Konstantin Zimenko

Andrey Polyakov

Denis Efimov

Wilfrid Perruquetti

Generalized Feedback Homogenization and Stabilization of Linear MIMO Systems*

Generalized homogenization of linear MIMO systems via linear feedback is introduced. The control algorithm for finite-time (or asymptotic) stabilization of linear MIMO systems via homogenization technique is developed. The robustness of the control algorithm with respect to system uncertainties and disturbances is studied. The theoretical results are supported by numerical examples.

I. INTRODUCTION

The homogeneity is a property, that describes sort of symmetry, i.e. an object remains consistent with respect to a group of transformations. This property is widely used for system analysis, control and observer design (see, for example, [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF]- [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF]). Such an interest to the homogeneity concept is based on very useful features of homogeneous systems. In particular, local stability of homogeneous system means the global one; if an asymptotically stable system is homogeneous with negative degree, then it is finitetime stable, etc. The present paper deals with generalized homogeneity [START_REF] Polyakov | On Homogeneous Distributed Parameter Systems[END_REF], [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF], which is based on groups of linear transformations (linear dilations).

The present paper addresses the problem of feedback homogenization, i.e. problem of a feedback design making the system homogeneous of a given degree, and control design problem for linear MIMO plants providing asymptotic or finite-time convergence depending on the homogeneity degree of the closed-loop system. Asymptotic stabilization via homogenization with positive degree can also attract attention since such systems have a number of interesting robust properties (see, for example, the results for generalized weighted homogeneous systems [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF]). Unlike existing solutions, the presented results do not require special forms of the system matrix and block decomposition of linear MIMO systems (as, for example, in [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF]), which in some cases can be accompanied by significant computational errors. The presented control law contains two terms: a linear feedback that homogenizes the system with a specified degree and a generalized homogeneous control law, which stabilizes the system in finite time (asymptotically) if degree of homogeneity is negative (positive).

The outline of this paper is as follows. Notation used in the paper is given in Section II. Section III introduces the problem statement and the basic assumptions. Section IV considers preliminaries used in the paper. After that the results on linear MIMO system homogenization and control design are presented in Section V. Two examples are considered in Section VI. Finally, concluding remarks are given in Section VII.

II. NOTATION

Through the paper the following notation will be used:

• R + = {x ∈ R : x > 0}, where R is the field of real numbers; • • denotes a norm in R n ; • A A = sup x∈R n Ax x
for A ∈ R n×n ; • S = {x ∈ R n : x = 1} is the unit sphere in R n ; • I n ∈ R n×n is the identity matrix;

• the minimal and maximal eigenvalues of a symmetric matrix P = P T are denoted by λ min (P ) and λ max (P ), respectively; • the inequality P > 0 means that the symmetric matrix P = P T ∈ R n×n is positive definite; • R(λ) denotes the real part of the complex number λ.

III. PROBLEM FORMULATION

Consider the system

ẋ(t) = Ax(t) + Bu(t) + d(t, x(t)), (1)
where x ∈ R n is the state vector, u ∈ R m is the vector of control inputs, A ∈ R n×n , B ∈ R n×m (the pair (A, B) is controllable, m ≤ n) are system and control gain matrices, d : R × R n → R n describes the system uncertainties and disturbances. The state vector x is assumed to be measured and can be used for feedback control design. To consider the case of discontinuous function d the theory of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] can be utilized.

The main goal of the paper is to propose a constructive stabilizing control algorithm based on generalized homogenization of the system (1). The control must guarantee asymptotic or finite-time stability, depending on the degree of homogeneity of the closed-loop system.

IV. PRELIMINARIES A. Stability Notions

Consider the following system

ẋ(t) = f (x(t)), x(0) = x 0 , t ≥ 0, (2)
where x(t) ∈ R n is the state vector, f ∈ R n → R n is a vector field, f (0) = 0. For any x 0 ∈ R n , a corresponding solution of ( 2) is denoted by x(t, x 0 ). The case of the discontinuous vector field f is not excluded. In this case the system solutions x(t, x 0 ) are understood in the sense of Filippov (see, for example, [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], [START_REF] Smirnov | Introduction to the Theory of Differential Inclusions[END_REF]).

According to [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] an absolutely continuous function x(t, x 0 ) is called a solution of the Cauchy problem associated to (2) if x(0, x 0 ) = x 0 and for almost all t > 0 it satisfies the following differential inclusion

ẋ ∈ K[f ](t, x) = co ε>0 µ(N )=0 f (t, B(x, ε)\N ), (3)
where co(M ) defines the convex closure of the set M , B(x, ε) is the ball with the center at x ∈ R n and the radius ε, the equality µ(N ) = 0 means that the measure of N ⊂ R n is zero.

Definition 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF] The origin of ( 2) is said to be globally finite-time stable if it is globally asymptotically stable and any solution x(t, x 0 ) of the system (2) reaches the equilibrium point at some finite time moment, i.e.

x(t, x 0 ) = 0 ∀t ≥ T (x 0 ) and x(t, x 0 ) = 0 ∀t ∈ [0, T (x 0 )), x 0 = 0, where T : R n \ {0} → R + is the settling-time function. It is fixed-time stable if in addition sup x0∈R n T (x 0 ) = T max < +∞.
Definition 2 [START_REF] Polyakov | Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems[END_REF] The set M is said to be globally finitetime attractive for (2) if any solution x(t, x 0 ) of (2) reaches M in some finite time moment t = T (x 0 ) and remains there ∀t ≥ T (x 0 ), T : R n → R + ∪ {0} is the settling-time function. It is fixed-time attractive if in addition the settlingtime function T (x 0 ) is globally bounded by some number T max > 0.

Theorem 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Laghrouche | Stabilization of perturbed integrator chains using Lyapunov-Based Homogeneous Controllers[END_REF] Suppose there exists a positive definite C 1 function V defined on an open neighborhood of the origin D ⊂ R n and real numbers C > 0 and σ ≥ 0, such that the following condition is true for the system (2)

V (x) ≤ -CV σ (x), x ∈ D \ {0}.
Then depending on the value σ the origin is stable with different types of convergence:

• if σ = 1, the origin is exponentially stable; • if 0 ≤ σ < 1, the origin is finite-time stable and T (x 0 ) ≤ 1 C(1 -σ) V 1-σ 0 ,
where V 0 = V (x 0 ); • if σ > 1 the origin is asymptotically stable and, for every ε ∈ R + , the set B = {x ∈ D : V (x) < ε} is fixed-time (independent on the initial values) attractive with

T max = 1 C(σ -1)ε σ-1 .
If D = R n and function V is radially unbounded, then the system (2) admits these properties globally.

B. Generalized Homogeneity

The homogeneity is a property that specifies sort of symmetry of an object with respect to a group of transformations (dilation operation). The type of homogeneity, dealing with linear transformations, is called generalized homogeneity.

Definition 3 [START_REF] Polyakov | On Homogeneous Distributed Parameter Systems[END_REF], [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF] A map d : R → R n×n is called dilation in the space R n if it satisfies:

• group property: d(0) = I n and d(t + s) = d(t)d(s) = d(s)d(t) for t, s ∈ R; • continuity property: d is a continuous map;

• limit property: lim s→-∞ d(s)x = 0 and lim s→+∞ d(s)x = +∞ uniformly on the unit sphere S. The dilation d is a uniformly continuous group [START_REF] Polyakov | On Homogeneous Distributed Parameter Systems[END_REF]. Its generator [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] is a matrix G d ∈ R n×n defined by

G d = lim s→0 d(s) -I n s .
The generator G d satisfies the following properties [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]

d ds d(s) = G d d(s) = d(s)G d , (4)
d(s) = e G d s = +∞ i=0 s i G i d i! , (5)
where s ∈ R.

Let us introduce the definitions on monotonicity of the dilation d(s).

Definition 4 [START_REF] Polyakov | On Homogeneous Distributed Parameter Systems[END_REF] The dilation d is said to be monotone if d(s) A < 1 for s < 0.

