Joey Reinders
email: j.m.f.reinders@tue.nl

Student Member, IEEE Mattia Giaccagli
email: mattia.giaccagli@univ-lyon1.fr

Bram Hunnekens
email: bram.hunnekens@demcon.com

Daniele Astolfi
email: daniele.astolfi@univ-lyon1.fr

Fellow, IEEE. Nathan Van De Wouw

Tom Oomen

Repetitive control for Lur'e-type systems: application to mechanical ventilation

Keywords: Mechanical ventilation, medical applications, Repetitive control, Lur'e-type system, circle criterion, nonlinear output regulation, convergent systems, harmonic regulation

Repetitive control (RC) has shown to achieve superior rejection of periodic disturbances. Many nonlinear systems are subject to repeating disturbances. The aim of this paper is to develop a continuous-time RC design with stability guarantees for nonlinear Lur'e-type systems. Approximate output tracking is achieved by combining an internal model, consisting of a finite number of linear oscillators with frequencies at the reference frequency and at its multiples, with a stabilizer that guarantees a convergence property of the closed-loop system. The developed RC approach is applied to a nonlinear mechanical ventilation system for Intensive Care Units (ICUs), which can be modeled as a Lur'e-type system. The experimental study confirms that the repetitive control scheme is able to successfully follow the desired target pressure profile to properly support the ventilation needs of an adult patient.

I. INTRODUCTION

Repetitive Control (RC) schemes are particularly suitable to achieve robust tracking of a periodic reference signal, see [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF], [START_REF] Longman | On the theory and design of linear repetitive control systems[END_REF], [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF]. Tracking of periodic signals is a common control problem in many relevant application fields, for example in healthcare. In this paper, the proposed analysis and controller design are motivated by the application of mechanical ventilation of patients on Intensive Care Units (ICUs). Mechanical ventilation is used to support the breathing of patients by providing the correct oxygen support and elimination of carbon dioxide [START_REF] Warner | Mechanical ventilation[END_REF]. In ventilation, tracking of a periodic signal, i.e., pressure target, is desired.

Several studies in literature have focused on achieving accurate tracking performance for mechanical ventilation, e.g., [START_REF] Borrello | Modeling and control of systems for critical care ventilation[END_REF], [START_REF] Scheel | Iterative learning control: An example for mechanical ventilated patients[END_REF], [START_REF] Reinders | Adaptive control for mechanical ventilation for improved pressure support[END_REF], and [START_REF] Li | Model predictive control for a multicompartment respiratory system[END_REF]. Particularly promising is the application of discrete-time frequency domain RC in [START_REF] Reinders | Improving mechanical ventilation for patient care through repetitive control[END_REF] and [START_REF] Reinders | Triggered repetitive control: application to mechanically ventilated patients[END_REF]. In this work, a significant reduction of the pressure tracking error is achieved. However, this is achieved under a linearity assumption of the considered ventilation system dynamics, whereas a mechanical ventilation system contains nonlinear system dynamics, see [START_REF] Reinders | Accurate pressure tracking to support mechanically ventilated patients using an estimated nonlinear hose model and delay compensation[END_REF].

The considered ventilation system is a nonlinear dynamical system, which can be modeled as a Lur'e-type system. Lur'etype systems consist of the interconnection of linear timeinvariant dynamics with a static nonlinearity in the feedback loop. These systems form a practically relevant sub-class of nonlinear systems, also for other application domains. To achieve robust tracking of the periodic pressure target of the ventilation system, a generically applicable RC scheme for Lur'e-type systems is developed.

The structural idea of RC is based on the internal model principle, namely on the fact that when a periodic signal with known period T must be tracked, a copy of the disturbance model generating such a signal must be included in the regulator [START_REF] Bin | About robustness of control systems embedding an internal model[END_REF]. This is generally done through a universal generator of a T -periodic signal. Such a generator is implemented using a memory loop with a delay of length T . This memory loop places an infinite number of poles on the imaginary axes at the fundamental frequency 2π/T and its multiples, see, e.g., [START_REF] Ghosh | Nonlinear repetitive control[END_REF], [START_REF] Hara | Repetitive control system: a new type servo system for periodic exogenous signals[END_REF]. Then, the extended system composed by the plant and such memory loop is stabilized with feedback control.

Because a delay is easily implemented in discrete-time, significant research efforts have been devoted to the development of discrete-time implementations of RC. In this approach, mostly linear systems are addressed from the theoretical point of view and good tracking performance is achieved in these systems, see, e.g., [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF], [START_REF] Costa-Castelló | A repetitive controller for discrete-time passive systems[END_REF], [START_REF] Onuki | Compensation for repeatable tracking errors in hard drives using discrete-time repetitive controllers[END_REF], [START_REF] Reinders | Improving mechanical ventilation for patient care through repetitive control[END_REF], [START_REF] Tomizuka | Discrete-time domain analysis and synthesis of repetitive controllers[END_REF], [START_REF] Tomizuka | Analysis and synthesis of discrete-time repetitive controllers[END_REF]. Unfortunately, the developed frequency analysis tools cannot be directly employed in the presence of nonlinearities. Therefore, typically no formal stability proofs are provided for repetitive control applied to nonlinear systems.

In existing literature, several studies have considered output tracking problems for continuous-time nonlinear systems. For instance, in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Byrnes | Nonlinear internal models for output regulation[END_REF], [START_REF] Marconi | Output stabilization via nonlinear luenberger observers[END_REF] the problem of output tracking for nonlinear systems that can be written in the canonical normal form is considered. In [START_REF] Pavlov | Incremental passivity and output regulation[END_REF], incremental passivity concepts are used for the design of global regulators and in [START_REF] Pavlov | Global nonlinear output regulation: Convergence-based controller design[END_REF] output regulation of Lur'e-type systems using convergent system properties is considered. However, a constructive design of the stabilizer and guaranteed harmonic regulation properties are not presented. The design in [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF] relies on statefeedback approaches, and the domain of attraction of the periodic solution is only local in the size of the reference [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF] or not discussed [START_REF] Ghosh | Nonlinear repetitive control[END_REF]. In [START_REF] Giaccagli | Sufficient conditions for global integral action via incremental forwarding for inputaffine nonlinear systems[END_REF], contractive feedback laws in tracking problems are developed for constant references, but not for periodic references. Finally, in [START_REF] Verrelli | Linear repetitive learning controls for robotic manipulators by padé approximants[END_REF], [START_REF] Verrelli | PI-generalizing saturated repetitive learning control for a class of nonlinear uncertain systems: Robustness w.r.t. relative degree zero or one[END_REF], and [START_REF] Verrelli | Adaptive learning control design for robotic manipulators driven by permanent magnet synchronous motors[END_REF] a learning control approach has been developed to achieve tracking in nonlinear systems with repetitive disturbances. These methods show a significant improvement in tracking performance. In the scope of the challenge taken on in the current paper, a drawback of the work in [START_REF] Verrelli | Adaptive learning control design for robotic manipulators driven by permanent magnet synchronous motors[END_REF]- [START_REF] Verrelli | Linear repetitive learning controls for robotic manipulators by padé approximants[END_REF] is that these algorithms do not apply straightforwardly to Lur'e-type systems with (uncertain) output nonlinearities.

Although significant progress on output regulation for nonlinear systems has been made, an intuitive RC-scheme for nonlinear Lur'e-type systems with a formal stability guarantee is not yet available. To achieve this, a finite-dimensional realization of the exact RC-scheme in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF], [START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF] is used. The reason to consider a finite-dimensional realization is the difficulty to analyze the interconnection of an infinite-dimensional system, i.e., the internal model, with nonlinear plant dynamics with nonlinear outputs. In this paper, we follow an approach that relies on the harmonic representation of the delay, see, e.g., [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF], [START_REF] Mattavelli | Repetitive-based control for selective harmonic compensation in active power filters[END_REF]. The RC-scheme is implemented by including a finite number n o of linear oscillators in the control loop. This results in n o poles on the imaginary axes at the frequency of the periodic reference and its multiples. Therewith, if the resulting closed-loop trajectories converge to a periodic solution, harmonic regulation of the tracking error is guaranteed. More precisely, the Fourier coefficients of the error signal corresponding to the frequencies embedded in the linear oscillators are zero, and the L 2 -norm of the error signal is sufficiently small if n o is large enough [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF]. To guarantee the existence of globally asymptotically stable periodic solutions, the theory of convergent systems is exploited, see, e.g., [START_REF] Pavlov | Convergent systems: nonlinear simplicity[END_REF], [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF], [START_REF] Pavlov | Convergent systems: analysis and synthesis[END_REF], [START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF]. To this end, we suppose that the static nonlinearity in the Lur'e-type system satisfies an incremental sector bound condition. Then, using the strictly positive real lemma, sufficient conditions for a stabilizing output-feedback law are established. From a practical point of view, such an approach is interesting because the conditions can be checked by visual inspection of the Nyquist plot and linear analysis tools (potentially using measured data only).

