Anette Green

Nielsen & Rasmus

Steen Pedersen
email: rpedersen@health.sdu.dk

Per Damkier Lene Noehr-Jensen

Kim Brosen

A G Nielsen

R S Pedersen

P Damkier

Two separate dose-dependent effects of paroxetine: mydriasis and inhibition of tramadol's O-demethylation via CYP2D6

Keywords: Paroxetine, Tramadol, Interaction, CYP2D6, Pupillometry, Metabolic ratios

Purpose To investigate paroxetine's putative dose-dependent impact on pupil reaction and inhibition of the O-demethylation of tramadol. Methods Twelve healthy CYP2D6 extensive metabolizers participated in this double-blinded randomized five-way placebo controlled cross-over study; they received placebo, 10, 20, 30, and 50 mg paroxetine as single oral doses at bedtime. Next morning the pupil was measured followed by oral intake of 50 mg of tramadol, and urine was collected for 8 h. Three hours after ingestion of tramadol a second measurement of the pupil was performed. Enantioselective urine concentrations of (+/-)-tramadol and (+/-)-O-desmethyltramadol (M1) were determined. Results With placebo, the median maximum pupil diameter was 6.43 mm (range 5.45-7.75 mm) before tramadol and 6.22 mm (4.35-7.65 mm) after 50 mg of tramadol (P=0.4935). Paroxetine resulted in a statistically significant, dosedependent dilatation of the pupil with a geometric mean difference of 1.17 (95% CI 1.10-1.24) after ingestion of 50 mg paroxetine (P<0.001). Likewise, a reduction in the relative constriction amplitude with a geometric mean difference of 0.81 (95% CI 0.71-0.92) (P<0.001) was seen. A dose-dependent inhibition of the metabolism of tramadol by an increase in the two urinary metabolic ratios (+)-tramadol / (+)-M1 [geometric mean difference 9.09, 95% CI 5.60-14.73 (P<0.001)] and (-)-M1 / (+)-M1 [geometric mean difference 2.84, 95% CI 2.15-3.77 (P< 0.001)] was also observed. Conclusions Paroxetine is a dose-dependent dilator of the pupil and as expected a dose-dependent inhibitor of (+)tramadol's O-demethylation.

Introduction

Chronic pain and depression show considerable co-morbidity [START_REF] Bair | Depression and pain comorbidity[END_REF], and each of the two conditions often requires treatment with drugs. Thus drug-drug interactions between analgesics and antidepressants are an important medical issue. Tramadol is a synthetic opioid used for mild to moderate pain [START_REF] Trescot | Opioid pharmacology[END_REF]. Tramadol is a racemate consisting of (+)-tramadol hydrochloride and (-)-tramadol hydrochloride. These compounds are metabolized to (+) and (-)-O-desmethyltramadol [(+)-M1 and (-)-M1] and both reactions, but in particular the former, are catalyzed by CYP2D6 [START_REF] Paar | The metabolism of tramadol by human liver microsomes[END_REF][START_REF] Paar | Polymorphic CYP2D6 mediates O-demethylation of the opioid analgesic tramadol[END_REF][START_REF] Poulsen | The hypoalgesic effect of tramadol in relation to CYP2D6[END_REF][START_REF] Laugesen | Paroxetine, a cytochrome P450 2D6 inhibitor, diminishes the stereoselective O-demethylation and reduces the hypoalgesic effect of tramadol[END_REF][START_REF] Pedersen | Enantioselective pharmacokinetics of tramadol in CYP2D6 extensive and poor metabolizers[END_REF][START_REF] Halling | CYP2D6 polymorphism in relation to tramadol metabolism: a study of Faroese patients[END_REF]. (+)-M1 is the active µ-opioid receptor agonist, while (+)-tramadol is an inhibitor of the 5-HT re-uptake and (-)-tramadol inhibits norepinephrine re-uptake [START_REF] Raffa | Opioid and nonopioid components independently contribute to the mechanism of action of tramadol, an atypical opioid analgesic[END_REF][START_REF] Lai | Tramadol, M1 metabolite and enantiomer affinities for cloned human opioid receptors expressed in HN9.10 neuroblastoma cells[END_REF][START_REF] Bamingbade | Actions of tramadol, its enantiomers and principal metabolite, O-desmethyltramadol, on serotonin (5-HT) efflux and uptake in the rat dorsal raphe nucleus[END_REF][START_REF] Halfpenny | Effects of tramadol stereoisomers on norepinephrine efflux and uptake in the rat locus coeruleus measured by real time voltammetry[END_REF].

CYP2D6 is involved in the metabolism of about 15% of the medical substances that are metabolized by the CYP enzymes [START_REF] Zanger | Functional pharmacogenetics/genomics of human cytochromes P450 involved in drug biotransformation[END_REF]. CYP2D6 polymorphism divides the population into two phenotypes: individuals without catalytic CYP2D6 activity, i.e., poor metabolizers (PMs), and individuals with present but variable catalytic CYP2D6 activity, i.e., extensive metabolizers (EMs). About 7-8% of the Caucasian population has the phenotype PM [START_REF] Ingelman-Sundberg | Influence of cytochrome P450 polymorfisms on drug therapies: pharmacogenetic, pharmacoepigenetic and clinical aspects[END_REF][START_REF] Ingelman-Sundberg | Genetic polymorphisms of cytochrome P450 2D6 (CYP2D6): clinical consequences, evolutionary aspects and functional diversity[END_REF]. The impaired O-demethylation of (+)-tramadol in the PM results in less (+)-M1 being formed and an impaired analgesic effect in this phenotype compared with the EM [START_REF] Poulsen | The hypoalgesic effect of tramadol in relation to CYP2D6[END_REF]. Paroxetine is a selective serotonin reuptake inhibitor (SSRI) that is metabolized by CYP2D6 [START_REF] Sindrup | Pharmacokinetics of the selective serotonin reuptake inhibitor paroxetine: nonlinearity and relation to the sparteine oxidation polymorphism[END_REF][START_REF] Sindrup | The relationship between paroxetine and the sparteine oxidation polymorphism[END_REF] and also an inhibitor of the enzyme [START_REF] Sindrup | Pharmacokinetics of the selective serotonin reuptake inhibitor paroxetine: nonlinearity and relation to the sparteine oxidation polymorphism[END_REF][START_REF] Sindrup | The relationship between paroxetine and the sparteine oxidation polymorphism[END_REF][START_REF] Brosen | Extremely slow metabolism of amitriptyline but normal metabolism of imipramine dnd Desipramine in an extensive metabolizer of sparteine, debrisoquine, and mephenytoin[END_REF][START_REF] Brosen | Inhibition by paroxetine of desipramine metabolism in extensive but not in poor metabolizers of sparteine[END_REF][START_REF] Skjelbo | Inhibitors of imipramine metabolism by human liver microsomes[END_REF][START_REF] Crewe | The effect of selective serotonin re-uptake inhibitors on cytochrome P4502D6 (CYP2D6) activity in human liver microsomes[END_REF]. Paroxetine seems to be a mechanism-based inhibitor of CYP2D6 [START_REF] Bertelsen | Apparent mechanism-based inhibition of human CYP2D6 in vitro by paroxetine: comparison with fluoxetine and quinidine[END_REF]. Paroxetine shows nonlinear saturation of CYP2D6 at steady-state plasma concentrations. At steady state, daily doses of 20-30 mg paroxetine result in a reduction of CYP2D6 activity by about 93% [START_REF] Venkatakrishnan | In vitro-in vivo extrapolation of CYP2D6 inactivation by paroxetine: prediction of nonstationary pharmacokinetics and drug interaction magnitude[END_REF].

