STATE-FEEDBACK-BASED FRACTIONAL-ORDER CONTROL APPROXIMATION FOR A ROTARY FLEXIBLE JOINT SYSTEM

Maher H. Al-Sereihy∗ , Ibrahim M. Mehedi∗,∗∗ , Ubaid M. Al-Saggaf ∗,∗∗ , and Maamar Bettayeb∗∗,∗∗∗

References

  1. [1] S. Saitou, M. Deng, A. Inoue, and C. Jiang, Vibration con-trol of a flexible arm experimental system with hysteresisof piezoelectric actuator, International Journal of InnovativeComputing, Information and Control, 6, 2010, 2965–2975.
  2. [2] M.A. Auwalu, Z. Mohamed, M. Mustapha, and A. Bature,Vibration and tip deflection control of a single link flexiblemanipulator, International Journal of Instrumentation andControl Systems, 3, 2013, 17–27.
  3. [3] B. Chen, J. Huang, and J.C. Ji, Control of flexible single-linkmanipulators having Duffing oscillator dynamics, MechanicalSystems and Signal Processing, 121, 2019, 44–57.
  4. [4] A.-C. Huang and Y.-C. Chen, Adaptive sliding control forsingle-link flexible-joint robot with mismatched uncertainties,IEEE Transactions on Control Systems Technology, 12, 2004,770–775.
  5. [6]–[8] is illustrated in Fig. 3.In a fractional-order scheme, the fractional integratoris expressed as 1/sα. A compensator K(s) is cascadedwith the fractional integrator as a static gain to obtaina closed-loop transient response similar to Bode’s idealtransfer function [21]. The state feedback Ks is responsiblefor stabilizing the plant.The compensator K(s) is expressed as follows [6]:K(s) =Δd(s)τcSN(s)1(1 + τf s)r(9)where Δd is the inner-loop characteristic polynomial, τc isthe time constant of Bode’s ideal transfer function, N(s) isthe numerator of the transfer function for a linear integersystem, and 1/(1 + τf s)ris a low-pass filter cascaded withK(s) to realize the compensator transfer function.The details and proof of (9) can be found in [6].3.2 Fractional Control ApproximationThis article focuses on three fractional-order approxima-tions for which there are existing MATLAB and Simulinktoolboxes.3.2.1 Commande Robuste d’Ordre Non EntierThe CRONE controller was designed by A. Oustaloup.The CRONE toolbox, developed by the CRONE team, isa special MATLAB and Simulink toolbox for a nonintegercontroller. An object-oriented version is also available.Some of the methods in the CRONE toolbox can beimplemented for multiple-input–multiple-output fractionaltransfer functions. Several other toolboxes depend on3and have been inspired by CRONE, e.g., ninteger andFOMCON [17].3.2.2 NintegerThe ninteger toolbox for MATLAB is intended to assistin the development of fractional-order controllers and theassessment of their performance [18].This toolbox uses integer numerical approximationsfor the fractional-order integrator and differentiator:C(s) = ksv, v ∈ RThree numerical approximation methods are providedby the ninteger toolbox: the CRONE method, Carlson’smethod and Matsuda’s method.Additionally, several Simulink blocks, such as “nid”and “nipid” blocks, are also included. Moreover, thistoolbox provides a user-friendly graphical user interface(GUI) for fractional-order proportional-integral-derivativecontroller design [16].The function in the ninteger toolbox that uses theCRONE approximation, called “crone1()” in MATLAB, isgiven as follows [22]:C(s) = kNn=11 + (s/ωzn)1 + (s/ωpn)(10)This function is a frequency-domain transfer function.In the above equation, ωzn and ωpn depend on the workingfrequency domain [ωh, ωl], and k is an adjusted gain.3.2.3 Fractional-Order Modelling and ControlFOMCON is a MATLAB toolbox developed by Tepljakov,Petlenkov and Belikov [16], [19], [23] that depends onthe FOTF mini-toolbox developed by Xue et al. Detailsregarding the FOTF mini-toolbox are given in [24]. FOM-CON provides GUIs, Simulink blocks and system iden-tification and control design functions. The relationshipbetween FOMCON and the other toolboxes mentionedabove is visualized in Fig. 4 [25].Figure 4. Relationship between FOMCON and the othertoolboxes discussed herein.4. Simulation ResultsIn this section, the state-feedback approach with fractionalintegral control is applied to the Quanser rotary flexiblejoint system, where Jl =0.02552 kg m2, Jeq =0.01625 kg m2,Beq = 0.65407 kg/s and Ks = 10.1227 N/m [19].When these system parameters are substituted intothe system state-space equations, these equations becomeA =⎡⎢⎢⎢⎢⎢⎢⎣0 0 1 00 0 0 10 671.7 −1.9192 00 −1, 098.8 1.9192 0⎤⎥⎥⎥⎥⎥⎥⎦B =⎡⎢⎢⎢⎢⎢⎢⎣00479.8052−479.8052⎤⎥⎥⎥⎥⎥⎥⎦C =⎡⎣0 1 0 01 0 0 0⎤⎦ D =⎡⎣0 00 0⎤⎦The open-loop transfer functions are illustrated inFig. 