Time Varying Attitude Control Strategies for the Myriade Satellites

Alexandru-Razvan Luzi, Jean-Marc Biannic, Dimitri Peaucelle and Jean Mignot

Abstract— This paper studies time-varying structures forsatellite attitude control systems. The considered satellites arefrom the Myriade family developed by the French SpaceAgency (CNES). Analysis of the existing control law andcontrol requirements show that model reference control issuitable provided that the reference model is time-varying. Amodel reference linear parameter varying (LPV) strategy isthus considered and a synthesis procedure ensuring closed-loop stability and tracking of a time varying model is given.A different time-varying scheme allowing model following issimple adaptive control (SAC). After recalling the stabilityconditions for this structure, the two control laws are comparedand applied to a simulation model of the DEMETER satellite,part of the Myriade program.

I. INTRODUCTION

The Centre National d’Etudes Spatiales (CNES) Myriadeseries is a new concept of modular microsatellites weighingless than 200 kg that afford quick, low-cost access to space.The aim of Myriade is to bring space research within easierreach of the scientific community, particularly in exploratoryfields where scientific results are not necessarily guaranteed.DEMETER (detection of electromagnetic emissions trans-mitted from earthquake regions) [1], [2], [3], the first satelliteof the Myriade series was launched in 2004 and it was takenoff service in the beginning of 2011, thus surpassing theinitial estimate orbit life by several years.

The on-orbit life of the Myriade satellites is decomposed,as for all satellites in general, into phases, allowing theconvergence from the separation state to the mission state,the orbit control, and the safe state in case of fault. Foreach phase or mode, a specific set of attitude pointingrequirements is to be met by using a specific set of actuatorsand sensors. Inside a mode, several attitude control andestimation algorithms can be designed, in order to meetincreasing performance levels requirements. Over the pastthirty years, the number of Attitude and Orbit ControlSystem (AOCS) modes has been reduced, so that genericarchitectures now typically consist of three modes (safehold,mission and orbit control mode). In the same time thesoftware complexity has increased to meet ever more de-manding requirements. Thus, the switching between differentAOCS modes requires permanently enhanced robustness to

This work was supported in part by the French Space Agency CNES.A.-R. Luzi and J.-M. Biannic are with ONERA, The French

Aerospace Lab, 2 avenue Edouard Belin, F-31055 Toulouse, Francealexandru.luzi@onera.fr, biannic@onera.fr

A.-R. Luzi, J.-M. Biannic and D. Peaucelle are with CNRS , LAAS , 7 avdu Colonel Roche, F-31400 Toulouse, France and Universite de Toulouse ,LAAS , F-31400 Toulouse, France peaucelle@laas.fr

J. Mignot is with CNES ; DCT/SB/PS ; BPI 1712, 18 avenue E. Belin,F-31401 Toulouse, France.

cope with more stringent kinematics conditions, while alsoimproving the pointing accuracy.

In this paper, the problem of the reaction wheel controlsoftware inside the mission mode of the Myriade satellites isconsidered. Due to the limitations of the satellite actuators,a classical LTI controller would not be adequate for attitudecontrol: the controller would either demand huge actuatorsolicitation at large pointing errors or generate a very slowclosed-loop response. Therefore, the desired closed loopbehaviour can only be obtained with a varying controller:when the pointing error is large the controller has to ensurea response with a limited speed and a rapid and preciseresponse is demanded when the pointing error is smaller.

The controller of the Myriade satellites satisfies this de-mand by switching between two control laws - at largepointing errors a speed tracking loop is used and at smallangles the controller switches to an attitude proportional- derivative law. A satisfying response time and a limitedcontrol activity (in particular a limitation of the reactionwheel rotation speed and a null torque for transient regimes,once the travelling speed is reached) are thus ensured. Adiscontinuity in the control torque can however appear ifthis travelling speed has not been reached at the switchingpoint. The switch also induces abrupt changes in the closed-loop plant and makes the stability analysis difficult. Theselimitations have been surpassed by the use of the LPVformulation in [4] where an LPV approximation of theswitching law and adequate tools for stability analysis weredeveloped.

