Francisco Lopez-Ramirez

Denis Efimov

Andrey Polyakov

Wilfrid Perruquetti

Fixed-Time Output Stabilization and Fixed-Time Estimation of a Chain of Integrators

A solution to the problem of global fixed-time output stabilization and estimation of a chain of integrators is proposed. A nonlinear scheme comprising a state feedback controller and a dynamic observer are designed in order to guarantee both fixed-time estimation and fixed-time control. Robustness with respect to exogenous disturbances and measurement noises is established and a parameter optimization algorithm is provided. The performance of the obtained control and estimation algorithms are illustrated by numeric experiments.

Introduction

Feedback stabilization of linear and nonlinear systems is one of the central problems in control systems theory. There are many methods to design a control [START_REF] Doyle | Feedback Control Systems[END_REF][START_REF] Van Der Schaft | L 2 -gain and passivity techniques in nonlinear control[END_REF][START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], which differ in the requirements imposed on the plant model and on the guaranteed performance of the closed-loop system. Among performance criteria the robustness with respect to external disturbances or measurement noises and the rate of convergence are the most popular. To assess robustness of nonlinear systems the input-to-state stability theory is frequently used [START_REF] Sontag | The ISS philosophy as a unifying framework for stability-like behavior[END_REF][START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]. The rate of convergence to the goal state or set can be asymptotic (e.g. exponential) or non-asymptotic, i.e. finite-time or fixed-time [START_REF] Moulay | Finite time stability of non linear systems[END_REF][START_REF] Roxin | On finite stability in control systems[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. In the latter case the system converges from any initial conditions of the state space in a uniform finite time (see more rigorous definitions below).

The present work studies the problem of output feedback design providing a fixed-time rate of convergence for a chain of integrators [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF]. The chain of integrators is a very versatile and well studied system since all linear controllable systems and many nonlinear ones can be transformed, through a linear coordinate transformation in the first case and through feedback linearization in the second, into this particular form [START_REF] Khalil | Nonlinear Systems. NJ 07458[END_REF][START_REF] Ben M Chen | Linear systems theory: a structural decomposition approach[END_REF]. Fixed-time stability, i.e. globally bounded convergence rate, is a relatively recent topic [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF][START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF][START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] and in [START_REF] Basin | Continuous finite-and fixed-time regulators[END_REF] a fixed-time feedback regulator for a chain of integrators is presented based on Lyapunov analysis. Here we present a full control scheme, endowed with a state and output feedback control as well as with a state observer, all of whom exhibit the fixed-time stability property. Instead of using a Lyapunov approach, the proposed solution is based on homogeneity theory [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF][START_REF] Bernuau | On homogeneity and its application in sliding mode[END_REF]; and contrarily to [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF], the presented control has an explicit form. Fixed-Time stability of the chain of integrators has also been achieved in [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF][START_REF] Ríos | A hybrid observer for fixed-time state estimation of linear systems[END_REF]. In both works, a switching in the homogeneous degree is applied to obtain fixed-time stability, and while [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF] uses a recursive controller to complete a homogeneous system, [START_REF] Ríos | A hybrid observer for fixed-time state estimation of linear systems[END_REF] uses the hybrid formulation. Nonetheless, to the best of our knowledge, there is not any work that fully discloses the parameter tuning and its relationship with the settling-time estimates. In contrast, in this article we provide a homogeneous non-recursive fixed-time output feedback controller and we provide a tuning algorithm, based on Linear Matrix Inequalities (LMI) and on the Implicit Lyapunov Function approach, that allows to assert fixed-time stability and to adjust the settling time. In addition, the obtained control and estimation algorithms do not have a complex computational structure usually inherited by the solutions based on the Implicit Lyapunov Function method.

A robustness analysis with respect to bounded external disturbances and measurement noises, based on previous results on homogeneous systems [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], is also carried out.

The outline of this paper is as follows: Notation and preliminary results are introduced in Sections 2 and 3. The precise problem statement is given in Section 4. The proposed algorithms are presented in Section 5 and the parameter optimization is treated in Section 6. Concluding remarks and the discussion appear in Section 7.

June 13, 2018 1

Notation

Throughout the paper the following notation is used:

• R + = {x ∈ R : x ≥ 0}, where R is the set of real numbers.

• | • | denotes the absolute value in R, . denotes the Euclidean norm on R n , x A = inf ξ∈A x -ξ is the distance from a point x ∈ R n to a set A ⊂ R n ; A 2 is the induced matrix norm for A ∈ R n×n .

• For a (Lebesgue) measurable function d : R + → R m , we use ||d|| [t0,t1) = ess sup t∈[t0,t1) d(t) to define the norm of d(t) in the interval [t 0 , t 1 ), then ||d|| ∞ = ||d|| [0,+∞) and the set of essentially bounded measurable functions d(t) with the property ||d|| ∞ < +∞ is further denoted as

L ∞ ; L D = {d ∈ L ∞ : ||d|| ∞ ≤ D} for any D > 0.
• A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. A continuous function β :

R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R + and lim t→+∞ β(s, t) = 0 for each fixed s ∈ R + .
• The notation DV (x)f (x) stands for the directional derivative of a continuously differentiable function V with respect to the vector field f evaluated at point x.

• A series of integers 1, 2, ..., n is denoted by 1, n.

Preliminaries

Consider the following nonlinear system

ẋ(t) = f (x(t), d(t)), t ≥ 0, (1)
where x(t) ∈ R n is the state, d(t) ∈ R m is the input, d ∈ L ∞ ; f : R n+m → R n ensures forward existence of the system solutions at least locally, f (0, 0) = 0. For an initial condition x 0 ∈ R n and input d ∈ L ∞ define the corresponding solution by X(t, x 0 , d) for any t ≥ 0 for which the solution exists. Following [START_REF] Roxin | On finite stability in control systems[END_REF][START_REF] Khalil | Nonlinear Systems. NJ 07458[END_REF][START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], let Ω be an open neighborhood of the origin in R n and D > 0.

Definition 3.1. At the steady state x = 0 the system (1) for d ∈ L D is said to be (a) uniformly Lyapunov stable if for any x 0 ∈ Ω and d ∈ L D the solution X(t, x 0 , d) is defined for all t ≥ 0, and for any > 0 there is δ > 0 such that for any x 0 ∈ Ω, if x 0 ≤ δ then X(t, x 0 , d) ≤ for all t ≥ 0;

(b) uniformly asymptotically stable if it is uniformly Lyapunov stable and for any κ > 0 and > 0 there exists T (κ, ) ≥ 0 such that for any x 0 ∈ Ω and d ∈ L D , if x 0 ≤ κ then X(t, x 0 , d) ≤ for all t ≥ T (κ, );

(c) uniformly finite-time stable if it is uniformly Lyapunov stable and uniformly finite-time converging from Ω, i.e. for any x 0 ∈ Ω and all d ∈ L D there exists 0 ≤ T < +∞ such that X(t, x 0 , d) = 0 for all t ≥ T . The function T 0 (x 0 ) = inf{T ≥ 0 : X(t, x 0 , d) = 0 ∀t ≥ T, ∀d ∈ L D } is called the uniform settling time of the system (1);

(d) uniformly fixed-time stable if it is uniformly finite-time stable and sup x0∈Ω T 0 (x 0 ) < +∞. The set Ω is called the domain of stability/attraction.

If Ω = R n , then the corresponding properties are called global uniform Lyapunov/asymptotic/finite-time/fixedtime stability of (1) for d ∈ L D at x = 0. Stability notions can be similarly defined with respect to a set, by replacing the distance to the origin in Definition 3.1 with the distance to an invariant set. For example, the global uniform finite-time stability with respect to a set A ⊂ R n is equivalent to the following two properties: (i) uniform Lyapunov stability: for any x 0 ∈ R n and d ∈ L D the solution X(t, x 0 , d) is defined for all t ≥ 0, and for any > 0 there is δ > 0 such that if x 0 A ≤ δ then X(t, x 0 , d) A ≤ for all t ≥ 0; (ii) uniform finite-time convergence: for any x 0 ∈ R n and all d ∈ L D there exists 0 ≤ T < +∞ such that X(t, x 0 , d) A = 0 for all t ≥ T .

Input-to-state stability

More details about this theory can be found in [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF].

Definition 3.2. The system (1) is called input-to-state practically stable (ISpS ), if for any input d ∈ L ∞ and any x 0 ∈ R n there are some functions β ∈ KL, γ ∈ K and c ≥ 0 such that

X(t, x 0 , d) ≤ β( x 0 , t) + γ(||d|| [0,t) ) + c ∀t ≥ 0.
The function γ is called nonlinear asymptotic gain. The system is called input-to-state stable (ISS) if c = 0.

Definition 3.3. A smooth function V : R n → R + is called ISpS Lyapunov function for the system (1) if for all x ∈ R n , d ∈ R m and some r ≥ 0, α 1 , α 2 , α 3 ∈ K ∞ and θ ∈ K:

α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), DV (x)f (x, d) ≤ r + θ( d ) -α 3 ( x ). Such a function V is called ISS Lyapunov function if r = 0.
Note that an ISS Lyapunov function can also satisfy the following equivalent condition for some χ ∈ K ∞ :

x > χ( d ) ⇒ DV (x)f (x, d) ≤ -α 3 ( x ).
Theorem 3.1. The system (1) is ISS (ISpS) if and only if it admits an ISS (ISpS) Lyapunov function.

Weighted homogeneity

Following [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF][START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], for strictly positive numbers r i , i = 1, n called weights and λ > 0, one can define:

• the vector of weights r = (r 1 , . . . , r n ) T , r max = max 1≤j≤n r j and r min = min 1≤j≤n r j ;

• the dilation matrix function Λ r (λ) = diag{λ ri } n i=1 ; note that ∀x ∈ R n and ∀λ > 0 we have Λ r (λ)x = (λ r1 x 1 , . . . , λ ri x i , . . . , λ rn x n ) T ;

• the r-homogeneous norm x r = (

n i=1 |x i | ρ r i ) 1 ρ
for any x ∈ R n and ρ ≥ r max , hence there exist some σ, σ ∈ K ∞ such that σ( x r ) ≤ x ≤ σ( x r ) ∀x ∈ R n .