5.The open-loop transfer functions for θ and α areGθ(s) =K(s2+ b)sΔ(s)(11)Gα(s) =−KsΔ(s)(12)where K = 61:6,326, b = 396:6 and Δ(s) = (s + 25.07)(s2+15.26s + 637.9).The poles of the closed-loop system are chosen to beP1 = −40 and P2 = −30. P1 and P2 are the complex rootsof the second-order polynomial P2+ 2ζω2n, where ζ = 0.7and ωn = 5 [20].Therefore, the state feedback is calculated as follows:Ks = [1.2274 − 10.0439 − 0.2389 − 0.8341]The closed-loop transfer functions between the refer-ence signal r(s) and the outputs θ(s) and α(s) areθ(s)r(s)=75.6475(s2+ 396.6)(s + 40)(s + 30)(s2 + 7s + 25)(13)α(s)r(s)=−75.6475 s2(s + 40)(s + 30)(s2 + 7s + 25)(14)From the open-loop system, the gain crossover fre-quency is ωn = 2.78 rad/s, and the phase margin isφm = 64.56◦.Figure 5. Transfer function representation of the rotaryflexible joint system.4Figure 6. Tracking performance for a square wave: pure state-feedback control (SF), state-feedback control with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE)) and state-feedback controlwith a fractional-order integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).Thus, in accordance with (15), τc = 0.3 s and λ = 0.1are chosen for the design of the fractional controller:λ =π − ϕmπ/2and τc =1ωλ+1c(15)Now, the fractional compensator K(s) is calculatedusing (9). In this equation, values of r = 3 and τf = 0.005 sare considered. The time constant of the closed loop, τc, islarger than the time constant of the filter, τf . Note that τfmust be greater than the time constant of the simulationor experiment (0.002 s).Two different numerical toolboxes are employed toapproximate the fractional integral operator 1/sα. Thechosen limits of the frequency domain are ωl = 10 andωh = 1, 000, and the adjusted gain k is 1.Both the CRONE toolbox and the FOMCON toolboxprovide integrator blocks in Simulink, making the simula-tion task easier.Figure 6 shows the tracking performance when a squarewave is used as the reference signal. This figure showsthe performance achieved using pure state feedback con-trol, state feedback control with a fractional-order inte-grator implemented using the ninteger toolbox (CRONEmethod) and state-feedback control with a fractional-orderintegrator implemented using the FOMCON toolbox.Figure 6 illustrates the effectiveness of using a cascadedfractional-order integrator in combination with state-feedback control. When the two numerical approximationsare used to implement the fractional-order integrator, theCRONE implementation shows a better tracking response,while the FOMCON implementation has a lower overshootvalue and less tip vibration.Figure 7 shows the tracking performance achieved witha sine wave as the reference signal; evidently, the angle θ(t)is smoother, and the tip vibration α(t) is reduced.5. Experimental ResultsThe experimental setup utilized herein was establishedusing laboratory systems and tools from Quanser
  6. [8] is illustrated in Fig. 3.In a fractional-order scheme, the fractional integratoris expressed as 1/sα. A compensator K(s) is cascadedwith the fractional integrator as a static gain to obtaina closed-loop transient response similar to Bode’s idealtransfer function [21]. The state feedback Ks is responsiblefor stabilizing the plant.The compensator K(s) is expressed as follows [6]:K(s) =Δd(s)τcSN(s)1(1 + τf s)r(9)where Δd is the inner-loop characteristic polynomial, τc isthe time constant of Bode’s ideal transfer function, N(s) isthe numerator of the transfer function for a linear integersystem, and 1/(1 + τf s)ris a low-pass filter cascaded withK(s) to realize the compensator transfer function.The details and proof of (9) can be found in [6].3.2 Fractional Control ApproximationThis article focuses on three fractional-order approxima-tions for which there are existing MATLAB and Simulinktoolboxes.3.2.1 Commande Robuste d’Ordre Non EntierThe CRONE controller was designed by A. Oustaloup.The CRONE toolbox, developed by the CRONE team, isa special MATLAB and Simulink toolbox for a nonintegercontroller. An object-oriented version is also available.Some of the methods in the CRONE toolbox can beimplemented for multiple-input–multiple-output fractionaltransfer functions. Several other toolboxes depend on3and have been inspired by CRONE, e.g., ninteger andFOMCON [17].3.2.2 NintegerThe ninteger toolbox for MATLAB is intended to assistin the development of fractional-order controllers and theassessment of their performance [18].This toolbox uses integer numerical approximationsfor the fractional-order integrator and differentiator:C(s) = ksv, v ∈ RThree numerical approximation methods are providedby the ninteger toolbox: the CRONE method, Carlson’smethod and Matsuda’s method.Additionally, several Simulink blocks, such as “nid”and “nipid” blocks, are also included. Moreover, thistoolbox provides a user-friendly graphical user interface(GUI) for fractional-order proportional-integral-derivativecontroller