Gildas Siggini

Sophie Demassey

Edi Assoumou

INVESTMENT CHOICES AND CAPACITIES AT RISK IN DECARBONIZING THE EU ELECTRIC SYSTEM

Keywords: optimization, long-term energy planning, decarbonization, interconnected electric system

In this paper, we analyze the implications of fully or nearly fully decarbonizing the European electric system by 2050. Future power mixes trajectories are computed with a 5 year time step at country level using eTIMES-EU, a bottom-up optimization model for the EU power sector. By comparing a near carbon neutral case (TGT90) with a fully decarbonized system (NEUTR), we discuss the additional costs, capacities and trading implications. Increasing the ambition from 90% mitigation goal to strict neutrality requires 32.6% more investments in 2050, 394 GW additional solar capacity in 2050 and more reliance on trade between countries.

INTRODUCTION

In line with the Paris agreement to keep temperature increase well below 2°C, the European Union aims at carbon neutrality by 2050. This will have strong implications for the power sector which was responsible for 24% of 𝐶𝑂 2 emissions in Europe in 2016. National action plans have been proposed that include phase out policies and targets for renewables. However in an interconnected system, the trajectories are interdependent and raise the questions of speed of change, burden sharing and trade.

Different studies in the literature have investigated the long term of the European electric system [START_REF] Eurelectric | Power Choices -Pathways to carbonneutral electricity in Europe by 2050[END_REF] [2] [3] [START_REF] Plessman | How to meet EU GHG emission reduction targets? A model based decarbonization pathway for Europe's electricity supply system until 2050[END_REF]. But to our knowledge, no study provided a comparison between investment decisions and system operation for different levels of decarbonization of the electric system. In this analysis, we use a bottom-up optimization model to explore three different configurations of the EU electric system in 2050: a reference case (REF), a system aiming at a reduction of 90% of the base year 𝐶𝑂 2 emissions (TGT90) and a carbon neutral system (NEUTR). Our emphasis is put on the impacts of increasing the ambition from a nearly decarbonized system to a fully decarbonized system and we quantify investment decisions that become irrelevant or that are only valid for the most stringent case. These impacts include in particular the volumes and directions of trades between countries. Section 2 presents our model and main assumptions. Section 3 then discusses the simulation results before concluding in section 4.

METHODOLOGY 2.1 The European electric sector model

We use a bottom-up optimization model of the EU power system from the MARKAL-TIMES models family to model technologies, production dynamics, regional potentials, and interconnections. eTIMES-EU thus builds up on full technical and economic details of all candidate technologies. An optimization stage then computes optimal investment and operation decisions to supply national electricity demands subject to sets of scenario specific constraints [START_REF] Loulou | Documentation for the TIMES Model[END_REF]. Competing generation options include: bioenergy (biogas, biomass, biomass with CCS 1 ), lignite, natural gas and coal with or without CCS, geothermal, hydro (dam, run of river, lake, pumping and storage), solar (PV, CSP), wind (onshore, offshore) and others (waste, ocean…). eTIMES-EU is a partial equilibrium model with exogenous electricity demands to be satisfied on each time slices.

Spatial and temporal resolution

The model comprises 29 countries distributed in 8 groups. To study the impact of long term energy and define optimal trajectories, the model runs by steps of 5 years between 2016, 2020 and 2050 (with a perfect foresight mode for this analysis). Within each period, 64 time slices are considered. These sub annual time steps model 4 seasons with one week day and one week-end day each subdivided in 8 time steps of three hours. Furthermore we use a peak reserve factor to alleviate the effect of the averaging on peak electricity demands and force the model to install more capacity to be able to cope with situations of higher demand. Austria (AT), Belgium (BE), Switzerland (CH), Germany (DE) , France (FR), Luxembourg (LU), Netherlands (NL) CEE 3 Czech Republic (CZ) , Poland (PL) Nordic & Wester Nordic Denmark (DK), Finland(FI), Norway (NO), Sweden (SE), Iceland (IS) NEE 4 Estonia (EE), Lithuania (LT), Latvia (LV) SEE 5 Bulgaria (BG), Greece (GR) , Croatia (HR), Hungary (HU), Romania (RO), Slovenia (SI), Slovakia (SK)

The calibration of the base year

The base year of the model is 2016. Installed capacities for each region were retrieved from the Platts Database and cross-validated with ENTSO-E and Eurostat data. Bioenergy data were taken from [START_REF]Bioenergy Europe, Statistical Report[END_REF]. The dynamics of base year production in the model were calibrated with ENTSO-E and EUROSTAT generation data. ENTSO-E 6 data are available at time steps down to 15 minutes low and were aggregated corresponding to the time slices considered in our model. A cross validation was made with EUROSTAT figures. The future electric demand considered in each region is calibrated with the EU Reference Scenario 2016 report [START_REF]EU Reference Scenario 2016-Energy, Transport and GHG emissions trends to 2050[END_REF].

Key country level assumptions

Renewable potentials and available technical options are differentiated by country. Where applicable, phase-out policies have also been considered. Wind and solar power plants have experienced a tremendous growth (+244 GW between 2000 and 2016) and it is estimated that installed capacity covers only 16% of the overall available potential [START_REF] Simoes | The JRC-EU-TIMES model-Assessing the long term role of the SET Plan Energy Technologies[END_REF]. Wind offshore offers a supplementary opportunity in the three sea basins across Europe [START_REF] Wind | Unleashing Europe's offshore wind potential -A new resource assessment[END_REF]. Regarding future hydro power plants, the remaining potential is expected to be of run of river type. Finally bioenergy, though suffering from competition for other usages could provide substantial resources via biomass and biogas [START_REF] Scarlat | A spatial analysis of biogas potential from manure in Europe[END_REF]. Renewable potentials have been extracted from [START_REF] Simoes | The JRC-EU-TIMES model-Assessing the long term role of the SET Plan Energy Technologies[END_REF] [11] [START_REF] Wind | Wind in power-2016 European statistics[END_REF]. Major coal plants phase out announcements were done across the continent and summarized in [START_REF]Beyond Coal Overview of national coal phase-out announcements[END_REF]. The situation of Nuclear power is also much contrasted with independently positions: Germany plans to phase-out nuclear by 2022 and France plans to reduce its nuclear activity to 50% in 2035 against 72.5% in 2016, while new development are planned in UK, Czech Republic, Finland, Hungary, Poland and Romania [START_REF][END_REF].

Scenarios

We investigate three scenarios with different levels of environmental concern: A Reference ("REF") scenario: This scenario serves as benchmark and describes the case of a moderate climate ambition in Europe. The 𝐶𝑂 2 penalty still rises from 5 € per ton in 2016 to 35 € per ton in 2030 and then remains constant, interconnections develop at the rhythm of TYNDP2016 recommendations and ACER opinions [START_REF] Acer | Opinion of the Agency for the cooperation of energy regulators[END_REF]. Current coal phase out policies are applied. The maximum installed capacity for coal in Germany and nuclear in France are also limited at the current capacities. The Target 90 (TGT90) scenario: In addition to the assumptions of the Reference scenario, we impose here a reduction of 𝐶𝑂 2 emissions by 90% between 2016 and 2050. The carbon neutrality (NEUTR) scenario: This scenario imposes a strictly 𝐶𝑂 2 neutral system in 2050.

Note that for the TGT90 and the NEUTR scenarios, no specific objective is fixed for the countries and the burden sharing is endogenously computed.

RESULTS

CO 2 emissions burden sharing and power mixes

Figure 1 depicts the aggregated evolution of CO 2 emissions by 2050 for all scenarios. In the reference scenario, emissions decrease by 13% over the horizon with a clear reduction before 2035 followed by a partial recovery after. In comparison, the TGT90 and NEUTR scenarios will respectively require an additional mitigation effort of 763 and 872 Mt CO 2 avoided.