Eventually, this proposed repetitive controller design is applied to the practical problem of mechanical ventilation. Existing literature, e.g., [START_REF] Reinders | Improving mechanical ventilation for patient care through repetitive control[END_REF] and [START_REF] Reinders | Triggered repetitive control: application to mechanically ventilated patients[END_REF], has shown that frequencydomain RC can significantly improve the tracking performance in ventilation systems. However, because these ventilation systems are nonlinear, formal stability guarantees for the closedloop system with repetitive control are missing. Therefore, the control approach developed in this paper is applied to this Lur'e-type ventilation system to improve its performance with formal stability guarantees.

Summarizing, the main contributions of this paper are:

• the development of an RC strategy for nonlinear Lur'etype systems including a formal stability analysis, and • the implementation and analysis of this RC scheme on the practical use-case of a nonlinear mechanical ventilator, including experimental validation.

This paper is organized as follows. In Section II, the problem statement is formalized. In Section III, the main results concerning the RC controller design are presented. Then, in Section IV, the RC paradigm is applied to the mechanical ventilation use-case. Finally, the main conclusions and recommendations for future work are presented in Section V.

Notations. Throughout this paper, s represents the Laplace variable. Given an n×n symmetric matrix P , we write P 0 (≺ 0) if P is strictly positive (negative) definite. Given a n×n matrix P , the operator blkdiag P . . . P represents a block-diagonal matrix with P as block-diagonal elements, the dimensions are specified case-wise. Furthermore, ẋ represents the continuous-time derivative of x. Finally, we define P T (r) as the set of C 1 T -periodic functions with bounded infinity norm and bounded infinity norm of its derivative. In particular, we say that r(t) | ṙ(t)| ≤ r for some non-negative real number r.

∈ P T (r) if r is C 1 , T -periodic,

II. PROBLEM STATEMENT

In the introduction, a gap in existing literature has been identified in the application of repetitive control to mechanical ventilation systems. More specifically, the commonly used frequency-domain RC does not provide formal stability guarantees when it is applied to Lur'e-type nonlinear ventilation systems. Therefore, in this section a formal problem statement is formulated to develop RC for Lur'e-type systems.

Consider a Single-Input Single-Output (SISO) Lur'e-type system of the form

ẋ = Ax + Bu + Ew y = M x + N w w = -ϕ(y) v = Cx + Dw, (1)
where x ∈ R n is the state, u ∈ R is the control input, w, y are in R, v ∈ R is the measured output, and A, B, E, M, N, C, and D are real matrices of appropriate dimensions. The static nonlinearity ϕ : R → R satisfies ϕ(0) = 0, and the following incremental sector bound condition:

φ ≤ ϕ(y 1 ) -ϕ(y 2 ) y 1 -y 2 ≤ φ ∀ y 1 = y 2 , (2)
for some known non-negative constants 0 ≤ φ ≤ φ. The control objective is to regulate the output v of the system (1) to a T -periodic bounded reference r ∈ P T (r). Hence, the output regulation error is defined as e(t) := r(t) -v(t) .

(

) 3
With the mechanical ventilation application in mind, where the full-state x is not available for feedback, we aim to design a dynamic output feedback controller for system (1), processing only the regulated output error e, such that harmonic regulation is achieved in the following sense.

Problem 1 (Harmonic regulation of order n o ). Consider system (1) with regulation error (3) and assume that the nonlinearity ϕ satisfies the incremental sector bound condition (2). Given any r > 0, n o > 0, determine a dynamic output feedback controller of the form

ż = ξ(z, e) u = ζ(z, e), (4)
such that for any reference r ∈ P T (r) and any initial condition x cl (0) := [x T (0), z T (0)] T ∈ R nx , the corresponding steadystate trajectory xcl := [x T , zT ] T of the closed-loop (1), ( 3), ( 4) is bounded, T -periodic, exponentially stable 1 , and the steadystate output error ē has no harmonic content at frequencies ω = k 2π T , k = 0, 1, . . . , n o . The repetitive control approach in [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF] is followed to achieve the harmonic regulation objective stated in Problem 1. The main idea is to include linear oscillators at the periodic reference frequency and its multiples in the regulator dynamics (4). This approach achieves, if the closed-loop stead-state trajectories are bounded and periodic, structural zeros at the frequencies kω with the blocking property of zeroing the Fourier coefficients of the output e corresponding to these frequencies. As a consequence, the strategy that we propose in this work is to: 1) design of the function ξ in (4) to include the linear oscillators; 2) design the feedback ζ in (4) to ensure the desired stability properties for the resulting closed-loop system; and 3) analyze the resulting trajectories and show that the harmonic content is zero at the desired frequencies. As a byproduct, we verify that if the number of oscillators included in the regulator are large enough, the asymptotic 2 -norm of the output e can be regulated to an arbitrarily small value.

III. REPETITIVE CONTROL OF LUR'E-TYPE SYSTEMS

In this section, the repetitive controller design for Lur'etype systems is presented. First, in Section III-A, the repetitive controller design is presented, the closed-loop dynamics are obtained, and a loop transformation is applied that allows the use of known results on exponentially convergent Lur'e-type systems. In Section III-B, known results on convergent Lur'etype systems are presented as a stepping stone to the stability analysis. Finally, in Section III-C, it is shown that the proposed controller design solves Problem 1.

A. Controller design

To provide a solution to Problem 1, the repetitive control approach in [START_REF] Ghosh | Nonlinear repetitive control[END_REF] is adopted by including n o linear oscillators in the control loop, at the reference frequency and its multiples, that process the output e to be regulated, as in standard output

1 a time-varying solution x(t) is called exponentially stable if x(x(0), t)- x(x(0), t) ≤ α x(0) -x(0) e -λt for some α, λ > 0.
regulation problems, see, e.g., [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], [START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF]. To this end, the control structure in ( 4) is defined as:

ż = ξ(z, e) := Φz + Γe (5) u = ζ(z, e) := Kz (6)
where

z = z 0 z T 1 . . . z T no T ∈ R (2no+1)×1 with z 0 ∈ R and z k ∈ R 2×1 for k = 1, . . . , n o
, and where the matrices

Φ ∈ R (2no+1)×(2no+1) , Γ ∈ R (2no+1)×1 , and K ∈ R 1×(2no+1)
are defined as

Φ := blkdiag(0 φ 1 . . . φ no ), Γ := γ 0 γ T 1 . . . γ T no T , K := [κ 0 κ 1 . . . κ no ] , (7)
where

φ k := k    0 2π T - 2π T 0    , k = 1, . . . , n o , (8)
with γ 0 = 0 the integrator gain such that z 0 embeds an integrator, and the matrix γ k ∈ R 2×1 is chosen such that the pair (φ k , γ k ) is controllable for any k = 1, . . . , n o . By construction, the pair (Φ, Γ) is therefore controllable. In this control structure, the z-dynamics represent the state-space representation of n o linear oscillators at the periodic reference frequency and its multiples. The number of oscillators n o represents a degree of freedom of the controller design as it defines the dimension of the chosen internal model and allows to suppress the first n o -harmonics of the steady-state trajectory which is formalized later.

Next, the closed-loop system consisting of the plant (1), (3) and the repetitive controller ( 5), ( 6) is written as a Lur'e-type system

ẋcl = A cl x cl + E cl w cl + d(t) y cl = M cl x cl + N cl w cl w cl = -ϕ(y cl ) (9)
where

A cl := A BK -ΓC Φ , E cl := E -ΓD , Q cl := 0 Γ , M cl := M 0 , N cl := N, (10)
where

x cl := [x T , z T ] T ∈ R nx , w cl := w, y cl := y, and
d(t) := Q cl r(t)
, is a periodic, with period time T , timevarying piece-wise continuous disturbance (induced by the periodic reference).