Motivated by the necessary time-variant nature of the de-sired closed-loop behaviour, a specific attention is devoted inthis paper to control methods enabling to track time-varyingreference models. Among various possibilities, one focuseson two different strategies: Linear Parameter Varying (LPV)control techniques and Simple Adaptive Control (SAC).

Initial research efforts on LPV control techniques in the1990’s ([5], [6], [7]) were essentially motivated by the needto surpass the main weakness of standard gain-schedulingapproaches which offer no guarantee when rapid changesoccur in the scheduling variables ([8], [9]). More recently,during the past decade, many efforts were devoted to variousimprovements in order to reduce the conservatism of standardLPV design approaches ([10], [11], [12]). Unfortunately,most of the proposed extensions introduce many additionalvariables and lead to (possibly non-convex) optimizationproblems which rapidly become intractable beyond threeparameters.

In this paper, since there is only one scheduling variable,the standard LPV control framework can be applied. More

2012 IEEE International Conference on Control Applications (CCA)Part of 2012 IEEE Multi-Conference on Systems and ControlOctober 3-5, 2012. Dubrovnik, Croatia

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precisely, the chosen design approach is based on a polytopicH∞ synthesis [6] for which a convex characterization of theLPV controllers is easily obtained and remains numericallytractable for rather high-order systems (with 20 to 30 states).To ensure such nice numerical properties, the approach isnecessarily based on the use of quadratic Lyapunov functionswhich are known to be rather conservative.

Another technique to be considered for the model trackingproblem is simple adaptive control. This adaptive schemeproposes to directly tune the control gains with the soleoutput measurements. The results presented in this paper arebased on the framework developed in [13], where closed-loop stability is proved for square systems (which havethe same number of inputs and outputs) based on passivityproperties. These results have been extended in [14], [15],[16] to ”almost stable” non square systems, making simpleadaptive control possible for all systems stabilizable by LTIcontrollers.

This paper is organized as follows. Section II gives abrief description of the Myriade AOCS control loop and ofthe quasi-LPV formulation of the switching controller. TheLPV model reference method and the model reference simpleadaptive control are exposed in Section III. In Section IV thetwo techniques are applied to the satellite attitude controlproblem. Simulation results are presented and compared.Finally, some conclusions are driven.

NotationsRm×p is the set of real m by p matrices. AT is the

transpose of the matrix A. Tr(A) is the trace of A. 1and 0 are respectively the identity and the zero matricesof appropriate dimensions. ||A||• =

√Tr(AAT ) is the

Frobenius norm of A.φ(Rm×p → Rm×p) is a function defined by φ(K) =

ψ(||K||2•) · K where ψ([0 νβ[→ [0 + ∞[) is a scalarfunction behaving as a dead-zone close to the origin ψ(0 ≤k ≤ ν) = 0 and a monotone increasing barrier function thatgoes to infinity as k converges to νβ. An example of suchfunction is ψ(ν ≤ k < νβ) = (k − ν)/(νβ − k).

II. MYRIADE AOCSThis section briefly presents the AOCS control loop of the

Myriade satellites, with an emphasis on the switching-basedcontroller, represented in Fig. 1. A full description of theDEMETER AOCS control loop is given in [3].