Figure 2-Repartition of the burden sharing between regions

Regions like CWE and CEE which concentrate more than 50% of the base year emissions make the largest efforts to attain carbon neutrality while in the Iberian peninsula existing hydropower potential and capex reduction for solar could significantly decrease emissions even in the reference case (Figure 2)

Achieving the levels of CO 2 reduction presented in Figure 1 induces profound modifications in the electricity mix. The decrease of emissions before 2035 in the REF case is for instance explained by coal to bioenergy and natural substitution while the subsequent increase is due to the demand growth effect. Alternative solutions such as natural gas with CCS are only implemented in the TGT90 scenario to reduce emission from fossil sources but not in neutral case. Finally output from renewables increases from 48% in the REF scenario in 2050 to 86.4% in the NEUTR scenario.

Capacity & investments

The total capacity installed in 2050 in the NEUTR scenario amounts to 1978 GW which corresponds to an increase of 91% compared to the base year capacity. In comparison the total capacity only increases by 4% in the REF scenario and by 46% in the TGT90 case. Renewables drive most of this augmentation and we compute the changes in terms of overall investment, selected technologies and trade between a completely decarbonized and a nearly decarbonized power system. These differences show investment decisions that are more risky as they are only valid for the most stringent case or inversely investments that could become stranded because they could be abandoned when the ambitions are tightened. The additional 10% of emissions reduction between the two scenario leads to an overall 32.4% extra investments in 2050 in the NEUTR scenario. Changes in capacity are also depicted These investment choices also affect trading patterns between countries as depicted in Figure 6. The overall activity of interconnections increases by 315 TWh over the period studied with negative net trades for several countries. This highlights a different type of policy risk as these countries will, in our cost competitive analysis, have to rely more heavily on effective implementation in other countries to satisfy their demand.

CONCLUSION

Reducing 𝐶𝑂 2 emissions in the electric system is a core preoccupation in European Union. In this context, this study investigates the major differences between a complete decarbonization of the EU electric system in 2050 and 90% reduction of CO 2 emissions between 2016 and 2050. Our results show in the NEUTR scenario a surge in the investments in 2050 (+32.6%) driven by supplementary solar capacity (394 GW) compared to TGT90 scenario. Trade activities are also impacted with 12% more electricity exchanged through the interconnected grid. We believe that such elements are not negligible and should be accounted for during decision making.

Figure 3 -Figure 4 -Figure 1 -

 341 Figure 3-Electric power generation per technology

Figure 5 -Figure 6 -

 56 Figure 5-Differences of capacities installed in the NEUTR scenario compared to the TGT90 scenario

Table 1 -Country groups

 1

	2

1

Carbon Capture and Storage


Emmanuel Delaleau
email: emmanuel.delaleau@isen.fr

A Proof of Stability of Model-Free Control

Cybernetics involves Control Theory and Control Practice. From its roots, Cybernetics has always been intimately to Control. The paper is devoted to the proof of an important theorem for the development of control: the closed loop stability of control laws that are calculated in the framework of modelfree control. Everyone knows the importance of control in the field of Cybernetics [1].

I. INTRODUCTION

This new framework has been recently introduced in France [START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF], [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] and has led to several convincing applications. The idea underlying this methodology is the estimation [START_REF] Fliess | Non-linear estimation is easy[END_REF] of some derivatives of measured variables in order to use them in the estimation of the not-modeled effects that influence the to-be-controlled system and to use this information in the feedback loop in order to cancel undesired effects. The control is based on a tracking of the response of an auxiliary model fed by algebraic estimations relying on measurements to which a given behavior is impressed.

Most nonlinear control laws are implemented using asymptotic observers. Contrarily to the linear case, the separation principle is not valid in the case of nonlinear control systems. However, most nonlinear control law methodologies have stability proves, most of them rely on standard tools coming from the pioneer work of Lyapunov. All these proofs need a model and that is a reason why there is no general proof of stability of model-free control.

The present paper aims at filling this gap and using advanced tools for stability analysis [START_REF] Hoppensteadt | Singular perturbations on the infinite interval[END_REF]- [START_REF] Khalil | On stability properties of nonlinear systems with slowly varying inputs[END_REF] will present a general proof of stability of model-free Control. These tools have already been used for the proof of stability and robustness of exact feed-forward linearization [START_REF] Hagenmeyer | Exact feedforward linearization based on differential flatness[END_REF]- [START_REF]Robustness Analysis with Respect to Exogenous Perturbations for Flatness-Based Exact Feedforward Linearization[END_REF]. We think that this result is important for the development of Cybernetics and the teaching of control.

The paper is organized as follow:

• Section II recalls the main aspects and ingredients of model-free control: the development of algebraic estimator (Sec. II-B) and the original development of model-free control (Sec. II-C).

• Section III exposes in details a continuous-time reformulation of model-free control which is the first contribution of this work. MFC is by essence a discrete-time control law although it is applied to continuous-time system. The continuous-time version of the model-free control law allow to draw a stability analysis of the tracking error.

• Section IV states the main result of the paper: it begins with a detailed stability analysis of the tracking error (Sec. IV-A) and ends up with the main theorem which relies on singular perturbation analysis and Kelemen's stability result (Sec. IV-B).

• A short summary of main notations is available at the end of the paper in Table I.

II. SUMMARY OF MODEL-FREE CONTROL

Model-free control is a very recent approach to nonlinear control that has been introduced in [START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF], (Refer to [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] for a thorough presentation). A first industrial and convincing application is reported in [START_REF] Join | Étude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF].

A. Brief Introduction to Mikusiński's Operational Calculus

A reader not familiar with advanced commutative algebra [START_REF] Van Der Waerden | Modern Algebra[END_REF] can skip the reading of this section and go through the end of the communication just with usual knowledge of Laplace transformation [START_REF] Doetsch | Introduction to the Theory and Application of the Laplace Transformation[END_REF]. This Section is devoted to a brief introduction to the operational calculus that has been introduced by Jan Mikusiński 1 .

a) Class K: A function 2 {f (t)} : [0, +∞[ -→ C is said to belong to class K if:
1) it has at most a finite number of points of discontinuity in every finite interval;

2) the integral b) Convolution: Given f, g two elements of K, their convolution is defined as

(f g) (t) = t 0 f (t -τ ) g(τ ) dτ = t 0 f (τ ) g(t -τ ) dτ
In the sequel, the " " symbol for convolution is omitted. c) Integral operator.: According to the definition of convolution we have:

{1} {f (t)} = t 0 f (τ ) dτ
And the function {1} plays the role of integral operator and can also be denoted as l. d) Cauchy formula: With the notations of operational calculus it is easy to establish the Cauchy formula:

t 0 tκ 0 • • • t3 0 t2 0 f (t 1 )dt 1 dt 2 . . . . . . dt κ-1 dt κ = t 0 (t -τ ) κ-1 (κ -1)! f (τ )dτ (1)
e) Construction of the operators: It is possible to show that (K, +, ) is a commutative ring, moreover, by Titchmarsh Theorem [START_REF] Mikusiński | Operational Calculus[END_REF], this ring is integral domain and it is possible to define "an operation inverse to convolution" and construct the field of fraction of this ring.

Namely, denote as a/b the solution of the convolution equation a = bc, a, b ∈ K. The solution c is not always a function of K and will be named an operator. f) Differential operator: Denote as s the operator defined as the inverse (w.r.t. to convolution) to l, i.e.

s = 1 l

This operator has the following property: If a function {f } ∈ K has a continuous derivative ḟ , then

s {f } = ḟ + f (0)
g) Operational form of a function: One calls operational form of a function its expression in terms of the differential operator s. Example:

{t} = 1 s 2 t k k! = 1 s k+1 {exp(-at)} = 1 s + a which mimics usual Laplace transforms.
Calculations of estimators rely on the operational calculus of Mikusiński (see [START_REF] Mikusiński | Operational Calculus[END_REF]).