Tramadol, like other opioids, causes a dose-dependent miosis through the active metabolite (+)-M1, which is caused by an excitatory effect on the parasympathetic nervous system [START_REF] Fliegert | The effects of tramadol on static and dynamic pupillometry in healthy subjects-the relationship between pharmacodynamics, pharmacokinetics and CYP2D6 metaboliser status[END_REF]. Through the inhibition by paroxetine of the formation of (+)-M1, one would expect that the pupil contraction caused by tramadol would diminish when administered together with paroxetine and that both effects would be dose-dependent.

Methods

Subjects

Twelve healthy volunteers, aged 22-32 years, participated in this study, including nine males and three females. All subjects were CYP2D6 EMs as assessed by previous phenotype studies [START_REF] Zwisler | The hypoalgesic effect of oxycodone in human experimental pain models in relation to the CYP2D6 oxidation polymorphism[END_REF][START_REF] Noehr-Jensen | Escitalopram is a weak inhibitor of the CYP2D6-catalyzed O-demethylation of (+)-tramadol but does not reduce the hypoalgesic effect in experimental pain[END_REF]. Female subjects conducted a negative pregnancy test prior to the beginning of the study and used safe contraception thoughout the trial. Routine laboratory tests (blood chemistry and hematology) were within normal values. Physical examination and medical history did not reveal signs of disease, drug abuse or alcohol dependence. Concomitant medication was not allowed.

Study design

The trial was registered in the European Clinical Trials Database (EudraCT no. 2008-001516-20) and in the US National Institutes of Health registry at http://www. clinicaltrials.gov (NCT 00785603). The protocol was approved by the Danish Medicines Agency (J. no. 2612-3730), the Local Ethics Committee for the Region of Southern Denmark (J.no. S-20080043) and the Danish Data Protection Agency (J.no. 2008-41-2277). The trial was carried out in accordance with Good Clinical Practice (GCP) guidelines and monitored by the GCP Unit, Odense University Hospital.

Sample size calculations were based on a primary outcome represented by interindividual differences in pupil diameter of subjects receiving 10 and 50 mg paroxetine respectively. It was estimated that a difference of one standard deviation could be detected given a two-sided level of significance (α) of 0.05 and a power of 80% using ten subjects. To allow for a possible drop out rate of 15%, 12 subjects were recruited.

The study was conducted as a double-blinded, randomized, five-way, placebo controlled, cross-over study. Subjects went through five phases receiving a single dose of 10, 20, 30, 50 mg paroxetine (Paroxetin, Copyfarm, Denmark) or placebo, respectively, before bedtime. The following day a measurement of the pupil size was performed followed by administration of 50 mg tramadol (Nobligan®, Grünenthal, Germany). Three hours later, the second measurement of the pupil was performed. The subjects collected urine for 8 h after ingestion of tramadol. The five phases were separated by a wash out period of minimum 2 weeks. Packing, blinding, and randomization of the drugs was conducted by Odense University Hospital Pharmacy.

Pupil measurement

The measurement of the pupil was performed with a handheld pupillometer PLR-100™, NeurOptics Inc. The light intensity was held constant at 25 cd/m 2 for all the pupil measurements, and the same eye of each subject was measured. The subjects adjusted to the light for 2 min and then focused on a spot on the wall 3 m away. After a focus period of 3 s, a measure period of 3.2 s followed during which a flash of light was emitted and the pupillary reaction was measured. The results of a measurement consisted of the maximum and the minimum pupil diameter. From these the more robust measure of relative constriction amplitude was calculated as (max. diametermin. diameter) / max. diameter.

Sample analysis

In all phases, total urine volumes were registered and aliquots were stored at -20°C until analysis. Concentrations of (+)tramadol and (-)-tramadol and the metabolites (+)-M1 and (-)-M1 in the urine were determined by enantioselective high performance liquid chromatography (HPLC) as described in detail in our previous method article [START_REF] Pedersen | Enantioselective HPLC method for quantitative determination of tramadol and Odesmethyltramadol in plasma and urine: application to clinical studies[END_REF]. We have previously shown that tramadol is a validated probe for CYP2D6 phenotyping [START_REF] Pedersen | Tramadol as a new probe for cytochrome P450 2D6 phenotyping: a population study[END_REF]. Two urinary metabolic ratios can be determined from urine collected over 8 h after ingestion of 50 mg tramadol. The first metabolic ratio (MR 1 ) is calculated as the amount of excreted (+)-tramadol divided by the amount of excreted metabolite (+)-M1. The second metabolic ratio (MR 2 ) is calculated as the amount of excreted (-)-M1 divided by the amount of excreted (+)-M1. MR 2 is the most valid measurement for determining the phenotype of CYP2D6 by using tramadol. EMs have an MR 2 <2.0, and PMs have an MR 2 ≥2.0 [START_REF] Pedersen | Tramadol as a new probe for cytochrome P450 2D6 phenotyping: a population study[END_REF].

Statistical analysis

Data are presented as medians with 25-75% percentile and range. Statistic inference was analyzed by means of paired t-test and repeated measures one-way ANOVA. ANOVA was followed by a Tukey's multiple comparison test. The data were transformed by the natural logarithm prior to statistical analysis in order to justify Gaussian distribution.

Results

Eleven subjects completed all five phases and one subject dropped out after four phases thereby missing the phase requiring ingestion of 30 mg paroxetine. The most frequently reported adverse events were fatigue (n=3), sleep disturbances (n=5), nausea (n=6), and anxiety (n=4).

The results from the subject that did not complete the last phase were not included in the repeated measures ANOVA and the Tukey's multiple comparison test but only presented in the figures illustrating the descriptive data.

The effect of tramadol on the pupil

The effect of tramadol on the pupil is seen by analyzing the difference between the results of the first and the second pupil measurement when 0 mg paroxetine (=placebo) was administered. The first measured median maximum pupil diameter was 6.43 mm (range 5.45-7.75 mm) and the second measured median maximum pupil diameter was 6.22 mm (range 4.35-7.65 mm) 3 h after ingesting 50 mg of tramadol. The paired t-test carried out on the logarithmic values showed no statistically significant difference between the two pupil measurements (P=0.4935). The first measured median relative constriction amplitude was 0.395 mm (range 0.252-0.464 mm) and the second measured median relative constriction amplitude was 0.375 mm (range 0.318-0.416 mm). No statistical significance was found (P=0.0954)

There was no statistically significant difference between the two measurements at any of the other phases with administration of concomitant paroxetine (data not shown). There was no correlation between the amounts of excreted (+)-M1 and pupil measurements either (data not shown). Overall, this shows that 50 mg tramadol has no effect on the pupil, and it was therefore not possible to observe a reduction in the miotic effect of tramadol with contemporary administration of different doses of paroxetine.