design [16].The function in the ninteger toolbox that uses theCRONE approximation, called “crone1()” in MATLAB, isgiven as follows [22]:C(s) = kNn=11 + (s/ωzn)1 + (s/ωpn)(10)This function is a frequency-domain transfer function.In the above equation, ωzn and ωpn depend on the workingfrequency domain [ωh, ωl], and k is an adjusted gain.3.2.3 Fractional-Order Modelling and ControlFOMCON is a MATLAB toolbox developed by Tepljakov,Petlenkov and Belikov [16], [19], [23] that depends onthe FOTF mini-toolbox developed by Xue et al. Detailsregarding the FOTF mini-toolbox are given in [24]. FOM-CON provides GUIs, Simulink blocks and system iden-tification and control design functions. The relationshipbetween FOMCON and the other toolboxes mentionedabove is visualized in Fig. 4 [25].Figure 4. Relationship between FOMCON and the othertoolboxes discussed herein.4. Simulation ResultsIn this section, the state-feedback approach with fractionalintegral control is applied to the Quanser rotary flexiblejoint system, where Jl =0.02552 kg m2, Jeq =0.01625 kg m2,Beq = 0.65407 kg/s and Ks = 10.1227 N/m [19].When these system parameters are substituted intothe system state-space equations, these equations becomeA =⎡⎢⎢⎢⎢⎢⎢⎣0 0 1 00 0 0 10 671.7 −1.9192 00 −1, 098.8 1.9192 0⎤⎥⎥⎥⎥⎥⎥⎦B =⎡⎢⎢⎢⎢⎢⎢⎣00479.8052−479.8052⎤⎥⎥⎥⎥⎥⎥⎦C =⎡⎣0 1 0 01 0 0 0⎤⎦ D =⎡⎣0 00 0⎤⎦The open-loop transfer functions are illustrated inFig. 5.The open-loop transfer functions for θ and α areGθ(s) =K(s2+ b)sΔ(s)(11)Gα(s) =−KsΔ(s)(12)where K = 61:6,326, b = 396:6 and Δ(s) = (s + 25.07)(s2+15.26s + 637.9).The poles of the closed-loop system are chosen to beP1 = −40 and P2 = −30. P1 and P2 are the complex rootsof the second-order polynomial P2+ 2ζω2n, where ζ = 0.7and ωn = 5 [20].Therefore, the state feedback is calculated as follows:Ks = [1.2274 − 10.0439 − 0.2389 − 0.8341]The closed-loop transfer functions between the refer-ence signal r(s) and the outputs θ(s) and α(s) areθ(s)r(s)=75.6475(s2+ 396.6)(s + 40)(s + 30)(s2 + 7s + 25)(13)α(s)r(s)=−75.6475 s2(s + 40)(s + 30)(s2 + 7s + 25)(14)From the open-loop system, the gain crossover fre-quency is ωn = 2.78 rad/s, and the phase margin isφm = 64.56◦.Figure 5. Transfer function representation of the rotaryflexible joint system.4Figure 6. Tracking performance for a square wave: pure state-feedback control (SF), state-feedback control with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE)) and state-feedback controlwith a fractional-order integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).Thus, in accordance with (15), τc = 0.3 s and λ = 0.1are chosen for the design of the fractional controller:λ =π − ϕmπ/2and τc =1ωλ+1c(15)Now, the fractional compensator K(s) is calculatedusing (9). In this equation, values of r = 3 and τf = 0.005 sare considered. The time constant of the closed loop, τc, islarger than the time constant of the filter, τf . Note that τfmust be greater than the time constant of the simulationor experiment (0.002 s).Two different numerical toolboxes are employed toapproximate the fractional integral operator 1/sα. Thechosen limits of the frequency domain are ωl = 10 andωh = 1, 000, and the adjusted gain k is 1.Both the CRONE toolbox and the FOMCON toolboxprovide integrator blocks in Simulink, making the simula-tion task easier.Figure 6 shows the tracking performance when a squarewave is used as the reference signal. This figure showsthe performance achieved using pure state feedback con-trol, state feedback control with a fractional-order inte-grator implemented using the ninteger toolbox (CRONEmethod) and state-feedback control with a fractional-orderintegrator implemented using the FOMCON toolbox.Figure 6 illustrates the effectiveness of using a cascadedfractional-order integrator in combination with state-feedback control. When the two numerical approximationsare used to implement the fractional-order integrator, theCRONE implementation shows a better tracking response,while the FOMCON implementation has a lower overshootvalue and less tip vibration.Figure 7 shows the tracking performance achieved witha sine wave as the reference signal; evidently, the angle θ(t)is smoother, and the tip vibration α(t) is reduced.5. Experimental ResultsThe experimental setup utilized herein was establishedusing laboratory systems and tools from Quanser [7]. The5Figure 7. Tracking performance for a sine wave: pure state-feedback control (SF), state-feedback control with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE)) and state-feedback controlwith a fractional-order integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).experimental setup is depicted in Fig. 8 and consists of thefollowing four components:• The QUARC Tool is a real-time control software pack-age based on MATLAB and Simulink.