Omar Alami
email: omar.alami@lcc-toulouse.fr

Regis Laurent
email: regis.laurent@lcc-toulouse.fr

Jean-Pierre Pierre Majoral
email: majoral@lcc-toulouse.fr

Nabil El

El Said

Anne-Marie Kazzouli

Caminade

Régis Laurent

Nabil El Brahmi
email: n.elbrahmi@ueuromed.org

Said El Kazzouli
email: s.elkazzouli@ueuromed.org

Anne- Marie Caminade
email: anne-marie.caminade@lcc-toulouse.fr

Copper complexes of phosphorus dendrimers and their properties

Keywords: Dendrimers, phosphorus, copper complexes, pyridine imine, diketone, catalysis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Dendrimers are highly branched macromolecules, constituted of concentric layers named "generations", which are synthesized step-by-step around a multifunctional central core. The number of terminal functions increases at the end of the synthesis of each generation. The number of terminal functions is most frequently multiplied by 2 at each generation [1], eventually by 3 [2], or even by 5 [3], depending on the number of functions at the branching points. Different types of dendrimers have been synthesized, either purely organic with nitrogen as branching points in most cases [1], or partly inorganic with either phosphorus or silicon as branching points [4]. Drawing dendrimers needs a large space when their size increases, thus they are most generally written in a linear way, with parentheses after each level of branching units. Both types of representations are shown in Figure 1 for a third generation dendrimer.

Most of the properties of dendrimers depend on the type of terminal functions they bear. A number of dendrimers have been functionalized with metal complexes [5], in particular phosphorus-containing dendrimers [6,7]. These dendrimers are essentially of type phosphorhydrazone. Many phosphorhydrazone dendrimers have been functionalized with palladium complexes, and used as catalysts for Suzuki-Miyaura [8][9][10], Stille [11,12] and Sonogashira [8,13] cross-couplings, for Heck reactions [8,13,14], and for allylic alkylations [15,16]. Phosphorhydrazone dendrimers functionalized with ruthenium complexes have been used as catalysts for Knoevenagel condensations and Michael additions [11], hydrogenation reactions [17], hydration of terminal alkynes [18][19][20] and isomerization of allylic alcohols [18,21,22]. Phosphorhydrazone dendrimers functionalized with rhodium complexes were used as catalysts for the isomerization of allylic alcohols [19], and [2 + 2 + 2] cycloaddition reactions [23], whereas scandium complexes have been used for catalyzing Friedel-Crafts acylations [24]. In this review, we will focus on another element, copper, complexed on the surface of phosphorhydrazone dendrimers, and on their properties as efficient catalysts.

Different advantages can be expected when using dendrimers as catalysts instead of "classical" monomeric catalysts. Due to their large size, dendrimers are generally slightly less soluble than small catalysts, thus it is generally easy to recover them by precipitation, using a solvent in which the products will be soluble, but not the dendrimers. Another possibility for recovery is ultra-filtration through a membrane, also possible thanks to the large size of dendrimers. Such recovery and reuse are desirable to decrease the cost associated to dendrimers. The presence of a large number of catalytic entities in close proximity on the surface of dendrimers might induce a synergy or a cooperativity, increasing the catalytic efficiency and/or the enantioselectivity. This may permit to decrease the temperature needed to perform the reaction, or to decrease the time needed for the reaction to go to completion. In some cases, a decrease in the metal leaching can be also observed.

All these particular properties of dendrimers are gathered under the concept of "dendrimer (or dendritic) effect" [25]. This effect is observed when a functional group behaves differently when it is alone or linked to a dendrimer, and it is a true non-linear effect, which can induce quite remarkable properties. The origin of such effect is not always fully understood, but has generally to be related to the concept of multivalency, in which stronger effects are observed when several functions are gathered in a single entity. A high local concentration of functions within well-defined nanoscopic reaction volumes can be attained with dendrimers, modifying the molecular interaction with their surrounding at the interface, which in turn affects the activity and selectivity. Such behavior cannot be replicated in randomly dispersed polymeric system. Of course, the efficiency between a monomer and diverse generations of dendrimers has to be compared for the same number of active sites. It must be emphasized that this dendrimer effect is not always positive, but there is a bias in the literature, as positive effects are more easily published than negative effects, especially concerning catalysis.

Figure 1.

Step-by-step synthesis of phosphorhydrazone dendrimers and structure of a 3 rd generation phosphorus dendrimer, drawn in two ways, either the full structure, or the linear structure with a parenthesis after each branching point. The red dots indicate the possible modifications of the surface [26,27].

Dendritic copper complexes for catalysis

In general, several generations of dendrimers are synthesized, using a two-step process for each generation, as shown in Figure 1 [26,27]. The synthesis starts with the reaction of six equivalents of 4hydroxybenzaldehyde in basic conditions, generally using cesium carbonate, on hexachlorocyclo triphosphazene (N3P3Cl6) used as core, to generate compound 1-G'0. The second step is the condensation of the phosphorhydrazide H2NNMeP(S)Cl2, obtained by reaction of methyl hydazine on P(S)Cl3 at low temperature. Such condensation affords the first-generation dendrimer 1-G1, having 12 Cl on the surface, to be compared with the core which had 6 Cl. Starting from dendrimer 1-G1, both steps are repeated, i.e. the nucleophilic substitution on the P-Cl functions with 12 HOC6H4CHO in basic conditions to afford compound 1-G'1, followed by the condensation with 12 equivalents of H2NNMeP(S)Cl2, to afford the second generation 1-G2. This two-step synthesis can be repeated many times, and was carried out up to the twelfth generation 1-G12, which is one of the largest dendrimers ever synthesized [28]. The presence of either P(S)Cl2 or aldehyde terminal functions offers a large palette of reactions on the surface of this family of dendrimers, to fulfil the desired properties [29].

Many reactions on the surface of these phosphorhydrazone dendrimers were carried out to graft ligands, to complex different metals, to study their catalytic properties and to compared their efficiency with that of the corresponding monomer. The experiments are generally carried out up to the third or fourth generation of the dendrimers. It must be emphasized that the quantity of metal is identical for experiments which are compared. This means that for the comparison between a monomer and a 3 rd generation dendrimer, if 1 equivalent of the third generation is used, its efficiency will be compared with that of 48 equivalents of the monomer, as the 3 rd generation dendrimer is functionalized with 48 terminal functions (Figure 1).

Copper is cheaper than "classical" transition metals used in catalysis and is generally believed to be less toxic, thus copper complexes are attractive for catalysis and their use has been previously reviewed [30]. The grafting of copper complexes onto the surface of phosphorhydrazone dendrimers had been successfully carried out a long time ago. Generations 1 to 3 dendrimers have been decorated with pyridine imine ligands, obtained by condensation of 4-aminophenol with 2pyridinecarboxaldehyde, then grafted to the dendrimers bearing P(S)Cl2 terminal functions. These pyridine-imine dendrimers were then used for the complexation of CuI, as shown in Figure 2. The three generations 2-G1 to 2-G3, as well as the corresponding monomer 2-M, have been used as catalysts in several coupling reactions [31]. A first example concerns the coupling of 3,5-dimethylphenol with phenyl iodide at 80°C in acetonitrile with K3PO4 as a base, using 10 mol% of Cu (Figure 2A). This loading was chosen so that the ratio N,N-chelate-to-copper is equal to 1 in all cases. In these conditions, the third generation 2-G3 was the most efficient catalyst, as it improved the catalytic efficiency, compared to the other generations and to the monomer 2-M (graph in Figure 2A). The third generation 2-G3 was still very active when lowering the temperature to 55°C, whereas the coupling with the monomeric catalyst 2-M was by far less efficient in these conditions. The same family of copper complexes has been used as catalysts in the arylation of pyrazole with aryl halides (phenyl iodide or bromide) at 80°C in acetonitrile with Cs2CO3 as a base in the presence of 10 mol% of catalyst (Figure 2B). Under these conditions, the monomer 2-M has no activity, but a quantitative conversion was observed with all dendrimers for the coupling of phenyl iodide with pyrazole. The less expensive and less reactive bromobenzene induced a strongly positive "dendritic effect" [25], the monomer being non-active, while the catalytic efficiency of the dendrimers increased when the generation increased (graph in Figure 2B). The next step with these catalytic copper complexes consisted in the vinylation of 3,5dimethylphenol and of pyrazole with E-α-bromostyrene, at very mild temperature (25°C). The vinylation of 3,5-dimethylphenol (Figure 2C) afforded the corresponding E-styrylarylether in 91% yield using the 3 rd generation dendrimer 2-G3 as catalyst, whereas a poor conversion of only 35% was obtained with the monomeric catalyst 2-M. The vinylation of pyrazole (Figure 2D) quantitatively occurred at 25 °C within 4 days when using the 3 rd generation dendrimer 2-G3, whereas the Estyrylpyrazole was obtained in only 70% yield when using the monomer 2-M in the same conditions. All these catalytic reactions were carried out in the mildest conditions ever reported at that time using copper complexes as catalysts [31]. The second example of copper complexes as terminal functions of phosphorus dendrimers from generation 1 to 3 concerned azabis(oxazoline) ligands. Two families were synthesized. In the first case, the bisoxazoline ligand was functionalized with a phenol, suitable for the reaction with dendrimers ended by P(S)Cl2 functions (1-Gn, n = 1-3) as shown for the synthesis of 3-G3A (Figure 3A). In the second case, the dendrimers with terminal functions (1-G'n) were first modified to get azide terminal functions. The next step was a copper-catalyzed azide-alkyne cycloaddition (CuAAC) reaction with a bisoxazoline functionalized by an alkyne, to form a benzyl triazole linker (3-G3B, Figure 3B). These dendrimers were used for the complexation of CuCl2 (one Cu per bisoxazoline ligand) and evaluated as catalysts in asymmetric benzoylations of diols. The maximum yield obtainable in these reactions starting from a racemic diol is 50% as only one isomer is expected to react. The first attempts were carried out with both families of dendrimers and the corresponding monomer 3-M using 5 mol% of CuCl2 in CH2Cl2 at 0°C for catalyzing the asymmetric benzoylation of racemic hydrobenzoin (Figure 3C). The best catalytic efficiency for the dendrimer of the 3-GnA family was observed with the second generation 3-G2A (41% yield, 80% ee), but all generations behaved almost identically. The best catalytic efficiency for the dendrimer having the triazole linker (3-GnB) was also observed with the second generation (28% yield, 86% ee), but for this family, the third generation has a very poor efficiency concerning both the yield and the enantioselectivity (ee), probably due to its lower solubility. One of the interesting properties of catalytic dendrimers is the possibility to recover and to reuse them many times by precipitation, in particular with hexane [10,24]. In this perspective, the triazole family was found superior to the tyramine family (see left graph in Figure 3), as yield and ee were almost identical at the first and third use. For this reason, the next experiment concerning the catalyzed benzoylation of trans-cyclohexanediol was only carried out with the triazole family 3-GnB (Figure 3D). The 1 st generation dendrimer 3-G1B was very efficient, with 99% ee even after the third run. The second generation afforded also 99% ee for the first and second runs, but a slight decrease was observed with the 3 rd run (ee = 95%). The third generation, here also, was less efficient (see the graph on the right in Figure 3). In a last series of experiments, the triazole-azabisoxazoline ligand was grafted to other supports, "MeO-PEG" and polystyrene, but these compounds were found to be far less efficient than the dendrimers, in particular in terms of ee, which were only 63 and 66%, respectively. This result emphasizes the positive role of dendrimers in catalysis over polymers, presumably due to a better accessibility of the reagents to the catalytic sites on the surface of dendrimers [32]. [32]. All the given yields correspond to isolated yields after column chromatography.