Next, a loop-transformation as described in [16, Chapter 7] is applied to the closed-loop dynamics. This looptransformation gives an equivalent Lur'e-type system where the transformed nonlinearity φ(y lt ) satisfies the incremental sector bound in (2) with φ = 0 and φ = ∞. This enables direct application of the known results on exponentially convergent Lur'e-type systems in Section III-B. This loop-transformation gives the following loop-transformed Lur'e-type system:

ẋlt = A lt x lt + E lt w lt + d(t) y lt = M lt x lt + N lt w lt w lt = -φ(y lt ) (11)
where

A lt := A cl -(E cl φ(M cl + N cl (1 + φN cl ) -1 φM cl )), E lt := E cl (1 - φD cl (1 + φN cl ) -1 ), M lt := φM cl -φN cl (1 + φN cl ) -1 φM cl , N lt := 1 + φN cl (1 + φN cl ) -1 (12)
where

x lt ∈ R nx , y lt ∈ R, w lt ∈ R, φ = φ - φ,
and φ(y lt ) satisfies the incremental sector bound in (2) with φ = 0 and φ = ∞. Furthermore, is assumed that the controller is designed such that Assumption 1 holds.

Assumption 1. The pair (A lt , E lt ) is controllable and the pair (A lt , M lt ) in (11) is observable.

Next, to solve Problem 1 it must be shown that the closedloop system exhibits a globally exponentially stable steadystate trajectory that is well-defined, bounded, T -periodic, and that the associated output error ē has no harmonic content at the frequencies included in the internal model. To show this, known results on exponentially convergent Lur'e-type systems are used. These results are provided next.

B. Exponentially convergent Lur'e-type systems

First, we provide the following definition of convergent systems, see, e.g., [START_REF] Pavlov | Convergent systems: nonlinear simplicity[END_REF], [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF], [START_REF] Rüffer | Convergent systems vs. incremental stability[END_REF], applicable to the Lur'etype systems of the form [START_REF]DEMCON macawi respiratory systems[END_REF] [START_REF] Pavlov | Convergent systems: nonlinear simplicity[END_REF], [START_REF] Pavlov | Frequency response functions for nonlinear convergent systems[END_REF], [START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF].

Property 1. Consider system [START_REF]DEMCON macawi respiratory systems[END_REF] and suppose it is exponentially convergent. If d is a periodic signal with period T > 0, i.e., d(t) = d(t + T ) for all t, then the corresponding steadystate solution xlt,d (t) is also periodic with period T .

To show that the closed-loop Lur'e-type system of the form (11) is a globally exponentially convergent system, let

H(s) = M lt (sI -A lt ) -1 E lt + N lt . (13)
Then, results from [START_REF] Va Yakubovich | Matrix inequalities method in stability theory for nonlinear control systems: I. absolute stability of forced vibrations[END_REF] [START_REF] Pavlov | Convergent systems: nonlinear simplicity[END_REF], [START_REF] Va Yakubovich | Matrix inequalities method in stability theory for nonlinear control systems: I. absolute stability of forced vibrations[END_REF]:

1) H(s) is Hurwitz; 2) Re{H(jω)} > 0 ∀ω ∈ R; and 3) H(∞) > 0 or H(∞) = 0 and lim ω→∞ ω 2 H(jω) > 0. Note that the SPR conditions on H(s) can be visually verified with the Nyquist plot, see [START_REF] Khalil | Nonlinear systems[END_REF]Chapter 7]. This makes these conditions particularly useful to verify in practical applications, see Section IV. Next, these results are used to design the feedback gain K in the feedback law [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF], such that Problem 1 is solved.

C. Harmonic regulation of Lur'e-type systems

The results of Lemma 1 enable the main theoretical result of this paper, which solves Problem 1. More specifically, it is shown that the presented repetitive controller achieves the desired harmonic regulation properties and convergence properties of the closed-loop system if the feedback gain K is designed properly.

If we can design the feedback law ( 6) such that the closedloop (1), ( 3), ( 5), ( 6) is globally exponentially convergent for any d(t), then by virtue of Definition 1 and Property 1, for every initial condition the solutions of ( 9) exponentially converge to a unique, bounded, and well-defined steady-state solution, which is T -periodic if the reference r(t) is Tperiodic. Then, using Lemma 2 below [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF], [START_REF] Ghosh | Nonlinear repetitive control[END_REF], it is shown that the Fourier coefficients, of the associated steady-state solution ē of the error e, corresponding to the n o frequencies embedded in the internal model [START_REF] Bin | About robustness of control systems embedding an internal model[END_REF], must be zero. Hence, harmonic regulation is achieved. Lemma 2. Let the steady-state solution (x, z) be a bounded trajectory of the cascade (1), (3), and (5), with corresponding steady-state output error ē(t). Suppose that ē(t + T ) = ē(t) for all t ≥ 0. Then, necessarily

T 0 cos(k 2π T t)ē(t)dt = T 0 sin(k 2π T t)ē(t)dt = 0 , (14)
for all k ∈ [0, 1, . . . , n o ]. Moreover, for any compact set C x ⊂ R n , for any r > 0, ū > 0, and ε > 0 such that x(t) ∈ C x , r ∈ P T (r), and |ū(t)| ≤ ū for all t ≥ 0, there exists n * o ≥ 1 such that the following holds:

ē(t) L 2 := T 0 |ē(t)| 2 dt 1 2 ≤ ε , ∀ n o ≥ n * o . ( 15
)
Note that a bound for ε can be computed following the proof of Proposition 3 in [START_REF] Astolfi | Approximate regulation for nonlinear systems in presence of periodic disturbances[END_REF]. This bound depends on the Lipschitz constant of the nonlinearity.

Then, combining these harmonic regulation properties and the convergent system properties of the loop-transformed closed-loop system solves Problem 1, which brings us to the result in Theorem 1.

Theorem 1. Consider the Lur'e-type system (1), [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF], with the nonlinearity ϕ(•) satisfying the incremental sector condition in (2), in closed loop with a dynamical controller (5), [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF]. Given an arbitrary integer n o > 0 and suppose that the matrix K is chosen such that the transfer function H(s) in Lemma 1, with the matrices A lt , E lt , M lt , N lt defined in [START_REF] Costa-Castelló | A repetitive controller for discrete-time passive systems[END_REF], satisfies Assumption 1 and the SPR conditions in Lemma 1. Then, Problem 1 is solved, namely harmonic regulation of order n o , as defined by [START_REF] Hara | Repetitive control system: a new type servo system for periodic exogenous signals[END_REF], is achieved. Proof. Consider system (1), (3) in closed-loop with ( 5), [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF], which can be written in the form of ( 9) and [START_REF] Costa-Castelló | A repetitive controller for discrete-time passive systems[END_REF]. Then, the loop-transformation can be applied to obtain [START_REF]DEMCON macawi respiratory systems[END_REF] and [START_REF] Ghosh | Nonlinear repetitive control[END_REF]. Since the conditions of Lemma 1 are satisfied and the loop-transformed system is equivalent to the original closedloop system, the closed-loop system is globally exponentially convergent. Hence by Property 1, if r(t) is periodic with period T > 0, there exists a bounded, globally exponentially stable solution x(t), z(t)) which is T -periodic. As a consequence, the resulting output steady-state trajectory ē is also bounded and T -periodic. By direct application of Lemma 2 it satisfies [START_REF] Hara | Repetitive control system: a new type servo system for periodic exogenous signals[END_REF]. This concludes the proof.

The statement of Theorem 1 establishes a set of sufficient conditions for the design of the regulator in ( 5) and [START_REF] Blanken | Multivariable repetitive control: Decentralized designs with application to continuous media flow printing[END_REF]. In particular, the matrices K and Γ should be designed such that the desired SPR conditions on H(s) are satisfied to ensure the satisfaction of the conditions in Lemma 1. The SPR conditions at the end of Section III-B can be supported by graphical checks in a Nyquist plot, similar to frequency-domain design techniques for linear controller design.