• The DAQ (Q2-USB) is a data acquisition device withan analogue output to command the motor and adigital input to receive encoder signals.• The amplifier (VoltPAQ-X1) is used to amplify thecommands to the motor voltage level (24 V).• The rotary flexible joint system is a rotary flexiblejoint module mounted on a rotary servo base unit. Therotary flexible joint system is used as the experimentalplant.Figure 9 shows the experimental results for a squarewave in the closed-loop system. The experimental resultsare the same as the simulation results for state-feedbackcontrol with a fractional integrator. In general, a fractional-order integrator improves the state-feedback control per-formance. Table 2 shows a performance comparison be-tween the two different fractional-order approximationFigure 8. Experimental setup.approaches. The FOMCON approximation is superior asthe vibrations and oscillations of the angle α(t) are smallerwith FOMCON than with CRONE.6Figure 9. Experimental results for a square wave in the closed-loop system: pure state-feedback control (SF), state-feedbackcontrol with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE))and state-feedback control with a fractional integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).Table 2Comparison between the Performance of CRONEand FOMCONCRONE FOMCONApproximation ApproximationRise Time Faster SlowerOvershoot Higher No overshootMinimization of Higher peak Lower peak valuevibration in α(t) value6. ConclusionIn this article, a fractional-order integrator is designedbased on a state-feedback approach. Fractional-ordercontrollers can be implemented using a wide range ofnumerical approximations. Two existing fractional MAT-LAB toolboxes are used to implement fractional control.To evaluate the performance of the proposed control designand the numerical approximation methods, simulation andexperimental results obtained for the Quanser rotary flex-ible joint system are reported. Different kinds of referencesignals are applied to compare the tracking performanceand the degree to which the tip vibration is minimized.The FOMCON approximation is found to be superior asthe tip vibration angle is smaller with FOMCON than withthe CRONE approximation. In addition, the proposedcontrol design can be further extended in several directions.One future research direction is to develop simple tuningmethods for fractional-order controllers. Fractional-ordercontrollers in industrial applications can be enhancedwith the help of autotuning methods and plug-and-playcontrollers.AcknowledgementThis project was funded by the Deanship of ScientificResearch (DSR), King Abdulaziz University, Jeddah, SaudiArabia under grant no. KEP-Msc-13-135-40. The authors,therefore, acknowledge with thanks DSR technical andfinancial support.References[1] S. Saitou, M. Deng, A. Inoue, and C. Jiang, Vibration con-trol of a flexible arm experimental system with hysteresisof piezoelectric actuator, International Journal of InnovativeComputing, Information and Control, 6, 2010, 2965–2975.[2] M.A. Auwalu, Z. Mohamed, M. Mustapha, and A. Bature,Vibration and tip deflection control of a single link flexiblemanipulator, International Journal of Instrumentation andControl Systems, 3, 2013, 17–27.[3] B. Chen, J. Huang, and J.C. Ji, Control of flexible single-linkmanipulators having Duffing oscillator dynamics, MechanicalSystems and Signal Processing, 121, 2019, 44–57.[4] A.-C. Huang and Y.-C. Chen, Adaptive sliding control forsingle-link flexible-joint robot with mismatched uncertainties,IEEE Transactions on Control Systems Technology, 12, 2004,770–775.[5] Y. Li, S. Tong, and T. Li, Adaptive fuzzy output feedbackcontrol for a single-link flexible robot manipulator drivenDC motor via backstepping, Nonlinear Analysis. Real WorldApplications, 14, 2013, 483–494.[6] U.M. Al-Saggaf, I.M. Mehedi, R. Mansouri, and M. Bettayeb,State feedback with fractional integral control design basedon the Bode’s ideal transfer function, International Journal ofSystems Science, 47, 2016, 149–161.[7] I.M. Mehedi, U.M. Al-Saggaf, R. Mansouri and M. Bettayeb,Stabilization of a double inverted rotary pendulum throughfractional order integral control scheme, International Journalof Advanced Robotic Systems, 16(4), 2019.[8] I.M. Mehedi, Full state-feedback solution for a flywheel basedsatellite energy and attitude control scheme, Journal of Vibro-engineering, 19(5), 2017, 3522–3532.7
  7. [9] I.M. Mehedi, U.M. Al-Saggaf, R. Mansouri and M. Bettayeb,Two degrees of freedom fractional controller design: Applica-tion to the ball and beam system, Measurement, 135, 2019,13–22.