A. Satellite modelA satellite model where the three axes are independent is

considered. Each axis is modelled by a double integrator:

Jiωi = Ti , θi = ωi (1)

where Ji is the axis inertia and Ti the total torque (controlaction and disturbances). To this model, several pairs of polesand zeros representing flexible modes are added. For the firstaxis of the DEMETER satellite, which will be used in thefollowing sections, the considered model is a fourth order:

Hsat,1(s) =0.03933s2 + 0.0005437s+ 0.2485

s4 + 0.01706s3 + 7.797s2(2)

Fig. 1. DEMETER AOCS loop

The controlled torque is delivered by accelerating the reac-tion wheel, represented in Fig. 2 where torque and speedsaturations can be observed :

HRW(s) =1.214s+ 0.7625

s2 + 2.40s+ 0.7625(3)

Fig. 2. Reaction wheel model

B. Flight software

The flight software is composed of an angular velocity es-timator and a nonlinear control law followed by a stabilizinglinear filter. The satellite angular velocity is estimated fromthe angular position via a high pass filter of time constant0.5s. The reaction wheel control system consists of a speedcontrol law for large pointing errors which switches to aproportional-derivative controller for small pointing errors:{

Tp = −(δω + ωdsign(δθ)) if |δθ| > θLTp = −(Kθδθ +Kωδω) if |δθ| ≤ θL

(4)

where δθ is the pointing error.Parameters θL and ωd allow to optimize a trade-off be-

tween convergence time and control activity. For DEMETER,their values are: ωd = 0.015 deg/s and θL = 0.3 deg. Thegains Kθ and Kω are chosen so that the produced torquein null, once the reference speed is reached, and continuousat the switching point. This implies KθθL = Kωωd and thenumerical values are: Kθ = 0.1 and Kω = 2. The stabilizinglinear filter was designed by a multi objective H2/H∞ LMIsynthesis with pole placement in an LMI region [1].

C. Quasi-LPV Description of the Closed Loop Plant

We present here the quasi-LPV approximation of thecontrol law obtained in [4]. The switching controller ischaracterized by the varying parameter λ:

Tp(λ) = λ(ρδθ)Kθδθ − (1 + λ(ρδθ)(Kω − 1))ω (5)

where ρ = Kω/θL and λ(v) is equal to 1 when |v| ≤ 1 and1|v| when |v| > 1.

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The nonlinear simplified closed loop (with the satellitemodelled as a double integrator) can thus be described as aquasi-LPV system that depends affinely on a parameter λ:(

θω

)=

(0 1

−KθλJ − 1+(Kω−1)λJ

)(θω

)+

(0KθλJ

)θr

(6)where θr is the attitude reference.

III. TIME-VARYING CONTROL STRUCTURES

The need for control strategies which could ensure a time-varying behaviour was presented in the Introduction. In theconsidered application the system is represented by an LTIand the problem proposed by this paper can therefore bestated as follows:

Problem 1: Given the system Σ with the state-space rep-resentation:

(Σ)

{xs = Asxs +Bsusys = Csxs

(7)

and the LPV reference model M affine with respect to thevarying parameter θ, evolving in polytope Θ:

(M)

{xm = Am(θ)xm +Bm(θ)umym = Cm(θ)xm +Dm(θ)um

(8)

find a controller K such that the system output ys followsthe model reference ym.

The following sections illustrate how the two consideredstrategies can be used for solving this problem. The maindifference between the presented techniques is the way thecontroller depends on the varying parameter: in the LPV casethis dependence is explicit while for the SAC structure theadaptation is based on system and model outputs.

A. LPV Control

In this framework, the parameter θ is considered availableto the controller whose state-space matrices vary dependingon its value. The control problem can then be written in thestandard form represented in Fig. 3:

Fig. 3. Standard form for LPV systems

The objective is to find a parameter-varying controllerK(s, θ) such that the following properties are respected:

Property 1: the closed-loop LPV plant of Fig.3 is stabi-lized for any admissible trajectory of the parameters in Θ.

Property 2: the L2 induced norm of the transfer from wto z is bounded by a parameter γ to be minimized.z = ys − ym denotes the error to be minimized and the

varying nature of the interconnection model P (s, θ) is onlyintroduced by the reference model.

