Carla De Souza

Valter Sophie Tarbouriech
email: sophie.tarbouriech@laas.fr

J S Leite

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

Valter J S Leite
email: valter@ieee.org

Co-design of an event-triggered dynamic output feedback controller for discrete-time LPV systems with constraints

Keywords: Linear parameter-varying systems, Event-triggered control, Saturation, Dynamic controller, Regional stability

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Introduction

In recent years, event-triggered control (ETC) has gained increasing interest due to its potential to reduce the usage of the communication and computational resources of control systems, which is quite important in communication networks with limited bandwidth and battery-powered wireless devices [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]. The main idea of event-triggering techniques consists of performing control tasks after the occurrence of an event, generated by some well-designed event-triggering mechanism, rather than the elapse of a certain fixed time interval, as in traditional sampled-data control. Consequently, ETC is capable of reducing the control tasks execution while guaranteeing stability and some performance index of the closed-loop system. Moreover, dynamic controllers can perform their tasks with reduced information about the controlled process. Two general categories allow the classification of the existing approaches, namely, emulation based [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF] and co-design based approaches [START_REF] Peng | Event-triggered communication and H ∞ control codesign for networked control systems[END_REF][START_REF] Li | Co-design of event-triggered H ∞ control for discrete-time linear parameter-varying systems with network-induced delays[END_REF][START_REF] Tarbouriech | Observerbased event-triggered control co-design for linear systems[END_REF]. In the context of the emulation-based approach, the design concerns only the controller or the event-triggering condition while the other part is given. In the co-design approach, both parts, the controller and the event-triggering conditions, are simultaneously designed.

Note that the co-design may lead to better closed-loop performance and more general and more flexible design conditions. Various ETC strategies can be found in the literature, see, for instance, [START_REF] Lunze | A state-feedback approach to event-based control[END_REF][START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF][START_REF] Wu | Suboptimal event-triggered control for time-delayed linear systems[END_REF][START_REF] Zhang | Event-triggered dynamic output feedback control for networked control systems[END_REF][START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback controllers[END_REF][START_REF] Wu | Event-triggered control for discretetime linear systems subject to bounded disturbance[END_REF][START_REF] Khashooei | Output-based event-triggered control with performance guarantees[END_REF][START_REF] Shen | Event-triggered output feedback H ∞ control for networked control systems[END_REF]. However, most previous works handle only linear time-invariant (LTI) and nonlinear systems, without considering both varying parameters and control constraints, such as saturating actuators.

LPV models concern a significant class of linear systems whose dynamics depends on a prior unknown but on-line measurable time-varying parameters [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF]. Due to their effectiveness in modeling and control time-varying and nonlinear systems, LPV models have been extensively studied in the literature [START_REF] Mohammadpour | Control of linear parameter varying systems with applications[END_REF][START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering & Control[END_REF]. However, the simultaneous design of both the event generator and feedback controller, i.e., the co-design, for LPV systems, has been little explored, mainly in the discrete-time context. In [START_REF] Li | Co-design of event-triggered H ∞ control for discrete-time linear parameter-varying systems with network-induced delays[END_REF], the authors propose an H ∞ eventtriggered control by jointly designing a mixed event-triggering mechanism and state-feedback controllers for discrete-time LPV systems under network-induced delays. A parameter-dependent state-feedback controller and an event generator are co-designed in [START_REF] Li | Event-triggered control for discrete-time uncertain linear parameter-varying systems[END_REF][START_REF] Golabi | Eventtriggered control for discrete-time linear parameter-varying systems[END_REF], stabilizing the LPV closed-loop system. Authors in [START_REF] Xie | Output-based event-triggered control for networked control systems: tradeoffs between resource utilisation and robustness[END_REF] propose a condition for the co-design of a mixed event generator and a parameter-dependent static output-feedback controller for LPV discrete-time systems.

Another important feature when dealing with stability analysis and control design is the presence of saturating actuators. Such a nonlinearity may cause performance degradation or even unstable behavior (see [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF] and references therein). Recently, the research on ETC has been extended to consider saturating actuators, both in the continuous-time setting [START_REF] Ni | Event-triggered control of linear systems with saturated inputs[END_REF][START_REF] Kiener | Actuator saturation and antiwindup compensation in event-triggered control[END_REF][START_REF] Seuret | LQ-based eventtriggered controller co-design for saturated linear systems[END_REF][START_REF] Liu | Event-triggered control for linear systems with actuator saturation and disturbances[END_REF][START_REF] Zuo | Dynamic event-triggered and selftriggered control for saturated systems with anti-windup compensation[END_REF][START_REF] Li | Event-based dynamic output-feedback controller design for networked control systems with sensor and actuator saturations[END_REF][START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] and in the discrete-time one [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF][START_REF] Zuo | Co-design of event-triggered control for discrete-time systems with actuator saturation[END_REF][START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF][START_REF] Ma | Event-triggered feedback control for discrete-time piecewise affine systems subject to input saturation[END_REF][START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]. Authors in [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] propose a procedure to design a state-feedback controller maximizing the region of attraction of a discretized system under saturating actuators for a given event-triggering condition. Another recent approach to minimize the region of attraction concerns discrete-time piecewise affine saturated systems [START_REF] Ma | Event-triggered feedback control for discrete-time piecewise affine systems subject to input saturation[END_REF]. However, the co-design is not addressed in these cases, which may lead to conservative results. The main difficulty in obtaining co-design methods lies in the nonlinear relations among the optimization variables involved. To overcome such an issue, authors in [START_REF] Zuo | Co-design of event-triggered control for discrete-time systems with actuator saturation[END_REF] suggest a cone complementarity linearization algorithm for solving the non-convex optimization problem yielding a method to design both an eventtriggering strategy and a state-feedback controller for LTI systems under saturating actuators. In [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], by using similarity transformations, other methods to the simultaneous design of static state-feedback gain and an event-triggering condition are found, ensuring the regional stability of saturated LTI systems.

Additionally, [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] also considers the co-design based on a dynamic state stabilizing controller.

Note that the discretization of continuous-time systems under time-varying parameters or variable sampling time may not be straightforward as discussed, for instance, in [START_REF] Braga | Discretization and discrete-time output feedback control of linear parameter varying continuous-time systems[END_REF][START_REF] Tóth | Discretisation of linear parameter-varying state-space representations[END_REF]. More sophisticated approaches may use sampleddata control handling the process changes between consecutive aperiodic samplings [START_REF] Gomes Da Silva Jr | Sampled-data lpv control: a looped functional approach[END_REF][START_REF] Palmeira | Aperiodic sampleddata control for LPV systems under input saturation[END_REF]. On the other hand, conventional discretization methods under periodic and small enough sampling may lead to nice discrete-time modelapproximations, where the discretization error is negligible [START_REF] Tóth | Crucial aspects of zeroorder hold LPV state-space system discretization[END_REF]. Additionally, as discussed in [START_REF] Emedi | Fixed-structure LPV discrete-time controller design with induced 2 -norm and H 2 performance[END_REF], LPV models obtained from identification methods are usually given in the discrete-time framework [START_REF] Verdult | Nonlinear system identification: State-space approach[END_REF][START_REF] Cerone | Input-output LPV model identification with guaranteed quadratic stability[END_REF]. Moreover, for processes that are naturally discrete on time, the discretization issues vanish. Since the focus of the current work relies on the ETM co-design for saturating LPV systems, the discretization procedure is not addressed here. Thus, we assume that the considered system in what follows is an already discretized model with time-varying parameters belonging to a polytopic set. Despite more involving techniques that combine ETM co-design with sampled-data control but without LPV characteristic, see for instance [START_REF] Donkers | Output-based event-triggered control with guaranteed L ∞ -gain and improved event-triggering[END_REF][START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], our approach provides useful achievements that overcome other similar conditions from the literature by an enhanced ETM proposal.

The simultaneous design of the event generator and the dynamic output feedback controller remains an open issue for LPV systems under saturating actuators. Therefore, using discrete-time LPV models, the contribution of this paper aims at providing some bricks to address this issue:

1. A convex procedure to design both a parameter-dependent dynamic outputfeedback controller with anti-windup action and an event-triggering condition;

2. The provided methodology allows the co-design considering optimization problems aiming at reducing the transmission activity, optimizing a certain level of performance, or maximizing the estimation of the basin of attraction of the origin;

3. The convex methodology can be simplified to design only an event-triggering condition for a given parameter-dependent dynamic output-feedback controller with anti-windup action;

To derive the convex formulation, the Lyapunov theory is used conjointly with S-procedure and the generalized sector condition, yielding a set of linear matrix inequalities (LMIs) that, if feasible, ensures the regional asymptotic stability of the closed-loop system and provides an estimate of the region of attraction of the origin. Furthermore, some optimization problems can be associated with the LMI constraints to deal with triggering activity or the size of the region of closed-loop stability.