  8. [10] I.M. Mehedi, State feedback based fractional order controlscheme for linear servo cart system, Journal of Vibroengineer-ing, 20(1), 2018, 782–792.
  9. [11] M. Bettayeb, R. Mansouri, U. Al-Saggaf, and I.M. Mehedi,Smith predictor based fractional-order-filter PID controllersdesign for long time delay systems, Asian Journal of Control,19(2), 2016, 587–598.
  10. [12] U.M. Al-Saggaf, I.M. Mehedi, R. Mansouri, and M. Bettayeb,Rotary flexible joint control by fractional order controllers,International Journal of Control, Automation and Systems,15(6), 2017, 2561–2569.
  11. [13] P. Shah and S. Agashe, Review of fractional PID controller,Mechatronics, 38, 2016, 29–41.
  12. [14] Z. Li, L. Liu, S. Dehghan, Y. Chen, and D. Xue, A reviewand evaluation of numerical tools for fractional calculus andfractional order controls, International Journal of Control, 90,2017, 1165–1181.
  13. [16]control toolbox2 Ninteger Fractional calculus and D.P.M.O. Valerio Valerio and da Costa
  14. [17]fractional-order controltoolbox3 CRONE Fractional-order The CRONE team The CRONE Team
  15. [18]control toolbox4 FOMCON Fractional-order modelling A. Tepljakov Tepljakov et al.
  16. [19]and controlIn this article, a state-feedback approach with frac-tional integral control is implemented using two differentapproximation methods: CRONE (Commande Robusted’Ordre Non Entier, meaning noninteger-order robustcontrol), developed by A. Oustaloup, and the FOMCON(fractional-order modelling and control) MATLAB tool-box. For comparison, pure state-feedback control and themodified technique are both implemented for simulationand experimentation in a rotary flexible joint system. Thetracking performance of the rotary arm and the degree towhich the tip vibration is minimized are evaluated andcompared between the pure state-feedback control schemeand the modified state-feedback-based fractional integralcontrol scheme.The remainder of this article is organized as follows.Section 2 describes the rotary flexible joint system and themathematical modelling of the system. In Section 3, thestate-feedback-based fractional integral control scheme isdesigned; in addition, a comparison of various fractional-order approximations is presented. In Sections 4 and 5,the simulation and experimental results are reported, re-spectively. Finally, in Section 6, conclusions regarding thefractional-order control approach are presented, followedby the references.2. Rotary Flexible Joint System2.1 System OverviewThe rotary flexible joint system developed by Quanser Inc.