The following theorem, based on the result of [6] givesnecessary and sufficient conditions for the existence of anoptimal polytopic H∞ controller:

Theorem 1: Let the polytopic representation of P (s, θ):

P (s, θ) ,N∑i=1

αi(θ)Pi(s)

with Pi ,

Ai B1i B2

C1i D11i D12

C2 D21 0

and where αi are such

that: θ =∑Ni=1 αi(θ)θi, αi(θ) > 0,

∑Ni=1 αi(θ) = 1,

the index i indicating the value of the associated matrix orparameter vector at each vertex of the polytope.

There exists a polytopic controller

K(s, θ) =

N∑i=1

αi(θ)Ki(s), Ki(s) =

(AKi BKiCKi DKi

)such that Properties 1 and 2 hold if and only if there existtwo symmetric positive definite matrices R and S such thatfor i = 1, . . . , N :

N TR

AiR+RATi B1i RCT1iBT1i −γI DT

11i

C1iR D11i −γI

NR < 0

N TS

AiS + SATi SB1i CT1iBT1iS −γI DT

11i

C1i D11i −γI

NS < 0

where:

NR = diag(NR,1), NS = diag(NS ,1),NR = Ker ( BT2 DT

12 ), NS = Ker ( C2 D21 ).Based on matrices R and S the controller matrices at each

vertex are obtained through LMI solving as indicated in [17].

B. Adaptive Control

The control law introduced in [13] is considered:

u = K(t)e+ L(t)xm +M(t)yr (9)

where yr is the output reference, xm and ym the state andoutput of the reference model and e the tracking error, e =ys − ym. The controller gains are adapted as follows:

K = −GeeTΓK − φ(K − F0)ΓKL = −GexTmΓLM = −GeyrΓM

(10)

where ΓK , ΓL and ΓM are constant positive definite matricalcoefficients that control the rate of adaptation.

The closed loop stability is guaranteed by the feedbackgain K, acting on the tracking error. An appropriate choiceof the G matrix and of the φ function parameters guaranteesthe stability of the close loop with K adapted according to(10). Conditions for finding such values are given in [16],under the assumption that there exists a stabilizing staticoutput feedback gain F0. The adaptive gain K then varies ina domain centred in F0 and bounded by the barrier function.

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The L and M terms of the adaptation law are added inorder to ensure model tracking and are part of the SACframework defined in [13]. They perform a steepest descentminimisation on the tracking errors ([18]) and asymptotictracking is guaranteed if there exist some ideal gains suchthat perfect tracking is achieved. If the reference model istime-varying the ideal gains are expected to have the sameproperty.

It should be noted that whenever the tracking error is notzero the gain K will evolve in the same direction, definedby the matrix G. Thus, after a sufficiently long period oftime over which e has had non zero values, K will reach thebound defined by the barrier function. This can for instancebe the case after applying a series of references ([19]) or inthe presence of any noise. In order to avoid this behaviour, aσ term ([20]) is added and the adaptation equation becomes:

K = −GeeTΓK − σK(K − F0)ΓK − φ(K − F0)ΓK (11)

With this adjustment, the gain K evolves in the directiondefined by G whenever the tracking error is important andreturns to the nominal values F0 when the error is weak.

With respect to the LPV scheme, the SAC structure hasthe advantage of not including the reference model in thesynthesis model. There is thus no constraint on its struc-ture, but the respect of the tracking hypothesis should beconsidered. The reference model should therefore be chosenbased on some apriori knowledge about the system in orderto guarantee the problem feasibility.

IV. RESULTS AND COMPARISON

The LPV and SAC schemes were applied on the DEME-TER AOCS (involving one flexible satellite axis, actuatorand sensor models and disturbances torques) and simulationresults are given in this section.

Before presenting the application of the synthesis proce-dures, let us consider the choice of the reference model.For both schemes, the closed loop representation defined by(6) is chosen: these equations describe the way the satelliteattitude and velocity should evolve after applying an attitudereference. This choice may seem confusing as it will leadto a controller generating a closed-loop behaviour similarto the one obtained with the LPV approximation of theswitching law. No significant improvement in the attitudecontrol system will thus be obtained. However, as the aim ofthe present publication is to validate a synthesis methodologyby applying it to an attitude control problem, the use ofthis method remains relevant despite the particular choice ofthe reference model. Once that specifications will have beenwritten in the form of equations for the desired behaviour,the approach presented here will be directly applicable.