The paper is organized as follows. In Section 2, the class of systems under constraints is described and the problem we intended to solve is formally stated. Section 3 is dedicated to preliminary results useful to develop the main conditions. In Section 4, the main results addressing both the emulation and the co-design cases are developed. The optimization problems evoked previously are also derived. In Section 5, numerical examples illustrate the usefulness of the proposed conditions, where simulations and comparisons are established with the related literature. The achieved results suggest our approach leads to fewer updates in the sensor channel. Finally, conclusions are given in Section 6

Notation: The sets of real numbers and non-negative real numbers are denoted by R and R + , respectively. The set of integer numbers belonging to the interval from a ∈ N to b ∈ N, b ≥ a, is denoted by I[a, b]. R m×n is the set of matrices with real entries and dimensions m×n. A block-diagonal matrix A with blocks A 1 and A 2 is denoted as A = diag{A 1 , A 2 }. The transpose of a vector or matrix A is denoted by A ⊤ and its ℓ th line is indicated by A (ℓ) . The matrix 0 stands for the null matrix of appropriate dimensions and I n corresponds to the identity matrix with dimension n × n. The symbol ⋆ stands for symmetric blocks within a matrix, • represents an element that does not influence on developments.

Problem Formulation

Consider the discrete-time saturated LPV system

x(k + 1) = A(θ k )x(k) + B(θ k )sat(u(k)), y(k) = Cx(k), (1)
where x(k) ∈ R n is the state vector, y(k) ∈ R p is the measurable output, u(k) ∈ R m is the control signal and sat(u(k)) is a symmetric saturation function given by

sat(u (ℓ) (k)) = sign(u (ℓ) (k)) min(|u (ℓ) (k)|, ū(ℓ) ) (2)
where ū(ℓ) > 0 denotes the symmetric level relative to the ℓ th control input.

The vector of time-varying parameters θ k , 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 with N known vertices defined by

Θ N i=1 θ k(i) = 1, θ k(i) ≥ 0, i ∈ I[1, N ] . (3)
The parameter-dependent matrices A(θ k ) ∈ R n×n and B(θ k ) ∈ R n×m can be written in the polytopic form as

A(θ k ) B(θ k ) = N i=1 θ k(i) A i B i . (4)
Note that it is usual to keep the general formulation of (1) to develop general conditions and to focus on the polytopic formulation (4) to obtain tractable numerical conditions. However, due to the controller structure assumed in the 120 sequence, we directly use the polytopic formulation along the text, simplifying the developments.

To regionally stabilize the system (1), we propose the design of the following parameter-dependent dynamic output feedback compensator with anti-windup action:

x c (k + 1) = A c (θ k )x c (k) + B c (θ k )ŷ(k) -E c (θ k )Ψ(u(k)), u(k) = C c (θ k )x c (k) + D c (θ k )ŷ(k), (5)
where x c (k) ∈ R n is the controller state, Ψ(u(k)) : R m → R m is a dead-zone nonlinearity defined by Ψ(u(k)) = u(k)sat(u(k)), and ŷ(k) is the last output measure updated, which was sent by the event-triggering mechanism (ETM). It is worth to say that the matrix E c (θ k ) ∈ R n×m is introduced to mitigate the windup effect caused by the saturating actuators. Therefore, the anti-windup acts only when saturation occurs, i.e., whenever Ψ(u(k)) = 0. Following the standard approach for LPV systems, in this work the information concerning the scheduling parameter is assumed online available for the controller.

In this paper, we are interested by the following ETM:

ŷ(k) :=    y(k), if f (ŷ, y, u) > 0, ŷ(k -1), otherwise. (6)
where f (ŷ, y, u) is the triggering condition defined by

f (ŷ, y, u) := ŷ(k -1) -y(k) 2 Qe -y(k) 2 Qy -u(k) 2 Qu > 0 (7)
with symmetric positive definite matrices

Q e , Q y ∈ R p×p and Q u ∈ R m×m .
The interest in such a structure of ETM resides in the use of more information from the closed-loop behavior, beyond the frequent employment of only y and sometimes only u, to decide to transmit or not the signals. Note that, it has already been proved effective in the emulation-based approach as proposed in [START_REF] Souza | Emulationbased dynamic output-feedback control of saturating discrete-time LPV systems[END_REF].

By considering the ETM in [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], if ( 7) is satisfied at instant k, then ŷ(k) is update to y(k), because the error ŷ(k -1)y(k) 2 Qe is too big to guarantee the stability and certain performance index for the closed-loop system. On the other hand, if [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] is not satisfied at instant k, then ŷ(k) is not updated, because the error is small enough to guarantee the stability and certain performance index for the closed-loop system. In the latter case, ŷ(k) maintains its value from the previous instant, ŷ(k -1). The matrices Q e , Q y , and Q u are used here to weigh the terms associated with the triggering condition. The choice of matrices Q e , Q y , and Q u directly impacts the event-triggering policy and, thus, on how much the data transmission rate can be reduced. This work aims to design these matrices conjointly with the ones of the controller (5) to yield a better event-triggered control scheme. The event-triggered control system of interest is depicted in Figure 1.

Plant ETM sat(u(k)) umin umax Controller u(k) y(k) ŷ(k)
Additionally, let us consider the following assumption: Assumption 1. The matrices of the controller (5) are supposed to have the following structure:

A c (θ k ) B c (θ k ) = 0.5 N i=1 N j=i (1 + ς ij )θ k(i) θ k(j) A cij B cij , (8)
C c (θ k ) D c (θ k ) = N i=1 θ k(i) C ci D ci , and E c (θ k ) = N i=1 θ k(i) E ci , with θ k ∈ Θ and ς ij = 1 if i = j and ς ij = 0 otherwise.
Note that any dynamic controller given in a polytopic constructive form can be described according to Assumption 1 by using the fact that θ k ∈ Θ and the following equivalence:

N i=1 θ k(i) N i=1 θ k(i) M i = 0.5 N i=1 N j=i (1 + ς ij )θ k(i) θ k(j) M ij , with ς ij = 1 if i = j
and ς ij = 0 otherwise. However, the converse is not always possible because formulation ( 8) is more general than the polytopic one.

Since there is a saturation in the loop, the system's global stability may not be guaranteed [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF]. In this case, the regional (local) stability must be studied and the region of attraction of the origin, denoted R A , is designed in terms of the augmented state vector ξ

(k) = x(k) ⊤ x c (k) ⊤ ⊤ ∈ R 2n .
The region R A is the set of all initial conditions yielding closed-loop trajectories that converge to the origin. As the exact numerical characterization of R A is, generally, a hard task, it is important to determine estimates with a well-fitted analytical representation (see, for example, [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF] for more details). By denoting 165 R E the estimate of the region of attraction, then we are interested in computing

R E ⊂ R A as large as possible.
From this, we intend to investigate the following problem: Problem 1. For the discrete-time saturated LPV system (1), design both the dynamic output-feedback controller (5) under Assumption 1 and the event-triggering condition, f (ŷ, y, u), that ensures the regional asymptotic stability of the closedloop system while reducing the number of data transmissions between the sensor/plant and the controller.

Preliminary Results

The closed-loop system (1)-( 5) can be rewritten in a compact form as follows:

ξ(k + 1) = A(θ k )ξ(k) -B(θ k )Ψ(u(k)) + E(θ k )e(k), u(k) = K(θ k )ξ(k) + D c (θ k )e(k), y(k) = Cξ(k), (9)
where

ξ(k) = x(k) ⊤ x c (k) ⊤ ⊤ ∈ R 2n is the augmented state and e(k) = ŷ(k) -y(k) ∈ R p
is the output error. The parameter-varying matrices, which also verify from Assumption 1

A(θ k ) E(θ k ) = 0.5 N i=1 N j=i (1 + ς ij )θ k(i) θ k(j) A ij E ij , and B(θ k ) K(θ k ) ⊤ = N i=1 θ k(i) B i K ⊤ i ,
with θ k ∈ Θ and ς ij = 1 if i = j and ς ij = 0 otherwise, are given by

A ij =   A i + A j + (B i D cj + B j D ci )C B i C cj + B j C ci B cij C A cij   , B i =   B i E ci   E ij =   B i D cj + B j D ci B cij   , K i = D ci C C ci , and C = C 0 .
If ( 7) is satisfied at instant k, then we have from ( 6) that e(k) = ŷ(k)y(k) = y(k)y(k) = 0, and if [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] is not satisfied at instant k, then we have from ( 6)

that e(k) = ŷ(k) -y(k) = ŷ(k -1) -y(k). So, the following inequality e(k) 2 Qe ≤ y(k) 2 Qy + u(k) 2 Qu ( 10
)
is always satisfied.

175

To investigate the regional asymptotic stability of the closed-loop system (9), we use the Lyapunov theory with the following Lyapunov candidate function

V (k) = ξ(k) ⊤ P -1 (θ k )ξ(k), (11)
where 11) is a Lyapunov function, then the estimate of the region of attraction of the origin for the closed-loop system is computed as

P (θ k ) = N i=1 θ k(i) P i , with 0 < P i = P ⊤ i ∈ R 2n×2n and θ k ∈ Θ. If (
R E = θ k ∈Θ E(P (θ k ) -1 , 1) = i∈I[1,N ] E(P -1 i , 1) (12)
with

E(P -1 i , 1) = {ξ(k) ∈ R 2n : ξ(k) ⊤ P -1 i ξ(k) ≤ 1, ∀i ∈ I[1, N ]}. (13)
Moreover, to deal with the actuator saturation, we use the following lemma directly derived from [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF].