In case the system in ( 9) is a minimum-phase system with unitary relative degree 2 , a systematic design of the gain K can be done by following [START_REF] Astolfi | Nonlinear robust periodic output regulation of minimum phase systems[END_REF]. The system in (9) can be put in this form, for instance when CB = 0 and D = 0. In such case, it can be put in the canonical normal form following [START_REF] Isidori | Nonlinear control systems[END_REF]Chapter 4]. Then, additional properties can be established. In particular, by selecting K such that the bound

KΓΓ T K T ≤ a (16)
holds with a a positive and bounded scalar which is independent of n o , it can be proved that the asymptotic L 2 -norm of ē can be made arbitrarily small by increasing the number of oscillators. More specifically, in such case in [START_REF] Isidori | Nonlinear control systems[END_REF] can be made arbitrarily small by increasing the number of oscillators n o . For instance, one can match the condition ( 16) by selecting the gains γ k in [START_REF] Borrello | Modeling and control of systems for critical care ventilation[END_REF] as γ k = k -(1+ ) γk with γk so that |γ k | ≤ γ for any k = 1, . . . , n o , for some γ > 0, and by selecting κ k in [START_REF] Borrello | Modeling and control of systems for critical care ventilation[END_REF] such that |κ k | ≤ κ for any k = 1, . . . , n o , for some κ > 0. Note that the condition [START_REF] Khalil | Nonlinear systems[END_REF] essentially establishes that the regulator (5), ( 6) has an L 2 -gain between the input e and the output u which does not depend on the number of oscillators n o . The complete proof of this property is given in [3, Lemma 3]. Intuitively, the main idea behind such a choice is to put decreasing weights on higher harmonics in order to 2 In this case, we refer to a system in normal form with stable zerodynamics. See for instance, (5) and Assumption 2 in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF]. Necessary and sufficient conditions under which a system of the form ( 9) can be written in canonical normal form are well known in the literature, see, e.g., [START_REF] Isidori | Nonlinear control systems[END_REF]. obtain a bounded gain when summing up the contribution of the L 2 -gain of all oscillators. This design philosophy will be pursued in the mechanical ventilation application to experimentally show the desired approximate L 2 output regulation objective [START_REF] Isidori | Nonlinear control systems[END_REF].

p out p aw p lung Q leak Q pat Q out p aw pilot line R hose R lung

IV. APPLICATION TO MECHANICAL VENTILATION

In this section, the RC strategy is applied to a nonlinear mechanical ventilation system, i.e., this section describes the second contribution of this paper. First, in Section IV-A, an overview of the considered ventilation system and the control goal for ventilation are described. Thereafter, the mathematical ventilation model and the actual mechanical ventilation setup are presented in Section IV-B. Then, repetitive controllers for mechanical ventilation are designed and stability of the closedloop system is analyzed in Section IV-C. Then, in Section IV-D, the experimental results are presented and analyzed. Thereafter, another ventilation use-case is briefly considered to analyze the conservatism of Theorem 1 in Section IV-E. Finally, a remark on RC design is made based on observations from the experimental case study.

A. Ventilation system overview and control goal

Mechanical ventilators are essential equipment in Intensive Care Units (ICUs) to assist patients who cannot breathe on their own or need support to breathe sufficiently. The goal of mechanical ventilation is to ensure adequate oxygenation and carbon dioxide elimination [START_REF] Warner | Mechanical ventilation[END_REF], and thereby sustaining the patient's life. Next, the considered ventilation system and corresponding control goal are described. 1) Ventilation system overview: A schematic overview of the considered ventilation system is depicted in Fig. 1. The main components of this system are the blower, the hosefilter system, and the patient. A centrifugal blower compresses ambient air to achieve the desired blower outlet pressure p out . The difference between p out and the airway pressure p aw results in the outlet flow Q out through the hose. This hose is modeled using a nonlinear hose model. The flow through the hose, i.e., the outlet flow Q out , is divided into a patient flow Q pat and a leak flow Q leak . The intended leak near the patient is used to flush CO 2 -rich air from the system. Finally, the patient's lungs are inflated and deflated by the patient flow.

2) Control goal: In this experimental use-case, Pressure Controlled Ventilation (PCV) is considered. A schematic example of PCV is depicted in Fig. 2. In PCV, the pressure near the patient's mouth, the airway pressure p aw , should track a desired pressure target p target , i.e., r := p target . On a preset periodic interval, of length T , the pressure level is increased to the Inspiratory Positive Airway Pressure (IPAP), and consequently lowered to the Positive End-Expiratory Pressure (PEEP). These varying pressure levels ensure the desired airflow in and out of the patient's lungs. The total breath length T consists of the inspiration time T i and expiration time T e , i.e., T = T i + T e . The control goal for PCV is to achieve a small tracking error e := r-p aw = r-v, where the reference r(t) is a time-varying signal that is perfectly periodic with an interval length T , i.e., r(t) = r(t + T ) for a known T > 0 and all t ≥ 0. Because of this periodicity property and the nonlinear nature of the hose model, the Repetitive Control (RC) strategy developed in this paper is particularly suitable for this application.

B. Mathematical model and experimental ventilation system

For controller design and the stability analysis, a mathematical Lur'e-type system model is derived. The ventilation model is based on [START_REF] Reinders | Accurate pressure tracking to support mechanically ventilated patients using an estimated nonlinear hose model and delay compensation[END_REF]. Thereafter, the actual experimental ventilation setup is presented.

1) Mathematical model of the ventilation system: In this section, first the separate models for the plant components are derived, i.e., blower model G b , hose model R hose , and patient-leak model G p . Thereafter, these models are combined to obtain the open-loop Lur'e-type ventilation system model for the controller design and associated stability analysis. The complete plant and the considered control strategy are visualized in the blockdiagram in Fig. 4.

The blower model G b is obtained by means of a sixthorder fit of a Frequency Response Measurement (FRF) of the actual blower dynamics [START_REF] Pintelon | System Identification: a frequency domain approach[END_REF]. This state-space model accurately describes the input-output relation of the blower, i.e., from the control signal p c to the blower output p out . The measured FRF and the blower model G b are depicted in Fig. 3, showing that G b is an accurate representation of the FRF measurement of the actual blower. The blower G b is modeled as the following state-space system:

ẋb = A b x b + B b p c p out = C b x b , (17)
with x b ∈ R 6 , p c ∈ R, p out ∈ R, and system matrices of appropriate dimensions. The hose is modeled by the nonlinear hose resistance R hose , as presented in [START_REF] Reinders | Accurate pressure tracking to support mechanically ventilated patients using an estimated nonlinear hose model and delay compensation[END_REF], which describes the relation between the flow through the hose Q out and the pressure drop over the hose ∆p := p out -p aw . From experiments it is concluded that the hose can be modeled as follows:

Q out : = R hose (∆p) = sign(∆p) -R 1 + R 2 1 + 4R 2 |∆p| 2R 2 , (18)
where R 1 and R 2 are the hose-resistance parameters.

Next, the combined patient-leak model G p describes the relation between the outlet flow Q out and the system output y = p aw . This patient model is described by the following first-order state-space model, based on the linear onecompartmental lung model in [START_REF] Bates | Lung Mechanics[END_REF]:

ṗlung = a p p lung + b p Q out p aw = c p p lung + d p Q out ( 19
)
with

a p = - 1 C lung (R leak + R lung ) , b p = R leak C lung (R leak + R lung ) , c p = R leak R leak + R lung , d p = R leak R lung R leak + R lung . ( 20
)
Finally, these separate models are combined to obtain the open-loop plant model, as depicted inside the dashed box in Fig. 4. Note that an additional term η∆p is added to the nonlinear hose-resistance, i.e., ϕ(∆p) := R hose (∆p) + η∆p, and subtracted in the parallel path; this is included to ensure that the linear dynamics of the open-loop plant in Lur'etype form are controllable and observable. The total system's dynamics, i.e., the full Lur'e-type ventilation system, are independent of the choice of η ∈ R.

To obtain the open-loop plant model, the blower, hose, and patient model are combined. This gives the open-loop model from p c to p aw in the form of ( 1). The open-loop ventilation system is defined by (1) with the system matrices

A = A b 0 -(1 -ηd p ) -1 ηC b b p a p + ηc p (1 -ηd p ) -1 b p , B = B b 0 , E = 0 -b p (1 -ηd p ) -1 , M = C b + d p (1 -ηd p ) -1 ηC b -c p -d p (1 -ηd p ) -1 ηc p T , N = d p (1 -ηd p ) -1 , D o = -d p (1 -ηd p ) -1 , C = -d p (1 -ηd p ) -1 ηC b c p + d p (1 -ηd p ) -1 ηc p , (21)
and the nonlinearity ϕ(y) := R hose (y) + ηy.

These open-loop system matrices and the nonlinearity in combination with the RC that is designed in Section IV-C are used to retrieve the closed-loop ventilation system and to guarantee that it solves Problem 1 using Theorem 1.

