Event-triggered dynamic output feedback control for networked control systems[END_REF].

	Design	average sampling time [sec]
	method	output	control
	Theorem 2 in [19]	0.15	0.18
	Theorem 1	0.2303	0.25
	Theorem 2	0.2303	0.375

Table 2

 2 Comparison of the updates rates -Example[START_REF] Zuo | Co-design of eventtriggered control for discrete-time systems with actuator saturation[END_REF].

	Design	Updates rates (%)
	Method	output	control
	[18, Th. 2] 34.50%	38.25%
	Theorem 2 30.50%	36.25%

Table 3

 3 Comparison of the number of samplings -Example (39).

				Design				update
				method			output control
		Theorem 3.1 in [6]		67	1000
		Theorem 4.1 in [6]		70	1000
			Theorem 2			85	232
	dure  2 with conditions of Theorem 2, and got the following
	ETM matrices
		Δ =	1.2955 0.9134 0.9134 0.6441	,	=	0.0758 0.0106 0.0106 0.0425	,
		Δ = 1.5369, and		= 0.0261,
	and the following dynamic controller matrices
	=	-6.3069 -29.0632 1.5869 7.3118	,		=	-0.0264 -0.0186 0.0067 0.0047	,
	=	-0.1133 0.0288	,	=	-6.6853 -3.1688	⊤ , and	=	1.3583 0.9577

⊤

.

+ ⊤ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ -1 2 ( + ) Δ ⋆ ⋆ ⋆ ⋆ ⋆ Δ ⋆ ⋆ ⋆ ⋆

-1 ( ) ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ Δ ⋆ ⋆ ⋆ ⋆ ⋆ Δ ⋆ ⋆ ⋆ ⋆ --1 ( ( ) -( )) ---1 ( ) 2 -1 ⋆

( ̂ + ̂ ) Δ ⋆ ⋆ ⋆ ⋆ ⋆ Δ ⋆ ⋆ ⋆ ⋆

⋆ This work has been supported by the Brazilian Agencies CAPES under the project Print CAPES-UFSC "Automation 4.0", CNPq (311208/2019-3 and 306927/2017-9); and by ANR under the project HANDY number 18-CE40-0010.