B. A Short Recall on Algebraic Estimators

Firstly, we recall basics of derivative estimation. Interested reader might refer to [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for a complete presentation. We consider a time signal υ that is available through a measurement υ m corrupted by some additive noise , i.e. υ m = υ + . The objective is to estimate some time derivatives of signal υ, up to a finite order of derivation, from its measurement υ m observed on a given time interval.

The Taylor expansion of υ around 0 reads:

υ(τ ) = ∞ n=0 υ (n) (0) τ n n! Approximate υ(t) in the interval [0, T ], T > 0, by the polynomial υ N (τ ) = N n=0 υ (n) (0) τ n n!
of degree N . The operational3 analogue (see [START_REF] Mikusiński | Operational Calculus[END_REF]) Υ N of υ N is given by:

Υ N = υ(0) s + υ(0) s 2 + • • • + υ (N ) (0) s N +1
It is possible to isolate each coefficient υ (i) (0) appearing in the previous expression by applying a convenient operator to Υ N (s) (See [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for details 4 . Indeed:

∀i = 0, . . . , N, υ (i) (0) s 2N +1 = (-1) i N !(N -i)! • 1 s N +1 • d i ds i • 1 s • d N -i ds N -i s N +1 Υ N (s) (2)
One obtain in the time domain the expression of υ (i) (0) as:

υ (i) (0) = T 0 P (µ; T )υ N (µ)dµ (3)
where P (µ; T ) is a polynomial in µ and T . Notice that (3) gives the calculation of y (i) (0) from an integral on the time interval [0, T ] for a given small T > 0.

As d i υ(t-µ) dµ i | µ=0 = (-1) i υ (i) (t)
it is possible to express υ (i) (t) as an integral which involves values of υ N on the time interval [t -T, t]:

υ (i) (t) = (-1) i T 0 P (µ; T )υ N (t -µ)dµ (4)
A simple estimator of the derivative υ (i) (t) is then obtained from the noisy signal υ m by:

υ (i) c T (t) = (-1) i T 0 P (µ; T )υ m (t -µ)dµ (5)
which is deduced from (3) by replacing υ N by υ m in (4). Note that the integral operation plays the role of low-pass filter and reduced the noise that corrupts υ m . The choice of T results in a trade-off: the larger is T , the smaller is the effect of the noise (the larger is T the better is integrals low pass filtering) and the larger is the error due to truncation.

In practice, the integral appearing in ( 5) is calculated with a numerical integration method (Simpson,...) and the estimator υ (i) c T (t) is thus evaluated at each sample time t k = k.T s , k = 0, 1, . . . Let denotes T s the sampling period and for any time function f denotes as f [k] its time-samples at period T s , i.e.:

f [k] = f (k.T s ), k ∈ Z
With these notations, the discrete-time approximation of the estimator ( 5) is simply a discrete sum that can be rewritten as:

υ (i) d Ts,ns [k] = ns j=0 w(j)P (jT s ; n s T s )υ m [k -j] (6)
with n s the number of samples used in the time window and n s T s = T and the w(j)'s are the weight associated with the used numerical integration method.

C. Model-Free Control 1) Problem statement: Assume we have a plant (see Fig. 1) and that one wants to control the output by mean of the control input, possibly in the presence of a disturbance d acting on the system (effect of a load, noise...). The control input is denoted by u, the output is denoted as y and the disturbance is denoted by d. The control objective can, in many situation be expressed in terms of a reference trajectory t → y , that represents a desired behavior of the output of the system Plant --u y Assume that we do not know any control model of the system of Fig. 1. This is a case in many situations such as:

• the model is not well developed or not writable in a usual manner to be tractable by well-established control framework (for example models described by partial differential equation are not tractable by most of control methodologies);

• the known model is not easily identifiable or the identification of coefficient gives results with low confidence;

• the system is affected with strong disturbance that are difficult to track in any model.

The tracking problem can be also expressed as the asymptotic convergence to 0 of the tracking error defined by: = y -y

for some non-zero initial condition (0) = 0 and under the action of various disturbance.

2) Original Presentation : the Discrete-Time Version of Model-Free Control: Remark 1: For the sake of simplicity, the presentation of this work is led with a single-input single-output system, i.e., u(t) ∈ R and y(t) ∈ R. The general multi-variable case will be developed in a future publication.

A quite recent work has proposed a completely new solution to the problem presented in Sec. II-C1 and have named it precisely model-free control as it does not relies on any modeling of the to-be-controlled plant. This new control strategy has been introduced for the first time in [START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF] and further developed, with many application examples, in [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF].

The control law that has been proposed by [START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF] can be summarized as follows: Introduce an auxiliary numerical model

y (ν) = F + α u (8)
where:

• α ∈ R is an non-physical constant design parameter;

• F represents all that is unknown in the input-output behavior of the system;

• the order ν is also a design parameter of the numerical model ( 8) that can be arbitrarily chosen by the designer.

Although this can be confusing at first sight, Eqn. (8) should not be mixed up with a "black-box" identified model of the plant of Fig. 1. In model-free control, Eqn. ( 8) is updated at each sampling time from the knowledge of the input-output behavior of the unmodeled plant in order to estimate the unknown quantity F and use it in the proposed control law (see below). With the notations introduced in Sec. II-B the estimation of F at sample k (i.e. at time t k = k.T s ) reads:

F [k] = y (ν) d Ts,ns [k] -α u[k] (9)
where y (ν) d Ts,ns [k] is the estimation of the derivative of order ν of the measured output of the plant and u[k] the control input applied to the system at sample time k.T s , k = 0, 1, . . . Based on the numerical knowledge F at each sample k of the quantity F , the control for sampling period k is calculated on (8) as a simple cancellation of the nonlinear terms F plus a closed loop tracking of a reference trajectory t → y (t):

u[k + 1] = - F [k] α NL Cancellation + y (ν) [k] -∆ ι ν-1 ( )[k] α
Closed loop tracking [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF] where:

• = y -y (and [k] = y[k] -y [k] for all k ∈ Z) is the tracking error;
• ∆ ι ν-1 is a closed-loop feedback controller based on the tracking error and that depends on the derivatives of up to the order ν -1 and on the integrals of up to the order ι ∈ N.

The shifting of one sample between the right-and left-hand side (this is indeed u[k + 1] and not u[k] in the left hand side of ( 10)) is necessary in order to avoid any algebraic loop.

Remark 2:

The controller is expressed by a functional in as it may involve integral terms. Consequently we write ∆ ι ν-1 ( )(t) to express its value at time t ∈ R and ∆ ι ν-1 ( )[k] for its value at sample time kT s , k ∈ Z.

Remark 3: The controller ∆ ι ν-1 can be of various forms, not necessarily linear. It is not useful to precise a detailed form at this stage of the presentation. See below for more details.

Remark 4: Note that the term -

F [k] α + y [ν) [k]
α is also the "nominal control" in the "flatness-based" control of (8) (See also [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF]- [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF]). When the closed loop controller is of "PID" type, model-free control can be named as "intelligent PID" (i-PID) (See [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF]). This control scheme is summarized in Fig. 2.

In order to be completely clear on the fact that modelfree control does not rely on any model, one can rephrase the presentation of model-free control as follows: One can think to solve the control problem exposed in Sec. II-C1 as finding a control law to the plant of Fig. 1 in order that its controlled behavior mimics the one of the following linear input-output system:

y (ν) = α u (11)
Keep in mind that ( 11) is not a model nor any try of modeling of the to-be-controlled plant, but a numerical intermediary in the development of the control law.