2) Experimental ventilation setup:

The main components of the experimental setup used in this case study are depicted in Fig. 5. The figure shows the Macawi blower-driven mechanical ventilation module [START_REF]DEMCON macawi respiratory systems[END_REF]. The dSPACE system (dSPACE GmbH, Paderborn, Germany) is used to implement the controls in MATLAB Simulink (MathWorks, Natick, MA). Furthermore, the ASL 5000 TM Breathing Simulator (IngMar Medical, Pittsburgh, PA) represents the patient. This lung simulator can be used to emulate a wide variety of patients with a linear resistance and compliance. Furthermore, a typical ventilation hose with leak is used to attach the ventilation module to the lung simulator. The system parameters that are used for the stability analysis are shown in Table I. The leak and hose parameters are obtained by a calibration and the patient parameters are the settings used on the mechanic lung simulator, i.e., patient emulator in Fig. 5.

The analysis in the following section is done using a continuous-time representation of the controller and plant model. However, the controller is implemented in dSPACE using a discrete-time representation of the continuous-time control strategy. The discrete-time controllers are obtained using the zero-order hold discretization scheme at a sampling frequency of 500 Hz. This sampling frequency is significantly higher than the relevant system dynamics, e.g., the blower shows strong roll-off at frequencies above 10 Hz. Furthermore, 500 Hz is significantly higher that the frequency content of the reference signal. Therefore, the continuous-time controller

C. Controller design for mechanical ventilation

Next, the RC controller design for mechanical ventilation is described and it is shown that it solves Problem 1 for this ventilation use-case.

1) Controller design: For the design of the feedback controller C in Fig. 4, the control strategy in ( 5), [START_REF] Borrello | Modeling and control of systems for critical care ventilation[END_REF], and (8) with feedback law (6) is followed. This means that the feedback controller C consists of an integrator and n o oscillators from the first up until the n th o harmonic of the breathing frequency ω b = 2π/T rad/s. Besides this feedback controller, a unity feedforward controller as depicted in Fig. 4 is used. The unity feedforward term is included to improve the overall regulation accuracy. Note that it does not affect stability since it is included in the closed-loop ventilation system through the disturbance term d in [START_REF] Califano | Stability analysis of nonlinear repetitive control schemes[END_REF]. The stability analysis is independent of this disturbance in view of the convergence properties of the closed-loop dynamics.

For the final RC design, different controllers are designed to analyze the effect of the number of oscillators, i.e., n o ∈ {0, 1, 5, 15, 20}. We select the integrator gain as γ 0 = 2π, and oscillator gains as γ k = 1 1 2 k 1+ with = 0.4, for k = 1, 2, . . . , n o . The feedback law is chosen as K ∈ R 1×(2no+1) with all entries 1. Note that the design of the gains Γ, K satisfies the condition [START_REF] Khalil | Nonlinear systems[END_REF]. Next, the stability properties of the closed-loop ventilation system with the RC controller are analyzed.

2) Stability analysis: To guarantee exponential convergence of the closed-loop ventilation system, and therewith showing that Problem 1 is solved, Theorem 1 is verified. First of all, the controlled system is written in the closed-loop form of ( 9), and the upper φ and lower φ sector bounds of the nonlinearity ϕ(y) in ( 22) are computed. Using these bounds, the loop-transformation is applied to obtain the system in [START_REF]DEMCON macawi respiratory systems[END_REF]. Thereafter, Lemma 1 is verified, which ensures that Theorem 1 holds.

The upper sector bound φ is defined by taking the derivative of ϕ(∆p) at the origin, where the slope of ϕ is the largest, see Fig. 6, which gives φ = 1 R1 + η. The lower sector bound φ is obtained from visual inspection, such that it holds on a finite domain of ∆p ∈ [-20, 20] mbar; this domain is sufficient for the practical application of ventilation. This leads to the sector ϕ ∈ [ φ, φ] = [80, 1 R1 + η] for the nonlinearity in [START_REF] Pavlov | Incremental passivity and output regulation[END_REF]. The nonlinearity and these sector bounds are visualized in Fig. 6.

Using these sector bounds, the loop-transformation is performed to obtain the system in [START_REF]DEMCON macawi respiratory systems[END_REF], and it is verified that the pair (A lt , E lt ) is controllable and the pair (A lt , M lt ) is observable for every n o , i.e., Assumption 1 holds. Thereafter, H(s) is constructed using the matrices of the loop-transformed system.

Then, it is guarantee that H(s) is SPR, it is first verified that for all n o ∈ {0, 1, 5, 15, 20} the transfer function H(s) is Hurwitz, which is verified by computing the poles and checking that they reside in the open left-half plane. Thereafter, it is graphically validated that Re(H(jω)) > 0 ∀ω ∈ [-∞, ∞]. This is validated in Fig. 7; it is clearly shown that for all considered values of n o the real part of H(jω) is strictly positive. Finally, it is verified that H(∞) > 0. This is also the case for all n o ∈ {0, 1, 5, 15, 20}.

From these results, Lemma 1, and Theorem 1, it is concluded that the nonlinear closed-loop ventilation system is exponentially convergent on a compact domain3 in state space for which ∆p ∈ [-20, 20] mbar and that this controller solves the repetitive control problem. Next, the performance of the different controllers is analyzed by means of experiments.

Remark 1. To ensure stability of systems with slowly varying parameters (which is typically the case for this mechanical ventilation case) an approach similar to [START_REF] Reinders | Improving mechanical ventilation for patient care through repetitive control[END_REF] could be followed. One could design a single controller for a nominal parametric setting and prove the stability property for a variety of (slowly, i.e., quasi-constant) varying parameters. This can be achieved by ensuring the SPR property for the linear dynamics of the Lur'e-type system for a set of parameters, e.g., lung compliances.

D. Experimental results for mechanical ventilation

The main experimental results are shown in Fig. 8 and9. The time-domain results of the 20 th breath with the integrator only, i.e., n o = 0, and the repetitive controller with 20 oscillators, i.e., n o = 20, are visualized in Fig. 8. The top plot shows the reference and the measured outputs and the bottom plot shows the tracking error for both controllers. The figure clearly shows that the tracking error is significantly reduced by the repetitive controller. The overshoot is eliminated and the rise-time is significantly shorter. Note that the residual error with repetitive control contains oscillatory behavior, especially during the expiration at PEEP level, i.e., between 82 and 84 seconds. These oscillations contain mostly frequency content higher than 20 times the breathing frequency, i.e., above 5 Hz. It is observed that the tracking error's frequency content at frequency above the n th o harmonic is increased. In Section IV-F, a remark and analysis of this phenomenon is included, since this phenomenon could potentially deteriorate the system's tracking performance. Concluding, all controllers show convergent behavior in the experiments, as expected by the analysis. Furthermore, the tracking error is reduced significantly, by more than a factor 3, by including repetitive control. The Fourier coefficients of the steady-state output error ē(t) are suppressed at the frequencies w = k 2π

T , k = 0, 1, . . . , n o .

E. Analysis of conservatism

To analyze how conservative the convergence properties of Theorem 1 are, an experimental use-case is presented where the SPR properties are violated. This is achieved by considering a ventilation use-case with lung parameters that represent a baby patient, i.e., C lung = 3 mL/mbar and R lung = 50 mbar s/L. The same hose and blower system as for the adult use-case are used, hence, the same sector conditions for the linearity can be used. Furthermore, the same RC design as for the adult use-case is followed for n o = 20. The transfer function H(s) is computed for this system and visualized in Fig. 10. This figure clearly shows that the second condition for SPR transfer functions is violated for n o = 20. Therefore, the desired convergence properties of the system cannot be guaranteed for this controller design with n o = 20 oscillators.

The resulting 2 -norm of the error per breath is shown in Fig. 11. This figure clearly shows that the system behaves unstable for n o = 20. Concluding, this use-case shows that the sufficient conditions in Theorem 1 have limited conservatism, which is a desirable property for practical controller design because it allows more design freedom.

F. Remark on repetitive controller design

In the experimental analysis, especially in the baby usecase, it is observed that the remaining error consists of oscillations at frequencies above the harmonics of the n th o oscillator. These oscillations in the error are increasing for an increasing number of oscillators, limiting the overall tracking performance. Especially in other use-cases it is observed that increasing the number of oscillators can significantly deteriorate the system performance. This effect can be explained by analyzing the sensitivity S re , i.e., transfer function from the reference r to the tracking error e, of a linearization of the closed-loop ventilation system. This linearized closed-loop system is obtained by replacing the nonlinearity in Fig. 4 by a linear resistance, i.e., R hose (∆p) is replaced by ∆p R lin with R lin = 2 φ+ φ , and η = 0. The resulting Bode magnitude plot of S re is shown in Fig. 12. This Bode magnitude plot clearly shows that the tracking error is zero at the harmonics of the breathing frequencies. However, it also shows an increase in magnitude at frequency above the oscillator frequencies. The magnitude at these frequencies is increasing for an increasing number of oscillators. This increase in magnitude causes the oscillations at these frequencies as shown in the experiments. Therefore, in future work, it should be analyzed how this increase in magnitude at these specific frequencies can be eliminated.