Thus, monotonicity means that d(s) is strong contraction for s < 0 (strong expansion for s > 0) and implies that for any x ∈ R \ {0} there exists a unique pair (s 0 , x 0 ) ∈ R × S such that x = d(s 0 )x 0 . The rate of contraction is introduced by the next definition.

Definition 5 [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF], [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF] The dilation d is said to be strictly monotone if ∃β such that d(s) A ≤ e βs for s ≤ 0.

Note [START_REF] Polyakov | On finite-time stabilization of evolution equations: A homogeneous approach[END_REF], that monotonicity property may depend on a norm • .

Theorem 2 [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF] If d is a dilation in R n , then

• the generator matrix G d is anti-Hurwitz, i.e. R(λ i ) > 0, i = 1, ..., n; • there exists a symmetric matrix P ∈ R n×n such that

P G d + G T d P > 0, P > 0. (6)
• the dilation d is strictly monotone with respect to the weighted Euclidean norm x = √ x T P x for x ∈ R n and P satisfying ( 6):

e αs ≤ d(s) A ≤ e βs if s ≤ 0, e βs ≤ d(s) A ≤ e αs if s ≥ 0, where α = 1 2 λ max P 1 2 G d P -1 2 + P -1 2 G T d P 1 2 , β = 1 2 λ min P 1 2 G d P -1 2 + P -1 2 G T d P 1 2
.

Definition 6 [START_REF] Polyakov | On Homogeneous Distributed Parameter Systems[END_REF] A vector field f :

R n → R n (a function g : R n → R) is said to be d-homogeneous of degree ν ∈ R if f (d(s)x) = e νs d(s)f (x), ∀x ∈ R n \ {0}, ∀s ∈ R. (resp. g(d(s)x) = e νs g(x), ∀x ∈ R n \ {0}, ∀s ∈ R.) (7)
A special case of homogeneous function is a homogeneous norm [START_REF] Kawski | Geometric Homogeneity and Stabilization[END_REF], [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF]

Theorem 3 [START_REF] Polyakov | Sliding Mode Control Design Using Canonical Homogeneous Norm[END_REF] An asymptotically stable d-homogeneous system ẋ = f (x), f : R n → R n of degree ν ∈ R is finite-time stable if and only if ν < 0.

C. On the Matrix Equation X Ā -ĀX = X

Let us consider the matrix equation

X Ā -ĀX = X. (9)
The following results hold true.

Proposition 1 [START_REF] Burde | On the Matrix Equation XA-AX = X P[END_REF] Let Ā ∈ R n×n . Then every matrix solution X ∈ R n×n of the equation X Ā -ĀX = X is nilpotent and hence satisfies X n = 0.

Note, that the linear matrix equation ( 9) is a special case of the Sylvester matrix equation. The general solution of the matrix equation is given in [START_REF] Gantmacher | The theory of matrices[END_REF].

In solving the inverse problem (search for the matrix Ā for given X), the following result is valid.

Proposition 2 For any nilpotent matrix X ∈ R n×n the equation ( 9) has a solution.

Proofs of all propositions are skipped.

V. MAIN RESULT

The following lemma gives the condition of the linear system to be d-homogeneous with the generator

G d . Lemma 1 Let G d ∈ R n×n be a generator of the dilation d(s) = e G d s , s ∈ R. Then the linear system ẋ = Cx, x ∈ R n , C ∈ R n×n is d-homogeneous of degree ν ∈ R if and only if CG d -G d C = νC. ( 10
)
Based on Proposition 1 and Lemma 1 it can be shown that the system (1) with d(t, x) = 0 can be homogenized by a linear control u = K 0 x.

Lemma 2 Let A + BK 0 be nilpotent for K 0 ∈ R m×n . Then for any ν ∈ R there exists a matrix G d such that the system (1) with d(t, x) = 0 and u = K 0 x is d-homogeneous of degree ν.

Further, these results are utilized for control design of the system [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF].

Theorem 4 (Disturbance-free case) Let d(t, x) = 0 and • for some ν ≥ -1 the system of matrix equations and inequalities

(A+BK 0 )X+X(A+BK 0 ) T +By+y T B T +αX ≤ 0, (11)
(A + BK 0 )L -L(A + BK 0 ) = (A + BK 0 ), (12)
(L -γI)B = 0, (13)
X > 0, (14)
is feasible, where the matrix

K 0 ∈ R m×n is such that the matrix A + BK 0 is nilpotent, L, X ∈ R n×n , y ∈ R m×n , α ∈ R + , γ ∈ R; • the control has the form u(x) = K 0 x + x ν(1+γ)+ d Kd(-ln x d )x, ( 15
)
where K = yP , P = X -1 and ∈ R + is chosen such that νL + νL T + 2 I n > 0, (16)
ηX ≥ νLX + νXL T + 2 X > 0 (17)
for some η ∈ R + . Then the closed-loop system (1), ( 15) is asymptotically (for ν > 0) / exponentially (for ν = 0) / finite-time (for 0 > ν ≥ -1) stable.

Remark 1 Obviously, the case ν = 0 is out of interest since in this case the task reduces to the standard linear control design problem.

Remark 2 According to Theorem 1 for 0 > ν ≥ -1 the settling time function is bounded as follows

T (x 0 ) ≤ - η αν V -ν 0 ,
for ν > 0 and any ε ∈ R + , the set B = {x ∈ R n : V (x) < ε} is fixed-time attractive with

T max = η ανε ν . Remark 3
The canonical homogeneous norm • d (Lyapunov function V ) is defined implicitly by [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]. To find • d the numerical procedures can be used, as, for example, in [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF] the bisection method has been applied to find implicitly defined Lyapunov function.

Note, that implicit Lyapunov function method was introduced by Korobov in [START_REF] Korobov | A general approach to synthesis problem[END_REF], [START_REF] Korobov | A general approach to the solution of the bounded control synthesis problem in a controllability problem[END_REF] for the first time.

Remark 4 The feedback matrix K 0 can be chosen using pole placement methods.

The following theorem presents the modification of the control scheme from Theorem 4 to reject some additive disturbances.

Theorem 5 (Disturbed case) Let ν ≥ -1 and

• the system ( 12)-( 14),

(A+BK 0 )X+X(A+BK 0 ) T +By+y T B T +αX +R ≤ 0, (18)
is feasible, where the matrix K 0 ∈ R m×n is such that the matrix

A + BK 0 is nilpotent, R, L, X ∈ R n×n , R > 0, y ∈ R m×n , α ∈ R + , γ ∈ R;
• the control has the form [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] with K = yP , P = X -1 and , η ∈ R + satisfying the inequalities ( 16), ( 17 Then the closed-loop system (1), ( 15) is asymptotically (for ν > 0) / exponentially (for ν = 0) / finite-time (for 0 > ν ≥ -1) stable.

Remark 5 The restriction on the system disturbances ( 19) is presented in the form, which is not appropriate for using in practice. It can be simplified using the result of Proposition 16 in [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF] if G d is presented in a diagonal form. Simplification and relaxation of the restriction ( 19) may be one of the directions for future research. Define the finite-time control u in the form [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] with the parameter ν = -0.5, where the matrix K 0 ∈ R 2×3 is chosen

K 0 = -2 -5.9442 -2.4721 -1 -4 - 2
that the matrix A + BK 0 is nilpotent and the matrices P ∈ R 3×3 , K ∈ R 2×3 are obtained from the inequalities ( 11)-( 14), ( 16), ( 17

B. Robust finite-time control of damper system

Consider a mass/spring/damper system [START_REF] Zhou | Essentials of Robust Control[END_REF] as shown in Fig. 3, where The parameters of the control in the form (15) were selected solving the system ( 12)-( 14), ( 18), ( 16), [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF] for T by the Euler method with the fixed step size h = 0.001. The paper presents homogenizing and stabilizing control algorithms for linear MIMO systems. It is shown that the system (1) can be homogenized with any degree via linear feedback. The settling time estimate is obtained for finitetime stabilized system. Tuning of control parameters is presented in the form of linear matrix equations and inequalities. The robustness of the control algorithm with respect to system uncertainties and disturbances is studied. Numerical examples demonstrate the effectiveness of proposed control.