The effect of paroxetine on the pupil

The result that shows the exclusive effect of paroxetine on the pupil is the first measurement of the pupil in each phase as the subjects had not yet ingested tramadol at this point. The median maximum pupil diameter with 25-75% percentile and range is shown for each phase in Fig. 1. The clear increase in the maximum pupil diameter with increasing doses of paroxetine was confirmed by repeated measures ANOVA (P<0.0001) and a subsequent Tukey's test as presented in Table 1. The ingestion of 50 mg paroxetine resulted in a change in geometric mean difference of 1.17 (95% CI 1.10-1.24) compared to the placebo phase.

Likewise, the relative constriction amplitude decreased significantly with increasing dose of paroxetine (P<0.0001) as shown in Fig. 2. The geometric mean difference was 0.81 (95% CI 0.71-0.92) in the phase with 50 mg paroxetine compared to the placebo phase.

The effect of paroxetine on metabolic ratios

The median metabolic ratio MR 1 with 25-75% percentile and range is shown for each phase in Fig. 3. A clear stepwise increase in median is seen with increasing dose of paroxetine. The repeated measures ANOVA was statistically significant (P<0.0001), and the subsequent Tukey's test including geometric mean difference with 95% CI carried out on the values of MR 1 is shown in Table 1. The MR 1 geometric mean difference between the placebo phase and the highest dose of 50 mg paroxetine was 9.09 with 95% CI of 5.60-14.73. In fact, all dosing steps resulted in statistically significant changes in MR 1 (P<0.05) apart from the step from 20 to 30 mg paroxetine.

As shown in Fig. 4 and Table 1, the MR 2 also increased with increasing dose of paroxetine. The repeated measures ANOVA shows likewise a significant increase in MR 2 (P<0.0001). In this case a geometric mean difference of 2.84 with 95% CI of 2.15-3.77 resulted in 10 out of 12 subjects changing phenotype from EM to PM by ingestion of 50 mg paroxetine according to the validated CYP2D6 phenotype marker MR 2 .

To summarize, the dosing steps from placebo and from 10 mg paroxetine to 30 and to 50 mg paroxetine resulted in statistically significant alterations in both the dynamic and kinetic parameters: maximum pupil size, relative restriction amplitude, MR 1 , and MR 2 .

Discussion

The aim of the study was to investigate the effect of an inhibition of CYP2D6 caused by paroxetine on the pharmacodynamics and pharmacokinetics of tramadol.

In this study we demonstrate that the maximum pupil diameter is increased and the relative constriction amplitude is decreased with increasing dose of paroxetine. Both are caused by an anticholinergic effect [START_REF] Johnson | An overview of the animal pharmacology of paroxetine[END_REF]. To our knowledge we are the first to report this clear dose-dependent mydriasis induced by a single dose of paroxetine. Dilatation of the pupil is stated to be an uncommon adverse event for paroxetine [START_REF]Product resume for Seroxat®[END_REF] and has been reported in the clinic [START_REF] Larson | Selective serotonin reuptake inhibitorinduced mydriasis[END_REF]. It has also previously been reported that a single dose of 20 mg paroxetine did not cause any significant effect on the pupil diameter [START_REF] Bitsios | Comparison of the effects of venlafaxine, paroxetine and desipramine on the pupillary light reflex in man[END_REF]. In the study by Bitsios, the pupil diameter was measured 3 h after ingestion of paroxetine [START_REF] Bitsios | Comparison of the effects of venlafaxine, paroxetine and desipramine on the pupillary light reflex in man[END_REF]. As paroxetine achieves a maximum plasma concentration 6 h after oral administration [START_REF]Product resume for Seroxat®[END_REF], a pupil measurement after 3 h is unlikely to reveal the full effect of paroxetine on the pupil diameter. In the current study the first pupil diameter measurement was performed approximately 9 h after administration of paroxetine, where an effect is biologically more plausible. Paroxetine plasma concentrations would perhaps have elucidated this issue. Furthermore, the correlation between paroxetine plasma concentrations and the endpoints could have been investigated. Unfortunately this was not included in the design of the study and therefore not feasible. On the other hand, the volunteers were a homogeneous group of healthy CYP2D6 EMs who probably display very little interindividual paroxetine PK. This assumption was fortunately supported by the clear correlation between paroxetine dose and both the dynamic and the kinetic end points.

We have recently shown that both a single and multiple doses of 10 mg of another SSRI, escitalopram, also caused a decrease in the relative constriction amplitude whereas no effect on the maximum pupil diameter was seen [START_REF] Noehr-Jensen | Impact of CYP2C19 phenotypes on escitalopram metabolism and an evaluation of pupillometry as a serotonegic biomarker[END_REF]. However a subsequent study showed that daily intake of 20 mg of escitalopram increased the maximum pupil diameter and decreased the relative constriction amplitude [START_REF] Noehr-Jensen | Escitalopram is a weak inhibitor of the CYP2D6-catalyzed O-demethylation of (+)-tramadol but does not reduce the hypoalgesic effect in experimental pain[END_REF]. This mydriatic effect is therefore probably a class effect of the SSRIs having affinity for muscarinic cholinergic receptors [START_REF] Larson | Selective serotonin reuptake inhibitorinduced mydriasis[END_REF].

This study did not show any effect of 50 mg single oral dose tramadol on the pupil diameter and subsequently a reduction in the pupil contraction caused by tramadol during concomitant administration of paroxetine could not be demonstrated. The most likely explanation is that a dose of 50 mg tramadol is too low to cause contraction of the pupil. It has previously been shown that the lowest dose of tramadol to cause miosis is 100 mg [START_REF] Fliegert | The effects of tramadol on static and dynamic pupillometry in healthy subjects-the relationship between pharmacodynamics, pharmacokinetics and CYP2D6 metaboliser status[END_REF]. Due to the demonstrated opposite effect of paroxetine on the pupil, it is unclear whether a dose of 100 mg tramadol in the present study would have resulted in a measurable pharmacodynamic change related to the pharmacokinetic interaction.

This study also clearly demonstrates that paroxetine is, as expected [START_REF] Jeppesen | Dose-dependent inhibition of CYP1A2, CYP2C19 and CYP2D6 by citalopram, fluoxetine, fluvoxamine and paroxetine[END_REF], a dose-dependent inhibitor of tramadol's O-demethylation via CYP2D6. This result also serves as yet another validation criterion of this CYP2D6 phenotype test. Any cytochrome P450 phenotyping metric should display sensitivity toward drug-induced inhibition [START_REF] Fuur | Appropriate phenotyping procedures for drug metabolizing enzymes and transporters in humans and simultaneous use in the "cocktail approach[END_REF], which is exactly what this study shows.

We have previously shown that daily intake of 20 mg paroxetine inhibited the formation of (+)-M1 and reduced the analgesic effect of tramadol [START_REF] Laugesen | Paroxetine, a cytochrome P450 2D6 inhibitor, diminishes the stereoselective O-demethylation and reduces the hypoalgesic effect of tramadol[END_REF]. By extrapolation we hypothesize that the reduction in tramadol's hypoalgesic effect caused by paroxetine is also dose-dependent, and the combination of the two drugs should be avoided.

Fig. 1 Fig. 2

 12 Fig. 1 Median maximum pupil diameter with 25-75% percentile and range for each dose of paroxetine measured at baseline before administration of tramadol

Fig. 4

 4 Fig. 4 Median MR 2 =amount of excreted (-)-M1 / amount of excreted (+)-M1 with 25-75% percentile and range for each dose of paroxetine

Table 1

 1 Geometric mean ratios (95% CI) between each dose of paroxetine Tukey's multiple comparison test Maximum pupil diameter Relative constriction amplitude MR 1

	a	MR 2 b

Acknowledgments We thank the Danish Research Council for Health and Disease; grant number 271-05-0266. We also thank Birgitte Damby Sorensen for analytical expertise.