One can legitimately ask why not to choose simply α = 1 in order to obtain a simple Brunovský canonical form [START_REF] Brunovský | A classification of linear controllable systems[END_REF] (a simple chain of ν integrators in series) instead of [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF]. Note that this can also be achieved by a simple rescaling of the control variable u. However, as it will appear later, α plays an important role in the the tuning of the model-free control law and it will be kept as a design parameter.

Recall that robust asymptotic tracking of the trajectory t → y to system [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF] can easily be achieved with the closed-loop control law of the form

u(t) = y (ν) -∆ ι ν-1 ( )(t) α (12)
where ∆ ι ν-1 is an appropriate controller, i.e. the closed-loop error equation:

(ν) + ∆ ι ν-1 ( ) = 0 (13)
impress lim t→+∞ (t) = 0, ∀ (0) (see below for a choice that ensures this convergence). Recall that ι ∈ N stands for the number of error integrals and ν ∈ N the number of derivatives of involved in the control law [START_REF]Robustness Analysis with Respect to Exogenous Perturbations for Flatness-Based Exact Feedforward Linearization[END_REF].

Remark 5: In the linear case, it is always possible to find a linear controller ∆ ι ν-1 that ensures global asymptotic stability, i.e. convergence to 0 of the tracking error from any initial condition As the real plant of Fig. 1 is far from being so trivial as the simple mono-variable linear system [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF], however, one can easily understand the model-free control as a generalization of (12) by adding an adaptive compensation term. To this end, consider the time-varying quantity Ω defined as:

Ω(t) = y (ν) (t) -α u(t) (14)
This quantity can be though as a "measure of difference" between the input-output behavior of the plant of Fig. 1 and the one of the dynamic system of model [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF]. Notice that moreover, as it is written in terms of u and y it can easily be estimated with the techniques recalled in Sec. II-B. Its estimation Ω, at sample time k.T s , reads:

Ω[k] = y (ν) d Ts,ns [k] -α u[k] (15)
Note that Ω is piecewise constant as it is estimated only once per sample period [k.T s , (k +1).T s [. Having this estimation on can add a compensation term in the control law [START_REF]Robustness Analysis with Respect to Exogenous Perturbations for Flatness-Based Exact Feedforward Linearization[END_REF] in order to help the to-be-controlled systems mimics the tracking of y by [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF].

The model-free control is thus

u[k + 1] = y (ν) [k] -∆ ι ν-1 ( )[k] -Ω[k] α (16)
In order to well understand the sequel of the development consider:

• The constant coefficient α ∈ R is a non-physical design parameter of the control law (see below);

• the time-varying quantity Ω represents all that must be compensated from input-output knowledge u → y in order to mimic [START_REF]Continuous-time non-linear flatness-based predictive control: an exact feedforward linearisation setting with an induction drive example[END_REF] after feedback. The compensation concerns the effect of perturbations, the ignorance of the model or the value of model parameters, the coupling with various other input or output of the system...

• the order ν ∈ N in ( 16) is also a design parameter that can be arbitrarily chosen.

The complete control scheme is depicted in Fig. 2. Notice that although the plant is a continuous-time one, the control law is a sampled-control or discrete-time control: the estimators of Ω and the control u are calculated once for each sample period.

Unmodeled plant

Estimator [START_REF] Doetsch | Introduction to the Theory and Application of the Laplace Transformation[END_REF] Feedback control ( 16) - 16) is quite similar to [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF], however the compensation term Ω of ( 16) has a clear signification in terms of what can be compensated from the input-output view of any measurements of control systems.

We will say that the reference trajectory t → y is quiet enough it there exist δ > 0, τ > 0 such that: ∀t 0, ∀i = 0, . . . , ν, 1 τ t+τ t y (i) (µ) dµ < δ

One says that the tracking of the trajectory t → y is asymptotically stable [START_REF] Demailly | Analyse numrique et quations diffrentielles[END_REF] if the tracking error is bounded, i. The stability result can now be stated as follows:

Theorem 1: Provided that:

• the trajectory t → y is quiet enough;

• the stabilizing controller ∆ ι ν ensures the robust tracking of (13) with sufficiently margin;

• the initial tracking error (0) is sufficiently small. Then the model-free control law ( 16) that the to-be-controlled system ensures the asymptotically stable tracking of the trajectory t → y .

The hypotheses of this theorem are quite standard compared with the ones in the huge literature on trajectory tracking.

Proof: The proof proceed with an application of Kelemen [START_REF] Kelemen | A stability property[END_REF]- [START_REF] Khalil | On stability properties of nonlinear systems with slowly varying inputs[END_REF] with a conjunct application of the singular perturbation theory [START_REF] Hoppensteadt | Singular perturbations on the infinite interval[END_REF], [START_REF] Fenichel | Persistence and smoothness of invariant manifolds for flows[END_REF], [START_REF] Lorenz | On the existence of a slow manifold[END_REF] on the differential system (24a)-(24i): The small parameter αT s appearing in the last equation (24i) ensures fast convergence of u to its "good value" u * . The latter, according to a steady state analysis, expresses as: u = -f (y , . . . , y (ν-1) ) b Stability of the tracking is easily proven by a standard application of Kelemen's result. Notice that in practice u * cannot be calculated as the model of the system is not known; the estimator involved in [START_REF] Doetsch | Introduction to the Theory and Application of the Laplace Transformation[END_REF] do the job in an indirect manner.

V. CONCLUSION

This paper establishes that model-free control produces an asymptotically stable trajectory tracking for nonlinear systems under mild hypothesis. This proof also gives insight to the tuning of model-free control laws. This result is easily transportable to multi-input multioutput systems.

Model-free control have been successfully applied in many practical situation before this proof was developed. This results should give some confidence to control practitioners for the applicability of this framework. One also hope that this could facilitate the introduction of this new methodology in modern teaching of control.

t 0 |

 0 f (τ ) | dτ has a finite value for every t > 0.

Fig. 1 .

 1 Fig. 1. To-Be-Controlled System

Fig. 2 .

 2 Fig. 2. Model-Free Control

  e. there exists o > 0 such that sup t∈[0,+∞[ | (t) | < o

  The author humbly thanks the thousands of volunteers who give their time to develop free softwares or open hardwares; they actively contribute to protect the intellectual heritage of Humanity. NOTATIONS f [k] Sample k of time function f υ c T Continuous-time estimation of υ υ d Ts,ns discrete-time estimation of υ T Time span used in continuous-time estimators Ts Sample period ns Number of sample involved in discrete-time estimators

TABLE I .

 I MAIN NOTATIONS

Jan Mikusiski (April 3, 1913 Stanisaww July

[START_REF] Fenichel | Persistence and smoothness of invariant manifolds for flows[END_REF] 1987 Katowice) was a Polish mathematician.[START_REF] Fliess | Vers une commande multivariable sans modèle[END_REF] We use here the notation introduced by Mikusiński himself to denote functions.

Reader not familiar with operational calculus can just think in terms of Laplace transform to understand the development of the derivatives estimators.

Note that those operators are not unique, we have chosen here to use the ones with the least order of integration for the sake of simplicity of the presentation.

In case of a linear controller, see[START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF], in nonlinear case just take the characteristic polynomial of the linearized controller around 0.

ACKNOWLEDGMENTS

The authors would like to thank A. S. Lucas for her helpful hints and benevolent encouragements during the preparation of this work.