Lemma 1. Consider a matrix G(θ k ) = N i=1 θ k(i) G i with G i ∈ R m×2n for all I[1, N ] and θ k ∈ Θ. If ξ(k) belongs to the set S(ū) defined by S(ū) {ξ(k) ∈ R 2n : |G(θ k )ξ(k)| ≤ ū}, ( 14
)
then the nonlinearity Ψ(u(k)) satisfies the following inequality:

Ψ ⊤ (u(k))M (Ψ(u(k)) -(K(θ k ) -G(θ k ))ξ(k) -D c (θ k )e(k)) ≤ 0, ( 15
)
or equivalently,

Ψ ⊤ (u(k))M (-sat(u(k)) + G(θ k )ξ(k)) ≤ 0, ( 16
)
for any diagonal positive definite matrix M ∈ R m×m .

Main Results

In this section, the emulation case is first considered, consisting of designing the event-triggering rule f (ŷ, y, u) with a given dynamic output-feedback controller ( 5). This result is then extended to design both the event-triggering rule f (ŷ, y, u) and the dynamic output-feedback controller. Finally, three optimization procedures are proposed to match different control objectives.

Emulation case 185

The following result focuses on designing the event-triggering rule f (ŷ, y, u)

when the dynamic controller is assumed given.

Theorem 1. Given the matrices A cij , B cij , C ci , D ci
, and E ci of the compensator in (5), consider that there exist symmetric positive definite matrices

P i ∈ R 2n×2n
, Q e , Qy ∈ R p×p , and Qu ∈ R m×m , positive definite diagonal matrix S ∈ R m×m , and matrices U ∈ R 2n×2n and

H i ∈ R m×2n , with i ∈ I[1, N ], satisfying                     CU 0 0 0 Qy ⋆ 1 2 (K i + K j )U 1 2 (D ci + D cj ) 0 0 0 Qu                     Mrij,1 > 0, r, i ∈ I[1, N ]; j ∈ I[i, N ]; (17)
and

  U + U ⊤ -P i ⋆ H i(ℓ) ū(ℓ) 2   > 0, i ∈ I[1, N ], ℓ ∈ I[1, m]. (18)
Then, the closed-loop system (9) subject to the event-triggering condition ( 6)- [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] with matrices Q e , Q y = Q-1 y , and

Q u = Q-1
u is regionally asymptotically stable and has a reduced number of data transmissions between the sensor/plant 190 and the controller. Moreover, the region R E , defined in ( 12)-( 13), is an estimate of the region of attraction of the origin for the closed-loop system.

Proof 1. The proof of Theorem 1 is presented in Appendix A.

Co-design

Before presenting the conditions for the co-design of the event-triggering rule f (ŷ, y, u) and the dynamic output-feedback controller (5), we introduce some matrices that are useful in the development of the results. Based on the approach proposed by [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF], let us define the following matrices

U =   X • Z •   , U -1 =   Y • W •   , and Ω =   Y I n W 0   . ( 19
)
with X, Y , W and Z ∈ R n×n .

195 Therefore, we have

U Ω =   I n X 0 Z   and Û = Ω ⊤ U Ω =   Y ⊤ F ⊤ I n X   , (20)
where, by construction

F ⊤ = Y ⊤ X + W ⊤ Z. (21)
Furthermore, using the partitioning

P i =   P i11 P i12 ⋆ P i22   , we obtain Pi = Ω ⊤ P i Ω =   Pi11 Pi12 ⋆ Pi22   , (22)
with Pi11 =

Y ⊤ P i11 Y + W ⊤ P ⊤ i12 Y + Y ⊤ P i12 W + W ⊤ P i22 W , Pi12 = Y ⊤ P i11 + W ⊤ P ⊤ i12 and
                 C CX 0 0 0 Qy ⋆ 1 2 Π 1ij 1 2 ( Dci + Dcj ) 0 0 0 Qu                  Mrij,2 > 0, r, i ∈ I[1, N ], j ∈ I[i, N ]; (23)
and

  Û + Û ⊤ -Pi ⋆ H i(ℓ) ū2 (ℓ)   > 0, i ∈ I[1, N ], ℓ ∈ I[1, m]; (24)
with

Π 1ij = ( Dci + Dcj )C Ĉci + Ĉcj , Π 2ij =   Y ⊤ (A i + A j ) + Bcij C Âcij A i + A j + (B i Dcj + B j Dci )C (A i + A j )X + (B i Ĉcj + B j Ĉci )   , Π 3ij =   Bcij B i Dcj + B j Dci   , Π 4ij =   -( Êci + Êcj ) -(B j + B i )S   , and Û =   Y ⊤ F ⊤ I n X   .
Then, by choosing non-singular matrices W and Z such that ( 21) holds, we have that the saturated LPV system (1) in closed loop with the dynamic outputfeedback compensator (5) defined by

D ci = Dci , C ci = ( Ĉci -D ci CX)Z -1 , B cij = (W -1 ) ⊤ ( Bcij -Y ⊤ (B i D cj + B j D ci ), (25)
A cij = (W -1 ) ⊤ ( Âcij -Y ⊤ (A i + A j + (B i D cj + B j D ci )C)X -W ⊤ B cij CX -Y ⊤ (B i C cj + B j C ci )Z)Z -1 , E ci = (W -1 ) ⊤ ( Êci S -1 -Y ⊤ B i ),
subject to the event-triggering condition ( 6)- [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF] with matrices Q e , Q y = Q-1 y , and

Q u = Q-1
u is regionally asymptotically stable and has a reduced number of data transmissions between the sensor/plant and the controller. Moreover, the 200 region R E , defined in ( 12)- [START_REF] Shen | Event-triggered output feedback H ∞ control for networked control systems[END_REF], is an estimate of the region of attraction of the origin for the closed-loop system.

Proof 2. The proof of Theorem 2 is presented in Appendix B.

For more information on how to choose matrices Z and W see, for instance, Remark 2.8 in [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF]. Also observe that Theorem 2 can also be used to design a 205 dynamic output-feedback controller when the event-triggering mechanism (6)-( 7) is fixed, i.e., when the matrices Q e , Q y and Q u are fixed.

Remark 1. Theorems 1 and 2 can be adapted to treat both precisely known and non-saturating systems. In the first case (known system), it is necessary to set r = i = j = 1, which leads to fixed matrices. In this case, the dynamic and input matrices of the controller, A c and B c , are recovered by [START_REF] Wu | Suboptimal event-triggered control for time-delayed linear systems[END_REF] as A c = 0.5A c11 and B c = 0.5B c11 , with A c11 and B c11 computed as in [START_REF] Li | Event-based dynamic output-feedback controller design for networked control systems with sensor and actuator saturations[END_REF]. In the second case (no saturation), one has to impose: i) the line and column 3 are deleted in the LMIs [START_REF] Golabi | Eventtriggered control for discrete-time linear parameter-varying systems[END_REF] and [START_REF] Liu | Event-triggered control for linear systems with actuator saturation and disturbances[END_REF], and ii) the LMIs (18) and (24) are discarded.

Remark 2. Particular cases of the event-triggering condition (7) can be considered by imposing certain structures on the matrices Q e , Q y , and Q u . A simple one is to set i) Q e = I p , Q y = σ y I p , and Q u = σ u I p . Its simplicity however can generate more conservative results. To add more degrees of freedom to the design, we can also assume less restrictive forms such as ii) Q e = I p , Q y = diag{σ y1 , σ y2 , . . . , σ yn }, and Q u = diag{σ u1 , σ u2 , . . . , σ un }.

Similar versions of both cases are considered, for example, in [START_REF] Groff | Observer-based eventtriggered control: A discrete-time approach[END_REF], where the event-triggering condition is based on a state observer and the control signal is not taken into account. By assuming that the event-triggering condition is given, an interesting choice that allows to evaluate the influence of the output and the control signal on the condition, is to set iii) Q e = I p , Q y = σ 2 µI p , and Q u = σ 2 (1µ)I m with µ ∈ I[0, 1], allowing the designer to adjust the weight of the control and output signals by tuning the parameter µ. Because the ETM parameters are given, the S-procedure can be applied to include the event-triggering condition. Such a case, considered in [START_REF] Souza | Event-triggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF], allows to include a positive constant κ multiplying the right-hand side of (A.4) (see the proof of Theorem 1 in Appendix A), being a degree of freedom in the optimization. On the other hand, the conditions are no longer LMIs, thus requiring a linear search on κ values.

Optimization procedures

In the sequel, we address three objectives to improve the closed-loop operation. We introduce convex procedures to optimize these objectives by using the conditions stated in Theorems 1 and 2. The first one concerns the design of the event-triggering condition to minimize the data transmission rate. The second one consists of minimizing a functional cost, ensuring the improvement of the closed-loop system's performance in an optimal sense. The last one refers to the maximization of the estimated attraction region of the closed-loop system.

Minimization of the update rate

In this case, the objective is to design both the dynamic controller ( 5) and the event-triggering condition f (ŷ, y, u) to minimize the transmission activity over the network, i.e., minimize the number of of output signal updates. Regarding the inequality [START_REF] Abdelrahim | Stabilization of nonlinear systems using event-triggered output feedback controllers[END_REF], we can reduce the update rate by reducing the weight on the error measure, i.e., by choosing Q e to shrink ŷ(k)y(k) Qe compared with the magnitude of the norm-sum of the output, y(k), and control, u(k), signals.