Competing interests

The head of the research group, K.B., is a member of the Advisory board of the Lundbeck Institute and has previously received funds for members of staff from H. Lundbeck A/S.


The PSOR-like preconditioner for CGS method List of authors: Toshihiro Nitta1 Toshiyuki Kohno 2Hiroshi Niki 3Let us consider a preconditioning for the iterative method for solving a nonsymmetric sparse linear system with the approximate inverse preconditioner P ,

P Ax = P b,
where A is a given n × n real matrix, b is a given real vector, and x is the solution vector which is to be determined. In 1991, Gunawardena et al. [START_REF] Gunawardena | Modified Iterative Methods for Consistent Linear Systems[END_REF]proposed the modified Gauss-Seidel method in which P = I + S, with

S = (s ij ) = -a ii+1 fori = 1, 2, • • • , n -1, j = i + 1 0 otherwise
In 2002, Kotakemori et al. [START_REF] Kotakemori | Accelerated iterative method for Z-matrices[END_REF] proposed to use P m = I + S m , where S m is defined by,

S m = (s m ij ) = -a ik i fori = 1, 2, • • • , n -1, j ≥ i + 1 0 otherwise k i = min j {j| max j>i+1 |a ij }, for i = 1, 2, • • • , n -1.
In addition, the relating preconditioners are discussed [START_REF] Kohno | A note on the preconditioner P m = (I + S m )[END_REF][START_REF] Kohno | Improving Modified Gauss-Seidel Method for Z-matrices[END_REF]. As you known, the combination of preconditioning and iterative method is effectiveness.

The Krylov subspace methods include popular methods such as Conjugate Gradients(CG), Bi-Conjugate Gradients(Bi-CG), Bi-CGstab, CGS and GMRES, etc. And there are the classical iterative method such as Jacobi, Gauss-Seidel and SOR. The preconditioner K approximates the coefficient matrix A under the assumption that K -1 v is solved more easily and faster than A -1 v, where v is a vector which use in Krylov subspace algorithm. The Incomplete LU decomposition (ILU) and Incomplete Cholesky decomposition (IC) are more widely and frequently used method for designing the preconditioner. Recently, the methods, which compute the linear system Kz = v by a classical iterative method at each iteration of the Krylov subspace method, have proposed by many researchers. These methods are so-called Hybrid algorithm.

In this paper, we propose the Hybrid algorithm which use the Preconditioned SOR-like method and a Krylov subspace methods.

Doctoral program in mathematical and environmental system science, Okayama University of Science, Japan

Faculty of Informatics, Okayama University of Science, Japan

Okayama University of Science, Japan


P Van Slingerland
email: p.vanslingerland@tudelft.nl

C Vuik
email: c.vuik@tudelft.nl

Exploiting the Nested Block Structure of DG Matrices: a Block ILU Preconditioner with Deflation and a Spectral Two-Level Strategy List of authors

The main advantages of Discontinuous Galerkin (DG) discretizations for elliptic problems are their flexibility in handling non-matching grids and in designing hp-refinement strategies. However, an important drawback is that the resulting linear systems are usually large (due to the large number of degrees of freedom) and ill-conditioned. As a consequence, standard Krylov methods typically require too much computational time to solve these systems.

To increase the practical applicability of DG discretizations, we propose a preconditioning technique that is based on a nested block incomplete LU factorization. Earlier, this type of preconditioner was applied successfully in the context of finite difference discretizations for oil reservoir simulation [START_REF] Appleyard | Nested factorisation[END_REF][START_REF] Meijerink | Iterative methods for the solution of linear equations based on incomplete block factorization of the matrix[END_REF]. Here, we combine their ideas with deflation [START_REF] Vuik | An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients[END_REF] and a spectral twolevel strategy to obtain a preconditioner that is suitable for our application, and that efficiently exploits the nested block structure of the coefficient matrix. Various symmetric test cases demonstrate that this technique can lead to uniform convergence of the conjugate gradient method.


Jeferson V Flores
email: jeferson.flores@ufrgs.br

Joâo Manoel Vieira

Gomes
email: jmgomes@ece.ufrgs.br.

Alexandre Silva Gomes Da Silva

J M Gomes Da Silva

A Seuret

Static anti-windup synthesis for linear systems with time-varying input delays

INTRODUCTION

Time-delay is a common phenomena in many physical systems such as aircraft stabilization, chemical reactors, communication networks and others. Due to the fact that the presence of delays in control systems can lead to unsatisfactory time-domain performances or even to instability, in the last years great attention has been paid to stability analysis and control of these systems (see [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], and references). Stability conditions for time-delay systems can be classified into delay-dependent and delay-independent conditions. Concerning the delay independent results, the stability is ensured no matter the size of the delay [START_REF] Shaked | Bounded real criteria for linear time-delay systems[END_REF]). On the other hand, in the delay dependent results, the size of the delay is directly taken into account and this fact can lead, especially when the time-delays are small, to less conservative results [START_REF] Fridman | Delay-dependent stability and H ∞ control: constant and time-varying delays[END_REF]).

Most of the works dealing with time-delays consider invariant delays. However, there is an increasing number of practical systems presenting uncertain and/or timevarying delays in the state and/or input variable. This is a common problem in communications systems where the presence of transmission delay variations is usually unavoidable [START_REF] Park | Anti-windup compensator for active queue management in tcp networks[END_REF]). Many approaches have been developed regarding the stability problem for systems with time-varying input delays: [START_REF] Niculescu | Delaydependent closed-loop stability of linear systems with input delay: an lmi approach[END_REF], [START_REF] Zhang | Delaydependent stabilization of linear systems with timevarying state and input delays[END_REF], [START_REF] Mahmoud | Improved robust stability and feedback stabilization criteria for time-delay systems[END_REF] and references therein. In [START_REF] Niculescu | Delaydependent closed-loop stability of linear systems with input delay: an lmi approach[END_REF], the delay was assumed to be a continuous bounded time-varying function and the analysis was presented considering a Razumikhin based framework and a frequency-filtering method. In [START_REF] Zhang | Delaydependent stabilization of linear systems with timevarying state and input delays[END_REF] a state/input-delay-dependent stabilization ⋆ The first two authors are supported in part by CNPq, Brazil. The third author is partially supported by European Comission through the FeedNetBack project.

condition was obtained considering a new integral inequality for quadratic terms and a Lyapunov-Krasovskii approach. Following the same ideas, in [START_REF] Mahmoud | Improved robust stability and feedback stabilization criteria for time-delay systems[END_REF] a L 2 -performance analysis is used to derive the solution for both nominal and polytopic uncertain models.

On the other hand, saturation is an ubiquitous problem in control systems, arising from the fact that actuators cannot delivery signals with unlimited amplitude or energy to the controlled plants. Neglecting actuator limitations can therefore be source of undesirable or even catastrophic behaviors for the closed-loop system (as the lost of the closed-loop stability). One of the most usual ways to mitigate the undesirable effects of the saturation is the anti-windup technique. This technique assumes that a controller was previously designed in order to guarantee some performance requirements and then this controller is modified in order to recover, as much as possible, the performance when saturation occurs (see, for example, the survey [START_REF] Tarbouriech | Anti-windup design: An overview of some recent advances and open problems[END_REF]).