A. LPV and SAC Model Reference Attitude Control for theDEMETER Satellite

In order to apply the LPV strategy presented in Theorem1, the system must be put in standard form (Fig. 3), bycombining the satellite and reference model. The aim isto find a controller that would minimise the transfer from

the position reference θr to the position and speed errors:εθ = θ − θm and εω = ω − ωm, where the subscript mindicates the outputs of the reference model. The influenceof the initial states θ0 and ω0 also has to be taken intoconsideration. The errors should only be minimized withina certain frequency range, defined by the desired bandwidthand the control activity should be limited.

The control problem can thus be written as a mixedsensitivity specification where the following fictitious filtersare added: first order low pass filters on εθ and εω , wherethe low frequency gain defines the desired accuracy and theirbandwidth the frequency domain of interest. A first orderhigh pass filter is placed on the control signal and constantgains weight the initial conditions. The detailed view of thisrepresentation is shown in Fig. 4.

Fig. 4. Standard form synthesis model

The controller’s input is y =[θ ω θm ωm

]Tand

its matrices are found by minimising γ in the LMI problemdefined by Theorem 1. γ < 1 guarantees the respect of thespecifications defined by the added filters.

The last element to be defined is the satellite model. Thesimplest representation, given by (1) is chosen in order tolimit the order of the controller, the robustness of the closedloop being guaranteed by the use of an H∞ synthesis.

Based on the chosen reference model, the varying pa-rameter is λ(ρ · δθ) defined in (5). Its variation interval isΛ = [0.006, 1], λ = 1 corresponding to a pointing errorinferior to 0.15 deg and λ = 0.006 to the maximal errorvalue of 25 deg, beyond which the safe mode is activated.Based on Theorem 1 stabilizing controllers over the largestvariation domains of λ are sought. Starting from each endof the complete interval, two domains are found for whicha finite γ exists : Λ1 = [0.035, 1] and Λ2 = [0.006, 0.644]which correspond to admissible pointing errors varying inthe interval ∆Θ1 = [0, 4.25] deg and respectively ∆Θ2 =[0.23, 25] deg. Two controllers are found guaranteeing γ < 1over the intervals:

Λγ1 = [0.06, 1] ∆Θγ1 = [0, 2.5]deg (K1)Λγ2 = [0.006, 0.526] ∆Θγ2 = [0.29, 25]deg (K2)

As a unique controller is sought for the entire interval Λ,each of the two controllers is linearly extrapolated to themissing extremity of Λ: 0.006 for K1 and respectively 1 forK2; the extended controllers are denoted K1 and K2. Thestability of each one of these two controllers is then testedat the added extremity. The controller K1(λ) stabilizes P (λ)for λ = 0.006 and guarantees the respect of the performances

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imposed by the weighting filters. This is illustrated for thefilter Wθ on Fig.5, where the singular values of the transferfunction from θr to εθ are plotted for different values ofλ. K2(λ) does not stabilize plant P (λ) for λ = 1 and thiscontroller is therefore not considered for implementation.

Fig. 5. Position reference to position error transfer

Simulation results obtained with K1 are presented next.First the response to strong initial conditions (ω0 = 4ωd =0.07 deg/s) is considered and at t = 700s a step referenceof 25 deg is applied. Fig. 6 shows that the satellite measuredposition and speed are very close to the reference modeloutputs. The actuator response can be seen on Fig.7: a largecontrol action is required by the strong initial speed, yetthe saturation values are not reached. As it can be seen, thecontroller K1 ensures a good closed-loop behaviour over thecomplete variation domain of λ, even for large values of thesatellite speed (and therefore of the variation speed of λ).