• the unit sphere and a ball in the homogeneous norm S r = {x ∈ R n : x r = 1} and B r (ρ) = {x ∈ R n :

x r ≤ ρ} for ρ ≥ 0.

Definition 3.4. A function g : R n → R is r-homogeneous with degree η ∈ R if ∀x ∈ R n and ∀λ > 0 we have:

λ -η g(Λ r (λ)x) = g(x).
A vector field φ : R n → R n is r-homogeneous with degree ν ∈ R, with ν ≥ -r min if ∀x ∈ R n and ∀λ > 0 we have:

λ -ν Λ -1 r (λ)φ(Λ r (λ)x) = φ(x)
, which is equivalent to the i-th component of φ being a r-homogeneous function of degree r i + ν.

The system (1) for d = 0 is r-homogeneous of degree ν if the vector field f is r-homogeneous of degree ν for d = 0.

The property of r-homogeneity can also be defined not for all x ∈ R n but for a subset of the state space (or it can be approximately satisfied [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF][START_REF] Efimov | Conditions of oscillations and multi-homogeneity[END_REF]).

An important advantage of r-homogeneous systems is that their rate of convergence can be evaluated qualitatively based on their degree of homogeneity. Lemma 3.1. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF] If (1) for d = 0 is r-homogeneous of degree ν and asymptotically stable at the origin, then it is (i) globally finite-time stable at the origin if ν < 0;

(ii) globally exponentially stable at the origin if ν = 0; (iii) globally fixed-time stable with respect to the unit ball B r (1

) if ν > 0. Define f (x, d) = [f (x, d) T 0 m ] T ∈ R n+m ,
it is an extended auxiliary vector field for the system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], where 0 m is the zero vector of dimension m. Theorem 3.2. [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] Let the vector field f be homogeneous with the weights r = [r 1 , . . . , r n ] T > 0, r = [r 1 , . . . , rm ] T > 0 with a degree ν ≥ -r min , i.e. f (Λ r (λ)x, Λ r (λ)d) = λ ν Λ r (λ)f (x, d) for all x ∈ R n , d ∈ R m and all λ > 0. Assume that the system (1) is globally asymptotically stable for d = 0, then the system (1) is ISS.

Therefore, for the homogeneous system (1), its ISS property follows from its asymptotic stability for d = 0 (as for linear systems [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]).

Implicit Lyapunov Function Method

Theorem 3.3. [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] If there exist two continuous functions Q i : R + × R n → R, i = {1, 2} such that:

C1 Q i (V, e) are continuously differentiable on R + × R n \{0}; C2 for any x ∈ R n \{0} there exists V > 0 such that Q i (V, x) = 0; C3 for Ω = {(V, x) ∈ R n+1 : Q i (V, x) = 0} we have: lim x→0 (V,x)∈Ω V = 0, lim V →0 + (V,x)∈Ω x = 0, lim x →∞ (V,x)∈Ω V = +∞;
C4 for all V > 0 and for all x ∈ R n \{0} the inequality -∞ < ∂Qi(V,x) ∂V < 0 holds;

C5 Q 1 (1, x) = Q 2 (1, x), for all x ∈ R n \{0}; C6 ∂Q1(V,x) ∂x f (t, x) ≤ c 1 V 1-µ ∂Q1(V,x) ∂V , for all t ∈ R, for all V ∈ (0, 1], for all x ∈ {z ∈ R n \{0} : Q 1 (V, z) = 0},
where c 1 > 0 and 0 < µ < 1 are some constants;

C7 ∂Q2(V,x) ∂x f (t, x) ≤ c 2 V 1+ν ∂Q2(V,x)
∂V , for all t ∈ R, for all V ≥ 1 and for all x ∈ {z ∈ R n \{0} : Q 2 (V, z) = 0}, where c 2 > 0 and ν > 0 are some constants.

Then the origin of (1) is fixed-time stable with the settling time estimate T 0 ≤ 1 c1µ + 1 c2ν .

Problem statement

Consider a chain of integrators:

ẋ(t) = A 0 x(t) + bu(t) + d(t), t ≥ 0, (2)
y(t) = Cx(t) + v(t), where x(t) ∈ R n is the state vector, u(t) ∈ R is the control input, y(t) ∈ R is the measured output; d(t) ∈ R n
and v(t) ∈ R are the exogenous disturbance and the measurement noise, respectively, (d, v) ∈ L ∞ and the matrices

A0 =        0 1 0 . . . 0 0 0 0 1 . . . 0 0 . . . . . . . . . . . . 0 0 0 . . . 0 1 0 0 0 . . . 0 0        , b =      0 . . . 0 1      , C = [1 0 . . . 0],
are in the upper-diagonal canonical form.

It is required to design a stabilizing dynamic output control u(t) that ensures the ISpS property of the system (1) for any (d, v) ∈ L ∞ ; and that for d = v = 0 provides global fixed-time stability of the closed-loop system at the origin.

Main result

The solution of the problem is divided in three steps. First, a state feedback is proposed ensuring the problem solution. Second, the equations of the observer are introduced. Third, a combined output feedback is presented and analyzed.

State feedback

For i = 1, n, x i ∈ R and α > 0 and denote x i α = |x i | α sign(x i ), then the control proposed in this work has the form:

u(x) = n i=1 a i x i αi(ν( x )) , (3)
α i (ν) = 1 + nν 1 + (i -1)ν , ν(ω) =      ν 1 if ω ≤ m, ν 2 if ω ≥ M, ν2-ν1 M -m ω + M ν1-mν2 M -m otherwise, (4)
where a = [a 1, . . . , a n ] ∈ R 1×n is the vector of control coefficients forming a Hurwitz polynomial, -∞ < ν 1 < 0 < ν 2 < +∞ and 0 < m < M < +∞ are the tuning parameters to be defined later. Denote

r i (ν) = 1 + (i -1)ν, i = 1, n, (5)
then it is straightforward to verify that for d = 0 the system (2), ( 3) is r(ν 1 )-homogeneous of degree ν 1 < 0 for x ≤ m and r(ν 2 )-homogeneous of degree ν 2 > 0 for x ≥ M . Let us show that for properly selected control parameters, the system (2), ( 3) is globally fixed-time stable at the origin.

Lemma 5.1. Let a ∈ R n form a Hurwitz polynomial, then for any 0 < m < M < +∞ there exists τ ∈ (0, n -1 ) such that if ν 1 ∈ (-τ, 0) and ν 2 ∈ (0, τ ) then the system (2), (3) for d = 0 is globally fixed-time stable at the origin.

Proof. Denote

A = A 0 + ba =      0 1 0 ... 0 0 0 0 1 ... 0 0 . . . . . . . . . . . . 1 . . . 0 0 0 ... 0 1 a1 a3 a3 ... an-1 an     
, then by the lemma conditions there are matrices P ∈ R n×n and Q ∈ R n×n such that

P = P T > 0, Q = Q T > 0, A T P + P A = -Q.
Following [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] consider for (2), (3) a Lyapunov function

V (x) = x T P x,
whose derivative admits a differential equation:

V (x) = DV (x)[A 0 x + bu(x)] = -x T Qx + 2x T P bδ(x), where δ(x) = n i=1 a i ( x i αi(ν( x )) -x i ). By construction, ν( x ) = 0 for x = µ = mν2-M ν1 ν2-ν1 . Since δ : R n → R is a continuous function and ν : R + → [ν 1 , ν 2 ]
, it is possible to select the values of ν 1 and ν 2 sufficiently close to zero such that |δ(x)| stays small enough, and therefore V (x) < 0, on any compact containing the level x = µ. Thus, there exists τ ∈ (0, n -1 ) (if τ ≥ n -1 then α i (ν( x )) may become non-positive) such that with ν 1 ∈ (-τ, 0) and ν 2 ∈ (0, τ ) for all x ∈ {x ∈ R n : m ≤ x ≤ M } we have V (x) < 0 for any selection of 0 < m < m < M < M < +∞. Using the arguments of [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] and Lemma 3.1, we can prove in this case that the system is r(ν 1 )-homogeneous of degree ν 1 < 0 and fixed-time stable at the origin from B r(ν1) (ρ 1 ) for B r(ν1) (ρ 1 ) ⊂ {x ∈ R n : x ≤ m}, and it is r(ν 2 )-homogeneous of degree ν 2 > 0 and globally fixed-time stable with respect to any ball B r(ν2 3) is globally convergent and it is globally fixed-time stable at the origin (the time that the system spent in the set {x ∈ R n : m ≤ x ≤ M } is finite).

) (ρ 2 ) such that {x ∈ R n : x = M } ⊂ B r(ν2) (ρ 2 ). The constants τ, m, M can be selected in a way that B r(ν1) (ρ 1 ) ⊂ {x ∈ R n : m ≤ x ≤ m} and B r(ν2) (ρ 2 ) ⊂ {x ∈ R n : M ≤ x ≤ M }, then (2), (
In order to analyze robust stability properties of the closed loop dynamics (2), (3) let us introduce

f ν (x, d) = A 0 x + b n i=1 a i x i + d1,i αi(ν) + d2 , where d = [ dT 1 dT 2 ] T ∈ R 2n
is the new disturbance input, d1 represents measurements noises and d2 = d.

Corollary 5.1. Let all conditions of Lemma 5.1 be satisfied, then the system (2), ( 3) is ISpS for any d ∈ L ∞ .

Proof. Consider the system (2), (3) for x ≥ M , and ẋ = f ν2 (x, d) is the corresponding approximating system, which is r(ν 2 )-homogeneous of degree ν 2 > 0 and globally fixed-time stable with respect to any ball B r(ν2) (ρ)

with ρ > 0 for d = 0. Take r = r(ν 2 ) r(ν 2 ) + ν 2 , then f ν2 (Λ r (λ)x, Λ r (λ) d) = λ ν Λ r (λ)f ν2 (x, d) for all x ∈ R n , d ∈ R 2n
and all λ > 0. Consequently, if all conditions of Lemma 5.1 are satisfied, then also all conditions of Theorem 3.2 are true and the system ẋ =

f ν2 (x, d) is ISS with respect to d ∈ L ∞ . Since ẋ = f ν2 (x, d) is the approximation of (2), (3) for x ≥ M , then (2), (3) (the system ẋ = f ν( x ) (x, d)) is ISpS.
Thus, the presented state control (3) solves the posed problem of robust global fixed-time stabilization for the system (2). Obviously, for any R ∈ R n×n , R = R T > 0 the system (2), (3) with ν( x R ), where x R = √ x T Rx, possesses the same properties as ( 2), (3) with ν( x ).