Phosphorhydrazone dendrimers, from generation 0 to generation 4, functionalized with β-diketones on their surface, were also able to complex copper, and they were used as catalysts. The first step of the synthesis consisted in the reaction of 4-hydroxyacetophenone with methyl pivalate to get the phenol diketone, which was then grafted to dendrimers with P(S)Cl2 terminal functions (Figure 4). The diketone moieties of these dendrimers 4-Gn (n = 0-4) were then used for complexing CuI. Either 2 or 4 diketone moieties per CuI were used; the generation 4 (4-G4) with two diketone moieties per Cu is shown in Figure 4, as well as the monomer 4-M. In a first attempt, the arylation of 3,5-dimethylphenol by bromobenzene was carried out in DMF at 110°C in the presence of Cs2CO3 as a base, and with 1 mol% of catalyst. The best results were obtained with 4 diketone moieties per CuI, as shown in the left graph of Figure 4. The reaction yields are almost identical when using the monomer (4-M) or generations 1 to 3 of the dendrimers 4-Gn, but a slightly better efficiency was obtained with the second generation 4-G2. The same reaction was then carried out at 120°C, using per CuI 4 diketone moieties of the monomer 4-M and all generations of the dendrimers 4-Gn, from 0 to 4 (right graph in Figure 4). The same type of reaction was then carried out with 4-bromoanisole and 4-bromoacetophenone, using the 2 nd generation dendrimer 4-G2 at 120°C in DMF to afford the corresponding coupled products in 56 and 96% yield, respectively. Unexpectedly, all attempts carried out to recover and reuse the dendrimers failed. A careful analysis of the outcome of the reaction revealed the presence of unexpected products, which were issued from the nucleophilic substitution of 3,5-dimethylphenoxyde onto the peripheral and internal P-O bonds of the dendrimers. Such substitution onto the P-O bonds was found to be fast, as the dendrimers were quantitatively disintegrated into monomers within 3 hours in the hard conditions of these catalytic reactions [33]. The absence of dendrimer effect and the impossibility to recover and reuse the dendritic catalyst is frequently observed and reported in the literature, but no explanation is generally given. Our work offers an explanation that might be expanded to many other cases in which recovery and reuse of the dendrimer was not possible.

A sequel of the use of dendritic copper complexes

In view of the ease with which the pyridine imine complexes shown in Figure 2 were obtained, this concept has been expanded for the synthesis of other types of pyridine-imine copper complexes. The first chelator was obtained by the condensation of tyramine with 2-pyridinecarboxaldehyde, affording N-(pyridin-2-ylmethylene) ethanamine functions, which were then grafted to the surface of dendrimers having P(S)Cl2 terminal functions 1-Gn, to afford the family of dendrimers 5-GnA (Figure 5A). The same process was applied to other ligands, starting from dendrimers 1-Gn. Tyramine was also used for the condensation with di(2-pyridyl) ketone, affording N-(di(pyridin-2-yl)methylene) ethanamine ligands on dendrimers 5-GnB (Figure 5B). Finally, the condensation reaction of 4hydroxybenzaldehyde with 2-hydrazinopyridine yielded 2-(2-methylenehydrazinyl)-pyridine as terminal functions of dendrimers 5-GnC (Figure 5C). In the three cases, the ligands were grafted to dendrimers of generations 1, 2, and 3. These dendrimers were then used for the complexation of CuCl2, using 1 Cu per chelator unit, then they were tested against several cancerous cell lines. The A family (5-GnA) was found the most efficient at 1 μM [34]. This family was shown the most stable in solution by comparative electron paramagnetic resonance (EPR) studies [35], and the biological mechanism of action has been elucidated [36].

Figure 5. Different types of pyridine-imine functions which have been linked to the surface of dendrimers, from generation 1 to generation 3, and their copper complexes, with one metal per pyridine imine function [34].

The pyridine imine ligands are able to complex different types of metals, and thus an attempt was made to complex AuCl3 instead of CuCl2. Surprisingly, using 1 equivalent of AuCl3 per terminal function resulted in the complexation of only half of these functions. The reason is that AuCl3 is not complexed, but AuCl2 + , with AuCl4 -as counter-ion, as shown in Figure 6 for dendrimer 6-G3A. A two-order of magnitude increase in the anti-cancer efficiency was observed on going from copper to gold. In view of this striking result, several examples of stochastic functionalization on the surface of the dendrimer have been attempted. An example is shown in Figure 6, in the case 20/10/10/8 (copper/free ligand/gold/PEG), which is one of the most efficient compounds [37]. In addition to the copper and gold complexes, the iron complex (FeCl2) of the AG3 dendrimer has been also synthesized and tested, but its antiproliferative activity is by far lower than that of copper [38]. Besides dendrimers, dendrons (dendritic wedges) were also functionalized with copper (or gold) complexes on their surface. They were obtained by the specific reactivity of one function of the cyclotriphosphazene core, different from the 5 others [39]. In the first step, five hydroxybenzaldehyde groups were grafted to N3P3Cl6, instead of six for the normal synthesis of dendrimers. The remaining Cl was then replaced by a phenol functionalized with a long alkyl chain. Two different chain lengths for the function at the core were used, C11H23 and C17H35. The next step was the classical reaction for the synthesis of dendrimers, i.e. the condensation of the aldehydes with H2NNMeP(S)Cl2. Such reaction was followed by the grafting of N-(pyridin-2-ylmethylene) ethanamine functions to the P(S)Cl2 terminal functions, as already shown in Figure 5. In the final step, two metals were complexed on the surface, either copper or gold, affording 4 different first generation dendrons, depending on the length of the chain at the core and the type of metal complexed on the surface 7a-G1Cu, 7b-G1Cu, 7a-G1Au, and 7b-G1Au (Figure 7). These dendrons were found active against several aggressive breast cancer cell lines [40].

Conclusion

We have shown in this review that copper complexes of phosphorhydrazone dendrimers have an interesting potential in catalysis. Different types of ligands were synthesized and grafted to the surface of dendrimers. It should be emphasized that our approach was not to find the best catalysts, but to demonstrate that even very poor monomeric catalysts can become highly efficient when grafted on the surface of dendrimers. Pyridine imine ligands complexing CuI were used as efficient catalysts in Oand N-arylations and -vinylations, and demonstrated a very positive dendritic effect. Azabis(oxazoline) complexes of CuCl2 were used as reusable catalysts in the asymmetric benzoylation of a linear diol or a cyclic diol in mild conditions. Dendrimers functionalized with diketone moieties for the complexation of copper were used as catalysts in the arylation of 3,5-dimethylphenol by diverse derivatives of bromobenzene. As a sequel of this work about catalysis, diverse types of pyridine imines complexing copper have been synthesized and then tested against various types of cancerous cell lines. In view of these results, it appears that dendritic copper complexes have still a large potential in catalysis, and may be also in biology.

Aknowledgements

The authors, and specially O.A. thank TOUBKAL/19/88: 41405WE for financial support.

Figure 2 .

 2 Figure 2. Synthesis of the 3 rd generation dendrimer 2-G3 functionalized with pyridine imine complexes of CuI. This dendrimer, as well as generations 1 and 2, and the corresponding monomer 2-M were used as catalysts in the coupling reactions shown in cases A and B. Only the monomer 2-M and dendrimer 2-G3 were used in cases C and D [31].

Figure 3 .

 3 Figure 3. Synthesis of two 3 rd generation dendrimers (3-G3A and 3-G3B) and the monomer 3-M functionalized with azabis(oxazoline) complexes of CuCl2, used as catalysts in the asymmetric benzoylation of a linear diol (C) and a cyclic diol (D). Doted lines indicate the maximum theoretical yield (50%). First experiments are represented in plain colors, whereas recycling experiments are represented in faint colors in graphs C and D, for both the yield and the enantioselectivity[32]. All the given yields correspond to isolated yields after column chromatography.

Figure 4 .

 4 Figure 4. Synthesis of dendrimer 4-G4 functionalized with diketone moieties for the complexation of Cu(I), and the corresponding monomer 4-M. Generations 0 to 4 were used as catalysts in the arylation of 3,5-dimethylphenol by bromobenzene (R = H) using either 2 or 4 diketone moieties per Cu at 110°C (graph on the left) or 120°C (graph on the right); only the cases where 2 diketone moieties per Cu were used are shown. Dendrimer 4-G2 was also used in the arylation of 3,5-dimethylphenol by 4bromobenzene, 4-bromoanisole and 4-bromoacetophenone [33].

Figure 6 .

 6 Figure 6. Third generation dendrimer complexing gold on all the terminal functions (6-G3A), and stochastic functionalization on the surface of generation 3 phosphorhydrazone dendrimers. Only one of the numerous possible isomers is shown [37].

Figure 7 .

 7 Figure 7. Chemical structure of first generation dendrons having a long alkyl chain (C11 or C17) linked to the core, and either copper (7-G1Cu) or gold (7-G1Au) complexes as terminal functions[40].


Angel Mercado-Uribe

Jaime A Moreno

Andrey Polyakov

| Denis Efimov

MIMO Homogeneous Integral Control Design using the Implicit Lyapunov Function Approach

Keywords: Homogeneous controller, nonlinear control design, robustness, high order sliding mode control, implicit Lyapunov function

In this paper, continuous and discontinuous integral controllers for MIMO systems are designed for a large class of nonlinear systems, which are (partially) feedback linearizable. These controllers of arbitrary positive or negative degree of homogeneity are derived by combining a Lyapunov function obtained from the Implicit Lyapunov Function (ILF) method with some extra explicit terms. Discontinuous integral controllers are able to stabilize an equilibrium or track a time-varying signal in finite time, while rejecting vanishing uncertainties and non-vanishing Lipschitz matching perturbations. Continuous integral controllers achieve asymptotic stabilization despite non-vanishing constant perturbations in finite-time, exponentially or nearly fixed-time for negative, zero or positive homogeneity degree, respectively.

The design method and the properties of the different classes of integral controllers are illustrated by means of a simulation example.