V. CONCLUSION AND FUTURE WORK

In this paper, a Repetitive Control (RC) scheme that achieves robust tracking for nonlinear Lur'e-type systems with stability guarantees is presented. The RC scheme is composed of a dynamical system consisting of n o linear oscillators at the reference's period and its multiples, which represents the internal model, processing the tracking error, and a pure integral controller guaranteeing the closed-loop system to be convergent. This convergence property ensures that the steadystate trajectory is periodic and therefore harmonic regulation is achieved at the frequencies included in the internal model. This RC scheme is successfully implemented in a mechanical ventilation system for Intensive Care Units, a medical application to support the breathing of patients. Through a stability analysis based on the Nyquist plot it is shown that this closed-loop ventilation system is convergent and hence the designed controller solves the repetitive control problem at hand. Additionally, it is also shown experimentally that by increasing the number of oscillators the asymptotic L 2norm of the regulated output is reduced. Furthermore, experiments show that the presented controller design is able to significantly improve pressure tracking when compared to pure integral action.

Fig. 1 .

 1 Fig. 1. Schematic representation of the blower-hose-patient system, with the corresponding resistances, lung compliance, pressures, and flows.

Fig. 2 .

 2 Fig. 2. Typical airway pressure for two breathing cycles of pressure controlled ventilation, showing the set-point () and the typical response ( ).

Fig. 3 .

 3 Fig. 3. Frequency response measurement () and 6 th order identified parametric model ( ) of the blower, i.e., from pc to pout.

Fig. 4 .

 4 Fig. 4. Block diagram of the full ventilation system, with G b the blower dynamics, C an arbitrary feedback controller, Gp the patient-leak dynamics, and ϕ(∆p) = R hose (∆p) + η∆p the nonlinear hose model.

Fig. 5 .

 5 Fig. 5. The experimental setup with the mechanic patient simulator, the respiratory module, ventilation hose, and dSPACE module.

Fig. 6 . 1 R 1 +

 611 Fig. 6. Visualization of the nonlinearity ϕ(∆p) ( ), and its sector bounds φ∆p ( ) and φ∆p ( ), showing that the incremental sector condition holds for [ φ, φ] = [80, 1 R 1 + η].

Fig. 7 .

 7 Fig. 7. Nyquist plot of H(s) for no = 0 ( ), no = 1 ( ), no = 5 ( ), no = 15 ( ), and no = 20 ( ). The figure shows that Re(H(jω)) > 0∀ω ∈ [-∞, ∞].

Fig. 8 .

 8 Fig. 8. The time domain results upon convergence for no = 0 () and no = 20 ( ), and the target pressure ( ). The figure shows that the error is significantly reduced by the repetitive controllers.

Fig. 9 . 2

 92 Fig. 9. The 2 -norm of the error for every breath for no = 0 ( ), no = 1 ( ), no = 5 ( ), no = 15 ( ), and no = 20 ( ). The figure shows that more oscillators results in a smaller error and the controllers converge in approximately 10 breaths.

Fig. 10 .

 10 Fig. 10. Nyquist plot of H(s) for no = 20 ( ) for the baby use-case. The figure shows that Re(H(jω)) > 0∀ω ∈ [-∞, ∞] does not hold, hence, convergence is not guaranteed.

Fig. 11 .

 11 Fig. 11. The 2 -norm of the error for every breath for no = 0 ( ), and no = 20 ( ) for the baby use-case. The figure shows unstable behavior for that the closed-loop system with no = 20.

Fig. 12 .

 12 Fig. 12. Bode magnitude plot of the sensitivity Sre for the linearization of the closed-loop system with no = 0 ( ), no = 1 ( ), no = 5 ( ), no = 15 (), and no = 20 ( ). The figure shows a magnitude increase at frequencies around 8 Hz, causing oscillations at these frequencies.

  , with D a set of piece-wise continuous, bounded disturbances. System (11) is called convergent for d ∈ D if it is convergent for any d(t) ∈ D see [27, Definition 2.16]. Note that for an exponentially convergent system, the steady-state solution is unique, see [27, Property 2.15]. Moreover, if the input d(t) is T -periodic, then, for exponentially convergent systems, xlt,d (t) is also T -periodic, as recalled in the next property, see, e.g.,

	Definition 1. Given d(t) ∈ D, the system (11) is said to be
	globally exponentially convergent if

• there exists a solution xlt,d (t) defined and bounded for all t ∈ R; • the solution xlt,d (t) is globally exponentially stable.

  Let Assumption 1 hold. If (2) holds and the transfer function H(s) is Strictly Positive Real (SPR), then the system (11) is globally exponentially convergent. Using Lemma 1, global exponential convergence of the closed-loop Lur'e-type system in (11) can be guaranteed by showing that H(s) is SPR. The transfer function H(s) is SPR if and only if the following conditions hold, see

and

[START_REF] Pavlov | Uniform output regulation of nonlinear systems: a convergent dynamics approach[END_REF] Chapter 5

] can be used. By combining the definition of a Strictly Positive Real (SPR) transfer function and the incremental sector bound condition (2), Lemma 1 is obtained that expresses sufficient conditions that guarantee that the system (11) is globally exponentially convergent, which is proved in

[START_REF] Khalil | Nonlinear systems[END_REF] Chapter 7]

.

Lemma 1.

TABLE I THE

 I RELEVANT SYSTEM AND EXPERIMENT PARAMETER FOR THE STABILITY ANALYSIS.

	Parameter Value Unit
	R leak	24	mbar s/L
	R 1	2.8	mbar s/L
	R 2	1.6	mbar s 2 /L 2
	R lung	5	mbar s/L
	C lung	50	mL/mbar
	η	-0.1	mL/s/mbar
	design and stability analysis is deemed relevant for this appli-
	cation.

Such domain can be explicitly formulated using a quadratic Lyapunov function following from the Kalman-Yakubovich-Lemma for the SPR transfer function H(s).


Giuseppe Peano

Paola Cantù

Erika Luciano

Keywords: Hilbert, Bernays, Wilson, Amaldi, Enriques, Veronese, Vivanti and Bettazzi)

Peano's axioms for arithmetic, published in 1889, are ubiquitously cited in the writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell's Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics is still to be achieved. In line with an increased interest of philosophy of mathematics in the investigation of mathematical practices, this thematic issue adds some contributions to a possible reconstruction of the philosophical views of the Peano School emerging from logical, mathematical, linguistic, and educational practices, as well as from the interactions with contemporary scholars in Italy and abroad (Cantor, Dedekind, Frege, Russell,

The Peano School

It is debatable whether the group can be classified as a "scientific school", and an exhaustive list of all the members be uniquely determined. The category of research mathematical school, explored in its distinctions and national features by David Rowe (2002), has recently been opposed to the category of a mathematical tradition. According to José Ferreiros (1999, pp. xxii-xxiii) a research mathematical school is "a group led normally by only one mathematician, localized within a single institutional setting, and counting on a significant supply of advanced students", whereas a mathematical tradition "implies that one can find a common research orientation in different actors that do not share a common institutional site, but are linked by traceable influences on each other". To settle the question whether the Peano group should be considered as a research school or as a mathematical tradition, one has to deconstruct several clichés occurring in the literature and clarify the nature of Peano's leadership, the circulation of knowledge within the Peano School, and the role of other collective enterprises beside the Formulario (e.g., the Rivista di Matematica, the journal Schola et Vita, the Dizionario, as well as other contemporary articles and teaching materials). Original contributions have been recently achieved through the exploitation of new archival sources: the discovery of new collaborators not previously mentioned, the distinction of different levels of decision-making in Peano's redaction of the Formulario, as well as new insights on the original contribution of each member to shared knowledge in the group [START_REF] Luciano | Characterizing a Mathematical School. Oral Knowledge and Peano's Formulario[END_REF].