State observer

To explain the observer structure, let us first consider the case d = v = 0, then the proposed observer takes the form (see also [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF][START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF][START_REF] Ríos | A hybrid observer for fixed-time state estimation of linear systems[END_REF]):

ż(t) = A 0 z(t) + bu(t) + k (ν(ζ(t)), y(t) -Cz(t)) , (6)
k i (ν, e) = L i e βi(ν) , β i (ν) = 1 + iν i = 1, n, ζ(t) = -0.5ζ(t) + p (ν(ζ(t)), y(t) -Cz(t)) , p (ν, e) = 4κ T (ν, e)P κ(ν, e), κ(ν, e) = Le -k(ν, e),
where z(t) ∈ R n is the state estimate, ζ(t) ∈ R + is an auxiliary time function; the function ν is given in (4) with -∞ < ν 1 < 0 < ν 2 < +∞ and 0 < m < M < +∞, are, as previously, the tuning parameters; L = [L 1 , . . . , L n ] T is the vector of coefficients of the observer providing Hurwitz property of the matrix A 0 -LC; P ∈ R n×n is a matrix solution of the equations

P = P T > 0, (A 0 -LC) T P + P (A 0 -LC) = -P.
In [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF] instead of using an auxiliary ζ-filter to commute the right-hand sides of ( 6) with negative and positive homogeneity degree, a time switching between two systems with positive and negative homogeneity degree is proposed. In [START_REF] Ríos | A hybrid observer for fixed-time state estimation of linear systems[END_REF], in order to switch between observers with negative and positive homogeneity degrees, a comparison of output estimation errors for these observers is used.

Lemma 5.2. Let A 0 -LC be a Hurwitz matrix for a given L ∈ R n×1 and assume that the solutions of (2) are defined for all t ≥ 0, then for any 0 < m < M < +∞ there exists τ ∈ (0, n -1 ) such that if ν 1 ∈ (-τ, 0) and ν 2 ∈ (0, τ ) then the system (2), ( 6) for d = v = 0 is globally Lyapunov stable and fixed-time convergent with respect to the set

A = {(x, z, ζ) ∈ R 2n+1 : x = z, ζ ≤ M } for all (x, z) ∈ R 2n , provided that ζ(0) > M is sufficiently big.
A more precise restriction on the value of ζ(0) is given in the proof of this lemma, it is not related with the initial conditions x(0), z(0), see [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF].

Proof. Denote e = x -z as the estimation error, then

ė = A 0 e -k (ν(ζ), Ce + v) + d = (A 0 -LC)e -k (ν(ζ), Ce + v) + L(Ce + v) + d -Lv = (A 0 -LC)e + κ(ν(ζ), Ce + v) + d -Lv.
Consider a Lyapunov function V (e) = e T P e, then

V = -V + 2e T P [κ(ν(ζ), Ce + v) + d -Lv] ≤ -0.5V + p (ν(ζ), Ce + v) + 4(d -Lv) T P (d -Lv).
For an auxiliary error variable ψ = V -ζ we obtain:

ψ ≤ -0.5ψ + 4(d -Lv) T P (d -Lv)
and ψ is exponentially converging to zero (ζ is converging to V ) if d = v = 0. In addition, if there is an instant of time t ≥ 0 such that ψ(t ) ≥ 0, then ψ(t) ≥ 0 for all t ≥ t .

Repeating the arguments of Lemma 5.1, for any 0 < m < M < +∞ there exists τ ∈ (0, n -1 ) such that if ν 1 ∈ (-τ, 0) and ν 2 ∈ (0, τ ) then the system

ė = A 0 e -k (ν, Ce) (7)
is globally asymptotically stable for any fixed value of ν ∈ [ν 1 , ν 2 ]. Moreover, for ν = ν 1 it is r(ν 1 )-homogeneous of degree ν 1 < 0 and globally finite-time stable at the origin, and for ν = ν 2 the system (7) is r(ν 2 )-homogeneous of degree ν 2 > 0 and globally fixed-time stable with respect to any ball B r(ν2) (ρ) with ρ > 0. In addition,

V (e) < 0 ∀e ∈ {e ∈ R n : m ≤ V (e) ≤ M } (8)
for any selection of 0 < m < m < M < M < +∞ and any (possibly time-varying

) value of ν(ζ(t)) ∈ [ν 1 , ν 2 ].
Denote by T M > 0 the uniform settling-time of convergence to the ball B r(ν2) (ρ M ) of ( 7) for ν = ν 2 , where

ρ M > 0 is such that {e ∈ R n : V (e) ≤ M } ⊂ B r(ν2) (ρ M ). Let T M < 2 ln(ζ(0) -M ), (9)
then ζ(t) ≥ M for t ∈ [0, t M ) with t M ≥ T M (t M can also be infinite), therefore ν(ζ(t)) = ν 2 for t ∈ [0, t M ) from ( 4) and the estimation error dynamics is r(ν 2 )-homogeneous and fixed-time stable with respect to the ball B r(ν2) (ρ M ) on this interval of time. Since t M ≥ T M , then the system [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] Corollary 5.2. Let all conditions of Lemma 5.2 be satisfied, then the system (2), ( 6) is ISpS with respect to the set A for any (d, v) ∈ L ∞ .

Proof. From the equation for ψ we have that:

V (t) ≤ ζ(t) + (V (0) -ζ(0))e -0.5t + 8χ(||v|| ∞ , ||d|| ∞ ),
where sup t≥0 (d(t) -Lv(t)

) T P (d(t) -Lv(t)) ≤ χ(||v|| ∞ , ||d|| ∞ ) = P 2 (||d|| 2 ∞ + L 2 2 ||v|| 2 ∞ ) and • 2 is the in- duced matrix norm. Consider the set Υ 0 = {(e, ζ) ∈ R n+1 : V (e) ≥ M +max{0, V (0)-ζ(0)}+8χ(||v|| ∞ , ||d|| ∞ )}.
From the inequality above ζ(t) ≥ M for e(t) ∈ Υ 0 . Thus, ν(t) = ν 2 if e(t) ∈ Υ 0 and the estimation error dynamics takes the form:

ė = f (e, d) = A 0 e -k ν 2 , Ce + d1 + d2 . Denote r = r 1 (ν 2 ) r(ν 2 ) + ν 2 , then f (Λ r (λ)e, Λ r (λ) d) = λ ν Λ r (λ)f (x, d) for all x ∈ R n , d ∈ R n+1 and all λ > 0.
Consequently, if all conditions of Lemma 5.2 are satisfied (the system ė = f (e, 0) corresponds to [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]), then also all conditions of Theorem 3.2 are true and the system ė = f (e, d) is ISS with respect to d ∈ L ∞ . Therefore, there exists an ISS Lyapunov function W : R n → R + (an alternative choice of W can be found in [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]) and some function of class

K ∞ such that W (e) ≥ (M + max{0, V (0) -ζ(0)} + 8χ(||v|| ∞ , ||d|| ∞ )) implies that e ∈ Υ 0 .
Then either an ISS estimate holds for W or W (e) ≤ (M + max{0,

V (0) -ζ(0)} + 8χ(||v|| ∞ , ||d|| ∞ ))
, which implies boundedness of W , and the same property for ζ. Consequently, all solutions of (6) are defined for all t ≥ 0 (x(t) is also defined for all t ≥ 0 by conditions of Lemma 5.2). Then the term (V (0) -ζ(0))e -0.5t can be skipped and the above consideration can be repeated for the set Υ = {(e, ζ) ∈ R n+1 : V (e) ≥ M + 8χ(||v|| ∞ , ||d|| ∞ )} in order to prove ISpS with respect to the set A of the system (2), [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. Obviously, the variable ζ(t) is also asymptotically bounded by M + 16χ(||v|| ∞ , ||d|| ∞ ) in this case.

Output feedback

The proposed dynamic output feedback consists in the application of the state feedback (3) with the state estimates generated by the observer (6):

u(z) = n i=1 a i z i αi(ν( z )) , (10)
then the dynamics of the closed-loop system (2), ( 6), ( 10) can be written in the coordinates x, e = x -z and ζ as follows:

ẋ = A 0 x + b n i=1 a i x i -e i αi(ν( x-e )) + d, ė = A 0 e -k (ν(ζ), Ce + v) + d, (11) ζ
= -0.5ζ + p (ν(ζ), Ce + v) .
The main result is a direct consequence of lemmas 5.1, 5.2 and corollaries 5.1, 5.2.

Theorem 5.1. Let the following conditions be satisfied:

(i) a ∈ R 1×n forms a Hurwitz polynomial;

(ii) A 0 -LC is a Hurwitz matrix for given L ∈ R n ;

(iii) ζ(0) > M is sufficiently big, then for any 0 < m < M < +∞ there exists τ ∈ (0, n -1 ) such that for ν 1 ∈ (-τ, 0) and ν 2 ∈ (0, τ ) the system (2), ( 6), ( 10 In Theorem 5.1, the same parameters m, M , ν 1 and ν 2 have been selected for the controller [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] and for the observer (6) in order to keep the notation compact, however, they can be chosen differently in applications and the result of Theorem 5.1 stays correct. then all conditions of Theorem 5.1 are satisfied. We test first the system without any perturbations; the results are depicted in the upper plot of Figure 1 where the initial conditions of the system are x 0 = (5, 10, 0) and those of the observer are z 0 = (0, 0, 0). The solid color lines represent the actual state x while the doted color lines represent the estimated state z. It can be seen how the estimated states converge rapidly to the actual states before converging both to zero. In the lower part of Figure 1 we can appreciate the elements of the control scheme, the upper and lower limits of the homogeneity degree M and n are shown as straight lines. The norm of the observed states z is depicted in purple, while this norm is between M and m the control's degree of homogeneity lies over the line ν2-ν1 M -m z + M ν1-mν2 M -m . In the case of the observer, the filter ζ(t) acts as the modulator of the observer's homogeneity degree. The control signal is shown in light blue. 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 without disturbances and with initial conditions x 0 = (5, 10, 0). 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = 10 3 (5, 10, 0).