In the anti-windup context for time-delay systems, we can cite [START_REF] Park | Dynamic anti-windup method for a class of time-delay control systems with input saturation[END_REF], [START_REF] Tarbouriech | Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems with saturating inputs[END_REF], [START_REF] Tarbouriech | Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs[END_REF], [START_REF] Zaccarian | L 2 antiwindup for linear dead-time systems[END_REF] and Gomes da Silva Jr. et al. [2006]. In [START_REF] Park | Dynamic anti-windup method for a class of time-delay control systems with input saturation[END_REF], a dynamic anti-windup strategy is considered for systems with input and output delays. In [START_REF] Tarbouriech | Delay-dependent anti-windup loops for enlarging the stability region of time-delay systems with saturating inputs[END_REF], the design of a static anti-windup compensator was studied with the aim of enlarging the region of attraction of the closed-loop system. In that work, a method for computing anti-windup gains for systems presenting only input delays is proposed leading to bilinear matrix inequalities (BMIs) conditions. In contrast to [START_REF] Park | Dynamic anti-windup method for a class of time-delay control systems with input saturation[END_REF] and [START_REF] Zaccarian | L 2 antiwindup for linear dead-time systems[END_REF], the proposed technique can be applied to both stable and unstable open-loop systems. In [START_REF] Tarbouriech | Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs[END_REF] and Gomes da Silva [START_REF] Gomes Da | Anti-windup design for time-delay systems subject to input saturation. an lmi-based approach[END_REF], based on a Lyapunov-Krasosvskii functional and on the application of a modified sector condition, global and local delaydependent stabilization conditions are derived. However, it should be pointed out that these works focus, respectively, on systems presenting output or state delays. Furthermore, in the above works the delays are supposed to be fixed.

In this work, the problem of anti-windup design for linear systems with time-varying input delay and actuator saturation is addressed. An anti-windup strategy considering two static compensators is considered: one responsible to correct the actual difference between the computed control signal and the control delivered to the plant and another responsible to correct the delayed difference between then, if the delay value is known at each time. Following the descriptor like approach considered in Gomes da Silva [START_REF] Gomes Da | Stabilisation of neutral systems with saturating control inputs[END_REF] and the modified sector condition from [START_REF] Tarbouriech | Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs[END_REF] to take into account the saturation effects, global and local stabilization conditions directly in the LMI form are derived to ensure both L 2 input to state stability and the internal stability of the closed-loop system. Provided that the maximal value of the time-varying delay and an upper bound for its derivative are known, convex optimization problems to compute the anti-windup gains are proposed. Simulation examples are presented to illustrate the the proposed solution.

Notation: The ith component of a vector x is denoted by x (i) . A (i) denotes the ith row of a matrix A ∈ R n×n and A T denotes its transpose. diag{x} denotes a diagonal matrix obtained from vector x, I m denotes the m-order identity matrix and 0 m×n denotes the m × n null matrix. λ(P ) and λ(P ) denote the maximal and the minimal eigenvalues of P, respectively. C h = C([-h, 0], R n ) is the Banach space of continuous vector functions mapping the interval [-h, 0] into R n with the norm φ c = sup -h≤t≤0 φ(t) . C ν h is the set defined by C ν h = {φ ∈ C h ; φ c ≤ ν, ν > 0}. * denotes symmetric block elements in a matrix.

PROBLEM STATEMENT

Consider the following linear time-delay system:

ẋ(t) = Ax(t) + Bu(t -τ (t)) + B w w(t) y(t) = C y x(t) z(t) = C z x(t) + D z u(t -τ (t)) (1) with initial conditions x(t 0 + θ) = φ(θ), ∀θ ∈ [-h, 0], t 0 ∈ R + , φ(θ) ∈ C ν h . In this case, x(t) ∈ R n is the state vector, u(t) ∈ R m is the control input, w(t) ∈ R l is the external disturbance vector, y(t) ∈ R p is the measured output, z(t) ∈ R p is the regulated output and τ (t) corresponds to a time-varying delay that satisfies 0 ≤ τ (t) ≤ h, τ (t) ≤ d < 1.
Matrices A, B, B w , C y , C z and D z are constant real matrices of appropriate dimensions, pairs (A, B) and (C y , A) are assumed to be stabilizable and detectable, respectively, and we suppose that the control input u is subject to amplitude limitations such that |u

(i) (t)| ≤ u 0(i) , u 0(i) > 0, i = 1, • • • , m.
The disturbance vector w(t) is assumed to be limited in energy, i.e. w(t) ∈ L 2 , and for some scalar δ, 0 ≤ 1 δ < ∞, the disturbance w(t) is bounded as follows:

w(t) 2 2 = ∞ 0 w T (t)w(t)dt ≤ 1 δ (2)
Considering system (1), we assume therefore that an n corder dynamic output stabilizing compensator ẋc

(t) = A c x c (t) + B c u c (t) y c (t) = C c x c (t) + D c u c (t), (3)
where x c (t) ∈ R nc is the controller state, u c (t) = y(t) ∈ R p is the controller input and y c (t) ∈ R m is the controller output has been designed in order to guarantee some performance requirements and the stability of the closedloop system in the absence of the control saturation. Matrices A c , B c , C c and D c are of appropriate dimensions.

In consequence of the control bounds, the control signal to be injected in the system is a saturated one, that is,

u(t) = sat(y c (t)) = sat(C c x c (t) + D c y(t)) (4) with each component of sat(y c (t)) classically defined as sat(y c(i) (t)) △ = sign(y c(i) (t)) min( y c(i) (t) , u 0(i) ), i = 1, . . . , m.
In order to mitigate the undesirable effects of windup caused by input saturation, two static anti-windup compensators

E c (sat(y c (t)) -y c (t)), E c ∈ R nc×m , and F c (sat(y c (t -τ (t))) -y c (t -τ (t))), F c ∈ R nc×m ,
can be added to the controller. Thus, replacing the control signal in (1) by its saturated version (4), considering the dynamic controller and the proposed anti-windup strategy, it follows that the closed loop system reads:

ẋ(t) = Ax(t) + Bsat(y c (t -τ (t))) + B w w(t) y(t) = C y x(t) z(t) = C z x(t) + D z sat(y c (t -τ (t))) ẋc (t) = A c x c (t) + B c y(t) -E c Ψ(u(t)) -F c Ψ(u(t -τ (t))) y c (t) = C c x c (t) + D c y(t), (5)
with Ψ(u(t)) = u(t) -sat(u(t)) being a decentralized deadzone nonlinearity.

Given h and d, we are interested in the synthesis of the anti-windup compensator gains E c and F c in order to address the following problems:

• L 2 -stabilization: all trajectories of the closed-loop system (5) remain bounded for any disturbance satisfying (2) for a certain δ, i.e. the input-to-state stability should be ensured. Moreover, the controller shall ensure an upper bound for the L 2 -gain between the disturbance w(t) and the regulated output z(t), which corresponds to the disturbance rejection problem. • Internal Stabilization: in the absence of disturbances, the internal asymptotic stability of the closed-loop system (5) shall be guaranteed for all initial conditions belonging to a certain set D 0 of functions φ ∈ C ν h , i.e. the set D 0 is included in the basin of attraction of the origin of the closed-loop system (5).