III. CONTINUOUS-TIME REFORMULATION OF MODEL-FREE CONTROL

A. Euler Method Recall

Consider the ordinary differential equation:

where x : I -→ X , f :

The simplest algorithm to approximate the solution to the Cauchy problem with initial condition (t o , x o ) associated with [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] is the Euler method that can be summarized as follows: Denote as {t 0 , t 1 , . . . , t N } a subdivision of the time interval I = [t o , t f ], i.e. a strictly increasing sequence

the successive steps of the algorithm. The approximated solution of ( 17) is denoted as x and is iteratively calculated at t = t k according to:

This algorithm is initialized with:

Results of convergence and stability of this algorithm are well-established in numerical analysis and we will not repeat them here. Roughly speaking, the properties of the algorithm mainly relies on the maximum step h = max k (h k ) (see [START_REF] Press | Numerical Recipes. The Art of Scientific Computing[END_REF] for details).

B. Continuous Time Counterpart of the Control Law (16)

The model-free control law ( 16) can be easily re-written in a different manner. To this end, recall that y = yso y (ν) = y (ν) - (ν) . Moreover from [START_REF] Van Der Waerden | Modern Algebra[END_REF], y (ν) (t) = Ω(t) + α u(t). Consequently:

Moreover

And finally the control reads:

and comparing with [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], one can interpret [START_REF] Hagenmeyer | Exact feedforward linearization based on differential flatness[END_REF] as the Euler algorithm associated to:

with initial condition u(0) = 0 and T = n s T s . One see the influence of α: it can always be chosen in order that the time αT s plays the role of a "small parameter", ensuring fast convergence of the control to the accurate value to achieve the control task. This will become more clear in the proof below.

IV. STABILITY ANALYSIS

To our best knowledge every results of stability of solution of differential system rely on a model given by equations (see e.g. [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF], [START_REF] Khalil | Nonlinear systems[END_REF] for a thorough survey). In order to analyze the stability of the closed loop we need to introduce a model of the system. Remember that this model is not used in the development of the control law [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF] or [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF]. For the sake of simplicity, we only consider a single input single-output system (SISO) and the general case will be developped in a future publication.

Moreover, we assume that the model of the plant is of the form:

where b ∈ R is a constant nonzero parameter and f a convenient smooth vector field. A more general model will be considered in a future publication. Note that this is not a limitation to consider the same order ν in [START_REF] Brunovský | A classification of linear controllable systems[END_REF] and in [START_REF] Join | Étude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF] as this order can be arbitrarily chosen during the control law design in [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF] or [START_REF]Robustness analysis of exact feedforward linearization based on differential flatness[END_REF]. The structure of model ( 21) is a controller-like form and this facilitate the error equation analysis.

Remark 7:

The structure of the model ( 21) evades some class of control systems, especially the difficult case of nonminimum phase ones which, as everyone knows are difficult to control, even with Model-Free control [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF].

A. Error Equation Analysis

The closed-loop ( 21)-( 20)-( 7) is given by:

For the stability analysis it is necessary to enter in the details of the controller ∆ ι ν-1 . For the sake of simplicity we exhibit one which is linear although there are many nonlinear choices that are quite convenient. It worth to give some precision on the choice of the closed-loop controller ∆ ι ν-1 that renders the closed-loop system asymptotically stable.

The tracking error is simply = y -y and denotes as χ i , i = 1, 2, . . . , ι, the i-st integral of , namely:

Among the many possible choices of stabilizing controller, linear ones exist and ∆ ι ν-1 can be chosen as:

One can renders the equation ( 13) globally asymptotically stable if the characteristic polynomial associated with this equation, namely: 23) is Hurwitz, i.e. has all roots with strictly negative real part.

Eliminating y = + y and all its derivatives in ( 22) leads to a first-order differential system that couples the variables: χ ι , χ ι-1 . . . , χ 1 , , ˙ , . . . (ν-1) and is excited by the quantities y , ẏ , . . . , y (ν) , and y (ν) c T . In full details, the equations reads:

Which is a differential system of equation,

The last equation (24i) exhibits a "small parameter", namely T s α, that ensures a fast convergence of u to the desired control. The complete stability analysis relies on the arguments of Kelemen's theorem [START_REF] Kelemen | A stability property[END_REF].

B. Proof of Stability

Before to state the main result of this paper, we need to introduce some more vocabulary.

We will say that the stabilizing controller ∆ ι ν-1 ensures the tracking of the ideal system (13) with a sufficient margin if there exits a > 0 such that all the roots λ l , l = 1, . . . , ν + ι of the associated 5 characteristic polynomial P ∆ ι ν-1 (see [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF]) satisfies Re(λ l ) < -a, l = 1, . . . , ν + ι


Badreddine El
email: badreddinelhaiek@gmail.com

Taha Zoulagh
email: zoulagh.taha@gmail.com

Abdelaziz Hmamed
email: hammedabdelaziz@yahoo.fr

El Mostafa El

POLYNOMIAL STATIC OUTPUT FEEDBACK H ∞ CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS VIA DESCRIPTOR APPROACH

Keywords: Robust static output feedback H ∞ control, Linear matrix inequalities (LMIs), Polytopic uncertainty, Homogenous matrices

This paper deals with the problem of the robust static output feedback H ∞ control (SOFC) for continuous linear systems with polytopic uncertainties. The controller has been gotten by the use of descriptor redundancy. Under this approach a sufficient condition is provided for the existence of a solution to the problem. Thus, the advantage of this method is to obtain more free matrices in the design condition, also the polynomial approach helps to have a less conservative result. In the end, the performance of the method is shown by several examples.

1. Introduction. The problem of the output feedback H ∞ controller design has been extensively studied in recent years for linear continuous systems with polytopic uncertainties using LMI-based convex conditions and has been programmed by means of simple interfaces (LMI control toolbox, YALMIP) and solved by efficient algorithms (LMI toolbox and SeDuMi) [START_REF] Lofberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF][START_REF] Gahinet | LMI Control Toolbox User's Guide[END_REF][START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF]. On the other hand, in control theory and practice output feedback control is very useful since it can be easily implemented with low cost [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]. As a specific case, the robust output feedback controller design for linear systems with polytopic uncertainties has been considered in many researches [START_REF] Peaucelle | An efficient numerical solution for H 2 static output feedback synthesis[END_REF][START_REF] Arzelier | Robust static output feedback stabilization for polytopic uncertain systems: Improving the guaranteed performance bound[END_REF][START_REF] Mehdi | Static output feedback design for uncertain linear discrete time systems[END_REF][START_REF] Nachidi | Static output-feedback stabilization for timedelay Takagi-Sugeno fuzzy systems[END_REF]. The design problem of such controller can be represented as a bilinear matrix inequality (BMI) problem, which is non-convex [START_REF] Kanev | Robust output-feedback controller design via local BMI optimization[END_REF][START_REF] Blondel | NP-hardness of some linear control design problems[END_REF].

Moreover, when the problem is to synthesize a robust controller for uncertain systems, Lypunov theory and LMI can also be applied. However, even synthesis problems that are convex for the precisely known case, as in the synthesis of state feedback stabilizing gain, become non convex when the considered system is uncertain; consequently, the LMI conditions are only sufficient. The synthesis of a robust static output feedback gain for uncertain systems is an even more involved problem, being the main subject of research in several papers [START_REF] Peaucelle | An efficient numerical solution for H 2 static output feedback synthesis[END_REF][START_REF] Arzelier | Robust static output feedback stabilization for polytopic uncertain systems: Improving the guaranteed performance bound[END_REF][START_REF] Mehdi | Static output feedback design for uncertain linear discrete time systems[END_REF][START_REF] Dong | Static output feedback control synthesis for linear systems with timeinvariant parametric uncertainties[END_REF][START_REF] Yaesh | Robust reduced-order output-feedback H control[END_REF]. Concerning uncertain systems with parameters in a polytopic domain, the analysis conditions for precisely known systems can be adapted and applied to the vertices of the polytopic, imposing the same matrix for entire set. Such approach, however, have results on conditions that are only sufficient and usually conservative. To reduce the conservativeness, the matrix variable may also be parameter-dependent [START_REF] Oliveira | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF][START_REF] Geromel | LMI characterization of structural and robust stability[END_REF]. Moreover, use homogenous polynomial parameter dependent (HPPD) variables with various degrees [START_REF] Benzaouia | Two dimensional systems: From the introduction to state of the art[END_REF][START_REF] Chesi | Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems[END_REF][START_REF] Agulhari | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF][START_REF] Bliman | Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex[END_REF][START_REF] Kririm | Parameter-dependent robust stability of time-delay systems[END_REF][START_REF] Hmamed | A new design of robust H ∞ filters for uncertain continuoustime stochastic systems[END_REF], in order to get less conservative results.