The application of the adaptive control law (10) is pre-sented next. The reference model being the same as theone considered for LPV control the tracking feasibility isguaranteed. The block diagram of the adaptive closed loopis shown in Fig. 8, where xm = ym = [ θm ωm ] arethe reference model outputs and state and ys = [ θs ωs ]are respectively the measured angular position and estimatedangular velocity (see Fig. 1).

The principle of this scheme is to construct the adaptivelaw starting from an existing LTI stabilizing configuration.For the case of our application, this is represented bythe stabilizing linear filter and the proportional-derivativecontroller (4), F0 = [ Kθ Kω ] = −[ 0.1 2 ], which

Fig. 6. Satellite and reference model position and speed - LPV Control

Fig. 7. Reaction wheel torque and speed - LPV Control

Fig. 8. Adaptive satellite attitude control

enter the synthesis conditions. The adaptive gains are thenexpected to allow better closed loop performances and inthe studied case, these performances are the following of atime-varying reference model.

An adaptive law based on an LTI controller is also pre-sented in [21], where SAC is applied for the altitude controlof an unmanned aerial vehicle (UAV). The strategy used inthe cited publication consists of augmenting the plant withan LTI filter in series and with a parallel feedforward blockA passivity property is thus obtained and it is the augmentedoutput that is fed to the controller. The approach presentedhere has the advantage of using the physical outputs ofthe system: it is thus these outputs that will track thereference and not the augmented ones. Furthermore, matrixG allows defining adaptive laws for non-square systems. Forthe presented application, this enables the use of both angularposition and velocity. This is not the case of [21] where onlythe of the position information is considered.

For applying the adaptive law presented in Section III, weuse the procedure given in [16], in order to find G and theparameters defining the function φ. The obtained numericalvalues for β = 1.1 are G = [ 27.1 − 470.5 ], ν = 0.0027.K will thus vary in the bounded set: ||K − F0||• ≤ 0.055.

The σ-modification (11) is added to the adaptation lawof K, so that the gain will tend to the nominal values F0

when the tracking error is small. The same term is added tothe adaptative equation of L and the numerical values weretuned to σK = 10−5 and σL = 10−3. The values of thematrical coefficients were tuned to: ΓK = diag {100 100},ΓL = diag {0.1 10}, ΓM = 10−3.

The simulation results for the same situation as for theLPV model reference case are presented in Fig. 9, where thetracking errors obtained with the two structures are plotted.The evolution of the adaptive gains is shown in Fig. 10,the barrier effect on the feedback gain K can be observedbetween t = 700s and t = 1500s.

As it can be seen on Fig. 9 the asymptotic tracking of thereference attitude and velocity is achieved with both control

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Fig. 9. Tracking errors for SAC and LPV model reference control

Fig. 10. Evolution of the adaptive gains

laws and the LPV structure gives better performances thanthe SAC controller. This can be explained by the fact that thereference model is explicitly taken into consideration whendesigning the LPV law. The variable parameter λ is also usedby this algorithm. Relying only on satellite and referencemodel outputs, the adaptive law gives lower performancesfor this application. The SAC framework offers howevermore freedom in the choice of the reference model than theLPV structure. In particular, it does not have to be affinewith respect to the varying parameter, as required by theimplemented LPV synthesis. Thus, the class of referencemodels which could be considered is bigger, but for thepresented application the cost of this degree of freedom is aless precise transient response.

Concerning the implementation aspects, the two structuresare very different. The LPV algorithm consists of a 7th order4 inputs one output controller, while the SAC law implements5 variable gains followed by a 4th order linear filter.

V. CONCLUSIONS

Motivated by the interest of time-varying model referencecontrol for satellite attitude control systems, a synthesisprocedure for finding controllers allowing to track LPVreference models has been presented. The designed LPV

controller was tested on a DEMETER satellite benchmark,allowing to validate the design procedure. The LPV controlalgorithm was compared with an adaptive law and a betterbehaviour was obtained with the LPV scheme.

Future work will be dedicated to extending the LPV refer-ence model framework, in particular to developing referencemodels allowing to increase the performances of the attitudecontrol system.

