Local model networks based mixed-sensitivity H-infinity control of CE-150 helicopters

Mohamed Redouane KAFI,Hicham CHAOUI,Suruz MIAH,Abderrazak DEBILOU

1.Laboratoire de Génie Electrique(LAGE),Université Kasdi Merbah Ouargla,Ouargla,Algeria;

2.Center for Manufacturing Research,Department of ECE,Tennessee Technological University,Cookeville,TN,U.S.A.;

3.Department of Electrical and Computer Engineering,Bradley University,Peoria,IL,U.S.A.;

4.Faculty of Sciences,Department of Electronics,Biskra University,Algeria

Local model networks based mixed-sensitivity H-infinity control of CE-150 helicopters

Mohamed Redouane KAFI1,4?,Hicham CHAOUI2,Suruz MIAH3,Abderrazak DEBILOU4

1.Laboratoire de Génie Electrique(LAGE),Université Kasdi Merbah Ouargla,Ouargla,Algeria;

2.Center for Manufacturing Research,Department of ECE,Tennessee Technological University,Cookeville,TN,U.S.A.;

3.Department of Electrical and Computer Engineering,Bradley University,Peoria,IL,U.S.A.;

4.Faculty of Sciences,Department of Electronics,Biskra University,Algeria

In this paper,a local model network H-infinity control is proposed for CE-150 helicopter stabilization.The proposed strategy capitalizes on recent developments on H-infinity control and its promising results in robust stabilization of plants under unstructured uncertainties.CE-150 helicopters are known for their varying operating conditions along with external disturbances.Therefore,local model networks are introduced for their adaptive feature and since they provide a powerful combination of fuzzy logic and conventional linear control techniques to control nonlinear systems without the added computational burden of soft-computing techniques.Using the fact that the system can be linearized at different operating points,a mixed sensitivity H-infinity controller is designed for the linearized system,and combined within a network to make transitions between them.The proposed control structure ensures robustness,decoupling of the system dynamics while achieving good performance.A comparison is carried-out against the well-known proportional-integral-derivative(PID)control technique.Results are presented to illustrate the controller’s performance in various operating conditions.

H-infinity control,helicopter,local model network,robust stabilization

1 Introduction

Helicopters have received a thorough attention and have been extensively used lately to demonstrate the effectiveness of different kinds of controllers.They are considered as a well-established benchmark challenge for many control problems and have been used in various applications such as transportation,and aboveground monitoring[1,2].Varying operating conditions along with structured and unstructured uncertainties,such as external disturbances,are among the numerous challenges that need to be addressed to successfully control such highly complex nonlinear unstablesystems.Unlike quadrotors that exhibit a good degree of decoupling,which makes them easier to control,helicopters are open-loop unstable systems and their dynamics is highly nonlinear,time-varying,uncertain,and strongly coupled.

Various flight control techniques have been proposed in the literature for the helicopter flight control problem[3–9],including robust adaptive control[6,10],state-dependent Riccati equation control[11],sliding mode control[12],trajectory tracking control[13,14],backstepping control[4,8,15],fuzzy control[16,17]and neural network control[18,19].In[10],robust nonlinear motion control of a helicopter is developed.In spite of the simplicity of control law design based on linearization around an operating point of the states,the control system’s performance and stability are achieved for only the approximated system and are not guaranteed for the overall system.On the other hand,fuzzy logic based controllers are incapable of incorporating any learning already acquired about the dynamics of the system in hand and neural network based controllers remain incapable of incorporating any human-like expertise.Moreover,these tools achieve outstanding performance at the expense of a heavy computation.Furthermore,they are based on heuristic which makes tuning not trivial[20,21].