Then, an intuitive method to reduce the transmission activity is to minimize the trace of Q e whereas the trace of Q u and Q y are maximized, or equivalently

O 1 :                  min tr(Q e + Qy ) + tr( Qu ) subject to            (17) and (18)
or ( 23) and ( 24)

with Qy = Q -1 y and Qu = Q -1 u . Let us stress that the data transmission activity is indirectly reduced thanks to the optimization procedure O 1 , which showed up to be effective in most of the tests performed by the authors. Whenever it is not the case, taking into account the generality of the objective function it might be interesting to impose additional constraints on some of the matrices, as done in [START_REF] Souza | Emulationbased dynamic output-feedback control of saturating discrete-time LPV systems[END_REF]. Moreover, since the matrices Q e , Q y and Q u are definite positives, the optimal value of the optimization procedure O 1 is ensured to be bounded.

Other alternatives, such as weighting the matrices and replacing the objective function in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] by tr(α e Q e + α y Qy ) + α u tr( Qu ) would be possible. Such kind of variation is not investigated here.

Optimal Linear Quadratic Cost

To ensure a certain level of control performance for the closed-loop system (9) under the event-triggering mechanism ( 6)-( 7), we associate with the closedloop system, the following linear quadratic cost function:

J ∞ = ∞ k=0 J(k) = ∞ k=0 x(k) ⊤ Qx(k) + u(k) ⊤ Ru(k), (27)
where Q ∈ R n×n and R ∈ R m×m are symmetric and positive definite matrices.

Such performance requirement associated with the previous requirements of the closed-loop system yields

∆V (k) -2Ψ(u(k)) ⊤ M (Ψ(u(k)) -(K(θ k ) -G(θ k ))ξ(k) -D c (θ k )e(k)) -e(k) ⊤ Q e e(k) + y(k) ⊤ Q y y(k) + u(k) ⊤ Q u u(k) < -J(k) ≤ 0. ( 28
)
Based on this assumption, we can reformulate the conditions in Theorems 1 and 2, ensuring both the regional asymptotic stability of the closed-loop system and a guaranteed cost J ∞ for the closed-loop system. The proof is omitted because it follows the same steps of the proofs presented in Appendix A and Appendix B, but considering [START_REF] Zuo | Co-design of event-triggered control for discrete-time systems with actuator saturation[END_REF]. In such a case, LMIs ( 17) and [START_REF] Liu | Event-triggered control for linear systems with actuator saturation and disturbances[END_REF] become

  M rij,× ⋆ 1 2 Π 5ij 1 2 Π 6ij 0 0 0 I n+m   > 0, (29)
where, for Theorem 1, M rij,× = M rij,1 , the matrix of the left-hand side of relation [START_REF] Golabi | Eventtriggered control for discrete-time linear parameter-varying systems[END_REF], and Π 5ij and Π 6ij are given by

Π 5ij = Q1 /2   2I K i + K j   U and Π 6ij = Q1 /2   0 D ci + D cj   ; ( 30
)
with I = I n 0 , and, for Theorem 2, M rij,× = M rij,2 , the matrix of the left-hand side of relation [START_REF] Liu | Event-triggered control for linear systems with actuator saturation and disturbances[END_REF], and Π 5ij and Π 6ij are given by

Π 5ij = Q1 /2   2I n 2X ( Dci + Dcj )C ( Ĉci + Ĉcj )   and Π 6ij = Q1 /2   0 Dci + Dcj   . (31)
In both cases, we have Q = diag{Q, R}.

Moreover, by summing [START_REF] Zuo | Co-design of event-triggered control for discrete-time systems with actuator saturation[END_REF] up from k = 0 to k = ∞, we have that J ∞ < ξ(0) ⊤ P -1 (θ 0 )ξ(0). Thus the upper-bound of the cost function J ∞ is related to the Lyapunov matrix P -1 (θ 0 ) and to the initial state ξ(0). Also, from ( 22) one gets P -1 = Ω P -1 Ω ⊤ , and by considering x c (0) = 0, we have that

J ∞ < x(0) ⊤ Y I P -1 (θ 0 ) Y I ⊤ x(0). (32)
Therefore, to ensure an optimal cost for the closed-loop system, we can minimize a scalar η ≥ 0 such that ηtr(P -1 (θ 0 )) ≥ 0, [START_REF] Gomes Da Silva Jr | Sampled-data lpv control: a looped functional approach[END_REF] or, still,

η -tr Y I P -1 (θ 0 ) Y I ⊤ ≥ 0. ( 34
)
With the aid of the Schur complement, the costs conditions ( 33) and [START_REF] Palmeira | Aperiodic sampleddata control for LPV systems under input saturation[END_REF] are expressed in the form of LMIs as follows:

  ηI 2n I 2n ⋆ P (θ 0 )   ≥ 0, (35a)
  ηI n Y I n ⋆ P (θ 0 )   ≥ 0, (35b)
respectively. Note that the LMIs (35a) and (35b) should be satisfied only for 255 P (θ 0 ) = N i=1 θ 0(i) P i and P (θ 0 ) = N i=1 θ 0(i) Pi , respectively, with θ 0 known. However, if θ 0 is not known a priori, we have to check the conditions for all P i and Pi with i ∈ I[1, N ] to ensure they are satisfied for any θ 0 .

Therefore, the optimization procedure can be summarized as:

O 2 :                  min η subject to            (29)
with ( 30), ( 18) and (35a), or ( 29) with ( 31), ( 24) and (35b).

(36)

Maximization of the estimation of the region of attraction R E

The objective here is to design the ETM or the ETM and the controller such that the estimate of the region of attraction is as large as possible. One way to do that is to maximize the volume of an ellipsoidal E(P 0 , 1), defined in the same way as in [START_REF] Shen | Event-triggered output feedback H ∞ control for networked control systems[END_REF], such that E(P 0 , 1) ⊆ R E , which can be ensured by

  P 0 I 2n ⋆ P i   ≥ 0, or still (37a)   P 0 Ω ⋆ Pi   ≥ 0, (37b)
with Ω given in [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF] and for all i ∈ I[1, N ]. However, the LMI (37b) is nonconvex due to the presence of W in Ω. To overcome such an issue, let us consider the partitioning P 0 =   P 011 P 012 ⋆ P 022   and x c (0) = 0, which allows us to dismiss the rows concerning the position of W in Ω. With that, the inequality (37b) can be rewritten as subject to ( 23), ( 24), (

  P 011 Y I n ⋆ Pi   ≥ 0, (38)
Remark 3. It is possible to combine the optimization procedures described in this section to have a trade-off between reducing the number of updates and increasing the estimate of the region of attraction of the origin for the closedloop system. In such a case, the objective function could be the weighted sum of each of the objective functions.

procedures given in Section 4.3 and compare the results with similar methods found in the literature. We explore both the LPV and the LTI cases, with and without saturating actuators.

System under saturating actuators

Consider the system (1) with matrices satisfying (3) and (4) described by

A 1 =   0.8040 0.0401 0.1602 0.8040   , A 2 =   1.2060 0.0601 0.2404 1.2060   , B 1 =   0.0401 0.0040   , B 2 =   0.0601 0.0060   , C = 1 1 , (40)
and a saturating actuator with symmetric saturation limits ū = 5. Our objective is to make the co-design for this system, i.e., we want to design the dynamic controller ( 5) and the ETM ( 6)-( 7), simultaneously, so that the update rate is minimized.

By using the optimization procedure O 1 given in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] with conditions of Theorem 2, we got the ETM with Q e = 18.2721, Q y = 0.5878, and Q u = 0.0626.

Figure 2 shows the achieved estimate of the region of attraction, R E , in the space of the system's state, where the cut of R E is marked with green dots (x c (0) = 0), and the projection of R E indicated by blue dots. A set of initial conditions is selected from the border of the cut of R E (green dots), points marked with * , and as expected, their trajectories converges to the origin without leaving the safe region. Additionally, we take some points outside R E : the ones marked in black lines still converge to the origin, despite not belonging to the estimate R E ; and those initial conditions in magenta lines generate divergent trajectories. For these cases, we choose θ k as a sequence that leads the open-loop system to have unstable modes. We measured the average update rate for the trajectories in the border of the green region, finding 36.19%.

Therefore, the proposed design allowed to reduce the updates in almost 2/3 of the samples with respect to the traditional (periodic) sample-data control, still ensuring regionally asymptotically stable trajectories. Note that, in this case, we are only concerned with reducing the update rate, which can result in a smaller attraction region as there is a trade-off between the attraction domain size and the transmission saving. In this sense, we can 295 use the topic iii) of Remark 2 to design an LPV dynamic controller for a given ETM specified by the parameters σ and µ, whose choice benefits the size of such a region, thus increasing the update rates obtained with co-design.

Batch reactor inspired LPV model

In this example, we explore a fourth order LPV system to illustrate the application of our approach under different control objectives. Consider the model ( 1)-( 4) with matrices:

A 1 =        
        , C =   1 0 1 -1 0 1 0 0   . (41)
Note that the pair (A, B) with A = (A 1 + A 2 )/2 and B = (B 1 + B 2 )/2 corresponds to the unstable batch reactor discretized with sampling time T s = 0.005 seconds investigated in [START_REF] Xiong | Stabilization of networked control systems with a logic 525 zoh[END_REF]. Here, we have adapted such a model to represent it as an LPV one.