Figure 2 shows the same setup with initial conditions x 0 = 10 3 (5, 10, 0), it can be seen that although the initial state is significantly larger, the settling time remains within the same interval, showing the expected uniformity w.r.t the initial state. In the lower part of the figure it can be seen that the control signal u grows considerably to cope with the conditions imposed. We next go back to the previous initial settings and introduce in the control scheme the disturbance

d(t) = sin(2t)+ 10 if t ∈ [30, 31] 0 otherwise . ( 12
)
The results are shown in Figure 3. We can notice that the system is robust against this disturbance and its effect in the control scheme elements are depicted in the lower plot of this figure. In particular we can see that the disturbance modifies both z and ζ(t) therefore changing the homogeneity degree of both the controller and the observer. 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = (5, 10, 0) and the disturbance term [START_REF] Efimov | Conditions of oscillations and multi-homogeneity[END_REF].

Remark 5.1. Since in Theorem 5.1, the ISpS property with respect to d is proven, then considering d as a function of x and assuming that the norm of such a function is less than the asymptotic gain function of the system for x sufficiently large, it is possible to prove fixed-time convergence to a zone and global boundedness of the system solutions for a nonlinear system also in the closed-loop with the proposed control [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] and the observer (6).

Parameter Tuning

In this section we provide effective algorithms to tune the parameters involved in the feedback controller. Based on the concept of Implicit Lyapunov Functions [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], these algorithms transform the design procedure to an LMI feasibility problem, which simplifies significantly the practical applicability of the proposed method. In addition, they will provide an upper bound, even if conservative, of the settling time.

Consider the following implicit Lyapunov function candidate

Q(V, x) = x T Λ r(ν) (V -1 )P Λ r(ν) (V -1 )x -1, ( 13
)
where V ∈ R + , x ∈ R n , P ∈ R n×n and P = P T > 0. Although this particular function possesses many properties1 , for the parametrization of the feedback controller and the state observer it will be sufficient to remark that Q(1, x) = 0 implies x T P x = 1 and that the implicitly defined function V (x) is r-homogeneous of degree 1.

Controller Parametrization

For brevity in the notation, the following representation of system ( 2), (3) will be used:

ẋ = A 0 x + ba x α(ν( x P )) , (14)
where

x α(ν) = (|x 1 | α1(ν) sign(x 1 ), . . . , |x n | αn(ν) sign(x n )) T .
Remark that without loss of generality, the usual norm of the argument of ν in control (3) has been replaced with a weighted one. Let us introduce the matrix H r(ν) = -diag(r 1 (ν), r 2 (ν), ..., r n (ν)). When ν takes a fixed value ν j denote, for brevity, the homogeneous weights r j = r(ν j ), H rj = -diag(r j,1 , r j,2 , ..., r j,n ), rj (λ) = diag(λ rj,1 , λ rj,2 , ..., λ rj,n ) and α j = α(ν j ). Accordingly,

Q j (V, z) = z T Λ rj (V -1 )P Λ rj (V -1 )z -1. ( 15
)
Theorem 6.1. Let for some ν 1 ∈ (-1 n , 0), ν 2 = -ν 1 , φ, β, κ, γ j > 0, β < φ and ∈ (0, 1) the system of matrix inequalities

A 0 X + XA T 0 + bY + Y T b T + φX + βbb T ≤ 0 (16a) -γ j X ≤ H rj X + XH rj < 0, ( 16b
)
ξI n ≤ X ≤ 1 κ I n , β 2 ξ zj 2 Y Y T X ≥ 0, (16c) zj,i =    g j,i (κ -1/2 ) + κ -1/2 pi if κ -1/2 ≤ α 1/(1-αj,i) j,i max{g j,i (α 1/(1-αj,i) j,i ), g j,i (κ -1/2 )} + κ -1/2 pi , if κ -1/2 > α 1/(1-αj,i) j,i , (17)
where g j,i : R → R, g j,i (z) = z αj,i -z , pi = (n -i + 1)ν 2 and α j,i = (

1+nνj rj,1 , ..., 1+nνj rj,n ), be feasible for some ξ > 0, X ∈ R n×n , X = X T > 0, Y ∈ R 1×n , with j = 1, 2, i = 1, n.
Then the system (2), ( 3) for d = 0, a = Y P and P = X -1 is globally fixed-time stable at the origin with

m = λ max (P -1 2 Λ r1 (1-)P Λ r1 (1-)P -1 2 ) < 1 < M = λ min (P -1 2 Λ r2 (1+ )P Λ r2 (1+ )P -1 2 )
and the settling time estimate

T max ≤ γ1 (φ-β)|ν1| + γ2 (φ-β)ν2 .
Before presenting the proof of this theorem, let us introduce a constructive procedure to calculate the controller's parameters. To this end, the system of LMI ( 16) can be solved using standard optimization tools such as MATLAB.

Constructive procedure to obtain the controller's parameters P , a, m and M .

1. Set the size of the chain of integrators n. 2. Fix a negative value for ν 1 . Start with values close to zero e.g. -0.001. 3. Fix positive values for and κ (a possible initial value is 0.5 for both). 4. Calculate the vectors zi,j using [START_REF] Moulay | Finite time stability of non linear systems[END_REF]. 5. Fix positive values for φ, β, γ 1 , γ 2 , β < φ (possible starting values are given in the examples). 6. Verify the feasibility of the system of inequalities [START_REF] Lopez-Ramirez | Finite-time and fixedtime observers design via implicit Lyapunov function[END_REF]. 7. If unfeasible, reduce the value of or modify the value of κ and repeat from step 3. If feasible, the value of |ν 1 | might be increased in step 2 until a desired value of T max is obtained without loosing feasibility (recall that in practice, this value might be conservative). 8. From the obtained matrices X and Y , calculate P , a, m and M as described in Theorem 6.1.

Note that the parameters that influence directly the settling-time are ν 1 , ν 2 , γ 1 , γ 2 , φ and β. The parameters and κ modify the bounds of the inequality (16c), so that its manipulation may relax the feasibility conditions of [START_REF] Lopez-Ramirez | Finite-time and fixedtime observers design via implicit Lyapunov function[END_REF].

Proof of Theorem 6.1. I The functions Q j (V, x), j = 1, 2 defined in [START_REF] Khalil | Nonlinear Systems. NJ 07458[END_REF] satisfy the conditions C1-C3 of Theorem 3.3. Indeed, they are continuously differentiable for all V ∈ R + and for all x ∈ R n . Since P > 0 then the inequalities λ min (P )

x 2 max{V 2minri , V 2maxri } ≤ Q j (V, x) + 1 ≤ λ max (P ) x 2 min{V 2minri , V 2maxri } , imply that for any x ∈ R n \{0} there exist V -∈ R + and V + ∈ R + : Q j (V -, x) < 0 < Q j (V + , x). Moreover, if Q j (V,
x) = 0 then, obviously, the condition C3 of Theorem 3.3 holds and there exists a Lyapunov function candidate V : R n → R + implicitly defined by the identity

x T Λ rj (V -1 )P Λ rj (V -1 )x = 1. II Since ∂Q j ∂V = V -1 x T Λ rj (V -1 )(P H rj + H rj P )Λ rj (V -1 )x,
taking into account (16b), we have that ∂Qj ∂V < 0 ∀V ∈ R + and x ∈ R n \{0}. So the condition C4 of Theorem (3.3) also holds and therefore Q j (V, z) = 0 implicitly defines a proper positive definite Lyapunov function candidate V : R n → R.

III By denoting ∂Qj (V,x) ∂x as the partial derivative of Q j along the trajectories of ( 14) we obtain

∂Q j (V, x) ∂x = 2x T Λ rj (V -1 )P Λ rj (V -1 )[A 0 x + ba x α(ν) ].
By adding and subtracting inside the brackets the auxiliary term V 1+nνj baΛ rj (V -1 )x and taking into account that Λ rj (V -1 )A 0 Λ -1 rj (V -1 ) = V -νj A 0 and that Λ rj (V -1 )b = V -1-(n-1)νj b, we simplify the derivative as

∂Q j (V, x) ∂x = Λ rj (V -1 )x d(V, x) T V -νj (P A 0 +A T 0 P +P ba+a T b T P ) P b b T P 0 Λ rj (V -1 )x d(V, x) , where d(V, x) = V -νj a(V -1-nνj x α(ν) -Λ rj (V -1 )x)
. By adding and subtracting the terms φV -νj x T Λ rj (V -1 )P Λ rj (V -1 )x and

1 β V νj d T (V, x)d(V, x) we obtain ∂Q j (V, x) ∂x = Λ rj (V -1 )x d(V, x) T Θ Λ rj (V -1 )x d(V, x) -φV -νj x T Λ rj (V -1 )P Λ rj (V -1 )x+ V ν j β d T (V, x)d(V, x), where Θ = V -νj (P A 0 +A T 0 P +P ba+a T b T P + φP ) P b b T P V ν j β .
Using the Schur complement, P -1 0 0 η T Θ P -1 0 0 η , for any η ∈ R, it is equivalent to the left-hand side of (16a) and we arrive to

∂Q j (V, x) ∂x ≤ -V -νj φx T Λ rj (V -1 )P Λ rj (V -1 )x + V νj β -1 d T (V, x)d(V, x).
The term d(V, x) can be rewritten as d(V, x) = V -νj d 0 (V, x) where d 0 (V, x) = a(V -1-nνj x α(ν) -Λ rj (V -1 )x), and we have that

∂Q j (V, x) ∂x ≤ -V -νj φx T Λ rj (V -1 )P Λ rj (V -1 )x + V -νj β -1 d T 0 (V, x)d 0 (V, x).
If the latter term is bounded as follows d T 0 (V (x), x)d 0 (V (x), x) < β 2 then we derive ∂Q j (V, x) ∂x ≤ -φV -νj + βV -νj , and using inequality (16b) we arrive to

∂Q j (V, x) ∂x ≤ φ-β γj V 1+νj ∂Q j ∂V .
for (V, x) : x T Λ rj (V -1 )P Λ rj (V -1 )x = 1. Then, by defining c j = φ-β γj , the conditions of Theorem 3.3 are satisfied.