On another aspect,local model network theory has received a thorough attention and an increasing interest from the control community[22].This is due to its simplicity since it is based on a combination of a set of linear controllers, where each of them corresponds to an appropriate operating point.Thus,the resulting control system is able to achieve good performance for a large operating range in the presence of plants nonlinearities and uncertainties.On the other hand,H∞control is considered as one of the promising robust control techniques.Its limitation is essentially a frequency domain optimization method for designing robust control systems.H∞r(nóng)efers to the space of stable and proper transfer functions.It has evolved since the initial seminal work of Zames[23].The Book by Francis records the progress in the initial development of the subject[24],much of which was concerned with solving the Nehari optimization problem.The state-space method of solving the H∞design problems is well-established as a very practical and a simple means of computing H∞controllers(see[25]).The polynomial approach for solving these problems has also been developed over the last few years[26,27]and seen recent advances through the use ofJspectral factorization algorithms[28,29].Besides,the control of a helicopter is a challenging problem since the system is multivariable,nonlinear,and unstable in open loop.In addition to uncertain parameters,and at least of the sixth order,depending on the modeling precision.All inputs and outputs are coupled.To effectively handle strongly coupled nonlinearities, model uncertainties and time-varying unknown perturbations,local model networks are combined in this paper with H∞control for helicopter stabilization.

The contribution of this paper is to propose a local model network based H∞controller for CE-150 helicopter stabilization problem.Local model networks provide a conceptually powerful combination of fuzzy logic and conventional linear control techniques providing an alternative approach for the control of nonlinear systems.Using nonlinear systems linearization at different significant operating points,H∞controller is designed for the linearized system,which is then combined in a local model network control structure.Therefore,decoupling of the system dynamics is achieved which is a key in obtaining good performance in the presence of uncertainties.The rest of the paper is organized as follows:Section 2 outlines the nonlinear model of helicopter simulator.Section 3 formulates the state-space model and outlines its linearization.The local model networks based H∞control synthesis is detailed in Section 4.In Section 5,simulation and experimental results are reported and discussed.Conclusion with few remarks and suggestions is also presented.

NotationsThroughout this paper,vectors and matrices will be denoted by lower case and upper case bold letters,respectively.Scalar quantities will be denoted by non-bold letters.We let R to denote the set of real numbers.

2 Helicopter model

Following[30],we consider a laboratory helicopter whose body is connected to a fixed base.Hence,two degrees of freedom of the helicopter are enabled where the elevation angle ψ(rotation around horizontal axis)and the azimuth angle ?(rotation around vertical axis)describe the motion of the helicopter body.The parameters describing the helicopter motion is depicted in Fig.1.The body is actuated by two DC motors which drive the main and tail propellers.The rotor axes acting on these propellers are orthogonal to each other.

Fig.1 Torques acting on the helicopter body in the vertical planes.

Suppose thatˉu1(t)andˉu2(t)represent the voltages driving the main and tail motors,respectively,at timet≥0.As such,the helicopter model can be treated as a two-input two-output nonlinear multi-variable system(Fig.2).Considering the forces acting on the vertical helicopter body,the dynamics of the elevation angle is given by

where

Imoment of inertia of the helicopter body around horizontal axis;

τ˙?centrifugal torque;

τmgravitational torque;

τGgyroscopic torque;

τf1friction torque(Coulomb and viscous);

τ1elevation driving torque(main propeller influence);

τω1main propeller angular velocity;

mmass;

g gravity;

l1distance fromz-axis to main rotor;

kω1main rotor constant;

kGgyroscopic coefficient;

Bψviscous friction coefficient(aroundy-axis);

CψCoulomb friction coefficient(aroundyaxis).

Fig.2 Two degrees of freedom,Helicopter CE-150.

Similar to the elevation dynamics,we consider the forces in the horizontal plain(see Fig.3)taking into account the forces acting on the helicopter body in the direction of the azimuth angle ?.The dynamics of ? is given by

where

Iψmoment of inertia around vertical axis;

τ2stabilizing motor driving torque;

τf2friction torque(coulomb and viscous);

τrmain motor reaction torque;

l2distance fromz-axis to stabilizing tail rotor;

kω2constant for the tail rotor;

ω2angular velocity of the tail rotors;

B?viscous friction coefficient aroundz-axis;

C?Coulomb friction coefficient aroundz-axis.

Fig.3 Torques acting on the helicopter body in the horizontal planes.