First, we consider a linear case, i.e., we assume actuators without saturation limits. Next, we take the saturation limits and provide designs including update rate and optimal cost minimizations.

Linear case with update rate minimization

We performed the co-design of the dynamic controller and the ETM parameters for the considered system without actuator saturation, and compared our results with those obtained by the approach proposed in [START_REF] Golabi | Eventtriggered control for discrete-time linear parameter-varying systems[END_REF], where a statefeedback controller is designed.

Thus, to minimize the update rate of the output signal, we use the optimization procedure O 1 given in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] with conditions of Theorem 2 and the changes Then, we simulate the closed-loop system response assuming the initial condition ξ(0) = -0.2430 -0.2178 -0.4239 -0.2182 0 1,4

⊤

and the parametervarying θ k = cos(0.01πk). Figure 3 shows the four states (top), the two control inputs (middle), and the inter-event intervals of the ETM (bottom). It is possible to see that the closed-loop system is asymptotically stable with 35 output updates on the simulation time-interval, which corresponds to an update rate as lower as 7%. Because this is a linear case, the initial condition can be as bigger as possible and there is no meaning to consider an estimate R E .

In the sequel, we compare our approach with Theorem 1 given in [START_REF] Golabi | Eventtriggered control for discrete-time linear parameter-varying systems[END_REF], from which we got the ETM parameter σ x = 5.8042 × 10 -4 and the state-feedback controller with gains The simulation of the closed-loop system response for the same time interval and initial conditions yielded an update rate of the control signal of 37.40%, which is almost 6 times bigger than ours, illustrating the better performance of our approach.

K 1 =   75

Saturating case with update rate minimization

In this case, we consider the saturation with ū = 0.6 0.6 By using the same initial condition as in Section 5.2.1 (linear case) to simulate the closed-loop response, we can check that the trajectories do not converge to the origin, i.e. the system is unstable. As expected, in the presence of saturation, it is not possible to guarantee the asymptotic stability of the system for the entire state space, but only for a set of initial conditions. In fact, such an initial condition does not belong to the region of attraction of the origin for the system (41) in presence of input saturation.

Then, we consider ξ(0) = -0.0540 -0.0484 -0.0942 -0.0485 0 1,4 ⊤ as the initial condition, which belongs to the region R E defined in [START_REF] Khashooei | Output-based event-triggered control with performance guarantees[END_REF]. The time-response was simulated, and Figure 4 shows the achieved results: state response (top), the control inputs (middle), and the inter-event interval of the ETM (bottom).

The asymptotic stability of the closed-loop is ensured despite the saturation of the second control signal (dashed green lines) during the interval 6 ≤ k ≤ 13, verifies an update rate of 6.60%.

LTI system under saturating actuators

Consider (1) as the discretized LTI model, with sampling time T s = 0.01 s, of an inverted pendulum also investigated in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF][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]. The system matrices are given by

A =   1.0018 0.01 0.36 1.0018   , B =   -0.001 -0.184   , C =   1 0 0 1   , (42)
and the symmetric saturation limit is ū = 1. In this example, we study both the co-design and the emulation and compare the achievements of our approach with the finds in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF][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] where state feedback controller and input saturation is considered. [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF] addresses the emulation case, with a given ETM and the design of the controller. The co-design is proposed in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF][START_REF] Groff | Event-triggered control co-design for discrete-time systems subject to actuator saturation[END_REF], and a dynamic state stabilizing controller is suggested in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF].

Considering the co-design approach provided by Theorem 2, we combine the optimization procedures O 1 and O 3 , and use the objective function given by tr(7.2P 0 ) + tr(Q e + 0.95 Qy ) + tr(0.05 Qu ), where the weights were adjusted by trial and error. As a result, we got the obtained ETM matrices: x c (0) = 0 0 ⊤ and the simulations take 1001 samples. Theorem 2 allows to reduce the number of updates between 16.42% and 74.3%.

Q e =   1
Next, we perform comparisons with the literature by using the emulation approach provide by Theorem 1. We made two emulation designs, one using the controller from Theorem 2, and the one given in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]. In both cases, we 355 simulated the closed-loop system under the designed ETM with the same initial conditions and simulations time previously used.

Assuming the controller of [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], we run a combination of the optimization For such case, we have found a number of samplings equal to 33. Therefore, we reduced the number of updates in 41.07% in relation to that obtained by the co-design (Theorem 2). Moreover, with respect to [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF][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], there was a reduction even more important, between 50.75% and 86.24%.

Conclusion

We proposed a methodology to co-design a dynamic output-feedback controller and an event-triggering mechanism for discrete-time LPV systems under saturating input. The method, based on LMI conditions, ensures the regional asymptotic stability of the closed-loop system for initial conditions belonging to an estimated region of attraction. We formulate some optimization procedures which allows us to conclude that (F ⊤ -Y ⊤ X) is also non-singular. As a result, it is always possible to choose non-singular matrices W and Z, such that [START_REF] Kiener | Actuator saturation and antiwindup compensation in event-triggered control[END_REF] is satisfied. This shows that the gains (25) are well-defined.

Moreover, by considering the matrices ( 19)-( 22) and the change of variables Âcij , Bcij , Ĉci , Dci and Êci according to [START_REF] Li | Event-based dynamic output-feedback controller design for networked control systems with sensor and actuator saturations[END_REF], pre-and post-multiply [START_REF] Liu | Event-triggered control for linear systems with actuator saturation and disturbances[END_REF] by diag{Ω -⊤ , I p , Ω -⊤ , I p , I m } and its transpose, respectively, to get (17) and, likewise, pre-and post-multiply ( 24) by diag{Ω -⊤ , 1} and its transpose, respectively, to get [START_REF] Xie | Output-based event-triggered control for networked control systems: tradeoffs between resource utilisation and robustness[END_REF]. Thus, from Theorem 1, these two equivalences allow to conclude the proof.

Figure 1 :

 1 Figure 1: Event-triggered mechanism for the LPV system (1).

for all i ∈ I[ 1 ,

 1 N ]. Thus, we have the following optimization procedure 260

Figure 2 :

 2 Figure 2: R E with event-triggering mechanism.

mentioned in Remark 1

 1 to disregard saturation. We got the following ETM matrices

Figure 3 :

 3 Figure 3:The closed-loop system response for θ k = cos(0.01πk) and ξ(0) = -0.2430 -0.2178 -0.4239 -0.2182 0 1,4

⊤ . Applying the optimization procedure O 1

 1 given in[START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] with conditions of Theorem 2, we did the co-design obtaining the ETM matrices:

Figure 4 : 4 ⊤.

 44 Figure 4:The closed-loop system response for θ k = cos(0.01πk) and ξ(0) = -0.0540 -0.0484 -0.0942 -0.0485 0 1,4

procedures O 1

 1 and O 3 , see Remark 3, and the objective function given by tr(3.6P 0 ) + tr(Q e + 0.95 Qy ) + tr(0.05 Qu ), where the weights were found by trial and error. The obtained ETM matrices are u = 0.0058. This emulation design achieves 62 updates, which is a better than the performance verified with [26, Theorem 4.1], reducing its updates in 11.43%.By using the controller obtained by Theorem 2, we run the optimization procedure O 1 with Theorem 1 adding a requirement: the cut of R E achieved in the co-design test must be guaranteed on plane x k , ensuring that it remains with the same region of admissible initial conditions for the system. The following ETM matrices were obtained:

  that allow the designer to minimize the data transmission rate from the sensor to controller, to maximize the size of the stability region, or to minimize the upper-bound of a quadratic linear cost function. The effectiveness of our proposal is illustrated and compared with the literature thought numerical tests, pointing out the classical trade-off between the different criteria of optimization.

  Pi22 = P i11 . Theorem 2. Consider that there exist symmetric positive definite matrices Pi ∈ R 2n×2n , Q e , Qy ∈ R p×p and Qu ∈ R m×m , a positive definite diagonal matrix

S ∈ R m×m and matrices H i , X, Y , F , Âcij , Bcij , Ĉci , Dci , and Êci of appropriate dimensions, with i ∈ I[1, N ] and j ∈ I[i, N ], such that the two following LMIs conditions are feasible.

Table 5 .

 5 3 provides the summary of the achievements of our co-design approach and those from[START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF][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]. The initial conditions are x(0) = 0.2 0.8

					
	.3048 0.3967 0.3967 0.1206	 , Q y =		0.1051 0.0608 0.0608 0.1001	 , Q u = 0.0064,

⊤

and 350

Table 1 :

 1 Comparison of the number of samplings.

	Design	number of
	method	updates
	Theorem 1 in [29]	90
	Theorem 3.2 in [27]	218
	Theorem 3.1 in [26]	67
	Theorem 4.1 in [26]	70
	Theorem 2	56

U + U ⊤ ⋆ ⋆ ⋆ ⋆ ⋆ -1 2 (P i + P j ) 0 Q e ⋆ ⋆ ⋆ ⋆ 1 2 (H i + H j -1 2 (D ci + D cj ) 2S ⋆ ⋆ ⋆ -K i U -K j U ) 1 2 A ij U 1 2 E ij -1 2 (B i + B j )S P r ⋆ ⋆

Û + Û ⊤ ⋆ ⋆ ⋆ ⋆ ⋆ -1 2 ( Pi + Pj ) 0 Q e ⋆ ⋆ ⋆ ⋆ 1 2 (H i + H j -Π 1ij ) -1 2 ( Dci + Dcj ) 2S ⋆ ⋆ ⋆ 1 2 Π 2ij 1 2 Π 3ij 1 2 Π 4ij Pr ⋆ ⋆

P (θ k ) ⋆ ⋆ ⋆ ⋆ ⋆ 0 Q e ⋆ ⋆ ⋆ ⋆ -S -1 (K(θ k ) -G(θ k )) -S -1 D c (θ k ) 2S -1 ⋆ ⋆ ⋆ A(θ k ) E(θ k ) -B(θ k ) P (θ k+1 ) ⋆

✩ This work has been supported by the Brazilian Agencies CAPES under the project Print CAPES-UFSC "Automation 4.0" and CNPq; and by ANR under the project HANDY number 18-CE40-0010.

availability of the varying parameter and a more general parameter dependence, as for example polynomial nature. Such a direction should impose to consider other methods as those based on sum-of-squares.