INTRODUCTION

The presence of external perturbations or internal model uncertainties constitutes a fundamental obstacle to attain good point stabilization or trajectory tracking. This makes the design of robust controllers an essential control task. If the effect of the disturbances/uncertainties does not vanish at the equilibrium point, i.e. they are non-vanishing, it is well-known that a continuous and static feedback controller is unable to stabilize it, and only practical stability can be achieved. However, a memoryless discontinuous (at the equilibrium) feedback controller can fully compensate for non-vanishing bounded matched perturbations. This fact makes the well-known (High-Order) Sliding-Mode Control (HOSMC) [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF][START_REF] Fridman | Sliding Mode in Control Engineering[END_REF][START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF] a popular strategy to achieve such insensitivity to exogenous disturbances or internal uncertainties for Single-Input Single-Output (SISO) systems employing discontinuity. These controllers have been extended to Multiple-Input Multiple-Output (MIMO) systems in [START_REF] Levant | Gain-scheduled high-order MIMO sliding mode control[END_REF][START_REF] Levant | Accelerated High-Order MIMO Sliding Mode control[END_REF]6 . On the other hand, an undesirable effect of discontinuous controllers is the "Chattering" phenomenon [START_REF] Fridman | Sliding Mode in Control Engineering[END_REF] .

An alternative approach to deal with non-vanishing perturbations is the use of dynamic controllers, being the integral control 7 the most classical one. But it is only able to fully compensate constant perturbations, which is by far a smaller class than a discontinuous controller can counterbalance. Therefore, an interesting idea is to design nonlinear integral controllers, which are able to reach the advantages of both approaches, i.e. using a continuous control signal to attenuate chattering while rejecting a large family of perturbations.

Since homogeneous systems posses several advantages [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] , imposing homogeneity (at least locally) on the closed-loop dynamics has been used frequently for the design of nonlinear controllers and observers. Homogeneous continuous or discontinuous controllers can be designed by different methods. In [START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode Design: a Lyapunov Approach[END_REF] the design is based on explicit Lyapunov functions, while in 10 the controllers are designed through an implicitly defined Lyapunov function. One attractive feature of this latter method is that the gain design can be reduced to an LMI problem.

For SISO systems with relative degree one, a classical discontinuous integral controller is the well-known Super-Twisting Algorithm (see e.g. [START_REF] Levant | Sliding Order and Sliding Accuracy in Sliding Mode Control[END_REF][START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF] ). In 12,[START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "Twisting" controller for secondorder sliding mode realization[END_REF] explicit Lyapunov functions have been proposed for the stability analysis and the gain design of the Super-Twisting. Recently, nonlinear continuous and homogeneous integral controllers have been proposed for mechanical systems [START_REF] Nakamura | High Precision Control of Robots Manipulators via Finite-Time P-PI Control[END_REF][START_REF] Nakamura | Homogeneous Integral Finite-Time Control and Its Application to Robot Control[END_REF] but without a formal stability proof. Distinct Continuous and Discontinuous homogeneous integral controllers have been presented for systems with different relative degrees in [START_REF] Zamora | Control Integral Discontinuo Para Sistemas Mecánicos[END_REF][START_REF] Moreno | Discontinuous Integral Control for Mechanical Systems[END_REF][START_REF] Kamal | Continuous Terminal Sliding-Mode Controller[END_REF][START_REF] Laghrouche | Higher Order Super-Twisting for Perturbed Chains of Integrators[END_REF][START_REF] Mercado-Uribe | Discontinuous Integral Control for Systems in Controller Form[END_REF][START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous integral action for arbitrary relative degree in sliding-mode control[END_REF][START_REF] Moreno | Discontinuous Integral Control for Systems with Arbitrary Relative Degree[END_REF][START_REF] Cruz-Zavala | Higher-Order Sliding Mode Control using discontinuous Integral Action[END_REF] , showing also diverse structural properties. In these works, various explicit Lyapunov functions have been developed to assure convergence and robustness on the one side, and to design the gains on the other side. For instance, the basic idea of the construction of the (non-smooth) Lyapunov function for relative degree one used in 12 has been extended for an arbitrary relative degree in [START_REF] Laghrouche | Higher Order Super-Twisting for Perturbed Chains of Integrators[END_REF] to obtain continuous and discontinuous integral controllers. All these methods have the drawback, that the gain selection becomes a highly nonlinear problem, making its application more difficult.

In [START_REF] Mercado-Uribe | Integral Control Design using the Implicit Lyapunov Function Approach[END_REF] an integral controller for SISO systems is presented, which is based on the Implicit Lyapunov Function (ILF) method. The ILF method was proposed originally in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF] to design static state-feedback controllers. We considered in [START_REF] Mercado-Uribe | Integral Control Design using the Implicit Lyapunov Function Approach[END_REF] homogeneous controllers of non-positive homogeneity degree, affected solely by time-varying matched perturbations. In the current paper, we extend this result to MIMO systems, of arbitrary (negative or positive) degree of homogeneity, and considering vanishing not matched perturbations and non-vanishing matched perturbations, which can both depend on state and time. The design of these homogeneous continuous or discontinuous integral controllers -for SISO and MIMO systems -is developed around the ILF method [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF] .

An important advantage of the proposed solution is the obtainment of constructive rules for tuning the control gains formulated in the form of LMIs, similar to linear time-invariant systems. Since a direct application of the ILF idea does not lead to a usable integral controller, we combine the ILF method for the design of a (rational) state feedback controller and an explicit Lyapunov function for the calculation of the integral part. This resembles the idea used for the Super-Twisting in 12 , and which is generalized for arbitrary order in [START_REF] Laghrouche | Higher Order Super-Twisting for Perturbed Chains of Integrators[END_REF] . This leads to a very useful method for designing homogenous integral controllers of an arbitrary positive or negative degree.

It is important to stress that continuous integral controllers can compensate only for constant perturbations (as for the classical integral action), while discontinuous controllers can reject Lipschitz perturbations. It is also worth noticing here that, since the control signal in the discontinuous integral action is obtained through the (time) integration of the signal produced by a discontinuous term, it is continuous and the effect of chattering can be in principle strongly reduced. The class of systems considered includes minimum phase partially feedback linearizable MIMO nonlinear systems, which are transformed to the Byrnes-Isidori normal form to make use of homogeneity.

Overview

The current work is organized as follows. In Section 2 we present some background on the required mathematical tools. In Section 3 we formulate the problem to be solved and the class of systems considered. In Section 4, we present the main result that consists of the homogeneous integral controller. In Section 5, we prove the main result by devising a strong Lyapunov function, resulting as a combination of the ILF proposed in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] and some extra terms, that ensures the robust global stability of the origin for the closed-loop system, and allows the design of the gains. Finally, in Section 6, we show some simulations using the controllers presented here to illustrate their properties.

Notation

R, R 0+ = {z ∈ R | z ≥ 0} and R + = {z ∈ R | z > 0}
stand for the real, the non negative and the positive real numbers, respectively. For a real variable z ∈ R and a real number p ∈ R the symbol z p = |z| p sign (z) represents the signed power p of z. According to this z 0 = sign (z). d dz z m = m |z| m-1 and d dz |z| m = m z m-1 . Note that z 2 = |z| 2 sign (z) = z 2 , and if p is an odd number then z p = z p and |z| p = z p for any even integer p. Moreover z p z q = |z| p+q , z p z 0 = |z| p and z 0 z p = |z| p .

Notice that we usually consider z 0 = sign (z) as a multivalued function, i.e.

z 0 = 1 if z > 0, z 0 ∈ [-1 , +1] if z = 0, z 0 = -1 if z < 0.
Likewise, let f a vector field, L f (z) represents the Lie derivative of z with respect to f . For a vector v ∈ R n , with components v i , i = 1, ..., n, v p and |v| represent the R n vectors defined by

v p = [ v 1 p , ..., v n p ] T and |v| = [|v 1 | , ..., |v n |] T , respectively. For n ∈ N, I n is the n × n identity matrix in R n×n , i.e. I n = diag{1, • • • , 1}. Finally, diag{v} stands for the diagonal R n×n matrix diag{v} =    v 1 • • • 0 . . . . . . . . . 0 • • • v n    .

PRELIMINARIES

For a given vector x = (x 1 , ..., x n ) T ∈ R n and any positive scalar ε > 0, the family of linear dilation operators is defined as Λ r (ε)x := (ε r 1 x 1 , ..., ε r n x n ), where r i > 0 are the weights of the coordinates and r = (r 1 , ..., r n ) is the vector of weights.

A function V : R n → R (respectively, a vector field f : R n → R n , or a vector-set field [START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF][START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] . A system is called homogeneous if its vector field (or vector-set field) is r-homogeneous of some degree. For a vector field f the Lie derivative of V along f is represented by L f V = ∂V ∂ x f . For a vector-set field F we denote by

F ⊂ R n ) is called r-homogeneous of degree m ∈ R if the identity V (Λ r (ε)x) = ε m V (x) holds for every x ∈ R n and ε > 0 (resp., f (Λ r (ε)x) = ε m Λ r (ε) f (x), or F(Λ r (ε)x) = ε m Λ r (ε)F(x))
L F V (x) = {y ∈ R | y = ∂ x V (x) • v , v ∈ F (x)} the
set of values taken by the Lie derivative of V along all vector fields contained in F. If V and F are r-homogeneous, then so is L F V .

The homogeneous norm is defined by x r,p :

= ∑ n i=1 |x i | p r i 1
p , ∀x ∈ R n , for any p ≥ max{r i }, and it is r-homogeneous of degree 1. The set S = {x ∈ R n : x r,p = 1} is the corresponding unit homogeneous sphere. Stability of homogeneous systems can be studied by means of homogeneous Lyapunov functions (HLFs) [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF] . Any continuous homogeneous system ẋ = f (x), with globally asymptotically stable (GAS) equilibrium point, admits a

C p (R n ) HLF of degree m if m > p • max{r i } for any p ∈ N 8 .
For continuous homogeneous systems the type of convergence is characterized completely by the homogeneity degree of the system. The following Lemma is well-known [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] Lemma 1. Let f be an r-homogeneous continuous vector field of degree l f . Let • r, p be any homogeneous norm. Let x = 0 be an equilibrium point of ẋ = f (x), and assume that it is GAS. Then

(i) If l f = 0 then x = 0 is Exponentially Stable, i.e., there exist positive constants b 1 , b 2 > 0 such that ∀x 0 ∈ R n all solutions x (t, x 0 ) satisfy x (t, x 0 ) r, p ≤ b 1 exp (-b 2 t) x 0 r, p , ∀t ≥ 0 .
(ii) If l f > 0 then x = 0 is nearly Fixed-time Stable (nFxTS), i.e., there exists a positive constant b > 0 such that ∀x 0 ∈ R n all solutions x (t, x 0 ) satisfy

x (t, x 0 ) r, p ≤ 1 1 + l f b x 0 l f r, p t 1 l f x 0 r, p , ∀t ≥ 0 .
Moreover, for any ρ > 0 there exists T ρ ∈ (0, +∞) such that x(t, x 0 ) ≤ ρ for all t ≥ T ρ and all x 0 ∈ R n .