A =     0 0 1 0 0 0 0 1 -k1 m1 k1 m1 -b1 m1 b1 m1 k1 m2 -k1+k2 m2 b1 m2 -b1+b2 m2     ,
K 0 = -1 1 -0.

  : a continuous positive definite d-homogeneous function of degree 1. Define the canonical homogeneous norm • d : R n → R + as x d = e sx , where s x ∈ R such that d(-s x )x = 1. Note that d(s)x d = e s x d and d(ln x d )x = 1.

  ); • the disturbance function d satisfy dT d T (-ln x d )R -1 d(ln x d ) d ≤ ζα x 2ν d ,(19) with ζ ∈ (0, 1).

  VI. NUMERICAL EXAMPLESA. Finite-time stabilization of linear MIMO system Consider the system (1) in disturbance-free case ( d(t, x) = 0) for n = 3,

  ) with the parameters α = 1, η = 3.4182: 0.25, = 1.1251. The numerical simulation of the closed-loop system has been done for x 0 = -1 5 -1 T by the Euler method with the fixed step size h = 0.01. To find values of • d the bisection method was used. The results of simulation are shown in Fig. 1, Fig. 2.

Fig. 1 .Fig. 2 .

 12 Fig. 1. System states versus time for finite-time control in disturbance-free case

Fig. 3 .

 3 Fig. 3. A two-mass/spring/damper system

2 0. 2 1

 2 -5 0.2 -0.3 , α = 1, η = 2.0858, R = I n : γ = 0.4, = 0.4414. Assume that the system operates in Ω = {x ∈ R n : x d ≤ 1}. In this case the inequality (19) can be rewritten as dT d ≤ ζ x -1.3572 d and it holds for ζ = 0.89. The numerical simulation shown in Fig. 4 has been done for x 0 = 1 -0.8 -2 -1.5

Fig. 4 .

 4 Fig. 4. System states versus time for finite-time control of the spring system

Andrey Polyakov, Denis Efimov and Wilfrid Perruquetti are with Non-A INRIA -LNE, Parc Scientifique de la Haute Borne 40, avenue


Carlos Baqueiro

Nelson Ibaseta
email: nelson.ibaseta@centrale-marseille.fr

Pierrette Guichardon

Laurent Falk

INFLUENCE OF REAGENTS CHOICE (BUFFER, ACID AND INERT SALT) ON TRIIODIDE PRODUCTION IN THE VILLERMAUX-DUSHMAN METHOD APPLIED TO A STIRRED VESSEL

Keywords: Micromixing, Villermaux-Dushman, Iodide iodate method, mixing characterization

This work studies how deeply the reagents choice influences micromixing characterisation by the Villermaux-Dushman method, when applying it to a 1 litre stainless steel standard vessel with two baffles, stirred by an inclined blade turbine. For the first time, borate and phosphate buffer are compared on their use in the method. It is observed that triiodide production is higher when borate buffer is used. Moreover, perchloric acid leads to higher triiodide production than sulphuric acid, when injecting the same concentration of both acids. Finally, the influence of the ionic strength is also studied, since there has been a great deal of controversy about it over the last years. The results show that the ionic strength affects triiodide production, although relatively slightly . Advice concerning the choice of the reagents is given in conclusion.

Graphical abstract

Highlights  The influence of the reagents choice in the Villermaux Dushman test is studied  Phosphate buffers lead to lower I3 -production that borate buffers  I3 -production is slightly influenced by the ionic strength  Both acids and buffers can be used for qualitative studies  The pKa of H2SO4 and H3PO4 should be considered in micromixing time estimations

Introduction

Micromixing [START_REF] Bałdyga | Turbulent micromixing in chemical reactorsa review[END_REF] is defined as mixing at the molecular scale, that is, below the Kolmogorov scale. It plays a main role when dealing with fast reactions: polymerization reactions, fast competitive-consecutive and fast parallel reactions, as well as reactive crystallizations (precipitations). In all these applications, the final product obtained depends on the micromixing rate. Hence, a special focus was given to developing micromixing characterization methods at the end of the 20 th century [START_REF] Bourne | An improved reaction system to investigate micromixing in high-intensity mixers[END_REF]Fournier et al, 1996a). The iodide iodate method, also called Villermaux-Dushman method, has probably become the main micromixing characterization method, thanks to a lower cost and easier product manipulation [START_REF] Guichardon | Experimental Comparison Of The Iodide-Iodate And The Diazo Coupling Micromixing Test Reactions In Stirred Reactors[END_REF]. Since it was proposed by Fournier et al (1996a) and modified by [START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF], the method has been widely used for selection of the injection point and comparison of the micromixing quality obtained using new devices or removing baffles [START_REF] Rousseaux | Micromixing Efficiency of a Novel Sliding-Surface Mixing Device[END_REF]Assirelli et al, 2002;Assirelli et al, 2005;Assirelli et al, 2008a) and determination of the local dissipative rate [START_REF] Schaer | Determination of local energy dissipation rates in impinging jets by a chemical reaction method[END_REF]Assirelli et al, 2008b;[START_REF] Nouri | Characterisation and comparison of the micromixing efficiency in torus and batch stirred reactors[END_REF][START_REF] Habchi | Mixing assessment by chemical probe[END_REF]. With the development of micromixers during the 2000s [START_REF] Hessel | Micromixersa review on passive and active mixing principles[END_REF], the method has also been adapted for their characterization [START_REF] Panić | Experimental approaches to a better understanding of mixing performance of microfluidic devices[END_REF][START_REF] Kölbl | The Iodide Iodate Method to Characterize Microstructured Mixing Devices[END_REF]Commenge and Falk, 2011). This method is based on the competition between two reactions: an instantaneous neutralization reaction (reaction 1, where B -is a base), and the Dushman reaction [START_REF] Dushman | The rate of the reaction between iodic and hydriodic acids[END_REF], which is a fast (but slower than reaction 1) comproportionation reaction (reaction 2).

H + + B -→ HB (reaction 1)

6H + + 5I -+ IO3 -→ 3I2 + 3H2O (reaction 2)
When some acid is fed in stoichiometric default in an aqueous basic solution containing an appropriate pre-mixture of iodide, iodate and B -anions, the products obtained depend on the quality of micromixing. In the case of perfect mixing conditions, all the protons will be consumed by the base according to reaction 1. In the case of imperfect mixing, there will be a local lack of base, protons will react with iodide and iodate anions, and some iodine will be produced.

A C C E P T E D M A N U S C R I P T
Iodine will then be involved in a quasi-instantaneous equilibrium with triiodide (reaction 3), as follows:

I -+ I2 ↔ I3 - (reaction 3)
Hence, triiodide concentration, which can be easily measured by UV-visible spectrophotometry, can be used to characterize the quality of micromixing.

The original method (Fournier et al, 1996;[START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF] used sulphuric acid and borate buffer. However, during the last ten years, different adaptations of the method have been proposed, most of them concerning the use of a different acid and/or buffer. [START_REF] Kölbl | The iodide iodate reaction method: The choice of the acid[END_REF] proposed replacing sulphuric acid with perchloric acid for quantitative treatment, arguing that sulphuric acid may not be completely dissociated. Hydrochloric acid has even been used [START_REF] Ehrfeld | Characterization of Mixing in Micromixers by a Test Reaction: single Mixing Unites and Mixer Arrays[END_REF], despite the catalytic effect of chloride anions on the Dushman reaction [START_REF] Schmitz | Kinetics and mechanism of the iodate-iodide reaction and other related reactions[END_REF]. On the other hand, [START_REF] Pinot | New protocol of the Villermaux-Dushman reaction system to characterize micromixing effect in viscous media[END_REF] proposed using a phosphate buffer (HPO4 -2 /H2PO4 -) instead of the borate buffer (H2BO3 -/H3BO3) because of the toxicity of borate. However, it is worth noting that the influence of these choices (especially the choice of the buffer) have not been deeply analysed. The present work aims at comparing results obtained using both acids and both buffers, and then proposing the most adapted combination for a given application.