Simulation results

In this section, we present numerical examples and simulations to validate our strategy and show its effectiveness. We apply the different optimization Appendix A. Proof of Theorem 1

By supposing the feasibility of ( 18), multiply its left-hand side by θ k(i) , and sum it up for i [START_REF] Geromel | H 2 and H ∞ robust output feedback control for continuous time polytopic systems[END_REF] to over-bound the block (1,1) by U ⊤ P -1 (θ k )U , and pre-and post-multiply the resulting inequality by diag{U -⊤ , 1}, to obtain

Finally, apply Schur complement and pre-and post-multiply the resulting inequality by ξ(k) ⊤ and ξ(k), respectively, to obtain

thus, ensuring that R E ⊂ S(ū), ∀k ≥ 0, i.e. any trajectory of the closed-loop 380 system starting in R E remains in S(ū).

By supposing the feasibility of ( 17), multiply its left-hand side by θ k+1(r) , θ k(i) and θ k(j) , and sum it up for r, i ∈ I[1, N ] and j ∈ I[1, N ]. Then, replace

bound the block (1,1) by U ⊤ P -1 (θ k )U , and pre-and post-multiply the resulting inequality by diag{U -⊤ , I p , S -1 , I 2n , I p , I m } and its transpose, respectively, to ⋆

Next, apply Schur complement three times and pre-and post-multiply the resulting inequality by ξ(k) ⊤ e(k) ⊤ Ψ(u(k)) ⊤ and its transpose, respectively. Finally, replace [START_REF] Zhang | Event-triggered dynamic output feedback control for networked control systems[END_REF], and ξ(k

Hence, the feasibility of ( 17) and ( 23) ensures the feasibility (A.2) and (A.4).

Then one has both the positivity of the function given in [START_REF] Wu | Event-triggered control for discretetime linear systems subject to bounded disturbance[END_REF] and the negativity of ∆V (ξ(k)). Therefore, one can conclude that R E given in ( 12) is an estimation of the region of attraction of the origin for the closed-loop system.

Appendix B. Proof of Theorem 2

By supposing the feasibility of (23), from block (1,1), it follows that Û + Û ⊤ > 0, consequently, Û is non-singular. Therefore, from (20), we have X and


Louis Couturier
email: louis.couturier@u-picardie.fr

Louis Bigo
email: louis.bigo@univ-lille.fr

Florence Levé
email: florence.leve@u-picardie.fr

ANNOTATING SYMBOLIC TEXTURE IN PIANO MUSIC: A FORMAL SYNTAX

Symbolic texture describes how sounding components are organized in a musical score. Along with other high-level musical components such as melody, harmony or rhythm, symbolic texture has a significant impact on the structure and the style of a musical piece. In this article, we present a syntax to describe compositional texture in the specific case of Western classical piano music. The syntax is expressive and flexible, unifying into a single text label information about density, diversity, musical function and note relationships in distinct textural units. The formal definition of the syntax enables its parsing and computational processing, opening promising perspectives in computeraided music analysis and composition. We provide an implementation to parse and manipulate textural labels as well as a bestiary of annotated examples of textural configurations.

INTRODUCTION

Texture commonly refers to a high-level feature of music which strongly relates to musical style and form, and to the auditory perception of the musical flow. In the context of this work, as in symbolic music analysis in general, texture describes how sounding components (voices, chords, notes) are organized. Variations in classical music are examples of compositional practices in which a given melody and an underlying harmony are realized or arranged through different possible symbolic textures. Song covers in popular music are also intuitive examples in which an original texture is changed by another one while conserving the original chords and melody. Although essential in musical scores, symbolic texture appears to be challenging to formalize, probably due to the variety of musical concepts it gathers. This study aims at proposing such formalization through a well-defined syntax for the specific case of Western classical piano music.

Musicological studies of symbolic texture. In the musicological field, the description of the textural dimension for Western classical music is rather recent, compared for example to the study of harmony that has been developed from Rameau's treatise in 1722 [START_REF] Rameau | Traité de l'Harmonie réduite à ses principes naturels[END_REF]. The notion of texture was mentioned in the 1920s as an essential element to the study of form in non tonal compositions [START_REF] Silva | Por uma definic ¸ão unificada de textura musical[END_REF]. It has been developed in parallel to the analysis of contemporary music through the 20th century, with analogies with fine art paintings analysis (see [START_REF] Perraudeau | La texture en musique: sa contribution pour la composition, l'apprentissage de la musique et ses effets sur la perception musicale et la cognition des enfants sourds implantés[END_REF] for usage of textural references by composers of late 19th and 20th). The term texture was only introduced in 1980 in the New Grove Dictionary of Music and Musicians, as referring to the vertical aspect of musical structure [. . . ] with regards to the way in which individual parts or voices are put together. Berry [START_REF] Berry | Structural functions in music: A Probing Exploration of Conceptual Bases and Techniques for the Analysis of Tonality, Texture, and Rhythm[END_REF] considers texture as one of the structural function in music, and several works have assessed its impact on the perception of structure [START_REF] Levy | Texture as a sign in classic and early romantic music[END_REF][START_REF] Duane | Texture in eighteenth-and early nineteenthcentury string-quartet expositions[END_REF]. When referring to the overall organization of instrumental or vocal parts, some compositions can be categorized into common textural types like monophony, homophony or polyphony [START_REF] Levy | Texture as a sign in classic and early romantic music[END_REF][START_REF] Huron | Characterizing musical textures[END_REF][START_REF] Benward | Music[END_REF].

The principle of texture is closely related to timbre and register, and those dimensions particularly arise when considering orchestral texture [START_REF] Piston | Orchestration[END_REF][START_REF] Nordgren | A measure of textural patterns and strengths[END_REF][START_REF] Guigue | The structural function of musical texture: Towards a computerassisted analysis of orchestration[END_REF]. In the case of piano music of the early 19th century, Hérold [START_REF] Hérold | Timbre et forme : La dimension timbrique de la forme dans la musique pour piano de la première moitié du dix-neuvième siècle[END_REF] dissociates the textural factors of timbre, as for example rhythmic or harmonic (horizontal or vertical) density, from its pianistic factors, which are more specific to the characteristics of the instrument.

Finally, a texture-based compositional approach can be found in the work of Moreira [START_REF] Moreira | Textural design: a compositional theory for the organization of musical texture[END_REF]. Textural configurations are thus described formally in the case of additive texture by stacking distinct textural parts. Different levels of abstraction and complexity are used, from ordered or unordered partitions of textural parts to more global categorizations in combinations of block and line elements.

MIR studies of symbolic texture. In the Music Information Retrieval field (MIR), the notion of texture is mostly studied in the audio domain where it is commonly associated with timbral properties [START_REF] Strobl | Sound texture modeling: A survey[END_REF] for tasks such as genre classification [START_REF] Nanni | Set of texture descriptors for music genre classification[END_REF][START_REF] Foleis | Texture selection for automatic music genre classification[END_REF].

In the symbolic domain, works on texture identification are still sparse, probably due to the difficulty to formalize this notion. However, a few attempts have been made, restricting the study to a specific repertoire. A machine learning approach has been used for the classification of texture in Italian madrigals of the 16th century [START_REF] Parada-Cabaleiro | Automatic recognition of texture in renaissance music[END_REF]. On classical string quartets, a syntax has been proposed, highlighting for each instrument its role (melody or accompaniment) and its relations to other voices [START_REF] Giraud | Towards modeling texture in symbolic data[END_REF][START_REF] Soum-Fontez | Symbolic textural features and melody/accompaniment detection in string quartets[END_REF]. Identifying musical roles through texture analysis has also been done for the specific case of guitar popular music, using audio [START_REF] Foulon | Automatic classification of guitar playing modes[END_REF] and symbolic data [START_REF] Régnier | Identification of rhythm guitar sections in symbolic tablatures[END_REF]. Detecting textural changes was also shown to be a promising approach to delimit successive sections of a piece in music structure retrieval [START_REF] Tenkanen | Detecting changes in musical texture[END_REF]. Alternatively, the formalization of some selected textural figures has been used to synthetically increase the training set of a machine learning model dedicated to roman numeral analysis [START_REF] López | Augmentednet: A roman numeral analysis network with synthetic training examples and additional tonal tasks[END_REF].