(iii) If l f < 0 then x = 0 is Globally Finite-Time Stable (GFTS), i.e., x(t, x 0 ) = 0 for all t ≥ T (x 0 ), ∀x 0 ∈ R n , where T : R n → R + is a settling-time function. Moreover, the settling-time T (x 0 ) can be estimated, using an r-homogeneous Lyapunov function of degree l V , as follows

T (x 0 ) ≤ 1 κ -l f l V V -l f l V (x 0 ) .
Let F : R 0+ × R n → R n be a vector-set field (multivalued map). It is said to satisfy standard assumptions if it fulfills the following conditions [START_REF] Filippov | Differential equations with discontinuous righthand side[END_REF][START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF] : (i) F(t, x) is a non-empty, compact, convex subset of R n for each t ≥ 0 and each x ∈ R n . (ii) F(t, x), as a set-valued map of x, is upper semi-continuous for each t ≥ 0. (iii) F(t, x), as a set-valued map of t, is Lebesgue measurable for each x ∈ R n . (iv) F(t, x) is locally bounded. For a differential inclusion (DI) ẋ ∈ F(t, x) satisfying these hypotheses (the solutions are understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous righthand side[END_REF] ), if 0 ∈ F(t, 0) for almost all t ≥ 0, then the definitions of strong stability notions coincide with ones given in Lemma 1 (strong means that all solutions issued in an initial condition possess the property). In addition, x = 0 is strongly GAS, iff there exists a C p (R n ) ∩C ∞ (R n \ {0}) (homogeneous) strong LF [START_REF] Nakamura | Smooth Lyapunov functions for Homogeneous Differential Inclusions[END_REF][START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF][START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF] . Moreover, if F is r-homogeneous of degree l < 0, then x = 0 is strongly globally finite-time stable (GFTS) and the settling time is continuous at zero and locally bounded [START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF][START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF][START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF] .

Finally, for the Implicit Lyapunov Function Method the following characterization of ILFs 10 is crucial.

Theorem 1. Consider a system described by a DI

ẋ(t) ∈ F(t, x(t)), t ∈ R + , x(0) = x 0 ∈ R n , (1)
where x ∈ R n is the state, F : R 0+ × R n → R n is a multivalued map satisfying standard assumptions 0 ∈ F(t, 0) , for almost every t ≥ 0 .

If there exists a continuous function

Q : R + × R n → R (V, x) → Q(V, x)
satisfying the conditions:

C1) Q is continuously differentiable outside the origin; C2) for any x ∈ R n \ {0} there exists V ∈ R + such that Q (V, x) = 0 ; C3) let Ω = {(V, x) ∈ R + × R n : Q(V, x) = 0} and lim x→0 V = 0 , lim V →0 + x = 0 , lim x →∞ V = +∞ ; ∀(V, x) ∈ Ω ; C4) ∂ Q(V, x) ∂V < 0 ∀V ∈ R + and x ∈ R n \ {0} ; C5) sup t∈R + ,y∈F(t,x) ∂ Q(V,x) ∂ x y < 0 ∀(V, x) ∈ Ω ;
then the origin of system (1) is globally uniformly asymptotically stable.

Remark 1. It is important to stress that under the restrictions imposed on the function Q in Theorem 1 a Lyapunov function V exists, that is defined implicitly by the equation Q(V (x), x) = 0. This can be seen from the required conditions. Condition C2 requires V to be positive definite for any x ∈ R n \ {0} and, together with C4, implies that V is a function. Condition C3 says that V is positive definite and radially unbounded. Finally, conditions C4 and C5 imply that V (x) is negative definite. The ILF V (x) is continuously differentiable for every x ∈ R n \ {0}. However, it is not assured to be differentiable at x = 0.

PROBLEM STATEMENT

Consider the following MIMO non-linear system

ξ = f (ξ ) + m ∑ i=1 g i (ξ )v i + ψ(t, ξ ) = f (ξ ) + g(ξ )v + ψ(t, ξ ) , σ 1 = h 1 (ξ ) , . . . σ m = h m (ξ ) , (2)
where ξ ∈ R s are the states, σ i are the outputs and v i are the control inputs. f and g i are smooth vector fields, h i are smooth real-valued output functions, and ψ is a smooth time dependent vector field representing some parameter or model uncertainties and/or external perturbations acting on the system. We write σ

= [σ 1 , • • • , σ m ] and v = [v 1 , • • • , v m ],
for the vector of outputs and control inputs, respectively. The outputs σ i can represent tracking errors or sliding variables in sliding-mode control as e.g. [START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF][START_REF] Levant | Weighted Homogeneity and Robustness of Sliding Mode Control[END_REF][START_REF] Levant | Quasi-Continuous MIMO Sliding-Mode Control[END_REF] . The aim of the control is to render the outputs σ i = 0 asymptotically or in finite-time despite of the acting perturbations/uncertainties ψ. When the (unperturbed) system (2) has a well-defined vector relative degree ρ

= [ρ 1 , • • • , ρ m ], with nonsingular matrix b(ξ ) =       L g 1 L ρ 1 -1 f h 1 (ξ ) • • • L g m L ρ 1 -1 f h 1 (ξ ) L g 1 L ρ 2 -1 f h 2 (ξ ) • • • L g m L ρ 2 -1 f h 2 (ξ ) . . . . . . . . . L g 1 L ρ m -1 f h m (ξ ) • • • L g m L ρ m -1 f h m (ξ )       ∈ R m×m , (3)
and some regularity assumptions on the distributions, it is well-known [START_REF] Isidori | Nonlinear Control Systems[END_REF] that the system can be transformed by a diffeomorphism of the states and a regular feedback v = α(ξ ) + β (ξ )u, with matrix β (ξ ) ∈ R m×m invertible, to the (Byrnes-Isidori) Normal form

η = q (η, x) + µ 0 (t, x, η) , (4)
ẋ = Ax + B (u + µ 1 (t, x, η)) + µ 2 (t, x, η) (5)
y 1 = x 1, 1 , . . . y m = x m, 1 ,
where the vector

x = col (x 1 , • • • , x m ) ∈ R n is composed of the partial state vectors x i ∈ R ρ i , n = ∑ m i=1 ρ i
, and u ∈ R m is the transformed control vector. We have also included the uncertainties/perturbations from ψ(t, ξ ). Subsystem (4) corresponds to the zero dynamics, with state η ∈ R s-n . Matrices A and B have the Brunovsky canonical form, i.e.

A =      A 1 0 • • • 0 0 A 2 • • • 0 . . . . . . . . . . . . 0 0 • • • A m      ∈ R n×n , A i =      0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 0      ∈ R ρ i ×ρ i , i = 1, ..., m . B =      b1 0 • • • 0 0 b2 • • • 0 . . . . . . . . . . . . 0 0 • • • bm      ∈ R n×m , bi =    0 . . . 1    ∈ R ρ i ×1 , i = 1, ..., m.
The decoupling matrix b(ξ ) (3) becomes in these coordinates the identity matrix, i.e.

b = I m . (6)
The uncertainties/perturbation vector ψ(t, ξ ) is decomposed in three terms: µ 0 (t, x, η) affects the zero-dynamics, µ 1 (t, x, η) is a matched term, acting on the control channel, and µ 2 (t, x, η) is a not matched term, acting on the main dynamics. In general [START_REF] Isidori | Nonlinear Control Systems[END_REF] , the representation (4)-( 5) is valid only locally. But for simplicity in the presentation we will assume that it is globally valid, and we will state the results in a global form. However, it is easy to recover the local statements, when the representation is valid only locally. We will also assume that the (perturbed) zero dynamics ( 4) is well-behaved, in the sense that its trajectories are globally bounded for all times, whenever (x(t), η(t)) is bounded. This allows us to concentrate on the main dynamics (5) to solve the output-zeroing problem.

Thus, in this set-up, the control objective is to robustly stabilize the origin of system (5), asymptotically or in finite time, despite the perturbations µ 1 , µ 2 . When both terms µ 1 and µ 2 are vanishing, i.e. µ 1 (t, 0, η) = 0 and µ 2 (t, 0, η) = 0, it is possible to design a memoryless state feedback control law u = φ (x) to stabilize the origin of the plant (5). However, when µ 1 (t, x, η) is non-vanishing, what is a reasonable assumption since it contains e.g. the derivatives of the signals to be tracked or external disturbances, a continuous controller is not able to render the origin x = 0 asymptotically stable. In this case, a discontinuous control law for u can solve the problem for any bounded but otherwise arbitrary perturbation µ 1 (t, x, η). This is the main ingredient of High-Order Sliding-Mode (HOSM) controllers [START_REF] Levant | Homogeneity Approach to High-Order Sliding Mode[END_REF][START_REF] Levant | Weighted Homogeneity and Robustness of Sliding Mode Control[END_REF][START_REF] Levant | Quasi-Continuous MIMO Sliding-Mode Control[END_REF][START_REF] Cruz-Zavala | Homogeneous High Order Sliding Mode Design: a Lyapunov Approach[END_REF] , and such a design can be obtained using the ILF Method, as it is presented in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] . The disadvantage of a discontinuous control law for u resides in the undesirable chattering effect, which is a high-frequency oscillation of the control signal, that causes the excitation of unmodeled fast dynamics in the plant and the rapid deterioration of the actuators.

MAIN RESULT: INTEGRAL CONTROLLER DESIGN USING THE IMPLICIT LYAPUNOV FUNCTION METHOD

In this paper, instead of using a discontinuous memoryless state-feedback we propose the use of the following dynamic control structure

u = u 1 (x) + z , ż = u 2 (x) , (7)
which resembles the classical PI-controller. This controller is composed of a (continuous) state feedback u 1 , which is able to stabilize the origin of the nominal system in the absence of the non-vanishing perturbation µ 1 , and an integral term u 2 , which is allowed to be discontinuous, and with the aim to estimate and compensate the perturbation term µ 1 . Note that even when the function u 2 is discontinuous, the control signal u is continuous, since it is the addition of the signal generated by the continuous state feedback u 1 and the time integral of the discontinuous signal u 2 . This fact may help in reducing the chattering effect. Due to the multiple properties of homogeneous systems, as e.g. achieving in a simple form finite-time stability and making use of a powerful mathematical apparatus, we want to design the controller (7) such that the closed-loop system (without perturbations) is homogeneous of degree ν, for positive or negative values of ν. The weights of the vectors (x 1 , ..., x m ) are given by r i = (r i, 1 , ..., r i, ρ i ), for i = 1, ..., m, with components r i, j+1 = r i, j + ν, j = 1, ..., ρ i . The vector of weights can be written as r = [r 1 , ..., r m ], and the dilation matrix is given by

Λ r (λ ) = diag{λ r } =    diag{λ r 1 } • • • 0 . . . . . . . . . 0 • • • diag{λ r m }    =               λ r 1, 1 • • • 0 • • • 0 • • • 0 . . . . . . . . . • • • . . . . . . . . . 0 • • • λ r 1, ρ 1 • • • 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . 0 • • • 0 • • • λ r m, 1 • • • 0 . . . . . . . . . • • • . . . . . . . . . 0 • • • 0 • • • 0 • • • λ r m, ρm              
.