Materials and methods

The following compounds were used without further purification in the preparation of the solutions: H2SO4 (VWR, GPR Rectapur), HClO4

[HClO4] [Na2SO4] [NaClO4] [NaH2BO3 ] & [H3BO3] [Na2HPO4 ] & [NaH2PO4 ] [KI] [KIO3] 1 0.5 - - - 0.1 each - 0.0116 0.00233 2 0.5 - - - - 0.1 each 3 - 1 - - 0.1 each - 4 - 0.5 - - 0.1 each - 5 - 1 - - - 0.1 each 6 - 1 0.5 - 0.1 each - 7 - 1 0.5 - - 0.1 each A C C E P T E D M A N U S C R I P T 4 - 0.5 - - - 0.1 each 0.5 - - 1 0.1 each - 0.5 - - 1 - 0.1 each - 1 - 0.5 0.1 each - 1 2.5 4 6.5 - 1 - 0.394 (*) 0.1 each - 0.904 (*) 1.41 (*) 1.92 (*)
The different sets of concentrations and species were appropriately chosen to examine the influence of sulphuric acid dissociation and the influence of the buffer at constant ionic strength, which is known to have an important influence of the kinetics of reaction 2.

The di-acid H2SO4 was then compared to the monoacid HClO4 to study the partial dissociation of sulphuric acid.

Different sodium salts, Na2SO4 and NaClO4, were also tested to study the potential basic behaviour of sulphate. And two buffer systems, NaH2BO3 and Na2HPO4, were also compared.

Experiments were carried out in a 1L stainless steel standard stirred vessel equipped with two baffles and an inclined blade turbine. Stirring speed was controlled with the aid of a Eurostat 20 Digital (IKA), and temperature was maintained at 25.0°C thanks to a double jacket connected to a Ministat 230 (Huber). An amount of 2 mL of acid solution was injected in the mobile discharge area, at the same height as the turbine and a radial distance of 0.8 times the vessel radius from the shaft. Feed was provided by a peristaltic pump (Masterflex L/S Digital Drive with a Masterflex L/S Easy-Load 3 head pump, Cole-Parmer) through an injection tube with an internal diameter of 1 mm. Absorbance at 353 nm, which can be directly correlated to the concentration in triiodide, was measured by UV-visible spectrophotometry (AvaSpec-ULS2048LTEC-USB2, with an AvaLight DH-S-DUV light source, both by Avantes). Samples were taken 5 minutes after the beginning of the injection.

If the injection flowrate is too high, the injection time will be smaller than the characteristic meso-and/or macromixing times [START_REF] Bałdyga | Turbulent micromixing in chemical reactorsa review[END_REF]. In that case, the reactions will be affected by meso-and/or macromixing, leading to greater triiodide productions for higher injection flowrates. Therefore, the injection flowrate has been gradually increased [START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF], without any significant variation of the absorbance (Fig. 1). It can therefore be concluded that mixing is only controlled by the last micromixing step. All the experiments were carried out at an intermediate injection flowrate of 0.040 mL•s -1 . Classical data analysis [START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF] in the Villermaux-Dushman method includes converting absorption at 353 nm into triiodide concentration by using the molar extinction coefficient at this wavelength, and then determining a segregation index, XS, and/or micromixedness ratio, α. However, the determination of both XS and α requires a precise estimation of the proton concentration available in the feed. As it will be seen in section 3.1, sulphuric acid is not fully dissociated when it is injected into the vessel, and the estimation of the proton concentration is not trivial. Hence, only absorbance (which can be directly related to triiodide concentration) will be considered here.

A C C E P T E D

M A N U S C R I P T

Results

Influence of the acid choice

The influence of the acid solution composition on the triiodide production (and thus, on the absorbance at 353 nm), when using a borate buffer is shown in Fig. 2 (left). In all cases, triiodide production decreases when increasing the stirring speed. As expected, a higher stirring speed increases the rate of energy dissipation and so decreases the micromixing time.

Results comparison for concentration sets 1 (0.5 mol/L H2SO4) and 3 (1 mol/L HClO4) shows a higher triiodide production for set 3. When comparing experiment sets 1 (0.5 mol/L H2SO4) and 4 (0.5 mol/L HClO4), it can be seen that triiodide production is lower for set 4. These results show the occurrence of the second dissociation of sulphuric acid in experiment set, even if it is not complete. They are in agreement with those of [START_REF] Kölbl | The iodide iodate reaction method: The choice of the acid[END_REF] who were the first to study the influence of the acid choice.

The behaviour of the sulphuric acid solutions can be better understood if we consider that, according to [START_REF] Bałdyga | Simplification of Micromixing Calculations. I. Derivation and Application of New Model[END_REF], engulfment controls micromixing for moderate viscosities (Sc << 4000). Large blobs of the acid solution are deformed and broken up, becoming long and thin slabs that expand engulfing the vessel fluid [START_REF] Bałdyga | Turbulent micromixing in chemical reactorsa review[END_REF]. Given the second dissociation constant of the sulphuric acid, pKa=1.99 [START_REF] Skoog | Fundamentals of analytical chemistry, 7th Edition[END_REF], the initial composition of these slabs is the following: 0.51 mol/L H + , 0.49 mol/L HSO4 -, 0.01 mol/L SO4 2-, at least as a first approximation (the actual pKa value is certainly different at this high electrolyte concentration). As the fluid from the vessel is incorporated, proton concentration decreases through both dilution and consumption (reactions 1 and 2). Thus, pH Let us now compare experiment sets 6 and 9. Both curves closely overlap, showing similar triiodide production. This similar behaviour has been confirmed by using a General Linear Model [START_REF] Saporta | Probabilités, analyse des données et statistique[END_REF] described in Appendix B. Given the high p-value of 0.88, the hypothesis that the two curves are actually the same cannot be dismissed. This is consistent with the fact that both experiment sets include the same concentration of each ion, even if they have been introduced in a different way. This means that sulphate anions act as a base in experiment 6, which confirms the observations made by Kölbl et al (2013a).

When comparing experiments 1 and 6, it can be observed that roughly the same amount of triiodide is produced, although slightly lower in experiment 1. This difference is statistically significant, as it can be deduced from the low p-value (9.4•10 -4 ). Although the initial proton concentrations are the same for both experiment sets (Appendix A), the ionic strengths are different. This seems to indicate a slight influence of the ionic strength, which is confirmed by the comparison of experiments 1 and 9. For both concentration sets, the same acid is at the same concentration, but concentration set 9 includes 1 mol/L of sodium perchlorate. The low p-value (9.6•10 -5 ) enables us to consider that the difference between both curves is significant.

As [START_REF] Pinot | New protocol of the Villermaux-Dushman reaction system to characterize micromixing effect in viscous media[END_REF] have proposed using a phosphate buffer instead of the borate buffer, the influence of the composition of the acid solution on triiodide production in the case of phosphate buffer use was also studied, and the results are shown in Fig. 2 (right). The trends are the same as when a borate buffer is: triiodide production decreases again when the stirring speed is increased; the hydrogen sulphate dissociation is not complete (experiments 2 and 5), but it is significant (experiments 2 and 8); the triiodide production is the same if the ion concentrations in the solution are the same, no matter how they are introduced (experiments 7 and 10; p-value = 0.65); and finally, sulphate anions have a basic character at low pH values (comparison of experiments 2 vs 10, and 2 vs 7). However, the influence of the ionic strength is not so obvious: experiments 2 and 7 are significantly different (p-value=0.025), whereas experiments 2 and 10 may not (p-value=0.12) Thus, the influence of the ionic strength will be

A C C E P T E D M A N U S C R I P T
further discussed in section 3.2, before considering the influence of the buffer choice in section 3.3.