Beyond musical analysis, the recent growth of the music generation field has also contributed to bring new challenges in symbolic texture modeling. This is the case for music style transfer where the texture of a first musical piece, commonly assimilated to its style, is mapped on the harmony and melody of a second one [START_REF] Cífka | Groove2Groove: one-shot music style transfer with supervision from synthetic data[END_REF]. The separation of harmony and texture is also found in [START_REF] Wang | Learning interpretable representation for controllable polyphonic music generation[END_REF], where these two dimensions are disentangled in a latent space for more controllable music generation.

Motivation and outline. We believe that texture analysis can improve our understanding of compositional practice by helping the study of both musical style and musical structure. We also think that such improvements will benefit to the field of music generation where these high level musical layers are known to be difficult to control.

This work introduces a syntax to annotate texture for symbolic piano music. In comparison to texture analysis in other repertoires, such as classical string quartets [START_REF] Giraud | Towards modeling texture in symbolic data[END_REF] in which each instrument mostly plays one voice at a time, a major challenge of piano music is the complex organization and high variability of voices and streams during the piece. Here, we focus on the specificities of piano music of Viennese classicism, whose major composers were Haydn, Mozart and Beethoven. By doing so, we ensure consistency and possibility of comparison with previous work on string quartets.

The contributions of this work are as follows. We identify the major characteristics of symbolic texture that are relevant for classical piano music based on existing studies (Section 2). In Section 3, we define a formal syntax to facilitate the computational manipulation of these characteristics. Finally, Section 4 presents a collection of annotated examples.

TEXTURAL CHARACTERISTICS

Based on musical and musicological resources, complementary aspects of texture have been taken into account in our study to describe the texture of a score region. Selected characteristics are detailed in this section and illustrated in Figure 1.

Diversity and density

The analysis of texture is related to the organization of atomic musical components -i.e. notes -into perceptu- ally coherent textural units. According to Huron [START_REF] Huron | Characterizing musical textures[END_REF], diversity and density respectively qualify the quantity and the (in)homogeneity of sounding components in an auditory scene. Similarly, Moreira [START_REF] Moreira | Textural design: a compositional theory for the organization of musical texture[END_REF] defines a textural configuration by the organization of n threads (quantitative aspect) into m textural parts (qualitative aspect) in a given passage of music. In our work, these textural parts are referred to as layers. One layer can be made of either one unique voice, as for example in the case of a single melodic line, or several voices gathered by some kind of similarity, as for example voices in homorhythmy (sharing the same rhythm). Three-voice inventions by Bach are typically composed of 3 single-voice layers (similarly to the top right representation in Figure 1), whereas a chorale is more likely to be made of one unique 4-voice layer, due in this case to the homorhythmy of the voices. Note that homorhythmic voices are not necessarily perceived as a single layer for example if their respective register, motion or articulation make them differ enough. Algorithms to group notes in voices or coherent groups (streams) have been proposed in [START_REF] Cambouropoulos | Voice and stream: Perceptual and computational modeling of voice separation[END_REF][START_REF] Makris | Visa3: Refining the voice integration/segregation algorithm[END_REF][START_REF] Chew | Separating voices in polyphonic music: A contig mapping approach[END_REF] but without specifically qualifying the texture of those groups.

Distinct layers (diversity)

The perception of distinct textural layers can emerge from several grouping mechanisms described in the works of McAdams and Bregman [START_REF] Mcadams | Hearing musical streams[END_REF][START_REF] Bregman | Auditory scene analysis: The perceptual organization of sound[END_REF], including both vertical and horizontal similarities (simultaneity and sequentiality). Onset-synchrony and semblant motions are core criteria for the fusion of notes or voices into layers, as highlighted in [START_REF] Huron | Characterizing musical textures[END_REF]. The number of distinct layers (named cardinality in [START_REF] Moreira | Textural design: a compositional theory for the organization of musical texture[END_REF]) corresponds to the diversity of the texture. In our work, based on these considerations, a primary step is to identify the distinct layers that would be most plausibly perceived as independent by a human listener. Note that we will not directly refer to the number of distinct layers, we will simply describe them one by one.

Number of voices (density)

The (vertical) density of a layer (named thickness in [START_REF] Moreira | Textural design: a compositional theory for the organization of musical texture[END_REF]) corresponds to the number of simultaneous notes heard at the same time in this layer. When considering a musical fragment with several concurrent layers, the global density of this fragment can be considered in most cases as equal to the sum of the densities of the constituting layers, as it is done in Moreira's textural spaces [START_REF] Moreira | Textural design: a compositional theory for the organization of musical texture[END_REF] or Gentil-Nunes Partitional Analysis [START_REF] Gentil-Nunes | Análise particional: uma mediac ¸ão entre composic ¸ão musical e a teoria das partic ¸ões[END_REF]. But in some more complex textural configurations, we allow for the global density to take a value different from the sum of the densities of its constituting layers. For example, the alternation of two singlevoice layers in call and response would lead to a global density value equal to one. In Figure 2, bar 1 has a global density of 1 (1 layer of density 1), bar 5 has a global density of 2 (2 layers of density 1), whereas bar 6 has a global density of 1 (since the bass layer is very short compared to the upper layer).

Melodic/harmonic/static functions

In [START_REF] Giraud | Towards modeling texture in symbolic data[END_REF], the distinction is made between melodies and accompaniment parts. Instead of characterizing such roles in the overall textural configuration, we propose to focus on the functional aspect of each layer taken individually, expressed by a combination of three labels: melodic, harmonic and static. These terms are named based on Benward and Saker's primary elements of musical texture [START_REF] Benward | Music[END_REF]. These three complementary dimensions respectively convey ideas of horizontality, verticality and stability.

Melodic A layer labeled as melodic (M) is shaped as a sequence of notes that expresses a strong sense of temporal continuity. It is not limited to dominant phrases. The scope of melodic layers includes primary, secondary or supporting melodies in Benward and Saker's primary elements [START_REF] Benward | Music[END_REF]. Counterpoint only has M-function layers, each voice existing independently. In most cases, a bass line could also be considered as an M layer. Furthermore, a melodic-function layer is not necessarily monodic -that is to say, with a density of one. It can be a sequence of harmonic thirds, sixths or even denser chords that form a melodic continuity. In this last case, it would additionally be labeled as harmonic. In the extract of Figure 2, the upper layer has systematically a melodic function and the left hand sometimes also has a melodic function, as from bar 5 to bar 8.

Harmonic Harmonic layers (H) qualify harmony-related layouts, such as chords or broken chords, which raise generally consonant combination of notes. For example, in bars 2 to 4 in Figure 2, the left hand plays a two-voice harmonic layer. Note that the density of a harmonic layer can be equal to one, as in the case of arpeggios.

Static Derived from Benward and Saker's static supporting parts [START_REF] Benward | Music[END_REF], our static label (S) encompasses layers structured with short-term repetition or sustained notes, like pedal notes or drones. Repeated notes might not necessarily be consecutive: ostinati are typically considered as static layers for they repeat short cells and are character- • (MS) A melody with short term repetition, expressing both continuity in the voicing and insistance on a given pitch, could have both melodic and static function;

• (MH) A chordal melody would have both melodic and harmonic function;

• (HS) A repeated Alberti bass, emphasizing harmony with a repetitive figure, would have both harmonic and static function, as would have a triple hammer blow [START_REF] Hepokoski | The medial caesura and its role in the eighteenth-century sonata exposition[END_REF] on a given chord;

• (MHS) In rare cases, a layer can involve at the same time the three dimensions M, H, and S. For example, a hammer blow which does not repeat exactly the same chord, but has a changing upper note, involves some melodic movement.

Internal organization of a layer

Voice relationships within a layer

In the case where several voices are gathered in the same layer because of similar rhythm or semblant motion, it appears important to specify which degree of similarity they share. Inspired by terminology and description of [START_REF] Giraud | Towards modeling texture in symbolic data[END_REF], we use three levels of description between voices of a single layer.

• Homorhythmy (h): all the voices or notes in the layer share a strongly similar rhythm. In the bars 2 to 4 in Figure 2, the left-hand harmonic part has its two voices played simultaneously, in homorhythmy. Note that octave motions are a specific case of parallel motions, which are also a subcase of homorhythmy. Because of these inclusive relations, we consider the most restrictive attribute to systematically induce the included ones. Single voice layers are naturally not concerned by these relations.

Characteristic musical figures

Each layer can be associated with specific attributes to precise how it is written. We defined some characteristic figures that seem relevant to the studied repertoire:

• sustained notes (t): the notes in the layer are sustained during most of a musical segment (see the left hand part in bar 5 and 13 in Figure 2).

• repeated notes (r): short-term repetition of the exact same pitches one after another. Each new pitch in the layer is repeated at least once (see Figure 2 bars 16 to 20).

• sparse ( ): layers with low horizontal density, locally. On an indicative basis, less than half time of studied section is filled with the notes of the layer (see Figure 2 bars 2 to 4).

• scale (s): ascending or descending sequence of short duration notes (see Figure 2 bars 13 and 14). Repeated ascending or descending short motives can also be considered (see Figure 2 bars 5 to 7).

• oscillation (b): fast alternation between two notes. An example can be found in the bestiary that is provided at the end of the paper, bars 38 and 41 to 43.

These last attributes tend to be more style-specific and are intentionally biased towards Western classical style, here. Room is left for future extensions.