Similar to the body dynamics in elevation,no connection between the speed of the side propeller and friction torque around vertical rotational axis has been introduced into the derivation of an analytical model of the helicopter dynamics[31].The torque τris significant and arises from the torque generated by the main motor acting on rotating body. Note that the propulsion system of the CE-150 helicopter model is mainly driven by two independent DC motors.Under certain assumptions on the DC motor dynamics as stated in[30],the DC motor and propeller dynamics are given by the following equations:

jis the motor index(j=1 for main motor andj=2 for tail motor),

ijarmature current;

ωjrotor angular velocity;

τjmotor torque;

τcjcoulomb friction load torque;

τpjair resistance load torque;

Rjarmature resistance;

kijtorque constant;

kbjback-emf constant;

Ijrotor and propeller moment of inertia;

Bjviscous friction coefficient;

Cjcoulomb friction coefficient;

Bpjair resistance coefficient(laminar flow);

Dpair resistance coefficient(turbulent flow),∈R.

Fig.4 complete system dynamics.

3 State-space model and linearization

The compact form of(12a)–(12h)can be expressed as

withx2=x4=x6=x8=0 andx3=α∈R is simply a constant.Clearly,ˉu1=x5andˉu2=x7.The solution forx5andx7can be obtained from(14a)and(14b)assuming the fact thatx1≡ψ(elevation angle)takes the value from[0,π].By doing so,we obtain

Forx1=0,5π/16,and 9π/16,the three different equilibrium points are

fora1=0.1165,a2=0.268,a3=0.1959,b1=0.062,b2=0.0408,b3=0.0202,T1=0.1,T2=0.25,I=184,I?=494.3,Bψ=0.08,B?=0.04,KG=0.3185 and τg=0.071.

4 Local model networks based H∞control

Local model networks(LMNs)operate based on the interpolation of the local models,weighted by their associated validity functions.The output of an LMN with? local models can be expressed as[32],

Usually,when the validity functions do not automatically sum up to 1,the partition of unity is achieved through normalization.This principle is illustrated in Fig.5.A soft-switching technique is used to combine all control outputs by a weighted(validity)function.As such,the output of global LMN controller is a weighted average of the local control outputs.This strategy enables a smooth transition of the control output between different operating points.

Fig.5 Local model network control.

The control law is given by

Φiis a function that depends on the operating point and must satisfy the condition(18).The functions Φiallow us to privilege each compensator in its functional domain.A simple choice of the functions Φiis based on the use of trapezoidal functions as indicated in Fig.6.

Fig.6 Switching.

In region(1),the compensatorK1is in operation,whereas in region(2)the system is controlled by linear combination ofK1andK2,however,the compensatorK2is used in region(3).For acceptable behavior of the system,the operating point is described by a variable which is slowly varying with time.

For simple cases,we can use the reference or the output as an indicator of the operating point.We note that the control principle presented in this paragraph comes closer to the principle of the adaptive control with advantage that parameter identification and estimation part is avoided,which yield less computational load and better time response.In this work,? is taken to be equal to 3.

On the other hand,mixed-sensitivity H∞control design consists of synthesising a controllerK(s)to minimize low frequency disturbances at the plant output and the high frequency control effort while providing robustness to additive uncertainty at high frequencies.Fig.7 shows a feedback control system with augmented plant,whereG(s)is a plant,andWs(s)andWT(s)are weighting matrices.In this paper,mixed-sensitivity H∞control is applied to CE-150 helicopter model.Therefore,the sensitivity functionS(s),and the complementary sensitivity functionT(s)are defined as follows[34]:

Fig.7 A mixed sensitivity configuration.

It is noteworthy from(20)that the minimisation ofT(s)at high frequencies leads to robustness to uncertainties.Therefore,designing a control law to meet the specifications consists of a proper selection of the weighting matricesWs(s)andWT(s),which capture the desired closed-loop dynamics.Then,the design of a stabilizing controllerK(s)is carried-out by minimizing the following cost function:

The selection process of the weighting matrices is repeated until satisfactory performance and robustness of the closed-loop system are achieved.The matrixWs(s)is computed as follows[35,36]:

Fig.8 Control structure for H∞local model network.