Influence of the ionic strength

Comparison between experiments 1 and 9, and experiments 2 and 10, respectively, seems to indicate a slight influence of the ionic strength on triiodide production, with higher ionic strengths leading to slightly higher triiodide productions. To confirm this trend, the ionic strength was modified by adding NaClO4 to the acid solution before injection (Experiments sets 11 to 15), and results are shown in Fig. 3. There is a slight increase in triiodide production for higher ionic strengths. This increase was verified by applying a one-way analysis of variance (ANOVA; [START_REF] Navidi | Statistics for Engineers & Scientists, 4 th Edition[END_REF]; the obtained p-value = 2.5•10 -10 enables us to confirm that the ionic strength in the acid solution influences the triiodide production. This conclusion opposes that of Kölbl et al (2013a), who did not observe a significant influence of the ionic strength. When NaClO4 was added to the pre-mixture solution (Experiments sets 16 to 19), a slight increase in triiodide production with increasing ionic strength was again observed. Once again, the one-way ANOVA confirmed this influence of the ionic strength (p-value=8.3•10 -7 ). This effect was not observed by Kölbl et al (2013a); one possible explanation is that the range of ionic strengths that they tested was narrower than ours.

The influence of the ionic strength might be attributed to two different mechanisms. First, the ionic strength affects the activity coefficients, thus altering both the dissociation constant of the boric acid and the self-ionization constant of the water; therefore, the H2BO3 -concentration in the pre-mixture solution might be slightly modified when the ionic strength in this solution is increased. Second, the evolution of the activity coefficients with the ionic strength impacts the Dushman reaction rate, although this influence might be not monotonic [START_REF] Barton | Kinetics of the Iodate-Iodide Reaction: Catalysis by Carboxylate and Phosphate Ions[END_REF]. Further comprehension of these mechanisms would thus require a fine modelling of the

A C C E P T E D M A N U S C R I P T
thermodynamics of these solutions and a reliable kinetic law for the Dushman reaction. This will be the purpose of an upcoming paper.

Influence of the buffer choice

The influence of the buffer choice is shown in Fig. 4. When comparing results for concentration sets 1 and 2, and concentration sets 3 and 5, respectively, it can be seen that triiodide production is significantly higher when a borate buffer is used. This result was actually unexpected since the same buffer concentration was used. Moreover, the H2BO3 -/H3BO3 buffer has a higher pKa value (9.24, see Table 2); iodine and triiodide are actually thermodynamically unstable in this buffer, although iodine disproportionation is slow enough for the experiments to be free of error [START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF]. On the other hand, H2PO4 -/HPO4 2-buffer has a lower pKa value (7.20, Table 2), which is much closer to the pH * at which iodine and triiodide are stable (Fournier et al, 1996a). Therefore, iodine and triiodide are much more stable in phosphate buffers, and the difference between both measurements could not be assigned to the thermodynamic stability of iodine.

Table 2: Dissociation constants at 25°C for boric and phosphoric acid [START_REF] Skoog | Fundamentals of analytical chemistry, 7th Edition[END_REF] Acid pKa,1 pKa,2 pKa,3 H3BO3 9.24 --H3PO4

2.15 7.20 12.35

One explanation might be given by a potential catalytic effect of the buffer. [START_REF] Barton | Kinetics of the Iodate-Iodide Reaction: Catalysis by Carboxylate and Phosphate Ions[END_REF] observed a catalytic effect of different bases, including HPO4 2-and H2PO4 -, on the Dushman reaction. Moreover, the catalytic rate constant seemed to follow a Brönsted law, meaning that the catalytic effect should increase when dealing with stronger bases (higher pKa values). One could then expect a catalytic effect of the H2BO3 -anions, stronger than that of the HPO4 -2 and H2PO4 -anions, since H2BO3 -is the strongest base. This would explain the enhanced triiodide production when using borate buffers. However, according to [START_REF] Schmitz | Kinetics and mechanism of the iodate-iodide reaction and other related reactions[END_REF], this catalytic effect is owed to a reaction between the base (B -) and I2O2, which is an intermediate species in the Dushman reaction mechanism. This seems unlikely in micromixing experiments, if we consider the mechanism of micromixing [START_REF] Bałdyga | Turbulent micromixing in chemical reactorsa review[END_REF]: as soon as the base B -is incorporated in the thin slabs containing the acid solution, the B -anions are consumed by the neutralization reaction (reaction 1). Thus, the Dushman reaction (reaction 2) can only take place in the case of an excess of protons, when all the B -has been consumed already. Thus, the catalytic effect of the buffer could probably be neglected in our case. Another explanation, which seems to us more likely, is that both H2PO4 -and HPO4 2-act as a base. Indeed, when the acid is injected, the local pH is low and smaller than the pKa for the third dissociation of phosphoric acid (pKa=2.15, Table 2). Thus, both anions act as a base at the beginning, leading to higher proton consumptions and thus reducing the iodine production (although later, when the pH becomes greater than 2.15 as a consequence of the engulfment, HPO4 2-turns out to be the only base of the system).

A C C E P T E D M A N U S C R I P T

Discussion

The experiments show that the acid and buffer choice has a great influence on iodine and triiodide production. However, the same trends (decreasing absorbance with increasing stirring speed, i.e., with decreasing micromixing time) are obtained in all cases. Thus, the influence of the acid and buffer choice on triiodide production should not be a criterion of choice when carrying out qualitative studies. Both phosphate and borate buffers could be used, but two considerations should be taken into account. On the one hand, iodine and triiodide are more stable in a phosphate buffer, since its pKa is closer than the pKa for borate to the pH * at which iodine and triiodide are thermodynamically stable. On the other hand, boric acid is reprotoxic. Therefore, a phosphate buffer would probably be a better choice if the Villermaux Dushman method is only used for qualitative studies.

For qualitative purposes, both sulphuric or perchloric acids may be used preferably, since the same trend is obtained for both (Fig. 2). However, it should be pointed out that perchloric acid and perchlorates are oxidants. Their oxidizing power in aqueous solutions is very little at room temperature because of kinetic activation barriers, but they become vigorous oxidants when heated (Greenwood and Earnshaw, 1997). Thus, perchloric acid use in the Villermaux Dushman method should be limited to room temperature, as it could oxidize iodide into iodine and iodine into iodate at higher temperatures.

If the Villermaux Dushman method is used for quantitative purposes (i. e., to estimate the micromixing time), the combination of perchloric acid and borate buffer is probably a more appropriate choice. Indeed, estimating micromixing times needs using a micromixing model [START_REF] Bałdyga | Simplification of Micromixing Calculations. I. Derivation and Application of New Model[END_REF][START_REF] Villermaux | A generalized micromixing model for initial contacting of reactive fluids[END_REF]Fournier et al, 1996b; Lemenand et

A C C E P T E D M A N U S C R I P T al, 2017
) and the kinetics of the Dushman reaction. Since sulphuric acid is an incompletely dissociated acid, the micromixing model should take into account its dissociation constant. The same reasoning applies to phosphate buffers: both H2PO4 -and HPO4 -2 have the role of a base in the first moments, but as long as the acid volume incorporates the vessel solution and pH increases, only H2PO4 -will continue to have this role. Therefore, at least the first dissociation constant of phosphoric acid should be considered in the model. This renders the calculations much more complicated. Moreover, the acid solution is quite concentrated (1 mol/L), meaning that the dissociation constants for ideal diluted solutions, which can be easily found in the literature, should not be used. The activity coefficients should be taken into account, rendering calculations even more tedious. A new model taking into account the dissociation constants of the sulphuric and phosphoric acids will be the goal of another paper.

Conclusion

The Villermaux-Dushman test for characterising micromixing effects through triiodide production gives qualitatively consistent results, whatever the acid and buffer chosen. In the case of quantitative purposes (micromixing time determination), the precise estimation of the real chemical operating conditions is of prime importance; perchloric acid and borate buffer are probably a wiser choice but a quantitative application also requires reliable kinetics data.

The kinetics of the Dushman reaction (reaction 2) obtained by [START_REF] Guichardon | Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: experimental procedure[END_REF] is often used in quantitative studies. However, it has been the object of discussions [START_REF] Bourne | Comment on the iodide/iodate method for characterising micromixing[END_REF][START_REF] Kölbl | Further comments on the Iodide Iodate Reaction Method for characterising micromixing[END_REF]Kölbl et al, 2013aKölbl et al, , 2013b)). Since the strengthening of the Villermaux-Dushman test deserves further investigation into the influence of the acid choice on the kinetics, it will be the main purpose of a next paper.