The philosophical interest

The school is generally considered as a phenomenon that appeared suddenly in all its splendor at the Paris congress of 1900 and then extinguished with the rapidity of a firework that leaves a vivid but indefinite memory. Given the long-lasting impression made on Russell and other participants in the 1900 Paris Conferences in Mathematics, Philosophy and Psychology by Peano's, Burali-Forti's, Padoa's, Pieri's, and Vailati's contributions, the literature has often searched for reasons to explain a general loss of philosophical interest in the Peano school in the first half of the 20th century. General explanations abound in the literature: the nonacademic nature of the group; the multiform topics of interest going from mathematical analysis to geometry, from linguistics to universal languages, from philosophical pragmatism to logicism [START_REF] Roero | Peano e la sua scuola fra matematica, logica e interlingua[END_REF]; the scarce attention given to the transformation of mathematics and to the development of set theory after 1910; a general belief that Peano was not really interested in the theory of inferential reasoning, or in the metalogical and metamathematical investigation of the properties of axiomatic theories (van Hejenhoort 1967).

Other philosophical explanations have also been suggested: Peano's utilitarian approach to logic (Grattan Guinness 2000); the lack of a shared and explicit epistemological framework concerning relevant logical and methodological issues such as functions [START_REF] Luciano | Characterizing a Mathematical School. Oral Knowledge and Peano's Formulario[END_REF][START_REF] Cantù | Origins and Varieties of Logicism[END_REF], logical identities [START_REF] Cantù | Il carteggio Padoa-Vailati. Un'introduzione alle lettere inviate da Chioggia[END_REF], definitions by abstraction [START_REF] Mancosu | Definitions by abstraction in the Peano school[END_REF], and questions of purity [START_REF] Arana | On the relationship between plane and solid geometry[END_REF]; a subdivision of labor that made Giovanni Vailati in Italy [START_REF] Arrighi | Logic and Pragmatism. Selected Writings of Giovanni Vailati[END_REF] and Louis Couturat in France [START_REF] Luciano | Archivio della corrispondenza degli scienziati italiani[END_REF] the chief philosophical spokesmen of the group; the belief that Peano's presentation of arithmetical axioms had less interesting philosophical implications with respect to logicism and structuralism than that of Dedekind (Ferreiros 2005); the interest of Peano's collaborators in pedagogical and political issues [START_REF] Giacardi | From Euclid as textbook to the Giovanni Gentile reform (1867-1923): Problems, methods and debates in mathematics teaching in Italy[END_REF][START_REF] Luciano | The Proposals of the School of Peano on the Rational Teaching of Geometry. In Dig where you stand[END_REF]).

The topic is reconsidered in a new light in this special issue, as the authors discuss the relationship between Dedekind's and Peano's axioms (Kahle, this volume), the absence of the universal quantifier among the primitive symbols of Peano's Formulario and its relation to the use of free variables (von Plato, this volume), the peculiarities of Peano's symbolic notation (Schlimm, this volume), the lack of recognition of Pieri's pedagogical remarks in Italy (Marchisotto and Millan Gasca, this volume), the early association in the USA with Russell's point of view (Lolli, this volume), the interaction between Peano's auxiliary international language project and the internationalization movement at the beginning of the century (Aray, this volume), the limits of Peano's proof of the impossibility of infinitesimals (Freguglia, this volume).

Logic and epistemology

Some of the usual explanations lose their effectiveness, not only because of some new specific results, but also because of the interdisciplinary and practical turn suggested by the intertwining of logic and epistemology, whereby the latter is taken to mean, as in neo-Latin languages, both the analysis of scientific knowledge and the critique of scientific theories. This perspective constituted the red thread of an international project (PICS INTEREPISTEME 2018-2020) co-funded by the French National Research Center and the Vienna Circle Institute and co-directed by Paola Cantù and Georg Schiemer in collaboration with Erika Luciano at the University of Turin. The objective was to compare three distinct collaborative and interdisciplinary epistemologies developed by the members of the Peano School, the editorial board of the Revue de Métaphysique et de Morale, and the Vienna Circle. The project showed various points of connection between collaborative and interdisciplinary approaches and educational and political aims, such as the vulgarization of scientific knowledge, and the criticism of disciplinary and national boundaries. But it focused on the origins and development of non-mainstream philosophical views that cannot be reduced to logicism or structuralism, and investigated the underestimated influence of Leibniz's philosophy [START_REF] Luciano | The influence of Leibnizian ideas on Giuseppe Peano's work[END_REF][START_REF] Cantù | The right order of concepts: Grassmann, Peano, Gödel and the inheritance of Leibniz's universal characteristic[END_REF]), 19th century positivism, empiricism, and neo-criticism on these standard views in philosophy of mathematics (Cantù and Schiemer, forthcoming).

The partial receiving of the specificity of the school's research program was due to the misunderstanding of the deep relation between education, linguistics, and axiomatics, but also to a simplistic association of Peano's ideas with Russell's philosophy. This tendency did not only emerge in van Hejenhoort's remarks on the lack of inference rules and metatheoretical investigations, or in the quick tendency to classify Peano as a logicist but was already evident in the early reception of Peano in the USA. Gabriele Lolli shows how the works of the Peano's school were discussed by Edwing B. Wilson already in 1904 in a review of two writings of Bertrand Russell, so that the two conceptions were eventually combined in "the Peano-Russell point of view".

The ability to discriminate subtle differences between the positions of Russell, Frege and Peano characterized a fine reader of Peano's writings: Kurt Gödel. If the philosophical notebooks (Max Phil) reveal a deep understanding of differences on the notions of function and definite description (Crocco et al. 2017, Cantù 2016a), the summary of the Formulario to be found in one of his Excerptenhefte shows the analysis of the rules of inferences used in deductive chains. The accurate summary of Peano's Arithmetices Principia written in the Gabelsberger shorthand on a loose paper at the time when he was preparing the article on Russell's logic (early 1943) and edited by Jan von Plato for this special issue, attests that Gödel read not only the Formulario, but also the Arithmetices Principia, focusing his comments on the formal character of proofs.

The philosophy implicit in the school's practices

The attention to mathematical practices showed that Peano not only had a strong impact on the writings by Frege, Russell, Carnap and Gödel, but developed a proper philosophical view that emerges from the logical investigation of definitions, the logical interpretation of the symbols of a formal language, the distinction between relations and functions, and the difference between primitive and derived terms or propositions in an axiomatic system. Peano's philosophical view, distinct both from logicism and from structuralism, emerges as a result of a joint investigation of logic, language and mathematics, considered both as theoretical and didactic practices. The interest in definitions and the analysis of language had significant effects on Peano's semantics, which differs from what is usually described as a conceptualist (or as a three-level: words / concepts / objects) semantics, because symbols refer to concepts only through the mediation of language. Like entries in a dictionary that get their meaning only when they are inserted in a given linguistic context, the symbols' meaning can be determined only through a preliminary substitution with linguistic sentences, and in each substitution the symbols refer to the concepts expressed by the corresponding words in ordinary mathematical language [START_REF] Cantù | Origins and Varieties of Logicism[END_REF].

This volume constitutes a further decisive step in the direction of reconstructing Peano's philosophical views from a detailed analysis of logical, mathematical, pedagogical but also linguistic practices. The essays gathered here focus on Giuseppe Peano's, Alessandro Padoa's and Mario Pieri's works, but the same method can be fruitfully applied to other members of the school, such as Giovanni Vailati [START_REF] Cantù | Life and Works of Giovanni Vailati[END_REF], Cesare Burali-Forti and Alessandro Padoa. Besides, the contributions of Peano and other members of the school are evaluated by comparison with contemporaries (Richard Dedekind, Gottlob Frege, Bertrand Russell, David Hilbert and Paul Bernays), resulting in a historically accurate analysis of some subtle but fundamental differences between their respective projects, which aimed aim to present, analyze or ground mathematics as a rigorous, deductive science. Three examples will be briefly mentioned in this introduction: axiomatics, linguistic symbolization and rigor. Different terms are often used to characterize the school's foundational enterprise: symbolization, formalization, axiomatization, reduction. A deep investigation of the Peano school's practices might help disentangle some of the differences between these fundamental notions, and shed new light on different ways to conceive generality, ideography, metatheoretical inquiries, and the role of notation, intuition, and rigor.

Axiomatics

General philosophical and historical reconstructions of the development of logic in the early 20 th century have accustomed us to think of Peano as one of the fathers of modern axiomatics because of his contribution to the formulation of the axioms of arithmetic, which still bear his name. Yet, a detailed analysis of the connections between logical, linguistic, mathematical and pedagogic writings of the Peano School might help reevaluate his contributions to logic and philosophy of mathematics and discover a specific approach to axiomatics. An axiomatization is a particular kind of presentation of a theory, where the logical and the mathematical content is specified by the respective axioms. Reinhard Kahle's contribution traces a history of the formulation of the properties of real numbers, considered as Sätze by Dedekind and explicitly formulated as axioms by acquiring a non-logical nature in Hilbert's works and a first-order formulation in Bernay's contributions. This is a historically fruitful example of how the investigation of different uses and presentations of the same mathematical properties of numbers can reveal very different conceptions of the axiomatization of arithmetic.