We also fix (without loss of generality) all the weights of the components of vector z to be equal to 1, i.e. r z, i = r z =

1 for i = 1, • • • , m.
From their relationship with the weights of x, given by r z, i = r i, ρ i + ν, we conclude that

r z, i = r z = 1 , r i, ρ i = 1 -ν , r i, j = 1 -(ρ i + 1 -j) ν , -1 ≤ ν < 1 max ρ i , i = 1, • • • , m , j = 1, • • • , ρ i . (8)
For homogeneity of the closed-loop system, function u 1 requires to be homogeneous of degree 1, while function u 2 needs to be homogeneous of degree 1 + ν. The homogeneity degree ν of the closed-loop system can be selected in the interval given in (8), because of the non negativity of the weights.

On the one extreme of the interval, when ν = -1, function u 2 is discontinuous, with homogeneity degree zero. In this case the right-hand side of the closed-loop system is discontinuous, the trajectories are to be understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous righthand side[END_REF] , and convergence is in finite-time (recall Lemma 1). A linear state feedback with linear integral control has homogeneity degree ν = 0, the right-hand-side is globally Lipschitz, and convergence is exponential. When ν > 0 the right-hand side of the closed-loop system is locally but not globally Lipschitz and convergence is nearly fixed-time.

In this section, we present the main result of the paper: we provide a procedure, based on the Implicit Lyapunov Function Method introduced in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] , to design a homogeneous integral controller (7) that robustly stabilizes the origin of system (5), despite non vanishing matched perturbations µ 1 , and vanishing unmatched perturbations µ 2 . Selection of the homogeneity degree ν determines the type of convergence (in finite-time, exponential or asymptotic) and the size of the allowable perturbations. The convergence is nearly fixed-time for ν > 0, exponential if ν = 0 and in finite-time when ν < 0. For ν > -1 the matched perturbation µ 1 needs to be constant to be fully rejected. Otherwise, if it is slowly time-varying, just practical stability will be achieved, i.e. the trajectories will reach after some finite time a neighborhood of the origin and will remain in it for all future times. This is the usual and expected behavior of a PI controller with respect to the perturbation. In contrast, and this is a quite surprising property, when ν = -1 the control law u 2 is discontinuous, the convergence is in finite-time and the matched perturbation µ 1 can be an arbitrary Lipschitz signal, i.e. µ 1 can be a growing signal, with bounded time derivative, and it will be fully rejected in the closed-loop. For µ 2 it is required in all cases, that it is vanishing, i.e. µ 2 (t, x, z) = 0 when x = 0. Its growth with respect to x will depend on ν, as it will be presented below. In general, the time derivative of µ 1 can also grow with the states, as detailed in the next subsection.

The ILF homogeneous integral controller

We first select some value of the homogeneity degree ν in the interval -1 ≤ ν <1 max ρ i , what determines the corresponding weights, as given by (8). By solving the following matrix inequalities, for some ε > 0 and some positive definite and constant matrix R ∈ R n×n > 0,

H r P + PH r < 0 , H r = -diag {r i } P (A -BK) + (A -BK) T P ≤ ε (H r P + PH r ) -R , ε > 0 , R > 0 , (9)
we find a constant, symmetric and positive definite matrix P = P T ∈ R n×n and a constant matrix K ∈ R m×n . It is shown in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] that inequalities (9) have always a solution P > 0 and K for any ν. Moreover, these matrix inequalities can be transformed to an LMI, which is manageable by using standard software. Furthermore, given P, it is also shown in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] that the equation

Q (V, x) x T Λ r V -1 PΛ r V -1 x -1 = 0 , (10)
defines implicitly a unique, continuous, homogeneous of degree 1, radially unbounded and positive definite function 1 V : R n → R + , such that Q (V (x), x) = 0 (see Theorem 1). Assume now that the perturbations µ 1 and µ 2 satisfy globally the following conditions:

dµ 1 dt ∞ = || μ1 || ∞ ≤ D 1 + D 2 x 1+ν r, p , D 1 , D 2 ≥ 0 (11) µ T 2 Λ r V -1 PR -1 PΛ r V -1 µ 2 ≤ β εV 2ν x T Λ r V -1 (-H r P -PH r ) Λ r V -1 x , 0 ≤ β < 1 . (12)
Define also the following function (which is the partial derivative of Q with respect to V )

Q V (V, x) := V -1 x T Λ r V -1 (H r P + PH r ) Λ r V -1 x .
Theorem 2. Consider the system (5) and the homogeneous integral controller

u = u 1 (x) + z = -V (x)KΛ r V -1 (x) x + z , ż = u 2 (x) = γ V ν (x) Q V (V (x) , x) B T PΛ r V -1 (x) x , γ > 0 , (13)
for some -1 ≤ ν < 1 max ρ i . Let ζ := z + µ 1 (t, x, η). Then for any β < 1, any D 2 ≥ 0 and a sufficiently large γ > 0, the point

(x, ζ ) = 0 of the closed-loop system is 1. GFTS if ν = -1, D 1 > 0 and D 1 +D 2 γ sufficiently small. 2. GFTS if -1 < ν < 0 and D 1 = 0. 3. Globally Exponentially Stable if ν = 0 and D 1 = 0. 4. nFxTS if 0 < ν < 1 max ρ i and D 1 = 0.
The proof of this result is deferred until the next Section 5. To show the convergence of the closed-loop with the integral controller (13) a (strong) Lyapunov Function will be used, which combines the ILF V obtained from (10) with an explicit term depending on the integral variable. The devised Lyapunov Function has the form (see (19)

below) V (x, z) = θ 1 2 V 2 (x) + 1 γ z T z 3-ν 2 - 1 γ x 1, ρ 1 , • • • , x m, ρ m z 2 , γ > 0 , θ > 0 . ( 14
)
Using this Lyapunov function and Lemma 1 (see the proof in Section 5) it is possible to estimate for ν = 0 the transit time 1. ν = -1 and D 1 > 0 or -1 < ν < 0 and D 1 = 0: The convergence time from an initial condition (x 0 , ζ 0 ), where ζ 0 = z 0 + µ 1 (0) includes the initial value of the perturbation, is upper bounded by

T i→0 (x 0 , ζ 0 ) = 3 -ν |ν| η V |ν| 3-ν (x 0 , ζ 0 ) ,
for some η > 0 depending on the parameters of the problem.

2. 0 < ν < 1 max ρ i and D 1 = 0: The transit time from an arbitrarily large initial condition (x 0 , ζ 0 ) → ∞ to a final condition different from zero x f , ζ f = 0, is upper bounded by

T ∞→ f x f , ζ f = 3 -ν νη 1 V ν 3-ν x f , ζ f ,
for some η > 0 depending on the parameters of the problem.

In conclusion, for ν < 0 the convergence to zero from an arbitrary initial condition is in finite-time. For ν > 0 the convergence to zero happens only asymptotically, but the convergence to an, e.g., small ball around the origin from "infinity" occurs in finite-time.

Note from the Theorem 2 that (assuming for simplicity that D 2 = 0) if D 1 > 0, i.e. the perturbation µ 1 is not a constant, then convergence to zero will be only possible if ν = -1, i.e. the controller u 2 is discontinuous. From Item (1) of the Theorem 2 it follows that a perturbation of any size D 1 > 0 can be fully compensated in this case by choosing a sufficiently large integral gain γ.

However, if ν > -1 and D 1 > 0 the trajectories of the system are globally uniformly ultimately bounded with bound b, that is, (see [START_REF] Khalil | Nonlinear Systems[END_REF] ) for every initial condition there is a finite time T (independent of the initial time) such that the trajectories will enter a neighborhood of zero of radius b and remain there for all future times. This is also equivalent to saying that for the closed-loop system the map μ1 → (x, ζ ) is ISS. This is basically the content of the following Lemma, also proved in the next Section 5. The first part of this result can be derived from [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] . Lemma 3. Consider the closed-loop system of Theorem 2 under the same hypothesis. The second item in the Lemma 3 is interesting, since it shows that the nearer the homogeneity degree ν is to the discontinuous case ν = -1, the smaller is also the effect of the perturbation µ 1 when it is not constant. This is, in some sense, intuitively appealing. Note however, that for this to be true it is required to have the ratio D 1 +D 2 γ , between the size of μ1 and the integral gain γ, small. If this is not the case, then the conclusion is false. Note that smallness of D 1 +D 2 γ can always be achieved by selecting γ sufficiently large.

Discussion of the results

We present some observations with respect to the results.

• For the implementation of the controller (13) it is necessary to find the actual value of V (x) by solving the (implicit) equation Q(V, x) = 0. This can be hardly done analytically, so that it has to be obtained numerically on-line. A numerical procedure is proposed in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF] .

• The asymptotic stability of the closed-loop, and moreover the existence of a smooth and strong Lyapunov Function [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF] , implies that the asymptotic stability is robust under rather general perturbations to the vector field, as e.g. small discretization errors, small delays in states or small noises acting on the variables. For homogeneous systems this robustness has some interesting forms (see e.g. 3,33 and 36 ). In particular, for the implementation of the control law derived using the ILF method, it is shown in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] that the discretization and numerical errors induced by the on-line solution of the implicit equation Q(V, x) = 0 does not destroy the stability properties, and ultimate boundedness of the solutions is attained.

• For -1 < ν < 1 max ρ i both control functions u 1 and u 2 in ( 13) are continuous everywhere. However, for ν = -1 although u 1 is continuous everywhere, u 2 is continuous for x ∈ R n \ {0}. At x = 0 it is discontinuous, but its value is bounded (see remark 13 in [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF]Remark 13 ). This kind of controllers is usually named Quasi-Continuous controllers in the High-Order Sliding-Mode literature [START_REF] Levant | Quasi-Continuous High-Order Sliding-Mode Controllers[END_REF][START_REF] Levant | Quasi-Continuous MIMO Sliding-Mode Control[END_REF] , in contrast to the Discontinuous ones, which have discontinuities also outside from x = 0.

• Note that in (11) the bound can also depend on z, i.e. || μ1

|| ∞ ≤ D 1 + D 2 (x, z) 1+ν
r, p , without any change in the proof. Moreover, when ν = -1 the bound becomes simply || μ1 || ∞ ≤ D 1 + D 2 .