[HSO4 -] = ca -x (Eq. A2)

[SO4 2-] = x (Eq. A3)

[H + ] = ca + x (Eq. A4)

where ca is the analytical concentration of the sulphuric acid in the solution, and x is the ratio of the extent of reaction to the solution volume. Combining Eqs. A1 to A4, a following quadratic equation is obtained, leading to the final expression of the proton concentration:

      2 4 2 a a a a a a a c K K c K c c H         (Eq. A5)
This solution is independent of a possible presence of perchlorate anions.

When perchloric acid is used, and a sulphate salt is added, the sulphate anions will play the role of a base. In this case, the following mass balances are obtained:

[SO4 2-] = cs -x (Eq. A6) [HSO4 -] = x (Eq. A7) [H + ] = ca' -x (Eq. A8)
where ca' and cs are the analytical concentrations of the perchloric acid and the sulphate salt. Combining Eqs. A1 and A6 to A8, the following expression for the proton concentration is obtained:

      2 ' 4 ' ' ' 2 s a a a s a a s a c c K c c K c c c H          (Eq. A9)
When comparing experimental sets 1 and 6 (or 2 and 7), it can be seen that ca=cs=ca'/2. In this particular case, Eq. A9 reduces to Eq. A5, that is, proton concentration is the same for both experimental sets. The equivalence of the two equations is a consequence of our choice of the acid and salt concentrations, and is independent of the dissociation constant, Ka.

Appendix B: General Linear Model for the comparison of experiments with different concentration sets.

The absorbance is a function of two variables: x1 (the stirring speed) and x2, which is a binary variable (for instance, when comparing concentration sets 6 and 9, x2=0 for the former and x2=1 for the latter). If we consider a linear model:

i i e x x y     2 2 1 1 0    (Eq. B1)
where yi is the absorbance obtained in a given experiment, βj are the model coefficients and ei is the error in experiment i.

The following vectors and matrix are defined:

Figure 1 :

 1 Figure 1: Influence of the injection flowrate on absorbance at 353 nm, for 2 mL of acid. [HClO4] = 1 mol/L; [H2BO3 -]=0.1 mol/L; [KI]=0.0116 mol/l; [KIO3]=0.0233 mol/l; N=180 rpm

  to further dissociation of the HSO4 -anions into protons and finally, further triiodide production.

Figure 2 :

 2 Figure 2: Influence of the stirring speed on absorbance at 353 nm, for different concentration sets using H3BO3/H2BO3 -(left) or H2PO4 -/HPO4 -2 (right) as buffer. (The interactive figure is available in the online article)

Figure 3 :

 3 Figure 3: Influence of the ionic strength on absorbance at 353 nm. ◊: Ionic strength change concerns the acid solution; ○: Ionic strength change concerns the pre-mixture solution. (The interactive figure is available in the online article)

Figure 4 :

 4 Figure 4: Influence of the buffer choice on absorbance at 353 nm. . (The interactive figure is available in the online article)

A

  j is the value of the variable xj in experiment i. Thus, all the n experiments can be summarised in the following equation: as an estimator of β.If there is not any influence of the concentration set on the absorbance, then β2=0 and the statistic z) is the probability density function of a Student's t distribution.ReferencesM. Assirelli, W. Bujalski, A. Eaglesham, A. W. Nienow. Study of Micromixing in a Stirred Tank Using a Rushton Turbine. Comparison of Feed Positions and Other Mixing Devices. Chemical Engineering Research & Design 80, 855-863 (2002) DOI: 10.1205/026387602321143390 M. Assirelli, W. Bujalski, A. Eaglesham, A. W. Nienow. Intensifying micromixing in a semi-batch reactor using a Rushton turbine. Chemical Engineering Science 60, 2333-2339 (2005) DOI: 10.1016/j.ces.2004.10.041 C. Fournier, L. Falk, J. Villermaux. A new parallel competing reaction system for assessing micromixing efficiency -Experimental approach. Chemical Engineering Science 51, 5053-5064 (1996a) DOI: 10.1016/0009-2509(96)00270-9 M.-C. Fournier, L. Falk, J. Villermaux. A new parallel competing reaction system for assessing micromixing efficiency -Determination of micromixing time by a simple mixing model. Chemical Engineering Science 51, 5187-5192 (1996b) DOI: 10.1016/S0009-2509(96)00340-5 N.N. Greenwood, A. Earnshaw. Chemistry of the Elements, 2 nd Edition. Butterworth-Heinemann, Oxford (1997)

Funding

This work was supported by the Ministère de l'Éducation Nationale, de l'Enseignement Supérieur et de la Recherche : DRRT Provence-Alpes-Côte d'Azur [programme 0172]

Appendix A: Estimation of the proton concentration in solutions containing sulphate and hydrogen sulphate anions

The proton concentration in the sulphuric acid solutions which are being injected can be estimated in the following way [START_REF] Skoog | Fundamentals of analytical chemistry, 7th Edition[END_REF]. First of all, let us consider the case where the injected acid is sulphuric acid. It can be considered as completely dissociated in hydrogen sulphate and protons:

Hydrogen sulphate can be further dissociated in sulphate anions and protons:

Reaction A2 is actually an equilibrium, with a characteristic equilibrium constant:

The mass balance of each species takes the following form:


Konstantin Zimenko
email: kostyazimenko@gmail.com

Andrey Polyakov
email: andrey.polyakov@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@inria.fr

Finite-time and fixed-time stabilization for integrator chain of arbitrary order*

In the present paper, homogeneous control laws are designed for finite-time and fixed-time stabilization of integrator chains of arbitrary order. Provided analysis is based on Lyapunov function method and homogeneity concept. Fixed-time convergence is achieved by use of hybrid control algorithm with homogeneity degree changing. Performance of the resulting finite-time and fixed-time feedbacks is illustrated by numerical simulations.

I. INTRODUCTION

Motivated by modern control applications, the finite-time and fixed-time stabilization feedback design is continuing to be the subject of numerous studies (see, for example, [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF]- [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]). Among these studies and many others, particular attention is paid to finite/fixed-time stabilization of integrator chains as a basic model for demonstration and further extension for multi-input multi-output linear plants. Interest to the disturbed integrator chain control systems is additionally motivated by different mechanical and electromechanical applications (see, for example, [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Chernousko | Control of nonlinear dynamical systems: methods and applications[END_REF]). However, most of existing control laws are discontinuous (as, for example, in [START_REF] Laghrouche | Stabilization of perturbed integrator chains using Lyapunov-Based Homogeneous Controllers[END_REF], [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode design: A Lyapunov approach[END_REF]), designed to stabilize the chain of integrators only of a certain order (for second order [START_REF] Orlov | Finite Time Stabilization of a Perturbed Double Integrator-Part I: Continuous Sliding Mode-Based Output Feedback Synthesis[END_REF], [START_REF] Sua | Robust finite-time output feedback control of perturbed double integrator[END_REF] for third order [START_REF] Trivedi | Finite-time stabilization of uncertain triple integrator with only switch and gain[END_REF], [START_REF] Sanchez | On a sign controller for the triple integrator[END_REF], etc.), or even depending directly on the initial conditions ( [START_REF] Sanchez | On a sign controller for the triple integrator[END_REF]).

The present paper is devoted to Lyapunov-based controller design which stabilizes the arbitrary order integrator chain in finite/fixed time. Presented control algorithms are obtained with use of Lyapunov function method and homogeneity concept.

The paper [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF] is related to development of robust finitetime and fixed-time stabilization controllers, which depend on implicitly defined homogeneous Lyapunov function. For practical implementation of these control algorithms it is required to use special computational procedures for calculating Implicit Lyapunov Function (ILF). In [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF] the finitetime control law has been modified to explicit form, where instead of ILF the homogeneous norm is used. Then the difference between ILF and homogeneous norm is considered as additional external disturbance, that has led to a more complex analysis and parameters tuning procedure. The present paper provides the analysis based on an explicitly defined Lyapunov function for the similar control law. This analysis allows to get simple parameters tuning procedure and obtain new estimates for settling-time function. It is also shown that fixed-time convergence can be achieved by changing the homogeneity degree in hybrid control algorithm.