Motivated by the fact that for the SOF controllers problems, the conditions guaranteeing the existence of H ∞ control law are given in terms of BMI, recently, the descriptor approach was an alternative solution to get good results [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF]. Thus, the use of the descriptor formulation may help to avoid appearance of crossing terms between the system matrices and the designed ones.

This paper tackles the static output feedback H ∞ control problem for linear continuoustime system based on polynomial approach. The interest is to design a static output feedback control for polytopic system, which guaranteed a prescribed H ∞ performance level. The main idea is to rewrite the closed loop system by the use of descriptor redundancy where the target is to separate system matrices and the controller gain. Different from [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF] that uses optimization solution "fminsearch" [START_REF] Coleman | Optimization Toolbox User's Guide, 3rd Edition[END_REF], an iterative LMI based procedure involving decision variables is proposed in this paper to improve the H ∞ SOF controllers, which gives opportunity to use more slack variables in order to reduce the conservatism. In the end, numerical examples are provided to make clear the efficiency of the proposed approach.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are presented in Section 2. In Section 3, we present two results. First, we obtain a new H ∞ performance analysis for the SOF control. Thereafter, based on this analysis, we propose a design method for H ∞ SOF control. Simulation studies are given to demonstrate the approach effectiveness in Section 4. Section 5 concludes this paper.

Notation: we use standard notations throughout this paper. The notation P > 0 (< 0) is used for positive (negative) definite matrices. * stands for the symmetric term of the diagonal elements of square symmetric matrix. I m denotes the identity matrix of dimension m. The superscript "T " and the notation sym(Y) represent respectively the transpose and Y + Y T .

System Description and Problem Statement.

Consider the following state-space representation for a class of uncertain continuous-time

   ẋ(t) = A(α)x(t) + E(α)w(t) + B(α)u(t) z(t) = C 1 (α)x(t) + F (α)w(t) + D(α)u(t) y(t) = C 2 (α)x(t) + H(α)w(t) (1)
where x(t) ∈ ℜ n is the state vector, y(t) ∈ ℜ p is the output measurement, u(t) ∈ ℜ m is the control input vector and w(t) ∈ ℜ f is the exogenous input in L 2 [0, ∞) and z(t) ∈ ℜ q the controlled output variable. The matrices A(α), B(α), E(α), C 1 (α), D(α), F (α), C 2 (α) and H(α) are the system matrices of appropriate dimensions. The time-invariant uncertain matrices belong to a polytopic domain given by:

M (α) = N ∑ i=1 α i M i , α ∈ Λ N Λ N = { α ∈ ℜ N : N ∑ i=1 α i = 1, α i ≥ 0, i = 1, . . . , N

}

Matrix M (α) represents any uncertain matrix given in [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], N is the number of vertices of the polytopic, M i = 1, . . . , N are the vertices and Λ N is the unit simplex. The output-feedback controller under consideration is of the form

u(t) = Ky(t) (2)
where K ∈ ℜ m,p is the controller gain to be designed.

In [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF], LMI based design for state feedback controller using the descriptor redundancy has been proposed to reduce computational cost. To take advantage of a descriptor redundancy formulation, (1) and ( 2) can be easily rewritten with the above defined notations respectively as:

{ Ẽ ξ(t) = Ã(α)ξ(t) + B(α)w(t) z(t) = C(α)ξ(t) + F (α)w(t) (3)
where

ξ(t) = [ x(t) T y(t) T u(t) T ] T and Ẽ =   I 0 0 0 0 0 0 0 0   , Ã(α) =   A(α) 0 B(α) C 2 (α) -I 0 0 K -I   , C(α) = [C 1 (α) 0 D(α)], F (α) = F (α), B(α) =   E(α) H(α) 0   .
Remark 2.1. Rewriting the closed loop system (1) by the use of descriptor redundancy [START_REF] Tanaka | A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions[END_REF] allows to avoid appearance of crossing terms between the system matrices and the designed ones (i.e., K). It makes it easier for the LMI formulation of synthesis conditions.

For the closed-loop system represented as in (3), we define H ∞ performance as follows.

• The closed-loop system (3) is asymptotically stable when w(t) = 0.

• The closed-loop system (3) has a prescribed level γ of H ∞ noise attenuation; i.e., the zero initial condition x(0) = 0 the condition

∥z(t)∥ 2 < γ∥w(t)∥ 2 (4)
is satisfied for any nonzero w(t) ∈ L 2 [0, ∞) over the entire polytope Λ N .

3. Main Result. In this section, we shall present a numerically efficient technique to find the SOFC gains in such a way to stabilize the system in closed loop, i.e., to ensure that (3) is asymptotically stable. We summarize the resulting LMI design conditions in the following theorem.

Theorem 3.1. Suppose that the controller gain K is known, the system in ( 3) is asymptotically stable if for a prescribed scalar γ > 0 of H ∞ performance, if there exists matrice X(α) such that the following conditions are satisfied:

Ψ =   ℵ(α) CT (α) X T (α) B(α) * -I F (α) * * -γ 2 I   < 0 (5) ẼT X(α) = X(α) T Ẽ ≥ 0 ( 6
)
where

ℵ(α) = ÃT (α)X(α) + X T (α) Ã(α).
Proof: Considering the following Lyapunov function candidate:

V = ξ T (t) ẼT X(α)ξ(t) (7)
where ẼT X(α) = X(α) T Ẽ ≥ 0 with X(α) having the following form:

X(α) =   X 11 (α) X 12 (α) X 13 (α) X 21 (α) X 22 (α) X 23 (α) X 31 (α) X 32 (α) X 33 (α)   (8)
in order to satisfy [START_REF] Oliveira | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF], it is clear that X 11 (α) = X 11 (α) T > 0; moreover, the matrices X 12 (α) and X 13 (α) should be equal to zero. We have the derivative of Lyapunov function (7) over the dynamic system (3) as follows:

V = ξT (t) ẼT X(α)ξ(t) + ξ T (t)X T (α) Ẽ ξ(t) Let J = V + z T (t)z(t) -γ 2 w T (t)w(t) = [ ξ T (t) ÃT (α) + w T (t) BT (α) ] X(α)ξ(t) + ξ T (t)X T (α) [ Ã(α)ξ(t) + B(α)w(t) ] + [ ξ T (t) CT (α) + w T (t) F T (α) ] [ C(α)ξ(t) + F (α)w(t) ] -γ 2 w T (t)w(t) by taking ζ(t) = [ ξ(t) T w(t) T ]
T ̸ = 0 then J becomes as the following:

J = ζ(t) T Φ ζ(t) with Φ = [ Φ 1 Φ 2 Φ T 2 Φ 3 ] where Φ 1 = sym ( ÃT (α)X(α) ) + CT (α) C(α), Φ 2 = X T (α) B(α) + CT (α) F (α), Φ 3 = F T (α) F (α) -γ 2 I, so J < 0 for any ζ(t) = [ ξ(t) T w(t) T ] T ̸ = 0 if Φ < 0.
by applying Schur complement and some row transformations on Φ, LMI (5) is obtained.