IV Proof of the estimate d T 0 (V, x)d 0 (V, x) ≤ β 2 . IV.a Let us introduce the sets Ω 1 = {x : m 2 ≤ x T P x ≤ 1} and Ω 2 = {x : 1 ≤ x T P x ≤ M 2 }. Considering first the set Ω 2 and using the change of variables y = Λ r2 (V -1 )x we obtain max (V,x):x T Λr 2 (V -1 )P Λr 2 (V -1 )x=1 and

x T P x∈[1,M 2 ] V = max (V,y):y T P y=1 and y T Λr 2 (V )P Λr 2 (V )y∈[1,M 2 ] V = 1 + .
Indeed, from inequality (16b) we know that ∂ ∂V y T Λ r2 (V )P Λ r2 (V )y > 0 and therefore min y:y T P y=1

y T Λ r2 (V )P Λ r2 (V )y > min y:y T P y=1 y T Λ r2 (1 + )P Λ r2 (1 + )y = M 2 if V > 1 + and 1 < x P ≤ M implies 1 ≤ V (x) ≤ 1 + . Similarly,
we show min (V,x):x T Λr 1 (V -1 )P Λr 1 (V -1 )x=1 and

x T P x∈[m 2 ,1] V = 1 -.
Hence, we immediately conclude that m ≤ x 2 P ≤ M ⇒ 1 -≤ V (x) ≤ 1 + . IV.b Recall the function ν is defined as follows

ν( x P ) =      ν 1 , if x P ≤ m, ν 2 , if x P ≥ M ν2-ν1 M -m x P + M ν1-mν2 M -m , otherwise .
Then we have that max

x∈Ω2 d T 0 (V (x), x)d 0 (V (x), x) = max x∈Ω2 δ (V (x), x) T a T aδ (V (x), x) ≤ aa T max x∈Ω2 δ (V (x), x) T δ (V (x), x), where δ (V, x) = V -1-nν2 ( x α(ν( x P )) -Λ r2 (V -1 )x).
Using again the change of variables y = Λ r2 (V -1 )x and considering that x ∈ Ω 2 =⇒ V ∈ [1, 1 + ] we have that

aa T max x∈Ω2 δ (V (x), x) T δ (V (x), x) ≤ aa T max V ∈[1,1+ ] y P =1 ψ(V, y) T ψ(V, y),
where ψ(V, y) = V -1-nν2 Λ r2 (V )y α(ν) -y , and in a component-wise expression we have

ψ i (V, y) = V -1-nν2+r2,iαi(ν) y i αi(ν) -y i = V -pi(ν) y i αi(ν) -V pi(ν) y i , s max1 1 s s g(ε 1/(1-ε) ) g(s)
g(s)

Figure 4: Plot of function g(s, ε) = s ε -s , with fixed ε ∈ R + . Remark that if s > 1, the maximum value of g(s) in the interval s ≤ s is the maximum between g(ε 1/(1-ε) , ε) and g(s, ε).

where p i (ν) = 1 + nν 2 -r 2,i α i (ν) ≥ 0. Hence, Lemma 6.1. The function g : R 2 + → R + defined as g(s, ε) = s ε -s , s, ε ∈ R + admits the following estimate

ψ i (V, y) ≤ max V ∈[1,1+ ] y P =1 y i αi(ν) -V pi(ν)
max s∈[0,s],ε∈[0,ε] g(s, ε) =    g(s, ε), for s ≤ ε1/(1-ε) max{g(ε 1/(1-ε) , ε), g(s, ε)}, for s > ε1/(1-ε) . (18)
Proof. The function g is depicted in Figure 4. It is easy to show that function g ε attains a local maximum at s max1 = ε 1/(1-ε) within the interval s ∈ [0, 1], therefore, if s ≤ s max1 then max s∈[0,s] g(s, ε) = g(s, ε) and if s > s max1 then max s∈[0,s] g(s, ε) = max{g(ε 1/(1-ε) , ε), g(s, ε)}. Taking into account that ∂(s ε -s) ∂ε = s ε ln s, we complete the proof. Since y T P y =⇒ |y i | ≤ λ -1/2 min (P ) = κ -1/2 then using Lemma 6.1 the term |y i | αi(ν) -|y i | can be bounded as follows:

y i αi(ν) -y i ≤ z2,i =    g α2,i (κ -1/2 ), for κ -1/2 ≤ α 1/(1-α2,i) 2,i max{g 2,i (α 1/(1-α2,i) 2,i ), g 2,i (κ -1/2 )}, for κ -1/2 > α 1/(1-α2,i) 2,i
.

Then for V ∈ [1, 1 + ] and y T P y = 1 one has

ψ i (V, y) ≤ z2,i + κ -1/2 ((1 + ) pi(ν) -1) ≤ z2,i + κ -1/2 ((1 + ) pi -1), where ν = ν( x P ) ∈ [0, ν 2 ], p i (ν) = 1 + nν 2 -(1 + (i -1)ν 2 ) 1+nν 1+(i-1)ν and pi = p i (0) = (n -i + 1)ν 2 .
Here, the fact that, for any x in the interval 1 ≤ x P ≤ M , p i (ν( x P )) ≤ pi has been used.

By applying the mean value theorem with h : R → R, h(θ) = θ pi , we obtain that h(1 + ) -h(1) = h (θ * ) , where θ * ∈ [1, 1 + ]. Noting that 0 < pi < 1, we have that (1 + ) pi -1 ≤ pi and we arrive to the following estimate ψ i (V, y) ≤ z2,i + κ -1/2 pi , for V ∈ [1, 1 + ] and y P = 1. Therefore, it has been proven that max

x∈Ω2 d T 0 (V (x), x)d 0 (V (x), x) ≤ a T a z2 2 .
where z2 = (z 2,1 , ..., z2,n ) T ∈ R n + and z2,i = z2,i + κ -1/2 pi . Accordingly, proceeding in the same fashion for the set Ω 1 , we obtain

max x∈Ω1∪Ω2 d T 0 (V (x), x)d 0 (V (x), x) ≤ a T a z2 2 .
Using the Schur complement, inequality (16c) becomes aP -1 a T ≤ ξβ 2 zj 2 and since aa T ξ ≤ aP -1 a T , we have that aa T ≤ β 2 zj 2 and we conclude that

d 0 (V (x), x) T d 0 (V (x), x) ≤ β 2 if x ∈ Ω 1 ∪ Ω 2 . IV.c Boundedness of d T 0 (V (x), x)d 0 (V (x), x) in R n \(Ω 1 ∪ Ω 2 ) If x ∈ R n \{x : x P ≤ M } then ν( x P ) = ν 2 and d 0 (V (x), x) = aV -1-nν2 ( x α(ν2) -Λ r2 (V -1 )x) = a( y α(ν2) -y)
where y = Λ r2 (V -1 (x))x and y T P y = 1 (since Q(V (x), x) = 0). Hence, ψ i (V, y) ≤ z2,i and the required estimate d 0 (V, x) T d 0 (V, x) ≤ β 2 is straightforward. Similar considerations can be provided for x : x P ≤ m.

Finally let us remark that following a similar procedure, along with the ideas presented in [START_REF] Lopez-Ramirez | Finite-time and fixedtime observers design via implicit Lyapunov function[END_REF], analogous algorithms can be developed for the observer. Example 6.1 (Controller Parametrization). We start by choosing n = 3, ν 1 = -0.02, ν 2 = -ν 1 , = κ = 0.5 and calculating zj ; following Theorem 6.1 we obtain z1 = (0.0715, 0.0662, 0.0605), z2 = (0.0721, 0.0689, 0.0661). We now choose the parameters φ = 0.5, β = 0.3, γ 1 = γ 2 = 3 and solve the set of LMIs (16a-16c) to obtain P c = , a = (-0.5952, -1.7576, -1.3889).

With this parameter choice, the maximum settling time of the controller is T max = 1500s. 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 without disturbances and with initial conditions x 0 = (0.5, 1, 0). Figure 6: Results of simulation of ( 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = 10 3 (0.5, 1, 0).

Figure 5 depicts the substitution of these values in the system (2), ( 6) with initial conditions x(0) = (5, 10, 0). It is possible to see a fast convergence to the real states before converging also rapidly to the origin. In Figure 6 the initial conditions have been changed to x(0) = 10 3 (5, 10, 0), it is possible to see that the settling time is not significantly modified and that in both cases the systems reach the equilibrium long before the settling time estimates. Finally, in Figure 7 the initial conditions were reset to x(0) = (5, 10, 0) and the disturbance term

d(t) = sin(2t) + 10 if t ∈ [30, 31] 0 otherwise (19)
was added. We can see that the convergence to zero is preserved and with a much better performance than in the Example 1. 2), ( 6) and [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = (5, 10, 0) and the disturbance [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

Example 6.2 (Comparative Example). This last example is meant to compare the fixed-time controller (2), (3), referred here as the non-recursive controller, with the finite-time and fixed-time ones described in [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF], referred accordingly as the recursive ones. The parameter choice for the tuning algorithm 6.1 is as follows: , z1 = (0.0047, 0.0056, 0.0064) and z2 = (0.0047, 0.0056, 0.0065). The corresponding parameters for both the finite-time and the fixed-time controllers in [START_REF] Harmouche | Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers[END_REF] are the same as the ones used in the example section of the cited article. Figure 8 shows the unperturbed states of the chain of integrators with initial conditions x 0 = (1, 1, 1). Clearly, the fixed-time controller outperforms the finite-time one. In Figure 9 it is possible to see how in the finite-time case, the variation of initial conditions affects the settling-time while the fixed-one remains within two seconds. Finally, Figure 10 shows an improvement of over two seconds in the settling time with respect to the recursive fixed-time controller, illustrating that with the tuning algorithm, the settling time can be adjusted. As shown in the interior plots, fixed-time stability is assumed whenever all the states enter a strip of magnitude 10 -4 (depicted between black lines in Fig. 10) and remain there for the rest of the simulation.

ν 1 = -1/400, ν 2 = -ν 1 , = 0

Conclusion

A state feedback control has been constructed for a chain of integrators, which ensures global convergence of all trajectories to the origin with an upper bound of the settling-time that is independent of the initial conditions (fixed-time stability). An observer has been proposed, which provides a global estimation of the plant state (global differentiation) with a fixed-time convergence rate. Both control and estimation algorithms are robust with respect to disturbances and noises. It has been shown that the combination of these algorithms results in a global fixed-time output stabilization control law. Effective tools to optimize the scheme's parameters, allowing the maximum settling time to be estimated have been presented and the efficacy of this scheme has been demonstrated in simulations. In the perturbed case, practical fixed-time stabilization is obtained, this is fixed-time stabilization to a ball containing the origin whose radius depends on the size of the perturbation.