The paper is organized in the following way. The problem formulation is presented in Section II. Section III recalls some basics on finite/fixed-time stability and weighted homogeneity. Section IV presents the main result on finitetime and fixed-time controllers design. Simulation results are shown in Section V for performance illustration of proposed control algorithms. Finally, concluding remarks are given in Section VI.

Through the paper the following notation will be used: R is the set of real numbers;

R + = {x ∈ R : x > 0}; diag{λ i } n
i=1 is a diagonal matrix with elements λ i ; for P ∈ R n×n the relation P > 0 (P < 0, P ≥ 0, P ≤ 0) means that the symmetric matrix P = P T is positive (negative) definite (semidefinite); a series of integers 1, 2, . . . , n is denoted by 1, n.

II. PROBLEM FORMULATION

Consider a chain of n integrators

ẋ = Ax + bu, (1)
where x ∈ R n is the state vector, u ∈ R,

A =        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 0 0 • • • 0        and b =        0 0 . . . 0 1        .
The objective of the present paper is to design Lyapunovbased controllers, which stabilize the arbitrary order system (1) to the origin in finite/fixed time.

III. PRELIMINARIES

A. Finite-Time and Fixed-Time Stability

Let us consider the system

ẋ = f (t, x), x(0) = x 0 , (2)
where x ∈ R n is the state vector, f ∈ R + × R n → R n is a vector field. If the vector field f is discontinuous with respect to x, then the solutions of (2) fall into the area of differential inclusions and need to be understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]. For any x 0 ∈ R n denote a corresponding solution by X(t, x 0 ) for the instants t ≥ 0 for which it exists, X(0, x 0 ) = x 0 . Definition 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Orlov | Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems[END_REF] The origin of ( 2) is said to be globally finite-time stable if it is globally asymptotically stable and any solution X(t, x 0 ) of ( 2) reaches the equilibrium point at some finite time moment, i.e., X(t, x 0 ) = 0, ∀t ≥ T (x 0 ), where T : R n → R + ∪ {0} is the settling-time function.

Definition 2 [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] The origin of ( 2) is said to be fixed-time stable if it is globally finite-time stable and the settling-time function

T (x 0 ) is bounded, i.e., ∃T max > 0 : T (x 0 ) ≤ T max , ∀x 0 ∈ R n .
Definition 3 [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] The set M is said to be globally finite-time attractive for (2) if any solution X(t, x 0 ) of ( 2) reaches M in some finite time moment t = T (x 0 ) and remains there ∀t ≥ T (x 0 ), T : R n → R + ∪ {0} is the settling-time function.

Definition 4 [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] The set M is said to be fixed-time attractive for (2) if it is globally finite-time attractive and the settling-time function T (x 0 ) is globally bounded by some number T max > 0.

Theorem 1 [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Laghrouche | Stabilization of perturbed integrator chains using Lyapunov-Based Homogeneous Controllers[END_REF] Suppose there exists a positive definite C 1 function V defined on an open neighborhood of the origin D ⊂ R n and real numbers C > 0 and σ ≥ 0, such that the following condition is true for the system (2)

V (x) ≤ -CV σ (x), x(t) ∈ D \ {0}.
Then depending on the value σ the origin is stable with different types of convergence:

• if σ = 1, the origin is asymptotically stable;

• if 0 ≤ σ < 1, the origin is finite-time stable and

T (x 0 ) ≤ 1 C(1 -σ) V 1-σ 0 , where V 0 = V (x 0 ); • if σ > 1 the
origin is asymptotically stable and, for every , the set B(0, ) = {x ∈ D : V (x) < ε} is fixedtime attractive with

T max = 1 C(σ -1)ε σ-1 .
If D = R n and function V is radially unbounded, then the system (2) is globally stable with respect to the equilibrium point.

B. Weighted Homogeneity

For r i ∈ R + , i = 1, n, ρ > 2 max i r i and λ > 0 define vector of weights r = r 1 , . . . , r n T , dilation matrix D r (λ) = diag{λ ri } n i=1 and homogeneous norm

x r = n i=1 |x i | ρ r i 1 ρ . (3)
Note that under introduced restriction on ρ the homogeneous norm is a continuously differentiable function out of the origin.

Definition 5 [19]

A function g : R n → R (vector field f :

R n → R n ) is said to be r-homogeneous of degree d if g(D r (λ)x) = λ d g(x) (f (D r (λ)x) = λ d D r (λ)f (x)) for fixed r, all λ > 0 and x ∈ R n .
Obviously, homogeneous norm is r-homogeneous of degree 1. Introduce the following compact set (homogeneous sphere) S r = {x ∈ R n : x r = 1}, then for any x ∈ R n there is y ∈ S r such that x = D r (λ)y for λ = x r .

Theorem 2 [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] Let f : R n → R n be defined on R n and be a continuous r-homogeneous vector field with degree d (d < 0). If the origin of the system

ẋ = f (x) (4)
is locally asymptotically stable then it is globally asymptotically stable (globally finite-time stable) and there exists a continuously differentiable Lyapunov function V which is r-homogeneous of degree v > -d. By definition of homogeneity there exist constants

c 1 , c 2 > 0 such that c 1 x v r ≤ V (x) ≤ c 2 x v r . (5)
The r-homogeneity presented in Definition 5 is introduced for some r and all λ > 0. Restricting the set of admissible values for λ the local homogeneity concept has been introduced in [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [9], [10].

IV. MAIN RESULT

For µ ∈ (0, 1] and the vector of weights r = (l + (n -1)µ, l

+ (n -2)µ, • • • , l) T , r max = l + (n -1)µ, r min = l denote the dilation matrix D r (λ) = diag{λ ri } n i=1 and diagonal matrix H = diag{-m + (n + 1 -i)µ} n i=1 , where l ∈ R + : l ≥ µ, m ∈ R + : m > nµ.
Theorem 4

If the system of matrix inequalities:

X > 0, (6)
α max ( HX + X H) + AX + XA T + by + y T b T ≤ -γX, (7)
-α max ( HX +X H)+AX +XA T +by+y T b T ≤ -γX, (8)
is feasible for some X ∈ R n×n , y ∈ R 1×n , γ ∈ R + and

α max = 1 + |k i | ρ ρ-r min ρ-r min r min r min ρ × 1 ri ρ rmax-r min rmax-r min ρ , (9)
P = X -1 , k = yX -1
, then the control of the form

u = x l-µ r kD r 1 x r x ( 10
)
stabilizes the origin of the system (1) in a finite time and the settling-time function estimate has the form

T (x) ≤ 2(m + l -µ)V µ 0 µγc µ 1 , (11)
where

V 0 = x T 0 H 1 x0 r P H 1 x0 r x 0 1 2(m+l-µ) , H(λ) = diag{λ -m+(n+1-i)µ } n i=1 and c 1 = min x r =1 (x T P x) 1 2(m+l-µ)
. Remark 1 The finite-time control law (10) for l = 1 coincides with the control presented in [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF]. In comparison with [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF], presented analysis allows to relax obtained LMIs and simplify the procedure of parameters tuning. Also, new estimates for settling-time function is obtained.

Remark 2 Since the inequalities ( 7), ( 8) are feasible at least for sufficiently small α (can be achieved by choosing sufficiently big l ∈ R + ) the presented control scheme implies simple control parameters tuning: it requires to solve LMIs (6) and

α( HX + X H) + AX + XA T + by + y T b T ≤ -γX, -α( HX + X H) + AX + XA T + by + y T b T ≤ -γX,
for some α ∈ R + and then choose such l for which α max ≤ α.

Remark 3 The parameter m can be chosen large enough to have HX + X H ≤ 0. In this case the inequality ( 7) can be reduced.

Remark 4 Parameters l, m, µ and γ allows the upper bound of the settling time function [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF] to be adjusted.