• The matched perturbation, i.e. the perturbation entering through the same channel as the control variable, has two terms: the component µ 2, n of the vector µ 2 , which is vanishing at x = 0, and µ 1 , which is non vanishing. If µ 1 depends only on an external perturbation, what we can represent as an exogenous time-varying signal µ 1 (t), then it has to be constant ( μ1 (t) ≡ 0) for a continuous integral term (ν > -1), but it can be an arbitrary Lipschitz continuous signal, i.e. | μ1 (t)| ≤ D 1 + D 2 , for the discontinuous integral term (ν = -1). In this latter case the exogenous signal µ 1 (t) can be time-varying and it does not require to be bounded, but its derivative has to be bounded. This is a much larger class of perturbations that can be fully compensated. When µ 1 is also a function of the states x, μ1 may also depend on u and z. If condition (11) is satisfied only locally instead of globally, then the stability result will be also local.

• The bound (12) for µ 2 imposes to each of the components of the vector µ 2 to satisfy |µ 2, i, j (t, x, η)| ≤ δ i, j x r i, j +ν r, p , for some δ i, j ≥ 0, and where the weights r i, j are given in (8). This requires µ 2, i, j has to be vanishing when x = 0, and to grow with the homogeneity degree corresponding to the variable x i, j+1 , i.e. of the component of the vector field of its channel. Since (12) depends on the value of the function V , which can be determined numerically on-line, the actual value of the allowed size of δ i, j , which also depends on K, can also be calculated on-line. This calculation is part of the process of design of the gain matrix K, which becomes a matrix inequality. Due to homogeneity it suffices to do that on the unit homogeneous sphere. In any case, asymptotic stability is preserved for sufficiently small values of δ i, j due to its intrinsic robustness.

• The problem of how to check the implicitely defined inequality (12) has been already addressed in the paper [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] .

Proposition 16 in [START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] provides a simple sufficient condition to verify (12) without the necessity of calculating the ILF V . It requires only the value of matrix P.

• In the proof of the Theorem 2 it is also shown that the integral variable z converges (nearly fixed-time, exponentially or in finite-time) to the value of µ 1 if D 1 = 0 for -1 < ν < 1 max ρ i , or when D 1 > 0 if ν = -1. This shows, as it is wellknown for the classical case, that the integral part of the controller reconstructs the perturbation and thus is able to fully compensate for it.

• For this controller, if stability is achieved for some value of the integral gain γ, say γ * , the stability is preserved for any γ ≥ γ * , without changing the gain K. This property can be understood from the passivity interpretation given in the proof of the Theorem in Section 5. The controller proposed in [START_REF] Laghrouche | Higher Order Super-Twisting for Perturbed Chains of Integrators[END_REF] posses this property, but it is not shared by other integral controllers presented in the literature as e.g. [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous integral action for arbitrary relative degree in sliding-mode control[END_REF] .

• In general, the gain selection here is easier than for the integral controllers designed using explicit Lyapunov functions, as e.g. the ones presented in [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Gutierrez-Oribio | Stabilization of the Reaction Wheel Pendulum via a Third Order Discontinuous Integral Sliding Mode Algorithm[END_REF] .

• We note some other differences between the controller (13) and other integral controllers presented e.g. in [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous Integral Control for Systems in Controller Form[END_REF][START_REF] Mercado-Uribe | Discontinuous integral action for arbitrary relative degree in sliding-mode control[END_REF][START_REF] Moreno | Discontinuous Integral Control for Systems with Arbitrary Relative Degree[END_REF] :

(i) In ( 13) the integral action u 2 depends on the full state x, while u 2 in [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous Integral Control for Systems in Controller Form[END_REF][START_REF] Mercado-Uribe | Discontinuous integral action for arbitrary relative degree in sliding-mode control[END_REF] can be a function of x 1 alone, or x 1 and a homogeneous function of any other states.

(ii) For ν = -1 the integral controller in ( 13) is discontinuous only at x = 0, so that it is of the quasi-continuous form. However, u 2 in [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous Integral Control for Systems in Controller Form[END_REF] can be discontinuous on homogeneous varieties larger than the set {x = 0}.

(iv) Since the ILF V is not smooth at x = 0 (see Remark 1), the Lyapunov function (14)(see next Section) for ( 13) is not smooth. In contrast, the Lyapunov functions for the controllers in [START_REF] Moreno | Discontinuous Integral Control for Systems with relative degree two[END_REF][START_REF] Mercado-Uribe | Discontinuous Integral Control for Systems in Controller Form[END_REF][START_REF] Mercado-Uribe | Discontinuous integral action for arbitrary relative degree in sliding-mode control[END_REF] are smooth. Moreover, the basic idea of the proof is completely different, and the specific properties are different.

PROOF OF MAIN RESULT: A LYAPUNOV FUNCTION APPROACH

We divide the proof of Theorem 2, including also Lemma 3, in three parts: i) First we consider the design of the state feedback controller u 1 using the ILF method. For this we assume that the matched perturbation µ 1 is absent. ii) Then we complete the ILF with an extra (explicit) term to build a weak Lyapunov Function to design the integral term u 2 . Since the previous weak LF does not allow to assert robustness with respect to e.g. perturbation µ 1 , we complete the weak LF with an extra cross-term in order to obtain a strong LF, which allows us to assure robustness and the type of convergence.

Recall that for a weak Lyapunov function V , sup t∈R + ,y∈ f (t,x)

∂V (x)
∂ x y ≤ 0, ∀x = 0, so that using standard Lyapunov arguments [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Orlov | Discontinuous Systems. Lyapunov analysis and robustness synthesis under uncertainty conditions[END_REF] only uniform stability can be assured. In contrast, for a strong Lyapunov function sup t∈R + ,y∈ f (t,x) ∂V (x) ∂ x y < 0, ∀x = 0 and uniform asymptotic stability follows. Note also that we are considering in this work only the concept of strong stability for a differential inclusion. This means roughly that the property is satisfied for all solutions, in opposition to the concept of weak stability, which applies for some solution [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF] .

We start by showing that the ILF Lyapunov function V , obtained as solution of (10), and the controller functions u 1 and u 2 are homogeneous of degrees 1, 1 and 1 + ν, respectively. Using the definition of the dilation matrix it is easy to see that Λ r (λV

) -1 = Λ r V -1 Λ -1
r (λ ). This implies that function Q, as defined in (10), is homogeneous of degree 0 with weight 1 for V , i.e.

Q (λV, Λ r (λ ) x) = x T Λ r (λ ) Λ -1 r (λ ) Λ r V -1 PΛ r V -1 Λ -1 r (λ ) Λ r (λ ) x -1 = Q (V, x
) . This implies that the function V , which is solution of the equation (10), is homogeneous of degree 1, i.e. V (Λ r (λ ) x) = λV (x).

The homogeneity degree of u 1 (x) is r z = 1, since

u 1 (Λ r (λ ) x) = -V (Λ r (λ ) x) KΛ r V -1 (Λ r (λ ) x) Λ r (λ ) x = -λV (x) KΛ r V -1 (x) x = λ u 1 (x) .
It follows also that function Q V (since it is the partial derivative of Q with respect to V ) is homogeneous of degree -1, i.e.

Q V (λV, Λ r (λ ) x) = (λV ) -1 x T Λ r (λ ) Λ r (λV ) -1 (H r P + PH r ) Λ r (λV ) -1 Λ r (λ ) x = λ -1 Q V (V, x) , while u 2 (x) is homogeneous of degree 1 + ν, that is u 2 (Λ r (λ ) x) = γV ν (Λ r (λ ) x) Q -1 V (λV, Λ r (λ ) x) B T PΛ r V -1 (Λ r (λ ) x) Λ r (λ ) x = γλ 1+ν V ν (Q V (V, x)) -1 B T PΛ r V -1 x = λ 1+ν u 2 (x) .

Design of the static feedback control u 1 (x) using an ILF

We first design a state feedback controller u 1 (x) for the system ẋ = Ax + Bu 1 (x) + µ 2 (t, x, z) . using the ILF Method, developed in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] . The LF V is defined implicitly by equation (10), i.e.

Q (V, x) = x T Λ r V -1 PΛ r V -1 x -1, P = P T > 0 .
Q satisfies all conditions C1-C5 of Theorem 1 (the details are given in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] ). Derivating Q with respect to time, we obtain

Q (V, x) = Q V (V, x) V + ∂ Q (V, x) ∂ x (Ax + Bu 1 + µ 2 ) = 0
where

Q V (V, x) ∂ Q (V, x) ∂V = V -1 x T Λ r V -1 (H r P + PH r ) Λ r V -1 x , ∂ Q(V, x) ∂ x = 2x T Λ r V -1 PΛ r V -1 ,
and H r = -diag {r i }. By hypothesis H r P + PH r < 0, so that condition C4 in Theorem 1 is satisfied, and we can obtain the derivative V from Q as

V = -2 (Q V (V, x)) -1 x T Λ r V -1 PΛ r V -1 (Ax + Bu 1 + µ 2 ) .
It can be easily shown that matrices A and B satisfy the following properties

λ ν Λ r (λ ) A = AΛ r (λ ) , Λ r (λ ) B = λ 1-ν B .
Using them in the previous expression of V , we achieve

V = -2 (Q V (V, x)) -1 x T Λ r V -1 P V ν AΛ r V -1 x +V -1+ν Bu 1 + Λ r V -1 µ 2 = -(Q V (V, x)) -1 x T Λ r V -1 V ν PA + A T P Λ r V -1 x + 2V -1+ν PBu 1 + 2PΛ r V -1 µ 2 .
Selecting the controller u 1 as

u 1 (x) = -V KΛ r V -1 x , we obtain V = -(Q V (V, x)) -1 Λ r V -1 x Λ r V -1 µ 2 T V ν P (A -BK) + (A -BK) T P P P 0 Λ r V -1 x Λ r V -1 µ 2 . ( 15
)
We assume that the perturbation µ 2 (t, x, z) satisfies the bound (12), that can be written as

Λ r V -1 x Λ r V -1 µ 2 T -β εV ν (H r P + PH r ) 0 0 -V -ν PR -1 P Λ r V -1 x Λ r V -1 µ 2 ≥ 0 , ( 16
)
where 0 ≤ β < 1. To show that when ( 16) is fulfilled then ( 15) is also satisfied we can use the well-known S-lemma. Alternatively, we use the following argumentation. Adding inequality (16) to the one of V , i.e. (15), we arrive at

V ≤ -(Q V (V, x)) -1 Λ r V -1 x Λ r V -1 µ 2 T V ν P (A -BK) + (A -BK) T P -β ε (H r P + PH r ) P P -V -ν PR -1 P Λ r V -1 x Λ r V -1 µ 2 .
If the matrix of this quadratic form is negative definite, then we can conclude that V < 0. Using Shur's complement this is the case if

V ν P (A -BK) + (A -BK) T P -β ε (H r P + PH r ) + R ≤ 0 . And thus, condition (9), together with the definition of Q V , implies that V ≤ -(1 -β )εV ν (Q V (V, x)) -1 x T Λ r V -1 (H r P + PH r ) Λ r V -1 x ≤ -(1 -β )εV ν+1 .