Yet, axiomatics cannot be reduced to the investigation of the axiomatic formulation of single theories. It is a back-and-forth between syntactic, semantic and pragmatic-linguistic level and their goals: 1) to make explicit the implicit assumptions of a theory (e.g. by stating all the hypotheses necessary to prove a given theorem); 2) to investigate the tacit assumptions of a theory, considering what happens if they are not implicitly assumed (e.g. by testing the possibility of creating non-standard models of a theory); 3) to define the scope and goals of a research program or discipline [START_REF] Woodger | Studies in the foundations of genetics[END_REF]. The linguistic analysis is a pillar of Peano's approach, and cannot be dissociated from epistemological goals, such as the search for the good order and the minimal number of concepts, and the questioning of the relation between mathematical practices and a rigorous mathematical language. The symbolization of logic, far from being exclusively aimed at the construction of axiomatic systems or the investigation of deductive inferences, rests on questions very similar to those that have developed in the social sciences: the need to distinguish the simple from the complex, the first for us from the first for itself, a canonical form from deviant forms, the definition of a term from the formation of a concept, the pragmatic consequences of a hypothesis from its theoretical role [START_REF] Cantù | Logica impura. Un contrappunto epistemologico[END_REF]).

Axiomatics heavily relies on complex practices of symbolization and formalization, practices that have a social and interactive nature, and that should be studied in their different components: phases of redaction (exam of the pertaining bibliography, construction of a hypothetic-deductive order of the collected results, codification in symbols), division of the tasks among group members, circulation of knowledge within the group in a hierarchical or peer context, and the construction of domains of shared knowledge, that need not be mirrored in the final version of publications [START_REF] Luciano | Characterizing a Mathematical School. Oral Knowledge and Peano's Formulario[END_REF].

Symbolization and language

Symbolization is a process that associates symbols to words, but symbols can play the role of schematic letters, as in Hilbert's formalization, i.e. as terms having a merely formal sense that allows for a variety of interpretations, or have a substantive role, as terms whose meanings have to be conveyed by elucidation [START_REF] Klev | Dedekind and Hilbert on the Foundations of the Deductive Sciences[END_REF]. Bertran San-Millan's contribution explains how Frege used the symbols of arithmetic as canonical names, i.e. as symbols with a specific and fixed meaning, so that the mathematical letters always have a specific domain, determined by the intended application. Peano shared a similar substantive understanding of mathematical symbols in his early writings but moved towards a view of undefined symbols as uninterpreted non-logical constants devoid of meaning, as soon as he investigated, together with Padoa, metatheoretical questions concerning the independence of the axioms.

The comparative investigation of logic, linguistics and notational practices offer further insights on the particular version of ideography that is developed in the Formulario: symbols mean ideas, but they are first introduced as names for terms of an interpreted mathematical language having those ideas as meaning, and then also considered as schematic variables that might receive different interpretations by substitution of different linguistic terms. The relation between mathematical symbols, words of mathematical language, mathematical concepts and mathematical objects is a complex history that a comparative investigation of Peano's contributions to logics, mathematics, linguistics, and symbolic notation might help disentangle.

The symbolization of mathematics is often discussed in the light of a reduction of mathematics to logic or as a translation that preserves the relevant mathematical meaning, but it cannot be fully understood without a detailed investigation of the design principles, the didactical and practical constraints that accompany the search for technical symbols in a new notation. The formalization is a way to distinguish the logical form from the non-logical content but can also be conceived as a method of conceptual analysis that identifies the relevant logical and mathematical ideas. As Dirk Schlimm shows in his contribution, this analysis might be used to determine the primitive terms and propositions, but also to check the adequacy of the analysis itself, thereby evaluating whether definitions are correct, and proofs are rigorous.

The distinction between symbolization and formalization is often difficult to trace, but the investigation of definitions and of metatheoretical issues of independence between axioms and an attentive investigation of the interactions with linguistics might be of help. There are several aspects of Peano's approach to the Interlingua that relate it to mathematical logic: in both cases a language in use (mathematical language and Latin) is taken as a starting point for the development of a universally understandable language (logical symbolism, Interlingua); secondly the two enterprises are based on collaborative networks; both are grounded in the heritage of Leibniz' characteristica universalis; finally they are combined in the last edition of the Formulario, written in Latin and symbolic language (Cantù 2016b). In her contribution, Basak Aray highlights another similarity: the connection between algebra and grammar developed in the Formulario, and suggests that the symbolization developed in Peano's mathematical practice guided the design of his proposal for an international auxiliary language: the Latino sine flexione. This algebraic understanding of grammar better explains how logic and language are both presented in an equational form, and generality is expressed using free variables instead of assuming a universal quantifier as a primitive logical term. Jan von Plato takes this to be the reason why Peano's axiomatic systems, like Schröder's algebraic logic, lacked some principles of reasoning with the quantifiers, even if they contained other rules of inferences.

Mathematical education and rigor

The interrelation between mathematical education and conceptual analysis offers further hints to understand the main traits of the Peano school's epistemology, the importance of rigor in scientific knowledge and education as well as the interpretation of axiomatics as a metatheoretical investigation based on a variety of alternative conceptual analyses leading to different axiomatic presentations and definitions of the mathematical concepts. The attention to the pedagogic component in the Peano School showed that to the sociological singularity of this research team (the only non academic-based group in the international panorama) corresponds a unique educational project to rigor that was deeply intertwined with the mathematical, philosophical, logical and linguistic views of the group, and that had nonnegligible effects on the evolution of mathematical teaching in Italy, and beyond. Rigor is not an accessory or external element that can be imposed on mathematical teaching, but a result of the development of rational mathematics and the evolution of all sciences towards the structure of axiomatic-deductive systems. Rigor is not primarily a foundational problem, but vice versa the foundational enterprise is intertwined with didactic concerns [START_REF] Luciano | A matter of Style and Praxis. On Peano's concept of rigor in mathematics education[END_REF].

Rigor is as a distinctive feature of the Peano School's style but also an essential topic in the Italian debate on mathematical pedagogic theory and teaching practice at the turn of the century. Peano's crusade in defense of rigor is not only a distinctive mark of his axiomatics, but a feature of the school's linguistic, mathematical and educational research programs. It was neither a negation of the importance of experimental methods in the early stages of mathematical education [START_REF] Luciano | A matter of Style and Praxis. On Peano's concept of rigor in mathematics education[END_REF], nor a simplistic negation of mathematical intuition, which was banished from the proofs of a theory but remained central in the choice of the axioms [START_REF] Rizza | Abstraction and intuition in Peano's axiomatizations of geometry[END_REF]. It was rather a didactical objective developed through exchanges with school teachers and their associations, the publication of new textbooks, and the participation to educational Governmental Committees [START_REF] Giacardi | From Euclid as textbook to the Giovanni Gentile reform (1867-1923): Problems, methods and debates in mathematics teaching in Italy[END_REF].

Elena Marchisotto and Ana Millan Gasca illustrate in their contribution Pieri's belief that an integration of sensible and rational intuition can deeply renew the teaching of geometry but also deploy a profound heuristic value. The analysis of Pieri's axiomatization of geometry exemplifies the almost symbiotic relation between axiomatics and pedagogy that is typical of the Peano school, as well as the partial and complex receiving of this idea in the works by Italian contemporary mathematicians, such as Enriques and Amaldi.

Yet, the very same idea of rigor gave rise to famous debates, whose philosophical objectives were sometimes obscured by putting forth educational motivations or formal demonstrations. The famous debate with Segre on rigor and intuition was not only a manifesto of Peano's style, but also an implicit criticism of Veronese's hyperspaces and the expression of the rivalry with the geometrical Italian school [START_REF] Luciano | A matter of Style and Praxis. On Peano's concept of rigor in mathematics education[END_REF].

Similarly, Freguglia claims that the famous proof of the impossibility of infinitesimals was not only developed to complete and rectify an untenable proof by Cantor-undergoing the similar mistake of presupposing an axiom that is equivalent to the Archimedean axiom and therefore incompatible with the existence of infinitesimals. It was also an implicit criticism of Veronese's theory of a geometrical non-Archimedean continuum and the occasion for hosting a scientific discussion of the topic in the newly founded Rivista di Matematica.