As with many control algorithms that focus on fast convergence rates, the control signal magnitude might grow significantly to cope with the time constraints imposed. In practice, these constraints can only be granted locally due to boundedness of the admissible control magnitude. However, in contrast to other control algorithms, the fixed-time controllers do not need to be retuned if the admissible control magnitude (and, consequently, the domain of fixed-time convergence) is increased. A related issue deals with the methods used for simulation, which have to be adapted to treat highly nonlinear systems [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF].

) is 1 ) 2 )

 12 fixed-time converging with respect to the set {(x, z, ζ) ∈ R 2n+1 : x = z = 0} for d = v = 0 and for any initial conditions (x(0), z(0)) ∈ R 2n , ISpS for any (d, v) ∈ L ∞ . Proof. The system (11) is a cascade of the (e, ζ)-and x-dynamics. If d = v = 0 then (e, ζ)-subsystem is autonomous and globally fixed-time converging with respect to the set {(e, ζ) ∈ R n+1 : e = 0} (with the uniform settling-time T o > 0) according to Lemma 5.2. During the interval [0, T o ] the system (2) has bounded trajectories due to the ISpS property with respect to measurement noises (estimation errors e) established in Corollary 5.1, and for t ≥ T o the x-subsystem is also autonomous and globally fixed-time converging at the origin by Lemma 5.1.For (d, v) ∈ L ∞ the ISpS property follows the results of corollaries 5.1, 5.2 and the cascade structure of[START_REF] Doyle | Feedback Control Systems[END_REF].

Example 5 . 1 .

 51 Let n = 3, L = 1.5 1.01 0.25 T , a = -2.5[1 1 1], m = 1, M = 5, ν 2 = -ν 1 = 0.1, ζ(0) = 5M and

Figure 1 :

 1 Figure 1: Results of simulation of (2), (6) and[START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 without disturbances and with initial conditions x 0 = (5, 10, 0).

Figure 2 :

 2 Figure 2: Results of simulation of (2), (6) and[START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = 10 3 (5, 10, 0).

Figure 3 :

 3 Figure 3: Simulation plots of (2), (6) and[START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = (5, 10, 0) and the disturbance term[START_REF] Efimov | Conditions of oscillations and multi-homogeneity[END_REF].

  y i = max y:y T P x=1{ |y i | αi(ν) -|y i | , |y i | αi(ν) -(1 + ) pi(ν) |y i | } = max y:y T P x=1 { y i αi(ν) -y i , |y i | αi(ν) -y i + y i -(1 + ) pi(ν) y i } ≤ max y:y T P x=1 |y i | αi(ν) -|y i | + |y i | 1 -(1 + ) pi(ν) .

Figure 5 :

 5 Figure 5: Results of simulation of (2), (6) and[START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 without disturbances and with initial conditions x 0 = (0.5, 1, 0).

Figure 7 :

 7 Figure 7: Results of simulation of (2), (6) and[START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF] for n = 3 with initial conditions x 0 = (5, 10, 0) and the disturbance[START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

. 1 ,

 1 φ = 8, β = 7, κ = 0.5, γ 1 = γ 2 = 3. And the obtained parameter values are a = -(384, 136.08, 16.94) and P c = 10 3

Figure 8 :

 8 Figure 8: Comparison results between the recursive finite-time observer and the nonrecursive fixed-time one for n = 3, without disturbances and with initial conditions x 0 = (1, 1, 1).

Figure 9 :

 9 Figure 9: Comparison results between the recursive finite-time observer and the nonrecursive fixe-time one for n = 3, without disturbances and with initial conditions x 0 = (10, 10, 10).

Figure 10 :

 10 Figure 10: Comparison results between the recursive finite-time observer and the nonrecursive fixe-time one for n = 3, without disturbances and with initial conditions x 0 = (10, 10, 10).

For a detailed explanation of the function[START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] and its properties, the reader is referred to[START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF].


Nicholas of Bar s Collection

Sylvain Piron [in Chris Schabel dir., Theological Quodlibeta in the Middle Ages. The Fourteenth Century, Leiden, Brill, 2007, p. 333-343.] The so-called "Collection of Nicholas of Bar" occupies a prominent place in Palémon Glorieux's repertory of quodlibetal questions. The 170 questions that it contains were first listed under the name of Nicholas in the 1925 volume, before being redistributed, in the second volume published in 1935, amongst the eighteen different authors involved in it. In the meantime, an article published in the third volume of the Archives d'histoire doctrinale et littéraire du Moyen Age in 1928 had justified the respective ascription of the questions to these little known late thirteenth century theologians. 1 Before discussing the interpretation he gave of this collection, one should recall the pioneering role played by Glorieux in the exploration of scholastic sources. Teaching at the time in Lille, Glorieux was connected with the French Dominican school lead by Pierre Mandonnet and Marie-Dominique Chenu, based a few kilometers away, at "Le Saulchoir," near Tournai, 2 and whose "Bibliothèque thomiste" hosted the two volumes of La littérature quodlibétique. In those years, as the editorial program of the Archives made clear, the priority was given to text editions and research based on first hand knowledge of unpublished materials. Taking his part in these efforts to tackle the mass of unexplored documentation pertaining to medieval scholasticism, Glorieux was also pursuing, so to speak, a peculiar hobby-horse, reconstructing the succession of masters occupying the chairs of theology within the University of Paris. 3 For that purpose, [334] he was searching to associate all University exercises he could come across -be they sermons or quodlibetal disputes -with precise dates, which would then serve as anchors for the global chronology of the theology faculty he was trying to construct. 4 With all the due respect one may have for his colossal achievements, it is fair to recall that many of his conclusions were prematurate. This eagerness for precise dates is evident in his approach to Nicholas of Bar's collection.

To quote Glorieux's own words: "The first impression one gets by reading this list [of the various masters' names found in the margins] is that one is confronted with quodlibeta that must have followed one another on a regular basis from year to year. 5 The rest of the demonstration serves only as a confirmation of this first impression, to which no objection is raised in the course of the article. The conclusions reached on the basis of that "first impression" has been generally accepted by scholars dealing with this material and has not been questioned so far. 6 However, the very statement of this result sounds rather unlikely: fragments of 29 different quodlibetal series would have been reproduced in a strictly chronological order, over a period of twenty years, running from 1285 to 1304. It suffices to add that these texts were copied by the same hand in only one go, for one to understand that a strict chronological ordering of the quodlibeta is highly implausible. It would require that the scribe himself would have shared Glorieux's obsession with the chronology of the theology faculty.

During the interwar years, another close associate of the Saulchoir, a Dominican friar himself although not residing at the convent, was Jean Destrez. His masterpiece volume on the diffusion of university manuscripts through the pecia system was published in 1935, in the same year [335] as the second volume of Glorieux's La littérature quodlibétique. 7 The line of approach exemplified by Destrez is another crucial feature of the "Saulchoir School, which achieved its most splendid results once a team of scholars educated along such principles was put in charge of reviving the Commissio Leonina in the early fifties. 8 A major lesson that has since then been widely accepted by the academic community as a whole is that at least a minimal level of attention to codicological matters is a necessary preliminary requirement in any study of medieval texts. Following this lesson, a few elements of Glorieux's brilliant treatment of the issue will be slightly amended here.

The name given to the collection derives from the fact that the only manuscript that 5 Glorieux, "Notices," p. 203-4: "La première impression que l'on ressent à la lecture de cette liste... est qu'on se trouve en présence... de quodlibets qui durent se succéder assez régulièrement, sans doute d'année en année.... La collection aurait été faite au fur et à mesure des soutenances annuelles, et suivant leur succession chronologique… Il semble parfaitement légitime de la considérer comme s'étendant de façon régulière sur une assez longue suite d'années." 6 Through confusion caused by the certitude displayed by Glorieux, some of his readers were mislead into thinking that the quodlibeta were actually dated on the manuscript itself. Such is the case of L. Cova, "Alcune questioni di Simone di Lens sul peccato originale," AFH 73 (1980) RSPT 98 (2005), pp. 31-110, insists on the continuity between the two "ages" of the Commission. Without diminishing the merits of the first team set up in the late 19th century, I feel it is necessary to stress more the human and cultural continuity between the second team and the Saulchoir school. As an irony of history, the same master general who had removed Chenu from teaching in 1942 then asked his disciples to take over the edition of Aquinas' opera omnia less than ten years later.

preserves it, Paris BnF lat. 15850, belonged to Nicholas of Bar-le-Duc, who, at his death in 1310, bequeathed it to the "poor masters of the Sorbonne, students in theology," as is made clear by two successive notes found on the back of the initial flyleaf. 9 In his reconstruction of the Sorbonne library, Léopold Delisle identified six Parisian volumes stemming from Nicholas' bequest to the College where he had once been a student. 10 Bishop of Mâcon since 1286, [336] Nicholas was apparently a rich man and generous towards his College. The Sorbonne obituary recalls that he "left us many books and a lot of money in rents." 11 As a matter of fact, his last will stipulated that all his belongings within the Paris diocese would go to the College, which should, in turn, support one master and two students from his kindred or his native land. The "mitigation" of his will, which reduced that number to only one student, listing all the properties handed over to the College, mentions a total of eighteen volumes. 12 According to the 1338 Sorbonne library catalogue, cod. lat. 15850 was located within the "Summe questionum" section. The flyleaf note describes it as containing "questions collected out of various quodlibeta." The codex consists of two parts that both fit such a description. The second and much longer part, having a medieval foliation of its own, is made up of a selection from two great masters of the quodlibetal art, Henry of Ghent (ff. 1r-202v) and Godfrey of Fontaines (ff. 205r-320r), followed by an uncomplete thematical index and a list of questions. 13 The first part of the codex comprises 44 folios. Its medieval foliation has been trimmed off on the occasion of a modern binding, so that it is difficult to make sure whether it had been conceived as a whole before being united together with the second part of the volume. These three blocks are as follows: A, a quire containing three disputed questions from the first Mendicant-Secular polemics by Bonaventura and William of Saint-Amour 14 (ff. 1ra-5vb); B, another quire containing on its two initial pages a selection of five quodlibetal questions by Henry of Ghent (ff. 6ra-7vb, the rest of the quire being blank); and C, the "collection" [337] itself, covering three quires (ff. 10ra-42va, f. 43 being blank). These units all have different page lay-outs, and were copied by different scribes on different occasions, the first one being presumably much older. Despite their distinct foliations, section 1-C was apparently produced in the same circumstances as part 2. The single hand that copied the whole of the collection was also involved in copying Godfrey's extracts (ff. 205r-233v). 15 The decoration of the initial letters in red and blue ink is similar in both sections, and it is also identical to what is found in cod. lat. 16158, another of Nicholas' volumes.