Design of the dynamic control feedback u 2 (x) using an explicit (control) Lyapunov Function

Consider now the full closed-loop system, where we define as state variable ζ = z + µ 1 (t, x, η), which is the addition of the integral state z and the matched perturbation µ 1 . Since µ 1 is unknown, this implies that the variable ζ is not available for feedback. Using as states (x, ζ ) the dynamics of the closed-loop system are given by

ẋ = Ax + B (u 1 + ζ ) + µ 2 (t, x, η) , ζ = u 2 (x) + d(t, x, η) d(t, x, η) := d dt µ 1 (t, x, η) .
The perturbation d is the total time derivative of the matched perturbation µ 1 . According to (11), it is assumed to be bounded as ||d|| ∞ ≤ D 1 + D 2 x 1+ν r, p . Our aim in this section will be to design u 2 (x). We note first that a direct utilization of the ILF method proposed in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF][START_REF] Polyakov | Robust stabilization of MIMO systems in finite/fixed time[END_REF] is unfeasible for the design of a usable integral term u 2 (x), since the ILF method would lead to a controller as u 2 (x, ζ ), and since ζ is not measurable, the control would not be implementable. Moreover, a function u 2 depending on (x, ζ ) is not a "true" integral action. For these reasons we have to combine the ILF V with some other (explicit) terms to arrive at a LF appropriate for the design of the integral controller. The following development is based on the idea used in 12,[START_REF] Moreno | Strict Lyapunov functions for the Super-Twisting algorithm[END_REF] to obtain a LF for the Super-Twisting algorithm, which has been generalized to an arbitrary order in [START_REF] Laghrouche | Higher Order Super-Twisting for Perturbed Chains of Integrators[END_REF] . Since the ILF V is not smooth at x = 0 (see Remark 1), it leads to a non-smooth Lyapunov function. This technical issue does not cause any serious problems with the proof, since the lack of differentiability can be overcome by using e.g. the idea presented in [START_REF] Moreno | Strict Lyapunov functions for the Super-Twisting algorithm[END_REF] for the Super-Twisting (for more details see [START_REF] Moreno | Discontinuous Integral Control for Systems with Arbitrary Relative Degree[END_REF] ). We will not repeat the argumentation in what follows.

A weak Lyapunov Function

We first construct a homogeneous and smooth (except at x = 0) but weak Lyapunov function for the integral control, of degree

2r z = 2 W (x, ζ ) = 1 2 V 2 (x) + 1 γ ζ T ζ , γ > 0 . (17)
Using the results of the previous section for the function V and the perturbation µ 2 , we arrive at the following expression for its derivative

Ẇ ≤ V -(1 -β )εV 1+ν -2V -1+ν (Q V (V, x)) -1 x T Λ r V -1 PBζ + 2 γ (u 2 + d) T ζ = -(1 -β )εV 2+ν -2V ν (Q V (V, x)) -1 x T Λ r V -1 PBζ + 2 γ u T 2 ζ + 2 γ d T ζ .
Note that the first term is negative (in x), while the second one is a cross-term, without definite sign. The third term depends on the selection of u 2 , and the last one is the effect of the perturbation d. If we forget about the term due to the perturbation, we can render Ẇ at least negative semi-definite by selecting u 2 (x) such that the cross-term is cancelled, i.e. with

u 2 (x) = γV ν (Q V (V, x)) -1 B T PΛ r V -1 x , (18)
where the parameter γ > 0 can be selected arbitrarily. With this selection we obtain

Ẇ ≤ -(1 -β )εV 2+ν + 2 γ d T ζ ,
which is negative semi-definite for d ≡ 0, and therefore W is a weak Lyapunov function. Using the extended LaSalle's invariance theorem, which is presented in [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov function[END_REF][START_REF] Orlov | Extended invariance principle for nonautonomous switched systems[END_REF] , we conclude that the origin (x, ζ ) = 0, in the absence of perturbation d = 0, is GAS. For ν = 0 the convergence is exponential, since the homogeneity degree is zero, while for ν < 0 the convergence is in finite-time, due to negative homogeneity degree of the vector field, and for ν > 0 convergence is nearly fixed-time, due to the positive homogeneity degree (see Lemma 1).

A Passivity Interpretation

The weak Lyapunov function ( 17) has a simple passivity interpretation: The system is a negative feedback interconnection of two passive systems, subsystem x and subsystem ζ . Subsystem x is a (strictly) passive system with V as storage function, input u 1 and output u 2 , as given in (18). Subsystem ζ is also passive, with storage function ζ T ζ , input u 2 and output ζ . W is the storage function of the interconnected system.

A strong Lyapunov function

The weak Lyapunov function W does neither allow us to establish the robustness of the closed-loop with respect to the perturbation d nor to estimate its convergence time, for example. It is advantageous to have a strong Lyapunov function, i.e. one with a negative definite derivative, instead of only a negative semi-definite one. In this section we obtain a strong LF by adding a cross-term to W as

V (x, ζ ) = θW α (x, ζ ) - 1 γ x T ρ ζ ω , γ > 0 , θ > 0 , α = 1 + ω -ν 2 , ω = 2 . (19)
Here, the vector

x ρ = (x 1, ρ 1 , • • • , x m, ρ m )
T is composed of the ρ i -th components of the state x, with time derivative given by ẋρ = u 1 + ζ + μ2 (since the decoupling matrix b presented in (6) is the identity), and the vector μ2 = (µ 2, ρ 1 , • • • , µ 2, ρ m ) contains the ρ i -th components of the perturbation vector µ 2 . Due to homogeneity we can conclude that V is positive definite for θ > 0 sufficiently large. Its time derivative along the closed-loop system is

V (x, ζ , d) ≤ αθW α-1 -(1 -β )εV 2+ν + 2 γ d T ζ - 1 γ (u 1 (x) + ζ + μ2 ) T ζ ω - 1 γ ωx T ρ diag{|ζ | ω-1 } (u 2 (x) + d) = -W (x, ζ ) + 2αθW α-1 ζ T -ωx T ρ diag{|ζ | ω-1 } 1 γ d - 1 γ μT 2 ζ ω , (20)
where

W (x, ζ ) θ α(1 -β )εW α-1 (x, ζ )V 2+ν (x) + 1 γ ωx T ρ diag{|ζ | ω-1 }u 2 (x) + 1 γ u T 1 (x) ζ ω + 1 γ ζ T ζ ω . Note that V (x, ζ ) is homogeneous of degree δ V = 1 + ω -ν, function W is homogeneous of degree δ W = 1 + ω, while the term 2αθW α-1 ζ T -ωx T ρ diag{|ζ | ω-1 } is homogeneous of degree δ d = ω -ν. Note that δ W = δ d if ν = -1, i.e. when u 2 (x) is a discontinuous function of homogeneity degree 0.
The derivative of the LF V in (20) has three terms. The second and third ones depend on the perturbations d and µ 2 , respectively. In absence of these perturbations, the derivative of V is negative definite. Proof. We note first that, although function u 2 is discontinuous at x = 0 for ν = -1, the function W is continuous (and homogeneous). Recall the following well-known property of continuous homogeneous functions (see e.g. 9, Lemma 12 ):

Let η : R n → R and ϕ : R n → R + be two continuous homogeneous functions, with weights r = (r 1 , ..., r n ) and degrees m, with ϕ(x) ≥ 0, such that it holds {x ∈ R n \{0} : ϕ(x) = 0} ⊆ {x ∈ R n \{0} : η(x) > 0}. Then, there exists a real number λ * such that, for all λ ≥ λ * for all x ∈ R n \ {0}, and some c > 0, η(x) + λ ϕ(x) > c x m r, p . The claim of the Lemma is a simple consequence of this property. The first term in W is non negative and it vanishes only when x = 0. The value of W for x = 0 is W (0, ζ ) = 1 γ ζ T ζ ω , which is positive for ζ = 0. And therefore W can be rendered positive definite selecting θ > 0 sufficiently large (for any γ > 0). Due to homogeneity, there exist positive constants 0 < η W < ηW and η d > 0 such that

η W V δ V +ν δ V (x, ζ ) ≤ W (x, ζ ) ≤ ηW V δ V +ν δ V (x, ζ ) , 2αθW α-1 ζ T -ωx T ρ diag{|ζ | ω-1 } ≤ η d V δ d δ V (x, ζ ) .
Moreover, from inequality (12) it follows that each component µ 2, i, j , for i = 1, • • • , m, j = 1, • • • , ρ i , of the the vector µ 2 is bounded by |µ 2, i, j | ≤ β εδV r i, j +ν , for some δ > 0. Since r i, ρ i = 1ν, this implies for the term μT

2 ζ ω | μT 2 ζ ω | ≤ β ε δ V δ V +ν δ V (x, ζ ) ,
for some δ > 0. Besides, from inequality (11) it also follows that (for some ηd > 0)

2αθW α-1 ζ T -ωx T ρ diag{|ζ | ω-1 } 1 γ d ≤ 1 γ η d D 1 V δ d δ V (x, ζ ) + ηd D 2 V δ d +1+ν δ V (x, ζ ) .
Accordingly, if we define β = β ε δ + ηd D 2 , inequality (20) can be written as

V (x, ζ ) ≤ -η W - 1 γ β V 3 3-ν (x, ζ ) + η d V 2-ν 3-ν (x, ζ ) 1 γ D 1 . (21)
Remark 2. Note that the solution to the scalar differential equation ξ = -κξ 3 3-ν , with ξ ≥ 0 κ > 0 and -1 ≤ ν < 1 max ρ i , is given by

ξ -ν 3-ν (t) = ξ -ν 3-ν (t 0 ) - ν ν -3 κ(t -t 0 ) , if ν = 0, or ξ (t) = exp(-κ(t -t 0 ))ξ (t 0 ) , if ν = 0 .
From these expressions we can calculate that the transit time T i→ f to go from an initial value ξ i to a final one ξ f , for ν = 0, is given by

T i→ f = ν -3 νκ ξ -ν 3-ν i -ξ -ν 3-ν f . If ν < 0 and ξ f = 0, then T i→0 = ν-3 νκ ξ -ν 3-ν i is finite. In contrast, if ν > 0 and ξ 0 → ∞, then T ∞→ f = 3-ν νκ ξ -ν 3-ν f
is finite. Note that this is related to the results of Lemma 1.

From inequality (21) we come to the following conclusions:

1. In the absence of the non-vanishing part of the perturbation d, i.e. D 1 ≡ 0, or equivalently, if µ 1 is an arbitrary constant plus a term vanishing with the state x: the origin (x, ζ ) = 0 is Globally Asymptotically Stable (GAS) for any value of -1 ≤ ν < 1 max ρ i if the perturbation satisfies (12) with β γ sufficiently small, i.e. 0 ≤ β γ ≤ η W . This can be always achieved selecting γ sufficiently large.

