ADAPTIVE BACKSTEPPING CONTROLLER DESIGN AND
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.orgADAPTIVE BACKSTEPPING CONTROLLER DESIGN ANDIMPLEMENTATION FOR A MATRIX-CONVERTER-BASEDIM DRIVE SYSTEMR.R. Joshi, R.A. Gupta and A.K. WadhwaniR. R. Joshi is with Department of Electrical Engineering, CTAE, Udaipur (Raj.), India;R. A. Gupta is with Department of Electrical Engineering, MNIT, Jaipur, (Raj.), India;A. K. Wadhwani is with Department of Electrical Engineering, MITS, Gwalior (MP), IndiaABSTRACTA systematic controller design and implementation for a matrix-converter-based induction motor drivesystem is proposed. A nonlinear adaptive backstepping controller is proposed to improve the speed andposition responses of the induction motor system. By using the proposed adaptive backsteppingcontroller, the system can track a time-varying speed command and a time-varying position commandwell. Moreover, the system has a good load disturbance rejection capability. The realisation of thecontroller is very simple. All of the control loops, including the current loop, speed loop and positionloop, are implemented by a digital signal processor. Several experimental results are given to validate thetheoretical analysis.commercially. This type of packaging canminimise the stray inductance and the size ofthe power devices [1]. Yaskawa Co. hasimplemented a commercial matrix converterand has shown it has many advantages.Currently, the cost of the matrix converter is alittle higher than the recti.er/DClink/ inverter.However, in the future, after the matrixconverter becomes more and more popular, thecost of the matrix converter will be reduced andwill compete with the traditional rectifier/DClink/ inverter [2].The matrix converter is superior to thetraditional PWM (Pulse width modulation)drives because of regeneration ability andsinusoidal input current [3-5].Therefore it meetsthe stringent energy-efficiency and powerquality requirements of the new century. Matrixconverter can be considered to be a directconverter, in this respect similar to acycloconverter, because, first, it doesn’t employa dc link and, secondly the output waveformsare composed of switched segments of the inputwaveforms. The matrix converter thereforepossesses the advantages of both thecycloconverter and the PWM drive, assummarized below :1.It can operate in all four quadrants of thetorque-speed plane because of itsregeneration capability.2.Its input current waveform is sinusoidaland the input power factor is unity1 INTRODUCTIONAn AC/AC converter is used to providesinusoidal output voltages with varyingamplitudes and frequencies, and to drawsinusoidal input currents from the AC source.The rectifier / DC-link / inverter is widely usedin conventional AC drive systems. The DC-linkrequires a large capacitor as an energy storagecomponent. The capacitor can be a criticalcomponent because it is large and expensive.The power factor and current harmonics of theinput AC side are not good enough. In addition,a braking resistance is required to absorb theenergy during braking the drive system. InAC/AC converter applications, the matrixconverter has become increasingly attractive inrecent years.The AC/AC matrix converter has severaladvantages. For example, the matrix converteris a single-stage converter. It does not requireany DC-link energy storage component. Inaddition, it has a high-power-factor sinusoidalinput current with a bidirectional power flow forthe whole matrix converter drive system. In thepast, the matrix converter, therefore, wasdeveloped in the research laboratory only andcould not be popularly used in industry. Thesituation has been changed. Recently, EupecCo. has developed a new technology forintegrating the whole matrix-converter powerdevice in a single package. In addition, theintegrated power modules are now available28
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.org3.4.5.6.7.8.9.To improve the tracking ability of thetime-varying command, this paper proposes anonlinear adaptive backstepping controlalgorithm for a matrix-converter-based IM drivesystem. Although the theoretical developmentof the nonlinear adaptive backstepping controlalgorithm has been well developed in [12], thereare only a few applications in motor drives [13].This paper, therefore, applies the nonlinearbackstepping controller for a matrix-converterbased IM drive system. According to theexperimental results, the transient responses,load disturbance responses and trackingresponses of the speed- and position-control IMdrive systems are effectively improved.In this paper, a novel matrix-converterbased IM control system is proposed. All of thecurrent, speed and position loops areimplemented by a digital signal processor. Afterthe adaptive backstepping speed- and positioncontrol algorithms are applied, a satisfactorycontrol performance can be achieved. Inaddition, the system can track both speed andposition time-varying commands well. To theauthors’ best knowledge, this is the first timethat the adaptive backstepping controller hasbeen applied in a high-performance matrixconverter-based IM system. Moreover, theimplementation of the control law is verysimple by using a digital signal processor andcan be applied for both speed and positioncontrol systems.The input power factor may be controlledacross the whole speed range.There is no dc link and therefore norequirement for energy storage devicesThe output voltage and input currentwaveforms can be controlled such thatthey are near sinusoidal in form.The harmonic distortion that is incurredis at high frequenciesIt may be used as direct frequencychanger, converting a fixed ac or dcsource into a variable ac or dc supply.Compact converter design.No dc-link components. Several publications on matrixconverters have dealt