More importantly, their contents also point in the same direction. Both selections of quodlibetal questions are almost exclusively concerned with practical moral cases. As is well known, such topics are considerably over-represented in quodlibeta. 16 Especially in the Lent session, in the season of the yearly compulsory confession, quodlibetal disputes provided an occasion to raise the most troublesome cases of conscience that a confessor might come across, be it through reporting real situations or making up fictitious ones. Understandably, the same focus could also determine the form under which quodlibetal questions would circulate. The case under consideration is by no means an isolated one. It may be useful to record that the only collection of quodlibeta by Servais of Mont-Saint-Eloi belongs to this type, where only a minimal number of questions (5 out of 85) are not directly pertaining to moral issues or to the administration of confession. 17 The formulation of Servais' questions provided by Glorieux can be misleading. For instance, a question that seemed to Jean-Luc Solère to be of the most speculative type 18 -can someone excommunicate himself? -in reality introduces a very specific case, asking whether an official, promulgating a sentence of excommunication against unnamed wrongdoers, could himself fall prey to his own sentence. 19 The restricted scope of the questions preserved [338] in this manuscript doesn't mean that Servais was unable to deal with more abstract topics or was not interrogated on such themes. It rather means that, in the absence of a proper "edition" of his Quodlibeta made available by the university stationers, his Quodlibeta circulated in such a format. In the same guise, at one point, the bishop of Mâcon added to his library a volume containing the "practical" extracts of both the great and the small masters of 15 A paleographical study of manuscripts produced around or within the Sorbonne in that period could produce interesting results. 16 See Elsa Marmursztejn's chapter in the first volume of this book. 17 P.Glorieux, "Les Quodlibets de Gervais du Mont-Saint-Eloi," RTAM 20 (1953), pp. 129-34, argues in favour of a division of the 85 questions by Servais found in Paris, BnF lat. 15350, ff. 269r-291, into eight different Quodlibeta, on the basis of some marginal signs and of the recurrence of these rare speculative questions that would mark the begining of a new Quodlibet. This division is supported by even more marginal signs. 18 See Solère's chapter in the previous volume. 19 Servatius de Monte Sancti Eligii, Quodlibet V, q. 7, Paris, BnF lat. 15350, f. 281v: Utrum aliquis possit excommunicare seipsum et ponitur talis casus, laicus perdidit uxorem suam, officialis autem loci tenet eam clausam, laicus petit ab officiali ut excommunicet omnes qui detinent eam, officialis autem auctoritate episcopi excommunicat omnes qui detinent eam, modo queritur utrum officialis sit excommunicatus. theology of the time. Some more observations can be made based on external criteria. The questions extracted from Godfrey, copied from the university stationary peciae, do not follow a chronological order.

Excerpts from Quodlibeta VII and VIII are placed before extracts of V and VI. The final items stem from Quodlibet XIII, which was disputed during the academic year 1297-98. 20 The absence of any question from Quodlibet XV, produced in 1303-04 at the earliest, suggests that the collection of excerpts had been created before that date. Another interesting fact is that the initial note of possession conveyed by the flyleaf is identical to the one found in cod. lat. 15985, containing the original manuscript of Thomas of Ireland's Manipulus florum. 21 This volume itself can be firmly dated to 1306, when Thomas, previously a Sorbonne member, provided the College with one copy and Nicholas with another. As Richard and Mary Rouse hypothesize, in doing so, he may well have been searching for a beneficial reward from the bishop of Mâcon. 22 At any rate, this incident shows the position Nicholas occupied in the first decade of the fourteenth century, as an informal patron of the Sorbonne circles. It may be in that connection, rather than for his own pastoral use, that he commissioned the quodlibetal miscellany we are dealing with.

The collection of minor masters was copied continuously by only one scribe, with no change of ink or size of writing that would suggest an interruption in the course of his work. This implies that any notion of a collection composed over a long period of time has to be ruled out from the start. Nevertheless, a few elements of discontinuity can be pointed out. At the beginning of the first quire, there is no rubric announcing [339] the general contents of what follows. Nor is there any ascription of the initial fourteen questions. Such ascriptions, starting with the fifteenth question, are, at first, noted by the copyist in the margin. For some time, he also reproduces the titles of the questions in the lower margin of the page, a fact that is acknowledged on the flyleaf. 23 In that respect, a change can be noticed on folio 35v. Henceforth, ascriptions are no longer located in the margins but within the page, as a rubric.

From that point on, an even more significant change regards the status of the questions themselves. On the previous pages, questions were rarely very long and elaborate. Some of them probably belong to the genre of reportationes, jotted down by listeners during the event.

The ten final quodlibeta (ff. 35v-42v) are unambiguously of that nature since the master's response is always introduced by verbal forms such as "dicebatur," "dictum fuit," "dixit" ("it 20 See John Wippel's chapter in volume I. 21 The "r" with a long descender in the word Barro is very characteristic. 22 R.H. Rouse and M.A. Rouse, Preachers, florilegia and Sermons. Studies on the Manipulus florum of Thomas of Ireland (Toronto 1979), pp. 94-5, 164-5. 23 Paris, BnF lat. 15850, f. 1v: Intitulationes quorundam questionum quere fol. xi. et .xii. These titles are found on ff. 10v-13v, 14v. This implies a continuous foliation of the first part of the codex, but doesn't reveal whether this was achieved after or before the first binding. was said," or "he said"). These questions are deprived of any set of initial arguments, and the magistral determination can be remarkably laconic, such as Eustache of Grandcourt in his Quodlibet II, q. 10: "Utrum sit licitum celebrare festum conceptionis Virginis Marie? Dictum est quod sic." 24 When the question is a complex moral case, it can even happen that the interrogation itself is longer than the answer, such as with the same Eustache in his Quodlibet III, q. 10. 25 This stylistic change is all the more obvious when one compares the slightly more elaborate style of Eustache's first Quodlibet, found earlier in the collection, with the later ones.

Even in the case of a very brief question, so short that it passed under Glorieux's eyes unnoticed, we still find a pro and contra structure before the determination of the question. 26 Answering the following [340] question, in a most spectacular fashion, Eustache begins discussing the ability to marry of a monster (a female with two head, four breasts, and one sexual organ) by recalling in the first person a memory of his own father having witnessed such an extraordinary birth -although it is not clear whether he was talking about a human being or another type of animal. 27 As absurd as it may sound, the use of the first person ("ego intellexi", "dico") would signify that a venerable master of theology did mention such personal memories at a public university event, and gave his stamp to its written circulation. The likelier explanation for the stylistic change witnessed within the collection is the following. Among the various materials the copyist had at his disposal, the final ten quodlibeta would have originated from the notes taken in a very summary way by someone who attended these events. It may be surmised that these quodlibeta, where the same names often recur, were not spread over a long period of time -a duration of three years being a reasonable guess. On such an hypothesis, Eustache of Grandcourt s Quodlibet II and Peter of Saint-Omer s Quodlibet II would belong to the first year, while the second would be filled by Eustache s Quodlibet III 24 Paris, BnF lat. 15850, fol. 36rb. The same question is tackled by Matthew of Aquasparta in 1279 (Quodl. II, q. 8), Henry of Ghent in 1291 (Quodl. XV, q. 13) and John Baconthorpe in 1330 (Quodl. III, q. 18), a sign of continuous interest in the issue. All these, and many more, are edited by A. Samaritani, "De beatae virginis immaculata conceptione Quodlibet XIII-XIV saec. primum edita," Marian Library Studies, 5 (1973) pp. 729-835. 25Eustacius de Grandicuria, Quodlibet III, q. 10, Paris, BnF lat. 15850, f. 39va: "Talis casus est: iste cognovit istam puellam affectu fornicatorio, et habet inde puerum; postea accipit eam in uxorem, et habent alium filium; modo queritur quis istorum, mortuo patre, debeat habere progenituram. Dictum fuit quod primo natus quia factus est legitimus per matrimonium, et ad hoc adducebat multa decreta." 26 Paris, BnF lat. 15850, f. 29ra: Sacerdos dicit nonam ante terciam ex ignoranciam et oblivione, postea recolit, queritur postquam talis dixerit terciam debeat iterare nonam. Videtur quod sic quia ordo ecclesie requiritur ut nona dicatur post terciam, ideo etc. Contra, qui semel solvit debitum non amplius tenetur ad solutum eius quia solucione eius quod debetur tollitur obligacio, ideo etc. Dicendum quod tenetur redicere nonam quia dicendo nonam ante tertiam corrumpit ordinem ecclesie et ideo non solvit sicut debet, ideo etc. 27 Eustacius de Grandicuria, Quodlibet I, Paris, BnF lat. 15850, f. 29rb: Quoddam monstrum fuit, habens a dyaphragmo superius duo capita, quatuor brachia et quatuor mamillas et duo corda; inferius unum instrumentum generationis et duos pedes, queritur utrum tale monstrum possit contrahere matrimonium… Dicendum quod ego intellexi a patre meo quod ipse vidit tale monstrum habens duo capita et quatuor mamillas et quod una illorum mulierum mortua fuit ante aliam per tres dies et altera fuit mortua per nimio fetore quam non potuit sustinere. Nunc dico quod possunt contrahere matrimonium et cum diversis, quia habent omnia per quem matrimonium potest esse perfectum quoniam habent consensum per quem matrimonium perficitur, habent etiam instrumentum per quid potest consumari… and Peter s Quodlibet III. Simon of Guiberville s Quodlibet I and Rénier of Clairmarais Quodlibet II could belong to the second or third year. The latter one would comprise Eustache s Quodlibeta IV and V, Guy of Cluny s Quodlibet II and Andrew of Mont-Saint-Eloi s Quodlibet I. Glorieux understands the mention "a magistro extacio" found in the margin as marking the beginning of a fifth Quodlibet, distinct from the fourth. This distinction is not necessary in my view, since the all the questions, before and after the note, display strong stylistic [341] similarities. The marginal note could be understood as a mere reminder of authorship.