  17. [20] consists of a rigid arm connected to a flexible joint thatis rotated by a DC motor. Speed feedback and positionfeedback are both obtained using an encoder sensor, asshown in Fig. 1.This system was designed to fit a servo plant (SRV02)that was also developed by Quanser. The rotary flexiblearm is connected through two springs anchored to the solidframe, allowing it to act as a flexible instrument. Thisplant is useful for evaluating the flexible joint control.A schematic diagram of the rotary flexible joint systemis shown in Fig. 2. The angle θ is the servo position angle,and the angle α is the angle of the vibration of the rotaryarm due to the flexible joint. It is challenging to controlFigure 1. Rotary flexible joint platform.Figure 2. Schematic representation of the rotary flexiblejoint system.the servo position angle θ while minimizing the vibrationangle α.2.2 System ModellingThe linear state-space equations are˙x = Ax + Bu (1)andy = Cx + Du (2)2The manipulated input command u(t) is the voltage(in units of volts) applied to the DC servo motor, and theoutput y(t) is the motor speed of the plant. Speed feedbackis measured by the encoder in radians per second.The system states and outputs for controlling therotary flexible joint system are defined in (3) and (4):xT= [θ α ˙θ ˙α] (3)yT= [x1 x2] (4)The measured outputs x1 and x2 are the servo positionangle and the tip vibration angle, respectively. Therefore,the matrices C and D in the output equation areC =⎡⎣0 1 0 01 0 0 0⎤⎦ D =⎡⎣0 00 0⎤⎦The equations of motion for the rotary flexible jointsystem are given by [20]¨θ = −BeqJeq¨θ +KsJeqα +1Jeqτ (5)¨α =BeqJeq˙θ −Jl + JeqJl × Jeqα +1Jeqτ (6)The parameters are defined as follows: τ is the torque,Beq is the viscous friction coefficient of the servo, Jeq is theinertia of the rotary arm, Jl is the inertia of the link, Ks isthe linear spring stiffness.By substituting the state x1 defined in (3) into theequations of motion, (5) and (6), one can find the matricesA and B defined in (1). From the state definition given in(3), ˙x1 = x3 and ˙x2 = x4. Substituting x into (5) and (6)yields˙x3 = −BeqJeqx3 +KsJeqx2 +1Jequ (7)and˙x4 =BeqJeqx3 − KsJl + JeqJl × Jeqx2 +1Jequ (8)The matrices A and B in the equation ˙x = Ax + Buare thusA =⎡⎢⎢⎢⎢⎢⎢⎢⎣0 0 1 00 0 0 10 KsJeq−BeqJeq00 −Ks(Jl+Jeq)JeqBeqJeq0⎤⎥⎥⎥⎥⎥⎥⎥⎦B =⎡⎢⎢⎢⎢⎢⎢⎢⎣001Jeq− 1Jeq⎤⎥⎥⎥⎥⎥⎥⎥⎦C =⎡⎣0 1 0 01 0 0 0⎤⎦ D =⎡⎣0 00 0⎤⎦Figure 3. State-feedback approach with fractional integralcontrol.Saturation of the servo actuator is a key concern inflexible joint control. Nevertheless, a fractional-order in-tegral control scheme can be used to avoid actuator sat-uration. In Sections 4 and 5, the results of a simula-tion and experiment, respectively, are presented to inves-tigate the tracking performance achieved with fractional-order control schemes using different types of numericalapproximations.3. Controller Design3.1 State Feedback with Fractional IntegralControlThe state-feedback approach with fractional integral con-trol developed in [6]–[8] is illustrated in Fig. 3.In a fractional-order scheme, the fractional integratoris expressed as 1/sα. A compensator K(s) is cascadedwith the fractional integrator as a static gain to obtaina closed-loop transient response similar to Bode’s idealtransfer function
  18. [21]. The state feedback Ks is responsiblefor stabilizing the plant.The compensator K(s) is expressed as follows [6]:K(s) =Δd(s)τcSN(s)1(1 + τf s)r(9)where Δd is the inner-loop characteristic polynomial, τc isthe time constant of Bode’s ideal transfer function, N(s) isthe numerator of the transfer function for a linear integersystem, and 1/(1 + τf s)ris a low-pass filter cascaded withK(s) to realize the compensator transfer function.The details and proof of (9) can be found in [6].3.2 Fractional Control ApproximationThis article focuses on three fractional-order approxima-tions for which there are existing MATLAB and Simulinktoolboxes.3.2.1 Commande Robuste d’Ordre Non EntierThe CRONE controller was designed by A. Oustaloup.The CRONE toolbox, developed by the CRONE team, isa special MATLAB and Simulink toolbox for a nonintegercontroller. An object-oriented version is also available.Some of the methods in the CRONE toolbox can beimplemented for multiple-input–multiple-output fractionaltransfer functions. Several other toolboxes depend on3and have been inspired by CRONE, e.g., ninteger andFOMCON [17].3.2.2 NintegerThe ninteger toolbox for MATLAB is intended to assistin the development of fractional-order controllers and theassessment of their performance [18].This toolbox uses integer numerical approximationsfor the fractional-order integrator and differentiator:C(s) = ksv, v ∈ RThree numerical approximation methods are providedby the ninteger toolbox: the CRONE method, Carlson’smethod and Matsuda’s method.Additionally, several Simulink blocks, such as “nid”and “nipid” blocks, are also included. Moreover, thistoolbox provides a user-friendly graphical user interface(GUI) for fractional-order proportional-integral-derivativecontroller design [16].The function in the ninteger toolbox that uses theCRONE approximation, called “crone1()” in MATLAB, isgiven as follows