Fig.9 Validity function and switching mechanism.

5 Simulation results

The purpose of this section is to show the tracking error performance of the azimuth angle ψ and the elevation angle ? for the linearized CE-150 helicopter model(16).The weight matricesW?(s),?=1,2,3,for three different operating points are chosen as

In order to keep a minimum steady-state tracking error,for azimuth angle control ?,a gain compensator is used,for each local controller(see Fig.5).The performance of the control law(19)are summarized in Figs.10–15.Comparison against a PID controller is also given.The step responses for reference azimuth and elevation angles(ψ?=0.25,??=0)and(ψ?=0,??=0.25)are shown in Figs.10 and 11,respectively.

Fig.10 Unit step response for reference ψ?=0.25 and ??=0.(a)Tracking performance.(b)Tracking error.

While the desired and actual tracking performances for these reference inputs are shown in Figs.10(a)and 11(a),a good coupling is observed through the tracking errors shown in Figs.10(b)and 11(b).It is noteworthy that while the PID controller shows a slower response in Fig.10,unacceptable overshoot is observed in Fig.11.We repeat this setup for the reference inputs(ψ?=0.5,??=0)and(ψ?=0,??=0.5)and the tracking performance of the azimuth and elevation angles(desired and actual),and their corresponding tracking errors are revealed in Figs.12 and 13,respectively.

Fig.11 Unit step response for reference ψ?=0 and ??=0.25.(a)Tracking performance.(b)Tracking error.

Fig.12 Unit step response for reference ψ?=0.5 and ??=0.(a)Tracking performance.(b)Tracking error.

Fig.13 Unit step response for reference ψ?=0 and ??=0.5.(a)Tracking performance.(b)Tracking error.

As can be noticed that as the settling time decreases,the performances in terms of oscillations are deteriorated because of the neglected nonlinear dynamics,as expected.Similar to previous simulations,the PID controller shows a slower response in Fig.12 while significant overshoot is observed in Fig.13.Unlike previous simulations,we choose(ψ?=0.5,??=0.5)in order to illustrate the controller’s ability to sustain the helicopter’s dynamic performance on the same azimuth and elevation angles(see Fig.14).

Finally,Fig.15 reports the controller’s performance due to a sudden change on the desired azimuth and elevation angles.

As it can be seen,the LMN controller achieves a good convergence.It is important to note that the ? angle convergence speed is relatively faster as opposed to the ψ angle.As illustrated in Figs.14 and 15,the LMN controller provides a faster step response convergence with less overshoot compared to the PID controller.The above simulation results reveal that the H∞controller coupled with the local model network has the ability to track a predefined trajectories of the azimuth and elevation angles regardless of their complexities.Furthermore,the proposed controller shows similar behavior and kept good performance in varying operating conditions.

Fig.14 Unit step response for reference ψ?=0.5 and ??=0.5.(a)Tracking performance.(b)Tracking error.

Fig.15 Time change step response for ψ?and ??.(a)Tracking performance.(b)Tracking error.

6 Conclusions

In this paper, a local model network basedH∞control technique is proposed to solve the stabilization problem of CE-150helicopters.Using the fact that the system can be linearized around a set of operating points,we have designed an H∞controller for the linearized system.For this,we have solved the so-called mixed sensitivity problem.The problem was transferred to a standard H∞problem and solved for the stabilizing gain that satisfies the desired criteria,next we embed them within a network.From simulation results,we notice that the obtained controller ensures the decoupling of the system dynamics,and good performance.Therefore,we can conclude that the local model networks based H∞control is suitable for the stabilization of the proposed helicopter simulator model.

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17 July 2015;revised 2 June 2016;accepted 2 June 2016

DOI10.1007/s11768-017-5073-x

?Corresponding author.