PRELIMINARIES

Define the augmented state ξ(t) = [x(t) T x c (t) T ] T ∈ R n+nc an the following matrices:

A = A 0 B c C y A c , A d = BD c C y BC c 0 0 , B w = B w 0 , B 1 = B 0 , B 2 = 0 I nc , B E = B 2 E c , B F = B 1 + B 2 F c , C z = [C z 0], C dz = [D z D c C y D z C c ], D z = D z and K = [D c C y C c ].
In this case, it is possible to represent the closed loop system (5) as follows: 6), it is possible to consider systems with both input and state delays, provided that the timevarying delay is exactly the same at each instant. In this case, the open-loop system (1) reads ẋ(t) = Ax(t) 7) and the augmented matrices A d and C dz in (6) must be replaced by

ξ(t) =Aξ(t) + A d ξ(t -τ (t)) -B E Ψ(Kξ(t)) -B F Ψ(Kξ(t -τ (t))) + B w w(t), (6) z(t) =C z ξ(t) + C dz ξ(t -τ (t)) -D z Ψ(Kξ(t -τ (t))) with initial conditions φ ξ defined on [-h, 0], i.e. φ ξ = ξ(θ), θ ∈ [-h, 0]. Remark 1. Based on (
+ A d x(t -τ (t)) + Bu(t -τ (t)) + B w w(t) y(t) = C y x(t) z(t) = C z x(t) + C dz x(t -τ (t)) + D z u(t -τ (t)) (
A d = A d + BD c C y BC c 0 0 and C dz = [ C dz + D z D c C y D z C c ].
Considering a matrix G 0 ∈ R m×(n+nc) and defining the polyhedral set

S 0 △ = {ξ ∈ R n+nc ; |(K (i) -G 0(i) )ξ| ≤ u 0(i) , i = 1, ..., m} (8)
the following Lemma, concerning Ψ(Kξ) can be stated. Lemma 2. [START_REF] Tarbouriech | Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs[END_REF]) Consider the deadzone nonlinearity Ψ(Kξ). If ξ ∈ S 0 then the relation

Ψ(Kξ) T T 0 [Ψ(Kξ) -G 0 ξ] ≤ 0 (9) is verified for any diagonal positive definite matrix T 0 ∈ ℜ m×m .

MAIN RESULT

The following theorem considers the anti-windup synthesis in a local context. In this case, the stability is ensured provided that the initial conditions of the closed loop system and the disturbances belong to certain admissible sets. Theorem 3. If there exist symmetric positive definite matrices n+nc) , Y E , Y F ∈ R nc×m and positive scalars γ and µ satisfying the matrix inequalities

Q 1 , X, U ∈ R (n+nc)×(n+nc) , positive definite di- agonal matrices S 0 , S 1 ∈ R m×m , matrices Q 2 , Q 3 ∈ R (n+nc)×(n+nc) , W 0 , W 1 ∈ R m×(
         Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 Σ 6 Σ 7 * Σ 8 0 W T 1 0 0 Q 1 C T dz * * -2S 0 0 0 0 0 * * * -2S 1 0 0 -S 1 D T z * * * * -2hQ 1 + hU 0 0 * * * * * -I 0 * * * * * * -γI          < 0 (10) Q 1 Q 1 K T (i) -W T 0(i) * µu 2 0(i) ≥ 0, i = 1, • • • , m (11)
Q 1 Q 1 K T (i) -W T 1(i) * µu 2 0(i) ≥ 0, i = 1, • • • , m (12)
µ -δ ≤ 0 (13)

where

Σ 1 = Q 2 + Q T 2 + X -U/h Q 1 A T -Q T 2 + Q 3 * -Q 3 -Q T 3 , Σ 2 = U/h A d Q 1 , Σ 3 = W T 0 -B 2 Y E , Σ 4 = 0 -B 1 S 1 -B 2 Y F , Σ 5 = h Q T 2 Q T 3 , Σ 6 = 0 B w , Σ 7 = Q 1 C T z 0 and Σ 8 = (d -1)X -U/h, then the anti-windup gains E c = Y E S -1 0 and F c = Y F S -1
1 ensure that:

(1) the trajectories of system (6) are bounded for every initial condition satisfying

β =[λ(Q -1 1 ) + hλ(Q -1 1 XQ -1 1 )]||φ ξ || 2 c + h 2 2 λ(Q -1 1 U Q -1 1 )|| φξ || 2 c ≤ µ -1 -δ -1 ; (14) (2) z 2 2 ≤ γ w 2 2 + γV (0);
(3) when w = 0, the closed-loop system origin is locally asymptotically stable and, for all initial conditions satisfying β ≤ µ -1 , the corresponding trajectories converge asymptotically to the origin.

Proof. Consider the following Lyapunov-Krasovskii functional, proposed in [START_REF] Fridman | An improved stabilization method for linear time-delay systems[END_REF] for dealing with time-varying delays:

V (t) =ξ T (t)P 1 ξ(t) + t t-τ (t) ξ T (s)M ξ(s)ds + 0 -h t t+θ ξT (s)R ξ(s)dsdθ
with P 1 , M and R being symmetric positive define matrices. From the definition of V (t), it follows that: 2) and ∀φ ξ such that β + δ -1 ≤ µ -1 . Hence, one gets ξ(T ) T P 1 ξ(T ) ≤ V (T ) ≤ µ -1 , i.e. for all T > 0 the trajectories of the system do not leave the set

V (0) ≤ξ T (0)P 1 ξ(0) + 0 -h ξ T (s)M ξ(s)ds + 0 -h 0 θ ξT (s)R ξ(s)dsdθ ≤[λ(P 1 ) + hλ(M )]||φ ξ || 2 c + h 2 2 λ(R)|| φξ || 2 c (15) Define now J(t) = V (t) -w T (t)w(t) + 1 γ z T (t)z(t). If J(t) < 0, it follows that T 0 J(t)dt =V (T ) -V (0) - T 0 w T (t)w(t)dt + 1 γ T 0 z T (t)z(t)dt < 0 (16) which implies that V (T ) ≤ V (0) + ||w(t)|| 2 2 ≤ β + δ -1 ≤ µ -1 (17) ∀w(t) satisfying (
E(P 1 , µ -1 ) = ξ ∈ R n+nc ; ξ(t) T P 1 ξ(t) ≤ µ -1 . Moreover, for T → ∞, (16) yields z 2 2 ≤ γ w 2 2 + γV (0).
In the sequel, we show that (10)-( 13) imply that J(t) < 0, provided that φ ξ is such that β + δ -1 ≤ µ -1 and w(t) satisfies [START_REF] Park | Dynamic anti-windup method for a class of time-delay control systems with input saturation[END_REF].