  19. [22]:C(s) = kNn=11 + (s/ωzn)1 + (s/ωpn)(10)This function is a frequency-domain transfer function.In the above equation, ωzn and ωpn depend on the workingfrequency domain [ωh, ωl], and k is an adjusted gain.3.2.3 Fractional-Order Modelling and ControlFOMCON is a MATLAB toolbox developed by Tepljakov,Petlenkov and Belikov [16], [19],
  20. [23] that depends onthe FOTF mini-toolbox developed by Xue et al. Detailsregarding the FOTF mini-toolbox are given in
  21. [24]. FOM-CON provides GUIs, Simulink blocks and system iden-tification and control design functions. The relationshipbetween FOMCON and the other toolboxes mentionedabove is visualized in Fig. 4
  22. [25].Figure 4. Relationship between FOMCON and the othertoolboxes discussed herein.4. Simulation ResultsIn this section, the state-feedback approach with fractionalintegral control is applied to the Quanser rotary flexiblejoint system, where Jl =0.02552 kg m2, Jeq =0.01625 kg m2,Beq = 0.65407 kg/s and Ks = 10.1227 N/m [19].When these system parameters are substituted intothe system state-space equations, these equations becomeA =⎡⎢⎢⎢⎢⎢⎢⎣0 0 1 00 0 0 10 671.7 −1.9192 00 −1, 098.8 1.9192 0⎤⎥⎥⎥⎥⎥⎥⎦B =⎡⎢⎢⎢⎢⎢⎢⎣00479.8052−479.8052⎤⎥⎥⎥⎥⎥⎥⎦C =⎡⎣0 1 0 01 0 0 0⎤⎦ D =⎡⎣0 00 0⎤⎦The open-loop transfer functions are illustrated inFig. 5.The open-loop transfer functions for θ and α areGθ(s) =K(s2+ b)sΔ(s)(11)Gα(s) =−KsΔ(s)(12)where K = 61:6,326, b = 396:6 and Δ(s) = (s + 25.07)(s2+15.26s + 637.9).The poles of the closed-loop system are chosen to beP1 = −40 and P2 = −30. P1 and P2 are the complex rootsof the second-order polynomial P2+ 2ζω2n, where ζ = 0.7and ωn = 5 [20].Therefore, the state feedback is calculated as follows:Ks = [1.2274 − 10.0439 − 0.2389 − 0.8341]The closed-loop transfer functions between the refer-ence signal r(s) and the outputs θ(s) and α(s) areθ(s)r(s)=75.6475(s2+ 396.6)(s + 40)(s + 30)(s2 + 7s + 25)(13)α(s)r(s)=−75.6475 s2(s + 40)(s + 30)(s2 + 7s + 25)(14)From the open-loop system, the gain crossover fre-quency is ωn = 2.78 rad/s, and the phase margin isφm = 64.56◦.Figure 5. Transfer function representation of the rotaryflexible joint system.4Figure 6. Tracking performance for a square wave: pure state-feedback control (SF), state-feedback control with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE)) and state-feedback controlwith a fractional-order integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).Thus, in accordance with (15), τc = 0.3 s and λ = 0.1are chosen for the design of the fractional controller:λ =π − ϕmπ/2and τc =1ωλ+1c(15)Now, the fractional compensator K(s) is calculatedusing (9). In this equation, values of r = 3 and τf = 0.005 sare considered. The time constant of the closed loop, τc, islarger than the time constant of the filter, τf . Note that τfmust be greater than the time constant of the simulationor experiment (0.002 s).Two different numerical toolboxes are employed toapproximate the fractional integral operator 1/sα. Thechosen limits of the frequency domain are ωl = 10 andωh = 1, 000, and the adjusted gain k is 1.Both the CRONE toolbox and the FOMCON toolboxprovide integrator blocks in Simulink, making the simula-tion task easier.Figure 6 shows the tracking performance when a squarewave is used as the reference signal. This figure showsthe performance achieved using pure state feedback con-trol, state feedback control with a fractional-order inte-grator implemented using the ninteger toolbox (CRONEmethod) and state-feedback control with a fractional-orderintegrator implemented using the FOMCON toolbox.Figure 6 illustrates the effectiveness of using a cascadedfractional-order integrator in combination with state-feedback control. When the two numerical approximationsare used to implement the fractional-order integrator, theCRONE implementation shows a better tracking response,while the FOMCON implementation has a lower overshootvalue and less tip vibration.Figure 7 shows the tracking performance achieved witha sine wave as the reference signal; evidently, the angle θ(t)is smoother, and the tip vibration α(t) is reduced.5. Experimental ResultsThe experimental setup utilized herein was establishedusing laboratory systems and tools from Quanser [7]. The5Figure 7. Tracking performance for a sine wave: pure state-feedback control (SF), state-feedback control with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE)) and state-feedback controlwith a fractional-order integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).experimental setup is depicted in Fig. 8 and consists of thefollowing four components:• The QUARC Tool is a real-time control software pack-age based on MATLAB and Simulink.