E-mail:kafi.redouane@univ-ouargla.dz.Tel.:+213662091913.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Mohamed Redouane KAFIreceived the B.Sc.degree in Electrical Engineering from the University Batna,Algeria,the M.Sc.degree from the Military Polytechnical School,Algeria.He is currently working towards his Ph.D.degree in Electrical Engineering from the University Biskra,Algeria.He is currently an Assistant Professor at the Electrical Engineering(EE)department at the University Kasdi Merbah Ouargla,Algeria.His research lies in the broad area of robust control design for nonlinear systems.In particular,he conducts research on UAV design,control systems and mechatronics.He is a founder member of the Electrical Engineering research Laboratory(LAGE)at the University Kasdi Merbah Ouargla,and a member of identification,command,control and communications laboratory at the University Biskra.Mr.KAFI is a author/co-author of technical papers,which are published in leading journals and conferences.E-mail:kafi.redouane@univ-ouargla.dz.

Hicham CHAOUI(S’01-M’12-SM’13)received the B.Sc.degree in Electrical Engineering from the Institut supérieur du Génie Appliqué(IGA),Casablanca,Morocco,in 1999,the M.A.Sc.degree in Electrical Engineering,the M.Sc.degree in Computer Science(with honors),the graduate degree in Project Management,and the Ph.D.degree in Electrical Engineering(with honors)all from the University of Quebec,Canada,in 2003,2005,2007,and 2012,respectively.His career has spanned both academia and industry in the field of intelligent control and renewable energies.Prior to his academic career,he held various engineering and management positions including Vice-President of Innovation and Technology Development at TDE Techno Design,Montreal,QC,Canada.He is currently an Assistant Professor at Tennessee Technological University,TN,U.S.A.and an Adjunct Professor at the Université du Québec Trois-Riviéres(UQTR),QC,Canada.His research interests include adaptive and nonlinear control theory,intelligent control,robotics,electric motor drives,and energy storage.His scholarly work has produced more than 75 journal and conference publications.Dr.Chaoui is a senior member of IEEE.He was a recipient of the Best Thesis Award(health,natural science,and engineering)and the Governor General of Canada Gold Medal Award for his doctoral dissertation in 2012.E-mail:hchaoui@tntech.edu.

Suruz MIAHhas his B.Sc.degree from Khulna University of Engineering and Technology(KUET),Bangladesh,and both M.Sc.and Ph.D.degrees from University of Ottawa,Canada.He is currently an Assistant Professor of the Electrical and Computer Engineering(ECE)department at Bradley University and also holds an Adjunct Professor position at the School of Electrical Engineering and Computer Science of the University of Ottawa.Before joining the ECE department at Bradley,he spent more than two years conducting research on multi-agent systems and control for Defence Research and Development Canada(DRDC)’s Centre for Operational Research and Analysis(CORA),Ottawa,Ontario,Canada.His research lies in the broad area of cyber-physical systems.In particular,he conducts research on mobile robot navigation,control systems,mechatronics,multi-agents systems and control.He is the founder of the Robotics and Mechatronics(RAM)Research laboratory at Bradley and a research member of the Machine Intelligence,Robotics,and Mechatronics(MIRaM)laboratory at the University of Ottawa.Dr.Miah is an author/co-author of more than 40 technical papers,which are published in leading journals and conferences.He has been an active member of the Institute of Electrical and Electronics Engineers(IEEE)since 2007,and has been serving as a reviewer of many prestigious journals and conference proceedings.E-mail:smiah@bradley.edu.

Abderrazak DEBILOUhas the DES degree from University Batna,Algeria,the M.Sc.degree from the University Bordeaux 1,France,and the Ph.D.degree from the University Bordeaux 1,France.He is currently a professor at Electrical Engineering(EE)department,and vice-president at the University Biskra,Algeria.His research lies in the broad area of robust control design for nonlinear systems.In particular,he conducts research on UAV design,control systems,robotics,and image processing.He is a founder of the identification,command,control and communications Laboratory,at the University Biskra.Dr.Debilou conducts and supervises multiple research projects.E-mail:debilou@univ-biskra.dz.