REFERENCES

[1] C. Pittet, J. Mignot, and C. Fallet. LMI based multi-objective H∞control of flexible microsatellites. In IEEE Conference on Decisionand Control, Sydney, Australia, 1999.

[2] F. Buisson. The DEMETER program: a pathfinder to a high perfor-mance microsatellite line. In AIAA USU conference on small satellites,pages 441–446, Logan, Utah, USA, 2003.

[3] C. Pittet and D. Arzelier. DEMETER: a benchmark for robustanalysis and control of the attitude of flexible microsatellites. In IFACSymposium on Robust Control Design, Toulouse, France, 2006.

[4] J.-M. Biannic, C. Ross, and C. Pittet. LPV analysis of switchedcontrollers for attitude control systems. Journal of Guidance, Controland Dyanmics, 34(5):1561–1566, 2011.

[5] P. Apkarian and P. Gahinet. A convex characterization of gain-scheduled H∞ controllers. IEEE Transactions on Automatic Control,40(5):853–864, May 1995.

[6] P. Apkarian and J-M. Biannic. Self-scheduled H∞ control of missilesvia Linear Matrix Inequalities. Journal of Guidance Control andDynamics, 18(3):532–538, May-June 1995.

[7] F. Wu, A. Packard, and G.J. Balas. LPV control design for pitch-axismissile autopilots. In Proceedings of the IEEE CDC, New Orleans,LA, USA, December 1995.

[8] J.S. Shamma and M. Athans. Analysis of gain scheduled control fornonlinear plants. IEEE Transactions on Automatic Control, 35(8):898–907, August 1990.

[9] J.S. Shamma and M. Athans. Guaranteed properties of gain scheduledcontrol for linear parameter-varying plants. IEEE Transactions onAutomatic Control, 27(3):559–564, March 1991.

[10] F. Wang and V. Balakrishnan. Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems. IEEETransactions on Automatic Control, 47(5):720–734, May 2002.

[11] F. Wu and K. Dong. Gain scheduling control of LFT systems usingparameter-dependent Lyapunov functions. Automatica, 42(1):39–50,January 2006.

[12] C.W. Scherer and I.E. Kose. Gain-scheduling synthesis with dynamicD-scalings. In Proceedings of the IEEE CDC, New Orleans, LA,USA, December 2007.

[13] H. Kaufman, I. Barkana, and K. Sobel. Direct adaptive controlalgorithms. Springer, New York, 1994.

[14] A.L. Fradkov. Passification of non-square linear systems and feedbackYakubovich-Kalman-Popov lemma. European J. of Control, 6:573–582, 2003.

[15] D. Peaucelle and A.L. Fradkov. Robust adaptive L2-gain controlof polytopic MIMO LTI systems - LMI results. Systems & ControlLetters, 57(11):881–887, 2008.

[16] D. Peaucelle, A. Drouot, C. Pittet, and J. Mignot. Simple adaptivecontrol without passivity assumptions and experiments on satelliteattitude control DEMETER benchmark. In IFAC World Congress,August 2011.

[17] P. Gahinet and P Apkarian. A linear matrix inequality approach toH∞ control. Int. J. of Robust and Nonlinear Control, 4:421–448,1994.

[18] I Barkana. Gain conditions and convergence of simple adaptive con-trol. International Journal of Adaptive Control and Signal Processing,19(1):13–40, 2005.

[19] D. Peaucelle, B. Andrievsky, V. Mahout, and A. Fradkov. Robustsimple adaptive control with relaxed passivity and PID control of ahelicopter benchmark. In IFAC World Congress, 2011.

[20] P. Ioannou and P. Kokotovic. Adaptive Systems with Reduced Models.Springer-Verlag, Berlin, 1983.

[21] I Barkana. Classical and simple adaptive control for nonminimumphase autopilot design. Journal of Guidance, control and Dynamics,28(4):631–638, 2005.

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