Noting that Aξ(t) -ξ(t) + A d ξ(t -τ (t)) -B E Ψ(Kξ(t)) -B F Ψ(Kξ(t -τ (t))) + B w w(t) = 0, it follows that the derivative of the Lyapunov-Krasovskii functional V (t) is given by: V (t) =ξ T (t)P 1 ξ(t) + ξT (t)P 1 ξ(t) + ξ T (t)M ξ(t)

-(1 -τ (t))ξ T (t -τ (t))M ξ(t -τ (t)) + h ξ(t) T R ξ(t) - 0 -h ξT (t + θ)R ξ(t + θ)dθ + 2ξ T (t)P T 2 [Aξ(t) -ξ(t) + A d ξ(t -τ (t)) -B E Ψ(Kξ(t)) -B F Ψ(Kξ(t -τ (t))) + B w w(t)] + 2 ξT (t)P T 3 [Aξ(t) -ξ(t) + A d ξ(t -τ (t)) -B E Ψ(Kξ(t)) -B F Ψ(Kξ(t -τ (t))) + B w w(t)].
Introducing the vector ξ(t) = [ξ T (t) ξT (t)] T and the matrix P = P 1 0 P 2 P 3 , we will follow the descriptor approach considered in Gomes da Silva Jr. et al. [2010].

In this case, J(t) can be expressed as:

J(t) = ξT (t)L ξ(t) + 2 ξT (t)P T 0 A T d T ξ(t -τ (t)) -2 ξT (t)P T 0 B T E T Ψ(Kξ(t)) -2 ξT (t)P T 0 B T F T Ψ(Kξ(t -τ (t))) + 2 ξT (t)P T 0 B T w T w(t) + ξ T (t)M ξ(t) -ξ T (t -τ (t))(1 -τ (t))M ξ(t -τ (t)) + h ξT (t)R ξ(t) - t t-h ξT (s)R ξ(s)ds -w T (t)w(t) + 1 γ z T (t)z(t) (18)
with

L = 0 I A -I ′ P + P ′ 0 I A -I .
Consider the sets S 0 defined in (8) and S 1

△ = {ξ ∈ R n+nc ; |(K (i) -G 1(i)
)ξ| ≤ u 0(i) , i = 1, ..., m}. Provided that ξ(t) ∈ S 0 and ξ(t -τ (t)) ∈ S 1 then, from Lemma 2, it follows that:

J(t) ≤J(t) -2Ψ(Kξ(t)) T T 0 [Ψ(Kξ(t)) -G 0 ξ(t)] -2Ψ(Kξ(t -τ (t))) T T 1 [Ψ(Kξ(t -τ (t))) -G 1 ξ(t -τ (t))] ( 19
)
where T 0 and T 1 are diagonal positive definite matrices.

Applying now the Jensen's inequality (see [START_REF] Gu | Stability of time-delay systems[END_REF] p.322), we have -

t t-h ξT (s)R ξ(s)ds ≤ -(ξ(t) -ξ(t - τ (t)) T R h (ξ(t) -ξ(t -τ (t)). Hence, it follows that J(t) ≤ η T (t)Γη(t) + 1 γ z T (t)z(t), with η(t) = [ ξT (t) ξ T (t - τ (t)) Ψ T (Kξ(t)) Ψ T (Kξ(t -τ (t))) w T (t)] T and Γ =        L + M -R/h 0 0 hR R/h 0 + P T 0 A d * (d -1)M -R/h * * * * * * G T 0 T 0 0 -P T 0 B E -P T 0 B F P T 0 B w 0 G T 1 T 1 0 -2T 0 0 0 * -2T 1 0 * * -I       
Suppose now that Γ < 0. Applying the Schur's complement to the terms 0 0 0 hR and 1 γ z T (t)z(t), it follows that

           L R/h 0 + P T 0 A d G T 0 T 0 0 -P T 0 B E * (d -1)M -R/h 0 * * -2T 0 * * * * * * * * * * * * -P T 0 B F h 0 I P T 0 B w C T z 0 G T 1 T 1 0 0 C T dz 0 0 0 0 -2T 1 0 0 -D T z * -hR -1 0 0 * * -I 0 * * * -γI            < 0 (20) with L = L + M -R/h 0 0 0 is verified.
Hence, the satisfaction of ( 20) results in J(t) < 0. Note also that if ( 20) is satisfied, one has L < 0, which implies that -P T 3 -P 3 is negative definite. Hence, since P 1 > 0, it follows that matrix P is invertible. Denote now the matrix

P -1 = Q = Q 1 0 Q 2 Q 3 and define a block diagonal ma- trix Ξ = diag{Q, Q 1 , T -1 0 , T -1 1 , I, I, I}.
By pre and postmultiplying ( 20) by Ξ T and Ξ respectively, one obtains:

           Q T LQ Q 1 RQ 1 /h A d Q1 Q 1 G T 0 -B 2 E c T -1 0 * (d -1)Q 1 M Q 1 -Q 1 RQ 1 /h 0 * * -2T -1 0 * * * * * * * * * * * * 0 -B 1 T -1 1 -B 2 F c T -1 1 h Q T 2 Q T 3 0 B w Q 1 C T z 0 Q 1 G T 1 0 0 Q 1 C T dz 0 0 0 0 -2T -1 1 0 0 -T -1 1 D T z * -hR -1 0 0 * * -I 0 * * * -γI            < 0 (21)
Consider the change of variables:

X = Q 1 M Q 1 , U = Q 1 RQ 1 , S 0 = T -1 0 , S 1 = T -1 1 , W 0 = G 0 Q 1 , W 1 = G 1 Q 1 , Y E = E c T -1 0 and Y F = F c T -1 1 . Noting that the development of (Q 1 -R -1 )R(Q 1 -R -1 ) ≥ 0 implies R -1 ≥ 2Q 1 -U , it follows that the LMI condition (10) is obtained. Suppose that φ ξ is such that (15) is satisfied. Hence it follows that λ(P 1 )||φ ξ || 2 c ≤ µ -1 ⇔ max θ∈[-h,0] λ(P 1 )φ ξ (θ) T φ ξ (θ) ≤ µ -1 .
On the other hand, ∀θ ∈ [-h, 0] one has

ξ(θ) T P 1 ξ(θ) ≤ λ(P 1 )ξ(θ) T ξ(θ) ≤ max θ∈[-h,0] λ(P 1 )φ ξ (θ) T φ ξ (θ) ≤ µ -1
and, if φ ξ verifies (15), we can conclude that ξ(θ) ∈ E(P 1 , µ -1 ), ∀θ ∈ [-h, 0]. Since ( 11) and ( 12) ensure that E(P 1 , µ -1 ) ∈ S 0 ∩ S 1 , from Lemma 2 it follows that 10) is verified, it follows that J(0) < 0 wich implies ξ(∆t) ∈ E(P 1 , µ -1 ), ∆t → 0. Following this reasoning, we can conclude that the relations above are indeed verified ∀t ≥ 0 and hence ( 10) and ( 13) ensures that J(t) < 0 and ξ(t

Ψ(Kξ(t)) T T 0 [Ψ(Kξ(t)) -G 0 ξ(t)] ≤ 0, Ψ(Kξ(t -τ (t))) T T 1 [Ψ(Kξ(t -τ (t))) -G 1 ξ(t -τ (t))] ≤ 0 for t = 0. Then, if (
) ∈ E(P 1 , µ -1 ), ∀t ∈ [-h, ∞],
provided that φ ξ satisfies (15) and w(t) 22 satisfies ( 2), with µ -1 ≤ β + δ -1 . This concludes the proof of the first and the second items of the Theorem 3.