If ( 5) is satisfied, we have J < 0 for any

ζ(t) = [ ξ(t) T w(t) T ] T ̸ = 0, which implies V (ξ(∞)) -V (ξ(0)) + ∫ ∞ 0 z T (t)z(t)dt -γ 2 ∫ ∞ 0 w T (t)w(t)dt < 0. With zero initial condition ξ(0) = 0 and V (ξ(∞)) > 0, we obtain ∫ ∞ 0 z T (t)z(t)dt < γ 2 ∫ ∞ 0 w T (t)w(t)dt for any nonzero w(t) ∈ L 2 [0, ∞).
This ends the proof.

Remark 3.1. The static output feedback H ∞ controller design was founded by many approaches [START_REF] Crusius | Sufficient LMI conditions for output feedback control problems[END_REF][START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF][START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF]. However, it was necessary to impose some equality constraints on the systems matrices [START_REF] Crusius | Sufficient LMI conditions for output feedback control problems[END_REF][START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF], also by making conditions about their rank. Moreover, they require to give some descriptions about the output matrix C 2 (α) if it belongs to the polytopic domain or not [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF]. Thus, in our study the constraints given [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF] have been considered and by avoiding the ones proposed in [START_REF] Crusius | Sufficient LMI conditions for output feedback control problems[END_REF][START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF], our approach presents suitable results for general case. Now, based on Theorem 3.1 we present our aim result on the static output feedback H ∞ controller design in the following theorem.

Theorem 3.2. The system in (3) is asymptotically stable for a prescribed scalar γ > 0 of H ∞ performance, if there exist matrices X 11 (α) > 0, X 21 (α), X 22 (α), X 23 (α), X 33 , T 1 , T 2 and Z satisfying the following matrix inequality:

      Π 11 * * * * Π 21 Π 22 * * * Π 31 Π 32 Π 33 * * Π 41 H T X 22 (α) H T X 23 (α) -γ 2 I * C 1 (α) 0 D(α) F T (α) -I       < 0 ( 9
)
where

Π 11 = A T (α)X 11 (α) + X 11 (α)A(α) + G T (α)X 21 (α) + X T 21 (α)G(α). Π 21 = -X 21 (α) + Z T T 1 + X T 22 (α)G(α). Π 22 = -X 22 (α) -X T 22 (α) + Z T T 2 + T T 2 Z 2 . Π 31 = B T (α)X 11 (α) + X T 23 (α)G(α). Π 32 = -X 33 T 2 -X T 23 (α) + Z. Π 33 = -X 33 -X T 33 . Π 41 = E T (α)X 11 (α) + H T (α)X 21 (α).
with

G(α) =    C 2 C T 2 C 2 , if C 2 (α) is fixed, i.e., C 2 (α) = C 2 and C 2 is of full row rank; C 2 , if C 2 (α) is fixed, i.e., C 2 (α) = C 2 and C 2 is of non-full row rank; C 2 (α), if C 2 (α) is non-fixed.
and the static output feedback H ∞ controller law is given by:

K = X -T 33 Z ( 10
)
Proof: Based on Theorem 3.1, the system in ( 3) is asymptotically stable with H ∞ performance γ, if conditions ( 5) and ( 6) are satisfied. Note that the system matrices in these conditions contain uncertainties. We assume that matrix X 11 (α) > 0 depends on the system uncertainties and to cast the condition (6) into LMI, we define matrices X(α) as:

X(α) =   X 11 (α) 0 0 X 21 (α) X 22 (α) X 23 (α) X 33 T 1 X 33 T 2 X 33   (11)
which makes an obligation to put X 11 (α) as follows X 11 (α) > 0 now by assuming that ( 5) and ( 6) hold and by considering the structure given of X(α), we replace each element by its equivalent in Theorem 3.1 then we obtain the following transformation: Z = X T 33 K (12) and from Π 33 we conclude that X 33 is invertible then ( 9) and ( 10) hold. This complete the proof. Remark 3.2. By using the descriptor redundancy formulation we have obtained a sufficient condition for the design of the controller law. The target is to separate system matrices and the controller gain by means of free matrices derives from the descreptor approach, which helps to reduce the conservatism. [START_REF] Agulhari | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF] in the aim to handle robust LMIs with parameters in the unit simplex can be applied. The parameter-dependent variables in Theorem 3.2 are supposed as homogenous polynomials of arbitrary degree g, although different degrees can be used producing results with distinct complexity and accuracy. Let Q g (α) be any parameter dependent variable in [START_REF] Arzelier | Robust static output feedback stabilization for polytopic uncertain systems: Improving the guaranteed performance bound[END_REF], of arbitrary degree g, denoted by:

Remark 3.3. To solve the robust LMI condition of Theorem 3.2, the technique given in

Q g (α) = ∑ k∈k(g) α k 1 1 . . . α k N N Q k , k = k 1 k 2 . . . k N ( 13
)
where α k 1 1 . . . α k N N , α ∈ Λ N , k i ∈ Z + , i = 1, .
. . , N are the monomials, and Q k ∈ R, ∀k ∈ k(g) are matrix-valued coefficients. k(g) is the N -tuples obtained as all possible combinations of non-negative integers K i , i = 1, . . . , N , such that k 1 + k 2 + . . . + k N = g. To illustrate this notation, consider a homogenous polynomial of degree g = 2 with N = 2. The set k is given by k = 02, 11, 20, corresponding to the generic from

Q 2 (α) = α 2 2 Q 02 + α 1 α 2 Q 11 + α 2 1 Q 20 .
This choice for the decision variables provides less conservative results with the increase of g at the price of greater complexity and computational effort. Robust LMIs with parameters in the unit simplex can be fully characterized by means of homogenous polynomial solutions, without loss of generality [START_REF] Bliman | Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex[END_REF]. That is, if a solution of degree ḡ exits, a sequence of LMIs providing sufficient conditions for the existence of homogenous polynomials of increasing degree g > ḡ can be used, with convergence assured for a large of enough g [START_REF] Agulhari | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF]. The LMI conditions, expressed only in terms of the vertices of the system, were obtained with the ROLMIP (Robust LMI Parser) toolbox available at http://www.dt.fee.unicamp.br//agulhari/rolmip/rolmip.htm. The toolbox is developed for Matlab and works jointly with YALMIP, returning the entire set of LMIs through simple commands that describe the structure of the matrices involved and the robust LMI conditions to be programmed. Remark 3.4. We mention that in [START_REF] Peaucelle | An efficient numerical solution for H 2 static output feedback synthesis[END_REF], there are products between the variables T i (i = 1, 2) and matrices Z and X 33 . In special case where T i (i = 1, 2) equal zero in [START_REF] Mehdi | Static output feedback design for uncertain linear discrete time systems[END_REF], Theorem 3.2 can be solved with good results. Thus, less conservative H ∞ bounds can be obtained by taking account of variables T i (i = 1, 2) as proposed in [START_REF] Peaucelle | An efficient numerical solution for H 2 static output feedback synthesis[END_REF] based on the algorithm [START_REF] Lacerda | Delay-dependent robust H ∞ filter design for state-delayed discrete-time linear systems via homogeneous polynomial matrices[END_REF] shown in the following step.

Thus, to solve the non linearities problem in [START_REF] Peaucelle | An efficient numerical solution for H 2 static output feedback synthesis[END_REF] caused by the product between T i (i = 1, 2) and Z, we propose the following algorithm [START_REF] Lacerda | Delay-dependent robust H ∞ filter design for state-delayed discrete-time linear systems via homogeneous polynomial matrices[END_REF]. Algorithm 1. [START_REF] Lacerda | Delay-dependent robust H ∞ filter design for state-delayed discrete-time linear systems via homogeneous polynomial matrices[END_REF]: Initialize matrices T i and scalar γ T and γ Z while

γ T -γ Z γ T

> ϵ and (Maximum number of iterations not reached) do

Solve Theorem 3.2 with T i , minimizing γ, γ T ←-γ find and store matrices X 33 and Z Solve Theorem 3.2 with X 33 and Z obtained in the previous step, minimizing γ, γ Z ←-γ, find matrices T i . end while.