Remark 5 For l = µ ≤ 1 the control (10) is continuous outside the origin and globally bounded for all x ∈ R n :

u 2 = x T D r 1 x r k T kD r 1 x r x ≤ ς x -2m r x T H 1 x r P H 1 x r x = ς x -2m r V 2m ≤ ςc 2m 2 , where ς ∈ R + : k T k ≤ ςP .
For µ < min{l, 1} the control (10) is continuous in the state variable x. If µ → 0 then the feedback (10) becomes a linear u = kx. If µ = l ≤ 1 the control is discontinuous at the origin and continuous outside. For practical realization of the control, the discontinuous feedback law can be replaced with a high-gain linear feedback if the system state is close to the origin as in [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF].

Remark 6 It can be easily checked that the system (1), ( 10) is homogeneous of degree -µ. Thus, the system has such qualitative stability properties of homogeneous systems as Input-to-State Stability with respect to additive perturbations in the right-hand side of (1) and measurement noises in the feedback.

Remark 7 The controller (10) can also be treated as the weighted homogeneous version of the unit sliding mode control [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Dorling | Two approaches to hyperplane design in multivariable variable structure control systems[END_REF].

Note that in presented control scheme instead of homogeneous norm (3) another homogeneous function of degree 1 can be used. In this case the main difference is in calculating the parameter α. Also if we choose instead of homogeneous norm implicitly defined homogeneous function equal to V , then for m = l = 1 we obtain implicitly defined Lyapunov function [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Robust Stabilization of MIMO Systems in Finite/Fixed Time[END_REF].

Q = x T D r (V -1 )P D r (V -1 )x -1 and control u = V 1-µ kD r (V -1 )x, V ∈ R + : Q(V, x) = 0 as in
Convergence within a fixed interval of time irrespectively of its initial conditions can be achieved by changing the degree of homogeneity in hybrid control algorithm. The following discontinuous fixed-time controller is based on time dependent switching scheme for the first time presented in [START_REF] Angulo | Robust exact uniformly convergent arbitrary order differentiator[END_REF] for an observer.

In this case for m 1 , m 2 ∈ R + :

m 1 = m 2 + 2µ, m 1 > nµ, l > (n -1)µ, µ ∈ (0, 1] denote vectors of weights r 1 = (l + (n -1)µ, l + (n -2)µ, • • • , l) T , r 2 = (l -(n -1)µ, l - (n -2)µ, • • • , l) T , homogeneous norms x r1 , x r2 and matrices D 1 (λ) = diag{λ r1(i) } n i=1 , H1 = diag{-m 1 +(n+ 1 -i)µ} n i=1 , D 2 (λ) = diag{λ r2(i) } n i=1 , H2 = diag{-m 2 - (n + 1 -i)µ} n i=1 , H 1 (λ) = diag{λ -m1+(n+1-i)µ } n i=1 , H 2 (λ) = diag{λ -m2-(n+1-i)µ } n i=1 . Define homogeneous functions of degree 1: V 1 (x) = x T H 1 1 x r1 P H 1 1 x r1 x 1 2(m 1 +l-µ) , V 2 (x) = x T H 2 1 x r2 P H 2 1 x r2 x 1 2(m 2 +l+µ) .
For V 1 and V 2 according to (5) define c 11 , c 21 and c 12 , c 22 correspondingly.

Theorem 5

If the system of matrix inequalities (6),

α 1 ( H1 X +X H1 )+AX +XA T +by+y T b T ≤ -γ 1 X, (12)
-α 1 ( H1 X + X H1 )

+ AX + XA T + by + y T b T ≤ -γ 1 X, (13)
α 2 ( H2 X +X H2 )+AX +XA T +by+y T b T ≤ -γ 2 X, (14) -α 2 ( H2 X + X H2 ) + AX + XA T + by + y T b T ≤ -γ 2 X, (15)
is feasible for some X ∈ R n×n , y ∈ R 1×n , γ 1 , γ 2 ∈ R + , were α 1 , α 2 are calculated according to (9) with r 1 and r 2 correspondingly, P = X -1 , k = yX -1 , then the control of the form

u =    x l+µ r 2 kD 2 1 x r 2
x f or t ≤ Ts,

x l-µ r 1 kD 1 1

x r 1

x f or t > Ts,

for some T s ∈ R + stabilizes the origin of the system (1) in a fixed time

T (x) ≤ 2(m 1 + l -µ)µ -1 c µ 22 γ 2 E µ + B µ γ 1 c µ 11 , (17)
where

E = 2(m1+l-µ)c µ 2 2 µγ2Ts 1 µ , B = max V2=E V 1 . (18)
Note, that the parameter T s ∈ R + can be arbitrarily chosen.

Similarly to finite-time case, remarks analogous to 2-6 can be presented for fixed-time control algorithm. For example, the fixed-time controller is constructed by switching homogeneity properties of the system: it is homogeneous with positive degree µ for t ≤ T s and with negative degree -µ for t > T s .

The convergence rate of proposed control algorithms can be accelerated via time-rescaling in similar way, as in [START_REF] Efimov | Supervisory Acceleration of Convergence for Homogeneous Systems[END_REF], [START_REF] Laghrouche | Stabilization of perturbed integrator chains using Lyapunov-Based Homogeneous Controllers[END_REF], [START_REF] Levant | Accelerated high-order MIMO sliding mode control[END_REF].

Remark 8 For λ > 1 and N = diag{λ 1-i } n i=1 rewrite the control ( 16) (or control (10)) as ū(x) = λ n u(N x). Then for x = N x, t = λt, Ts = T s /λ we obtain ẋ = Ax( t) + bu(x( t)) and the following estimates of settlingtime function: T (x 0 ) ≤ T1+Ts λ for the control [START_REF] Trivedi | Finite-time stabilization of uncertain triple integrator with only switch and gain[END_REF] and

T (x 0 ) ≤ 2(m+l-µ)V (x0) µ µγc µ 1 λ
for the control (10).

V. SIMULATION RESULTS

Let us demonstrate the performance of presented control laws using the system of triple integrator with initial conditions x 1 (0) = 10, x 2 (0) = -30, x 3 (0) = 0. Using Theorem 4 the finite-time stabilizing controller is designed for this system in the form (10) with the parameters l = 20, m = 1, µ = 0.3, ρ = 41.201, where the matrix P ∈ R 3×3 , P > 0 and the vector k ∈ R 1×3 are obtained from the inequalities ( 6)-( 8 The parameters of the fixed-time stabilizing controller [START_REF] Trivedi | Finite-time stabilization of uncertain triple integrator with only switch and gain[END_REF] were selected solving the LMI (6), ( 12)-( 15 Simulations for finite-time and fixed-time convergence accelerated with λ = 2 have also been carried out. All simulations results are presented in Fig. 123456. The paper is focused on finite-time and fixed-time control for integrator chains of arbitrary order. Obtained finite-time control design method consist in appropriate modification of control law presented in [START_REF] Zimenko | Stabilization of Chain of Integrators with Arbitrary Order in Finite-time[END_REF] and use of analysis based on explicitly defined Lyapunov function. Based on this result simple procedure of parameters tuning and new estimates for settling-time function were obtained. The fixed-time control algorithm for chain of integrators is presented. It is shown that fixed-time convergence can be achieved by changing the homogeneity degree in hybrid control algorithm. The performance of obtained control algorithms is demonstrated through simulations.

A detailed study of the presented control algorithms on robustness analysis with respect to disturbances, uncertainties, delays and extension of these results on wider class of systems goes beyond the scope of the paper providing the subject for a future research.

  7.0626 -10.5577 -3.9899 .

  ) with the parameters l = 40, m 1 = 0.91, m 2 = 0.31, µ = 0.3, ρ 1 = ρ 2 = 81.7: 20.2539 -21.1388 -6.8414 . The parameter T s has been chosen equal to 1.5.

Fig. 1 .

 1 Fig. 1. System states versus time for finite-time control

Fig. 2 .Fig. 3 .

 23 Fig. 2. Control law (10) versus time

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Control law (16) versus time

*This work is supported by the Russian Science Foundation under grant 17-19-01422.