Sublayer decomposition

In addition to the decomposition of a texture into simultaneous layers, a single layer can be decomposed itself into several sublayers. A sublayer decomposition can be useful to indicate relationships between a subset of voices included in the same layer. In a homorhythmic chorale for example, this would be done by splitting the top-level MH layer into multiple M sublayers. Sublayer decomposition can also be useful in case of compound melodies [START_REF] Lester | Compositional theory in the eighteenth century[END_REF] or two-part accompaniments such as stride piano. In the latter case, it enables to describe bass notes and chord notes in two distinct sublayers.

Top-level textural layers are described rather independently, whereas sublayers are interconnected and help describing the content of a given single layer. The subdivision of layers into sublayers occurs recursively and handles vertical separation at different levels of precision. An example of sublayer decomposition is shown Figure 4.

A SYNTAX FOR TEXTURE ANNOTATION

In this section, we develop a general syntax for notating symbolic piano texture in flat text that is both musically intuitive and flexible but at the same time rigidly structured. The syntax enables to depict into a single text label the five characteristics of texture described in Section 2 and illustrated Figure 1.

Formal structure

In this work, and similarly to [START_REF] Giraud | Towards modeling texture in symbolic data[END_REF], it is proposed to annotate texture at the bar level. We estimate that a description of texture with shorter time span would indeed limit the representation of high level textural concepts previously described. On the other hand, notating the texture of several consecutive bars by a single annotation might yield to labels that would be either overloaded in text or too approximative regarding the musical surface. It is not uncommon to see a drastic change of texture in the middle of a bar, for example at the transition between two different structural sections. To overcome such cases, the syntax enables to indicate two consecutive textures separated with a comma.

Topologically, we represent a textural configuration by a data structure of forest, i.e. a set of trees, in which each tree corresponds to a top-level layer. It transcribes the sublayer description in section 2.4 by splitting a layer description (a node) into the individual descriptions of their sublayers (children nodes). The depth in the tree represents a level of precision: roots of a textural tree portrays surface description of the top-level layers whereas leaves can detail a finer voice-specific description. An example is given in Figure 4.

Flat text syntax

The syntaxic formalism we propose is described in Backus-Naur form in Figure 5. A texture is either composed of a unique textural layer or a set of layers. In the latter case, the layers are ordered from high to low register and separated by the character /. Each layer is expressed individually by the combination of its musical functions (M, H, S, MH, MS, HS, MHS or N for None) and its density (as a positive integer value), followed by all the relevant attributes describing the internal organization of the layer: M1, MH4h or S2ot for example.

In case a sublayer decomposition is needed, it has to be inserted after each corresponding layer by defining the ordered set of sublayers in parenthesis: A/B(B a /B b ). Finally, an integer indicating the global density is added by encompassing the whole label with brackets []. This notation is only necessary when the global density does not correspond to the sum of all layer densities: M1/M1/S2 is a shortcut notation of 4[M1/M1/S2], which is perfectly equivalent. At this global level, additional tags may also be added, like in the second measure of Figure 2, 3h[M1/H2h], which links two separated layers with homorhythmy. A Python implementation of this structure can be found at http://algomus.fr/code. It allows the parsing of the syntax by converting flat text texture labels into structured objects.

Influence of the context

Even though texture is annotated at the granularity of the musical bar, it is worth mentioning that the context (i.e. neighbor bars) plays an important role in the identification of the local texture. We illustrate how it impacts the annotation by using three examples from Figure 2.

• pattern repetition: the inclusion of the scale attribute (s) in bar 5 to 7 takes into account the repetition of the melodic cell on the three bars. The motive is consecutively transposed in a descending manner. Bar 5 alone is not sufficient to detect this pattern.

• layer continuity: the parallel attribute (p) in the upper part of bar 16 results from a continuity of the layer M2p (two-voice parallel melodic layer) started in bar 15. • layer separation: Bars 2, 3 and 4 could have been annotated MH3h in another context due to the onset synchrony, which tends to merge notes into the same stream [START_REF] Huron | Characterizing musical textures[END_REF]. However, the choice has been made to annotate it with two distinct top-level layers, to preserve the continuity of the melodic part that has been heard just before. It assumes that the perception of streams is in this case more influenced by the sequential grouping of notes rather than vertical grouping. The homorhythmy is then indicated as a global attribute, resulting in the label 3h[M1/H2h].

ANNOTATED EXAMPLES

To accompany the presentation of our syntax, we provide a Bestiary of musical textures (see Figure 6), which aims at illustrating some useful and common examples of textural configurations.

The bestiary does not intend to provide an exhaustive description of all possible combinations of layers that can be found in classical piano music. Instead, it focuses on the illustration of typical layers. It constitutes not only a useful tool and reference to understand textural annotation, but also a catalog of textural ideas created to be extended in the future.

In addition to the synthetic fragments gathered in the bestiary, the syntax was experimented on musical extracts from the classical piano repertoire. The first measures of the third movement of Mozart Piano Sonata n°2 (K. 280/189 e ) are shown in Figure 2. The annotation of the full piece is available in text format at http://algomus.fr/code.

CONCLUSIONS

In this article, a new syntax has been defined and illustrated for the annotation of symbolic texture in piano music of the Western classical-era. Based on musicological reflexions, the textural elements we presented and their organization provide a practical framework for texture analysis. By integrating them in a unified syntax, we facilitate their computational manipulation for automated analysis and generation tasks.

Future works include the elaboration of a distance function that could systematically be applied to evaluate the similarity between two textural labels. The similarity between two labels should reflect the similarity between the corresponding symbolic textures. Besides texture analyses and comparison between composers or eras, this project also aims at studying the evolution of texture within a single piece of music. Strong links are to be found between texture and the structure of the piece, for instance in the case of sonata form. Future work will also include the creation of a consistent dataset with texture annotations. Resulting data would be usable for the training of supervised models for automatic texture estimation in classical piano scores. Additionally, our texture model is a step towards the interpretation of arbitrary texture categories that would be discovered using unsupervized methods. It would finally contribute to further improve the understanding and the control of this musical dimension in generative tasks.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

Figure 1 .

 1 Figure 1. Schematic representation of main textural characteristics. Diversity: three voices to form respectively one, two or three layers. Density: a unique layer of density 1, 2 or 3. Musical functions: melodic, harmonic, static. Additional elements are used to qualify the internal organization of each layer, to indicate characteristic musical figures or specific relationships between their voices.

Figure 2 .

 2 Figure 2. First bars of the Presto from Piano Sonata n°2 by W. A. Mozart (K. 280/189 e ), annotated with texture labels using proposed syntax. Textural labels are given in their detailed expression which begins with their global density value. However, this information can be omitted when the global density is equal to the sum of the density of each layer. The label of bar 5 can be shortened this way: M1s/M1t instead of 2[M1s/M1t]. On the contrary, 1[M1s/M1 ] in bar 6 cannot.

Figure 3 .

 3 Figure 3. Example of textural layers with combined melodic M, harmonic H and static S musical functions. From Mozart K.280/189 e mm. 1-2; K.279/189 d mm. 6, K.265/300 e variation 6 mm. 1-3 and K.309/284 b , mm. 58.

•

  Parallel motions (p): all the voices in the layer share (almost) the same rhythm and the same intervallic consecutive motions. With third or sixth doubling, for example, major or minor intervals are tolerated. See the melodic layer in measures 17, 18 and 19 in Figure 2. • Octave motions (o): all the voices in the layer have parallel motions plus the same pitch classes. In Figure 2, bars 13-14, the left-hand echoes the motion in bars 5-6, this time with octave doubling.

Figure 4 .

 4 Figure 4. Textural annotation and tree structure of the first measure of the Allegro moderato from Partita in D major for piano, attributed to J. Haydn (Hob.XVI.14, 1767). The bar is labeled 2[M1/H1(S1/M1)]. Two main layers with density of one are separated: a melodic (M1) and a harmonic (H1) part. Furthermore, the lower part can be decomposed into two sublayers: a melodic stream can be distinctly heard (M1: D, E, F sharp), in alternation with the repeated note A with static function (S1).

  <layer> ['/' <layer-list>] <layer> ::= <layer-function><description> ['(' <layer-list> ')'] <description> ::= <density> [<relation>][<attribute-set>] <attribute-set> ::= <attribute> [<attribute-set>] <layer-function> ::= M | H | S | MH | MS | HS | MHS | N <relation> ::= h | p | o # homorhythmy, parallel motions, octave motions <attribute> ::= s | t | b | r | _ # scale, sustained, oscillation, repetition, sparse <density> ::= .. integer in N * ..

Figure 5 .

 5 Figure 5. Syntax of Texture Notation in Backus Naur Form. As an example, the 2[M1/H1(S1/M1)] label from Figure 4 contains a <layer-list> of two elements. The first <layer> (M1) has a melodic <layer-function> M and a <density> of one. The second <layer> has a nested <layer-list> S1/M1, to describe its sublayers. The value of 2 corresponds to the global <density> in the detailed <description>.

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Figure 6 .

 6 Figure 6. Bestiary of musical textures. The texture of each bar is annotated with the syntax defined in Figure5, illustrating various textural layers and combination of layers that could be found in piano music. The file is available at http: //algomus.fr/code.

Acknowledgments

This work has been partly funded by Région Hauts-de-France and CPER MAuVe.