Using the comparison principle and the Remark 2, we can obtain from (21) the convergence time from an initial condition (x 0 , ζ 0 ) to the origin for ν < 0 to be

T i→0 = ν -3 ν η W -β γ V -ν 3-ν (x 0 , ζ 0 ) ,
or from an initial condition at infinity to a final condition (x f , ζ f ) when ν > 0 to be

T ∞→ f = 3 -ν ν η W -β γ V -ν 3-ν x f , ζ f .
2. In presence of a non-vanishing time-varying matched perturbation µ 1 , i.e. D 1 > 0, we consider two situations, associated to the relation between the powers of V in ( 21):

(a) ν = -1: In this case the two powers are equal, (21 [START_REF] Levant | Gain-scheduled high-order MIMO sliding mode control[END_REF] and the origin (x, ζ ) = 0 is Globally Finite-Time Stable if the perturbation is sufficiently small (or γ sufficiently large), i.e.

) becomes V ≤ -η W -β γ -η d 1 γ D 1 V 3
1 γ D 1 < η W -β γ η d . (22)
The convergence time from an initial condition is given by

T i→0 = 4 η W -β γ -η d 1 γ D 1 V 1 4 (x 0 , ζ 0 ) . (b) -1 < ν < 1 max ρ i :
In this case, the powers in ( 21) satisify 2-ν 3-ν < 3 3-ν , so that the term due to the perturbation d dominates near the origin (x, ζ ) = 0 and V can be positive in a neighborhood of zero. However, far from the origin the negative term is dominating, and V < 0 at points far from zero, i.e. choosing some 0

< λ < 1, V ≤ -λ η W - β γ V 3 3-ν -(1 -λ ) η W - β γ V 3 3-ν + η d V 2-ν 3-ν 1 γ D 1 = -λ η W - β γ V 3 3-ν -(1 -λ ) η W - β γ V 1+ν 3-ν -η d 1 γ D 1 V 2-ν 3-ν -λ η W - β γ V 3 3-ν , ∀V (x, ζ ) ≥   η d 1 γ D 1 (1 -λ ) η W -β γ   3-ν 1+ν
.

From this latter inequality we conclude that the trajectories of the closed-loop system are ultimately and uniformly bounded, and that they will arrive at the set

V (x, ζ ) ≤ η d 1 γ D 1 (1-λ ) η W -β γ 3-ν 1+ν
in finite-time, and they will remain there for all future times. This is also equivalent to saying that the system is ISS with respect to d.

Note that when D 1 satisfies (22), we can select λ such that

η d 1 γ D 1 (1-λ ) η W -β γ < 1.
Since lim ν→-1 + 3-ν 1+ν = +∞ we conclude that the final bound for the trajectories shrinks to zero as ν → -1, i.e. lim ν→-1

+ η d 1 γ D 1 (1-λ ) η W -β γ 3-ν 1+ν = 0.

SIMULATION RESULTS

In this section we illustrate the behavior of the integral controllers of different homogeneity degrees developed in the paper. For this we perform a simulation study on the following academic example

ẋ =          ẋ1,1 ẋ2,1 ẋ2,2 ẋ3,1 ẋ3,2 ẋ3,3          =          0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0          x +          10 -5 3 0 0 0 2 7 3 0 0 0 0 0 0 2 2 8          (u + µ 1 ) + µ 2 y =   x 1,1 x 2,1 x 3,1   . (23)
This is a MIMO system, having 6 states, 3 control inputs and 3 outputs. It is assumed that is already in the normal form. Although the matrix B is not in the Brunovsky form, it can be easily transformed to it by just multiplying the control input by an invertible matrix M, i.e. v = Mu. The subsystems have relative degrees 1, 2 and 3, respectively. For the simulation we use as non-vanishing matching perturbation µ 1 , given by

µ 1 (t) =   0.5 + 0.05 sin(t) 0.25 + 0.1 cos(t) 0.1t   .
µ 1 is time-varying and the third component is a ramp. The vanishing non matched perturbation µ 2 is state-dependent and is given for the simulation as

µ 2 (x) =          µ 2,1 µ 2,2 µ 2,3 µ 2,4 µ 2,5 µ 2,6          =               0.3 x 1,1 1 r 1,1 + 0.2 x 2,2 1 r 2,2 0.2 x 2,1 r 2,2 r 2,1 + 0.1x 2,2 0.2 x 1,1 1 r 1,1 + 0.2 x 2,1 1 r 2,1 0.5 x 3,1 r 3,2 r 3,1 0.1 x 3,1 r 3,3 r 3,1 + 0.1x 3,3 0.3 x 3,2 1 r 3,2               .
Note that the powers correspond to the weights of homogeneity associated to the variables. This is chosen in this way, because the growth of the vanishing perturbation which can be compensated depends on the homogeneity degree of the integral control designed.

We design 4 integral controllers, given by equation ( 13), with homogeneity degrees ν = {-1, -1 2 , 0, 1 4 }. In all cases we find the Lyapunov matrix P and the state-feedback gain matrix K by solving an LMI problem derived from equations (9). For this YALMIP of MATLAB TM with the SeDuMi solver, is used. A value of ε = 0.5 was used. The following matrices are obtained, for each of the controllers Subindex L stands for the linear integral controller (with homogeneity degree ν = 0), H represents the homogeneous continuous controller (with homogeneity degree ν = -1 2 ), D corresponds to discontinuous case (with homogeneity degree ν = -1), while PH symbolize the integral controller with positive homogeneity degree (ν = 1 4 ). As integration method for the simulation a fourth-order Runge-Kutta method of fixed step is used. The sampling time was 1 × 10 -5 [s]. In all cases an integral gain of γ = 5 is implemented. During the simulation, the actual value of the (implicit) Lyapunov function is obtained numerically on-line using the method presented in [START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF] .

P L =          0.
         , K T L =          -1.
         , K T H =          1
         , K T D =          -1.5880 -
         , K T PH =          1
The simulation results are organized in two groups.

(i) Figures 1 to 3, present the results for the integral controllers L, H and D. They illustrate mainly the behavior in steadystate, since negative homogeneity degrees are particularly good performing near the equilibrium point. In particular, we emphasize the high precision of the discontinuous controller D, despite of time-varying perturbations. For these simulations a small initial condition x 0 = x(0) = 0.2 0.2 0 0.2 0 0 T is selected.

Figure 1 contains the time behavior of the states for the linear L (Figure 1a), the homogeneous H (Figure 1b) and the discontinuous D (Figure 1c) integral controllers, respectively. As expected, the steady-state behavior of the discontinuous D controller is much better, i.e. the error is smaller, than that of the homogeneous H and of the linear L ones. Moreover, the smaller the homogeneity degree, the smaller also the final error. Since the initial condition is small, it is also noticeable that the D controller converges faster than the other ones.

Figure 2 shows the three control signals generated by the L, H and D integral controllers. As is characteristic of the integral action, all are continuous. Note, moreover, that in steady-state they all converge to the inverse of the perturbation, since they aim to compensate for it. As shown in the previous figures, the lower the homogeneity degree, the better is the compensation and the nearer the control signal is to the perturbation.

It has been shown in the main Theorem, that the signal z + µ 1 converges to zero. This is a characteristic of the integral action, being able to estimate the non-vanishing perturbation in order to counteract its influence in the system. The time evolution of z+ µ 1 is depicted in Figure 3 for the three integral controllers L, H and D. Again, the discontinuous controller D is able to force this signal to zero in finite-time, showing that it can estimate exactly the perturbation, while the H and the L controllers are only capable to perform an approximate estimation. Once again, the smaller the homogeneity degree the smaller is also the estimation error.

(ii) Fig. 4 shows the behavior of the PH integral controller, compared to the linear (L) and the discontinuous (D) ones. Since a remarkable characteristic of controllers with positive homogeneity degree is its high velocity of convergence for large initial conditions, a much larger initial state x(0) = 3, 000 × [1, 1, 1, 1, 1, 1] was selected. For these simulations we also eliminated the perturbations, i.e. µ 1 = 0, µ 2 = 0, since they are more relevant for the steady-state behavior.

Figure 4 presents the Euclidean norm x (t) 2 of the states for the integral controller with positive homogeneity degree PH. For comparison, the corresponding norms for the linear L and the discontinuous D integral controllers are also shown. It is apparent that the convergence velocity to a neighborhood of the equilibrium for the PH controller is much higher than that for the linear and the discontinuous controllers.

CONCLUSIONS

We have shown that it is possible to combine an Implicit Lyapunov function, obtained using the ILF method, with some extra terms to design integral controllers of arbitrary positive or negative degree. This includes the discontinuous integral action, which stands out because it is able to fully compensate non-vanishing Lipschitz matched perturbations. Continuous integral controllers are in contrast able to counteract just constant perturbations. They reach the equilibrium point in finite time when the homogeneity degree is negative, exponentially for homogeneity degree zero or nearly fixed-time, when the homogeneity degree is positive. This latter type of convergence is very weak near the equilibrium, but it is very strong for initial conditions very far from it. They reach a neighborhood of the equilibrium in a time independent of the initial condition. A natural task to do is to combine controllers of different degrees of homogeneity. For integral controllers this combination is not so straightforwardly as for memoryless state feedback. The class of systems considered includes MIMO nonlinear systems which are partially feedback linearizable and minimum phase. They are transformed to the Byrnes-Isidori normal form and (partially) linearized in order to take full advantage of the (c) Homogeneity degree ν = -1. weighted homogeneity property, which is coordinate dependent. In the original coordinates, although the homogeneity (in strict sense) is lost, the convergence and robustness properties remain. Compared to other methods to design integral controllers the ILF-based method proposed here is attractive because the gain design can be converted to an LMI problem. This is in contrast to other methods, where the finding of appropriate gains is a highly nonlinear problem.

2 γ

 2 (i) If -1 < ν < 1 max ρ i and D 1 > 0 the closed-loop system is Input-to-State Stable (ISS) from the input μ1 to the state (x, ζ ). (ii) Let b be the ultimate bound. If D 1 +D

Lemma 4 .

 4 W (x, ζ ) > 0 for θ > 0 large enough.

  (a) Homogeneity degree ν = 0. (b) Homogeneity degree ν = -1 2 .

FIGURE 1

 1 FIGURE 1 Time evolution of all states for Integral controllers L (1a), H (1b) and D (1c)

( a )

 a Input signal u 1 . (b) Input signal u 2 .(c) Input signal u 3 .

FIGURE 2

 2 FIGURE 2 Time evolution of the three control signals u 1 (2a), u 2 (2b) and u 3 (2c), generated by the Integral controllers L, H and D.

As noted in Remark 1, the ILF V is not assured to be differentiable at x = 0. For example, for ν = 0 the ILF obtained from[START_REF] Polyakov | Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Method[END_REF] is V (x) = √x T Px, which is clearly not differentiable at x = 0.

ACKNOWLEDGEMENTS

The authors would like to thank the financial support from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica), project IN110719; CONACyT (Consejo Nacional de Ciencia y Tecnología) CVU 705765, by the Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.