Consider now w(t) = 0. Then, J(t) < 0 implies that V (t) < -1 γ z(t) T z(t) < 0 provided that ξ(t) ∈ S 0 and ξ(t -τ (t)) ∈ S 1 . Hence, from (15) if φ ξ is such that β < µ -1 , we have ξ(t) T P 1 ξ(t) ≤ V (t) ≤ V (0) ≤ β ≤ µ -1
which means that we get ξ(t) ∈ E(P 1 , µ -1 ), for all t ≥ 0. Because LMIs ( 11) and ( 12) are satisfied, it follows ξ(t) ∈ S 0 and ξ(t -τ (t)) ∈ S 1 , for all t ≥ 0. Thus, for any initial condition such that β ≤ µ -1 , it follows effectively that V (t) < 0, which concludes the proof of the third item of Theorem 3.

Theorem 3 considers the local (or regional) stabilization, in the sense that the computed gains E c and F c ensure asymptotic stability just for the initial conditions satisfying ( 14) and any disturbance satisfying [START_REF] Park | Dynamic anti-windup method for a class of time-delay control systems with input saturation[END_REF]. Provided the open-loop system is asymptotically stable, it can be possible to compute globally stabilizing gains, i.e. the input-to-state stability is ensured for any w(t) ∈ L 2 and the origin of the system is globally asymptotically stable. The next result, which can be seen as a particularization of Theorem 3, allows to address this problem. Corollary 4. If there exist positive definite matrices Q 1 , X, U , positive definite diagonal matrices S 0 , S 1 , matrices Q 2 , Q 3 , Y E , Y F of appropriate dimensions and a positive scalar γ satisfying (10) with W 0 = W 1 = KQ 1 , then then there exist anti-windup gains

E c = Y E S -1 0 and F c = Y F S -1
1 which ensures that:

(1) the trajectories of system (6) are bounded for every initial condition φ ξ ∈ C ν h and w(t) ∈ L 2 ; (2) y (3) when w = 0, the closed-loop system origin is globally asymptotically stable.

Proof. The proof mimics the one of Theorem 3. In this case, if G 0 = G 1 = K it follows that the modified sector condition ( 9) is verified ∀ξ(t) ∈ R n+nc and the global asymptotic stability follows.

Remark 5. In the case where the delay is not mensurable at each instant, it suffices to consider F c = 0 (which corresponds to fix Y F = 0 in LMI ( 10)) in Theorem 3 to ensure the L 2 -stabilization and/or the internal stabilization of the closed loop systems (6).

OPTIMIZATION PROBLEMS

In this section we show how the proposed theoretical conditions can be casted into LMI-based optimization problems to determine the anti-windup gains E c and F c . In particular, for given bounds h and d on τ (t) and τ (t), respectively, the following three criteria are considered:

• Maximization of disturbance tolerance: The idea is to maximize the L 2 norm bound on the disturbance for which it can be ensured that the system trajectories remain bounded. Considering that the initial condition is null (i.e., φ ξ = 0, ∀θ ∈ [-h, 0]), this can be accomplished by the following convex optimization problem: min µ subject to (10) to ( 12). ( 22)

• Maximization of disturbance attenuation: For a nonnull bound on the L 2 norm of the admissible disturbances (given by µ -1 = δ -1 ), the idea is to minimize the upper bound for the L 2 gain of w(t) on z(t).

Considering that the initial condition is null, this can be obtained from the solution of: min γ subject to (10) to ( 13).

(23)

• Maximization of the set of admissible initial conditions:

We consider here the disturbance free case, i.e. w(t) = 0. In order to ensure the stability of system (1) by using the Theorem 3, the admissible initial conditions must verify condition ( 14 = κ 2 and note that as the maximal eigenvalues of Q

-1 1 , Q -1 1 U Q -1 1 and Q -1 1 XQ -1 1
decreases, the value of κ 1 and κ 2 for which ( 14) is verified increases. Hence, the problem of finding E c and F c leading to the maximization of the region of stability of the closed-loop system can be achieved by minimizing these maximal eigenvalues. Consider

λ Q1 I I * Q 1 ≥ 0, λ X I I * 2Q 1 -X ≥ 0, λ U I I * 2Q 1 -U ≥ 0. (24)
From the fact that R

-1 ≥ 2Q 1 -U and M -1 ≥ 2Q 1 -X it follows that these LMIs are respectively equivalent to λ Q1 ≥ λ(Q -1 1 ), λ X ≥ λ(Q -1 1 XQ -1 1 ) and λ U ≥ λ(Q -1 1 U Q -1 1
). Hence, the following optimization problem can be considered:

min 11), ( 12) and (24) (25) where β 1 , β 2 and β 3 are weights that should be tuned in order to satisfy some trade-off between κ 1 and κ 2 .

β 1 λ Q1 + β 2 λ X + β 3 λ U subject to (10), (

NUMERICAL EXAMPLE

Consider system (1) given by the following matrices:

A = 0.1, B = B w = 1, C y = C z = -5, D z = -20, u 0 = 1
and a stabilizing PI controller described by:

A c = 0, B c = -0.2, C c = 1, D c = -2
Since the open loop system is unstable, we apply the local stability results proposed in Theorem 3.

In Table 1 is presented the result from the solution of Problem ( 22) with d = 0.1 and different values of h considering two situations: the value of τ (t) is known at each time and can be used to calculate Ψ(Kξ(t -τ (t))) (in this case F c = 0); the value of τ (t) is unknown at each time, i.e. it is not possible to compute Ψ(Kξ(t -τ (t))) (in this case F c = 0). It is easy to see that the solution with F c = 0 is less conservative since the minimal value of µ to which is possible to solve Problem ( 22) is smaller than the one obtained for F c = 0. h µ @ Fc = 0 µ @ Fc = 0 0.0100 8.3942 × 10 -13 5.

CONCLUSIONS

The problem of anti-windup design for linear systems with time-varying input delay and actuator saturation has been addressed. An anti-windup structure with two static compensators has been considered. Based on a Lyapunov-Krasovskii functional and a modified sector approach, global and local stabilization conditions directly in the LMI form were derived to ensure both the external and the internal stability of the closed-loop system in the presence of L 2 -bounded disturbances. Provided that the maximal value of the time-varying delay and its derivative are known, convex optimization problems were proposed to compute the anti-windup gains aiming at three different control objectives. Simulation examples are provided to illustrate the effectiveness of the proposed solution.

The extension of the results to consider systems with both input and output time-varying delays are in study. This case is particularly interesting in networked control applications.

  ) with β ≤ µ -1 . Consider ||φ(θ)|| 2 c = κ 1 and || φ(θ)|| 2 c

Table 1 .

 1 Results for Problem (22)In Table2is presented the result from the solution of Problem (23) with h = 0.1 and d = 0.1 for different values of 1 δ = 1 µ . Also in this case, less conservative results for γ are obtained with F c = 0 since the minimal value of µ (which implies in larger disturbance bounds) to which is possible to solve Problem (23) is smaller than the one obtained for F c = 0.

			2113 × 10 -11
	0.0500	0.0845	0.1978
	0.1000	0.1375	0.2609
	0.5000	0.1986	0.9686
	0.5995	0.8906 × 10 3	3.2892 × 10 3
	µ	γ @ Fc = 0	γ @ Fc = 0
	0.1371	9.3049 × 10 5	-
	0.2607	386.2742	7.6351 × 10 5
	1	386.1440	430.1776
	10	386.1427	386.4972
	100	386.0567	386.3698
	1000	386.0510	386.3686

Table 2 .

 2 Results for Problem (23)