It is now time to turn back to the list of masters found in the margins or the rubrics of the collection. Once the "chronological" hypothesis has been discarded, the elements of certainty are rather limited. 28 A few names, given in full, are identified beyond doubt. They are those of masters who remained regents for a long period of time. Nicholas du Pressoir held his first Quodlibet in 1272 and was active in 1286, but nothing is known of his later career. Eustache of Grandcourt, regent master in 1272, had stopped teaching by 1303. Peter of Saint-Omer was already a master in 1289, chancellor in 1296, and active until 1302. Two other names offer no difficulty in identification, but do not shed any further light on the collection either, since nothing else is known about their university career: Renier of Clairmarais' name is given in full, while "Guido de Cluniaco" can hardly be other than the Benedictine theologian Guy of Pernes, master in theology, and abbot of Beaulieu in 1303, but the notion that his activity as a regent master would have started as early as 1287 is nothing but an effect of Glorieux's fantasy. More precise dates are provided by the two Dominican masters explicitely named, who taught in Paris for briefer periods. Oliverus predicator is necessarily Olivier of Tréguier, regent in 1291-92. [START_REF] Kaeppeli | Scriptores Ordinis Praedicatorum medii aevi[END_REF] The Quodlibet raimondi predicatorum can securely be attributed to Raimond Guilha and dated in 1293-95. 30 The Franciscans are not so easily identified. The most likely candidate for the Quodlibet fratris iohannis de ordine minorum is John of Murrovalle, regent in 1289-90, and later Minister General of the Order and then cardinal. The time of his regency happened to fir perfectly with Glorieux's global chronological scheme. Yet this identification is by no means certain, since many other Friars Minor named John were around in those years. For instance, one can point to the case of the almost forgotten John of Pershore. [START_REF]Spicilegium Pontificii Atheneai Antoniani[END_REF] Vital du Four [342] produced, as a reminder (memoralia), an abbrievated version of eighteen of his questions, now preserved in Todi, Biblioteca Comunale 95, a volume that contains most of the literary output of the Gascon theologian. These questions deal mostly with metaphysical and trinitarian issues, often directed against Peter John Olivi's positions. The miscellaneous nature of the final four, touching upon practical moral cases or problems of cognition and grace, gives weight to the notion that we are facing a quodlibetal dispute. Pershore is known to have been active in Oxford around 1290, but the anti-Olivian orientation of this Quodlibet and its relation to Vital render a continental origine of these questions more plausible. The Quodlibet could have been disputed in Paris, while Vital was a student there (1288-92). Its contents display no obvious resemblance to the Franciscan John present in Nicholas of Bar's collection, however. If the "S. minor" has to be identified with Simon of Lens, his Quodlibet should be associated with his regency in 1282, which sounds at odds with the dates obtained so far that rather point towards the first half of the 1290s. Therefore, the hypothesis that this initial S could represent a "Stephanus" or a "Simon" that we know nothing about cannot be totally discarded. Some other names remain little more than shadows. Who was Gonterus or Ganterus? What was the exact christian name of "R. of Arras"? Here, as with the case of the anonymous Friar Minor to whom are attributed the questions 15-20 of the collection, the disappearance of Glorieux's chronology brushes away the little knowledge available about them. Two other characters, whose identity was reconstructed by Glorieux, are still in a uncomfortable situation.

Twice, the marginal notes refer to a Cistercian master, clearly labeled as "abbatis sancti bernardi" the second time. Out of his chronological calculations, Glorieux concluded that he should be identified, "with no hesitation," with John of Waarde, a Cistercian master who died, still active, in 1293. The difficulty here is that the first reference to a theologian of Saint-Bernard in the collection, a few pages above, describes him with a problematic abbreviation that can be developed in various ways -the most litteral reading would imply a name such as "Rilus" or "Nilus" -but which certainly doesn't fit with "Johannes." While discussing the identity of the character described as "M. de Walle Scolarium," Glorieux decides in favour of John of Châtillon only on the basis that, at the time in which this Quodlibet would fit within his chronology, another possible candidate, Lawrence of Poulangy, was then away from Paris.

More recent research on the Val-des-Ecoliers would suggest that John of Bray [343] was actually a master regent in Paris for a longer period than Châtillon, who apparently taught as regent master for a very brief period, in 1297-98, between the regencies of Poulangy and Bray. 32 Two masters appearing only in the final seventy-six briefer questions limit the time span of that section: Simon of Guiberville was active as a master from 1296 on, and became chancellor in 1302, while Andrew of Mont-Saint-Éloi's regency didn't begin before 1303-04. That Peter of Saint-Omer is described as chancellor in the same section, a fact strongly emphasized by Glorieux, would confirm a date posterior to 1296. Taking into consideration the presence of Andrew, the hypothesis of a three-year overall duration may have to be slightly extended, for Andrew's inception might have taken place after Eustache's withdrawal from teaching. In such a case, these quodlibetal cases could represent the state of debate on moral issues during the years 1300-04.

Before closing this chapter, the initial fifteen questions, which convey no trace of idenfication at all, have to be dealt with. Glorieux had proposed Bernard of Trilia OP as a candidate for the questions 7-14, on the basis of a complex reasoning. In the first place, it relied on the ascription to Bernard of a justificatory memoir that was later recognised as being penned by John Quidort. [START_REF] Glorieux | The history of the debate is synthetised by P. Künzle, "Prolegomena," in Bernardus de Trilia, Quaestiones disputatae de cognitione animae separatae (Bern 1969), 16*[END_REF] Although nothing connects Bernard to Nicholas' collectionanymore , some scholars have nevertheless been misled by Glorieux's certainties. [START_REF] Veraja | Le origini della controversia teologica sul contratto di censo nel XIII secolo[END_REF] By removing this mistaken ascription, nothing links the early parts of the collection to the 1280s. There is apparently no reason to conceive that the initial fourteen questions were produced by more than one author. In order to shed more light on his profile, a detailed study of his positions on various issues would be required. All in all, the safer conclusion that can be expressed for the time being is that the initial parts of Nicholas of Bar's collection would represent a selection of cases discussed in quodlibetal disputes during the 1290s, and probably for most of them during the first part of that decade., while the final section contains reportationes of questions from the early 1300s.

  32 C. Guyon, Les écoliers du Christ. L'ordre canonial du Val des Écoliers (1201-1539) (St Etienne 1998), pp. 231-233.

P. Glorieux, "Notices sur quelques théologiens de Paris de la fin du XIIIe siècle," AHDLMA 3 (1928), pp.

201-38. 2 Although a detailed account of the impact of the Saulchoir school on medieval studies is still lacking, a more general picture can be found in the studies accompanying the edition, by G. Alberigo, of M.-D. Chenu, Une école de théologie: le Saulchoir (Paris 1985). See also J.-P. Jossua, "Le Saulchoir: une formation théologique replacée dans son histoire," Cristianesimo nella storia 14 (1993), pp. 99-124; A. Duval, "Aux origines de l'Institut historique d'études thomistes au Saulchoir (1920 et sq.)," RSPT 75 (1991), pp. 423-48. 3 Glorieux, Répertoire I-II; idem, "D'Alexandre de Halès à Pierre Auriol. La suite des maîtres franciscains de Paris au XIIIe siècles," AFH 26 (1933), pp. 257-81. 4 On the use of university sermons by Glorieux, see L.-J. Bataillon, "Guillaume de la Mare. Note sur sa régence parisienne et sa prédication," AFH 98 (2005), pp. 367-422 (esp. pp. 368-74).

Paris, BnF lat. 15850, f. 1r: "Istud liber est magistri N. de barro ducis (cancelled); In hoc volumine continentur questiones collecte de diversis quodlibet, ex legato magistri Nicolai de barro ducis domui pauperum magistrorum de Sorbona in theologia studentium."

L. Delisle, Cabinet des manuscrits de la bibliothèque nationale (Paris, 1874), vol. 2, p. 163. The other codices are lat. 15351 (Thomas Aquinas, Quodlibeta), lat. 15818 (Thomas Aquinas, Summa contra Gentiles), lat. 15985 (Thomas of Ireland's Manipulus florum), lat. 16158 (philosophical commentaries by Peter of Auvergne, Giles of Rome and Albert the Great), and lat. 16160 (more philosophical commentaries, mostly by Thomas Aquinas).

"Dies 16 maii obiit magister Nicolaus de Barro Ducis, socius hujus domus, anno Domini 1310, qui nobis legavit multos libros et multam pecuniam in redditibus," in Le registre de prêt de la bibliothèque du collège de Sorbonne, ed. J. Vieillard and M.-H. Jullien de Pommerol (Paris 2000), p. 568. This register doesn't record any borrowing of cod. lat. 15850. The obituary was already edited in P. Glorieux, Aux origines de la Sorbonne. I. Robert de Sorbon (Paris 1966), p. 165.

P. Glorieux, Aux origines de la Sorbonne. II. Le cartulaire (Paris 1965), pp. 519-23. Surprinsingly, Nicholas is presented as filius Ponceti de Barro, Tullensis dyocesis. There is, indeed, a city called Bar about fifteen kilometers north of Tulle. But since the students he wanted to support should stem, in order of priority, from his kindred, the city of Bar-le-Duc, Saint-Mihiel, and the diocese of Verdun, a Lorraine origin is beyond doubt.

See the details of the questions in J. Hoffmans and A. Pelzer, Etude sur les manuscrits des Quodlibets (Les philosophes belges,

14) (Louvain 1937), pp. 177-80 and R. Macken, Bibliotheca Manuscripta Henrici de Gandavo, (Leuven-Leiden 1979), vol. 1, pp. 634-44. 14 Bonaventura, De perfectione evangelica, q.2 a. 2, Opera omnia 5 (Quaracchi 1891), pp. 134-155 ; A. G. Traver, "William of Saint-Amour's Two Disputed Questiones De Quantitate eleemosynae and De valido mendicante," AHDLMA 62 (1995), pp. 295-342.