Remark 3.5. This algorithm provides less conservative results and helps to have extra free weighting matrices

T i (i = 1, 2)
. Thus, this method provides better results than ones in [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF].

4. Numerical Examples. This section is devoted to numerical evaluation of the conservatism of the proposed design methodology as compared with other method in literature. All the simulations given were implemented in Matlab, version 7.6.0 using the tool boxes Yalmip [START_REF] Lofberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] and SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF].

Based on Theorem 3.2 and the algorithm given in the previous section by initializing T i (i = 1, 2) for each example with appropriate dimension, γ T = 2, γ Z = 1.5, ϵ = 0.00000001 and the maximum number of iterations in Algorithm 1 has been fixed in 40. Other initial conditions for T i (i = 1, 2) could provide different results. ϵ

( γ T -γ Z γ T > ϵ
) is chosen in the form to get the closed values between γ T and γ Z , which means that the obtained γ is stationary.

Example 1.

Consider the continuous-time system [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF] with the vertices number being N = 2 given as the following:

A 1 =     -2.98 -0.57 0 -0.034 -0.99 -0.21 0.035 -0.0011 0 0 0 1 0.39 -5.5550 0 -1.89     , A 2 =     -2.98 2.43 0 -0.034 -0.99 -0.21 0.035 -0.0011 0 0 0 1 0.39 -5.5550 0 -1.89     B 1 = B 2 =     0.032 0 0 1.6     , E 1 = E 2 =     0 0 0 1     C 11 = C 12 = [ 1 0 0 2 ] , D 1 = D 2 = 1, F 1 = F 2 = 0, C 21 = C 22 = [ 0 0 1 0 0 0 0 1 ] , H 1 = H 2 = [ 0.5 -1 ]
and T i (i = 1, 2) has been initialized by

T 1 = [1 1 1 1] and T 2 = [1 1].
Table 1 shows the H ∞ performance bounds obtained for different value of g. As can be seen, the proposed approach provides less conservative results than the ones in [START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF]. Note that, N D increases by increasing the polynomial degree. Due to that, by increasing degrees the conservatism decreases even if the problem becomes more complicated. Now we show the output controlled z(t) and the ratio r(t) = √ ∫ t 0 z T (i)z(i)di ∫ t 0 w T (i)w(i)di in Figure 1 and Figure 2 respectively with w(t) = sin(t) t 1.5 +1 and initial condition of the state x(t) are null, under the obtained control law value K = [-5.4885 -3.3617].

Figure 2 shows that the ratio tends toward a constant value 0.5847, which is less than prescribed value γ min = 0.6163.

Figure 3 depicts the evolution of γ Z and γ T in Algorithm 1 proposed, where the improvements obtained through the iterative procedure in terms of smaller bounds are apparent and tend toward constant values.

Example 2.

In this example, make a comparison between Theorem 3.2 and the approaches given in [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF][START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF], where the system parameters are in the following form with the vertices number being N = 2: 2 for different degrees g. It can be mentioned that through this example the effectiveness of the proposed approach is clear, specially by increasing the polynomial degree g. Compared to [START_REF] Dong | Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties[END_REF][START_REF] Chang | Robust static output feedback H ∞ control design for linear systems with polytopic uncertainties[END_REF], the H ∞ performance values obtained for different degrees are always the best.

A 1 =   -0.
  B 1 =   -97.78 0 30   , B 2 =   -85.09 0 30   , E 1 = E 2 =   0 1 1   , C 11 = C 12 = 1, D 1 = 0, D 2 = 0, C 21 = [ 1 0 0 0 1 0 ] , C 22 = [ 1 0 0 0 1 1 ] , H 1 = H 2 = 0
In Figure 4 we present the computational result of transfer function (TF1 and TF2) for the two vertices derived from the system (1) compared with the prescribed value γ min . In this example we present a real application of the H ∞ SOF control to show the interest of the proposed method in reality. Consider the continuous-time system given in [START_REF] Oliveira | LMI relaxations for reduced-order H ∞ control for continoustime uncertain linear systems[END_REF] which represents 2 mass-spring model borrowed from [START_REF] Iwasaki | Robust performance analysis for systems with structured uncertainty[END_REF] as the following:

  A(α) B(α) E(α) C 1 (α) D(α) F (α) C 2 (α) H(α) 0   =            0 0 1 0 0 0 0 0 0 1 1 0 -q 1 (α)+q 2 m 1 q 2 m 1 -C 0 (α) m 1 0 1 m 1 q 2 m 2 -q 2 m 2 0 -C 0 (α) m 2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0           
with m 1 = 2, m 2 = 1, q 2 = 0.5, 1 ≤ q 1 ≤ 4 and 1 ≤ c 0 ≤ 4, yielding a polytope of N = 4 vertices and T i (i = 1, 2) has been initialized by

T 1 = [1 1 1 1] and T 2 = [1 1].
The approaches from [START_REF] Yaesh | Robust reduced-order output-feedback H control[END_REF][START_REF] Oliveira | LMI relaxations for reduced-order H ∞ control for continoustime uncertain linear systems[END_REF] and Theorem 3.2 in this paper are applicable for designing the robust static output feedback H ∞ controller (2) for this example, the minimum values of the H ∞ performance γ are given in Table 3. The computation results show that the proposed method for different value of g in this paper provides a better alternative design for this example. 1, Table 2 and Table 3 we want to clarify the complexity side of our approach and to put some comments on computational efficiency. Through this work, we were able to add some free matrices via descriptor redundancy approach in order to reduce the conservatism, but it increases the complexity N D and the dimension of our LMIs. This constraint may be considered as the cost to be paid to obtain the best results.

Conclusion.

In this paper, the problem of static output-feedback H ∞ controller for continuous-time linear systems has been studied. Sufficient condition for controller design has been derived via solving a set of linear matrix inequalities based on an iterative algorithm and the polynomial approach. In particular, it has been proved that the new proposed conditions are more relaxed than the existing ones which reduce the conservatism. Finally, numerical examples have been considered to illustrate the effectiveness of the proposed approach.

  HAIEK, T. ZOULAGH, A. HMAMED AND E. M. EL ADEL

948B.Figure 1 .

 1 Figure 1. Response of the controlled output z(t)

Figure 2 .

 2 Figure 2. Comparison between the ratio r(t) and the prescribed value γ min

Figure 3 .

 3 Figure 3. Behavior of γ Z , γ T and γ T -γ Z γ T along the iteration of algorithm for Example 1 and g = 1

Figure 4 . 3 .

 43 Figure 4. The computational result of transfer function

Theorem 3 .

 3 2 g = 3 Theorem 3.2 g = 2 Theorem 3.2 g = 1 [15] [13] state x(t) and the output controlled z(t) in Figure 5 and Figure 6 respectively with w(t) = sin(t) t 1.5 +1 and initial condition of the state x(t) are null with the controller gain (2) K = [-2.6277 1.9311] .

Figure 5 .Figure 6 . 1 Remark 4 . 1 .

 56141 Figure 5. The state behavior of the system

Table 1 .

 1 Comparison results of H ∞ performance

		Theorem 3.2 g = 2 Theorem 3.2 g = 1 [24]
	γ min	0.6163	0.6163	0.6581
	N D	76	52	27

with N D being the decision variable number

Table 2 .

 2 Comparison results of H ∞ performance

		Theorem 3.2 g = 2 Theorem 3.2 g = 1 [24]	[23]
	γ min	2.8007	2.8026	6.6836 7.0362

Table 3 .

 3 Comparison results of H ∞ performance with literature