• The DAQ (Q2-USB) is a data acquisition device withan analogue output to command the motor and adigital input to receive encoder signals.• The amplifier (VoltPAQ-X1) is used to amplify thecommands to the motor voltage level (24 V).• The rotary flexible joint system is a rotary flexiblejoint module mounted on a rotary servo base unit. Therotary flexible joint system is used as the experimentalplant.Figure 9 shows the experimental results for a squarewave in the closed-loop system. The experimental resultsare the same as the simulation results for state-feedbackcontrol with a fractional integrator. In general, a fractional-order integrator improves the state-feedback control per-formance. Table 2 shows a performance comparison be-tween the two different fractional-order approximationFigure 8. Experimental setup.approaches. The FOMCON approximation is superior asthe vibrations and oscillations of the angle α(t) are smallerwith FOMCON than with CRONE.6Figure 9. Experimental results for a square wave in the closed-loop system: pure state-feedback control (SF), state-feedbackcontrol with a fractional-order integrator implemented using the ninteger toolbox (CRONE method) (SF + FOI(CRONE))and state-feedback control with a fractional integrator implemented using the FOMCON toolbox (SF + FOI(FOMCON)).Table 2Comparison between the Performance of CRONEand FOMCONCRONE FOMCONApproximation ApproximationRise Time Faster SlowerOvershoot Higher No overshootMinimization of Higher peak Lower peak valuevibration in α(t) value6. ConclusionIn this article, a fractional-order integrator is designedbased on a state-feedback approach. Fractional-ordercontrollers can be implemented using a wide range ofnumerical approximations. Two existing fractional MAT-LAB toolboxes are used to implement fractional control.To evaluate the performance of the proposed control designand the numerical approximation methods, simulation andexperimental results obtained for the Quanser rotary flex-ible joint system are reported. Different kinds of referencesignals are applied to compare the tracking performanceand the degree to which the tip vibration is minimized.The FOMCON approximation is found to be superior asthe tip vibration angle is smaller with FOMCON than withthe CRONE approximation. In addition, the proposedcontrol design can be further extended in several directions.One future research direction is to develop simple tuningmethods for fractional-order controllers. Fractional-ordercontrollers in industrial applications can be enhancedwith the help of autotuning methods and plug-and-playcontrollers.AcknowledgementThis project was funded by the Deanship of ScientificResearch (DSR), King Abdulaziz University, Jeddah, SaudiArabia under grant no. KEP-Msc-13-135-40. The authors,therefore, acknowledge with thanks DSR technical andfinancial support.References[1] S. Saitou, M. Deng, A. Inoue, and C. Jiang, Vibration con-trol of a flexible arm experimental system with hysteresisof piezoelectric actuator, International Journal of InnovativeComputing, Information and Control, 6, 2010, 2965–2975.[2] M.A. Auwalu, Z. Mohamed, M. Mustapha, and A. Bature,Vibration and tip deflection control of a single link flexiblemanipulator, International Journal of Instrumentation andControl Systems, 3, 2013, 17–27.[3] B. Chen, J. Huang, and J.C. 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Mehedi, Full state-feedback solution for a flywheel basedsatellite energy and attitude control scheme, Journal of Vibro-engineering, 19(5), 2017, 3522–3532.7[9] I.M. Mehedi, U.M. Al-Saggaf, R. Mansouri and M. Bettayeb,Two degrees of freedom fractional controller design: Applica-tion to the ball and beam system, Measurement, 135, 2019,13–22.[10] I.M. Mehedi, State feedback based fractional order controlscheme for linear servo cart system, Journal of Vibroengineer-ing, 20(1), 2018, 782–792.[11] M. Bettayeb, R. Mansouri, U. Al-Saggaf, and I.M. Mehedi,Smith predictor based fractional-order-filter PID controllersdesign for long time delay systems, Asian Journal of Control,19(2), 2016, 587–598.[12] U.M. Al-Saggaf, I.M. Mehedi, R. Mansouri, and M. Bettayeb,Rotary flexible joint control by fractional order controllers,International Journal of Control, Automation and Systems,15(6), 2017, 2561–2569.[13] P. Shah and S. 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