with modulation strategiesto improve the performance of the matrixconverter [6–8]. Some papers discussed theprotection issues of the matrix converter [9].Unfortunately, only very few publications haveconsidered the controller design and positioncontrol methods for the matrix-converter-basedAC drive system [10]. Owing to its possiblepopularity in the future of the matrix converterdrive system, the controller design is verychallenging. The major reason is that theswitching frequency of a matrix converter islower compared to that of the rectifier/DC-link/inverter. As a result, the matrix converterproduces larger current harmonics and torquepulsations than the AC motor, which causesserious uncertainty in the dynamic model andmakes the controller design more difficult. As aresult, it is very challenging to achieve a highperformance speed or position drive system byusing a matrix converter.With respect to controller design, aproportional–integral (PI) controller has beenwidely used in many industrial drives becauseof its simple and reliable characteristics.However, this PI controller cannot obtain both agood command tracking ability and a rapidrecovery of the external load disturbance. Toresolve this problem, a two degree- of-freedomcontroller was designed [11]. By using the twodegree-of-freedom controller, the transientresponse and load disturbance rejectionresponse can be individually controlled. As aresult, better performance can be obtained for astep input command and load disturbance.Unfortunately, when a time-varying commandis applied, the linear control system produces asteady-state amplitude error and phase delay inoutput responses.2 SYSTEM DESCRIPTION2.1 Matrix Converter Drive SystemThe block diagram to be considered inthis paper is shown in Fig. 1a. The hardware ofthe system consists of a three-phase IM andload, a nine-switch matrix converter, and adigital signal processor system. The motorchosen for the study is a three-phase, Yconnected, IM. Its rated speed is 1500 rpm. Aresolver is mounted on the motor shaft forvelocity and position sensing. Two Hall effectcurrent sensors are used to detect the statorcurrents of the motor because the drive systemis a three-phase balanced system. The drivesystem works as follows. First, the position(speed) command is compared with thefeedback position (speed). Next, the position(speed) controller generates the q-axis currentcommand and then transfers into three-phasecurrent commands. After that, the three-phasecurrent commands are compared with the threephase29
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.orgThe virtual DC-bus voltage is obtained byselecting the switches of the AC/DC converter.In addition, the virtual DC-bus voltage can besuitably adjusted if one selects differentswitching states of the AC/DC converter. Thevirtual DC/AC inverter is used to regulate thethree-phase output currents. Take the A-phaseas an example. When the A-phase currentcommand is larger than the A-phase current, theupper switch of the A-phase leg is turned on. Asa result, the A-phase current increases to trackthe current command. On the other hand, whenthe A-phase current command is smaller thanthe A-phase current, the lower switch of the Aphase leg is turned on. Then, the A-phasecurrent decreases to track the current command.The B-phase and C-phase current regulators areoperated using the same method. Finally, bymapping the switching states of the AC/DCconverter and the DC/AC inverter into the realmatrix converter, one can determine theswitching condition of the matrix converter [14,15].IMIMFig. 1. Drive system configuration (a) Blockdiagram of whole drive (b) Virtual equivalentcircuitfeedback currents, which thus determines theswitching states of the matrix converter. Thedetails of the switching algorithm of the matrixconverter are discussed in the followingSection. Finally, the motor rotates, and a closedloop drive system is thus achieved.3 CONTROLLER DESIGNThe adaptive backstepping controller isdesigned for speed control and position control.In this Section, first, the mathematical model ofthe uncontrolled plant is discussed. Next, thenonlinear speed controller is discussed. Finally,the nonlinear position controller is presented.2.2 Switching Algorithm of Matrix ConverterThe switching patterns of the nineswitches of the matrix converter should satisfythe basic requirements. For instance, theswitches of the matrix converter cannot shortcircuit the input voltage sources. In addition, theswitches of the matrix converter cannot openthe output load currents because the load is aninductive load. Generally speaking, there aretwo major switching methods for the matrixconverter: direct and indirect. The direct methodprovides the output currents or voltages byusing only six switching patterns. By suitablycontrolling the duty cycle of the nine switchesunder the six switching patterns, the requiredoutput voltage can be obtained. This method,however, requires a lot of computations todetermine the switching states. The indirectswitching method is based on the basic principleof the switching strategy of the conventionalconverter and inverter. The switching algorithmof the indirect switching method can beexplained using a virtual equivalent circuit. Thevirtual equivalent circuit, shown in Fig. 1b,consists of an AC/DC converter and a DC/ACinverter. In addition, the AC/DC converter andDC/AC inverter are controlled independently.3.1 Mathematical Model of Uncontrolled PlantThe uncontrolled model of a IM with a currentregulated inverter can be described by thefollowing differential equations [12]:(1)(2)where J is the motor shaft inertia, B is the motorviscous coefficient, kt is the torque constant, u isthe control input, d is the total value ofequivalent load disturbance and systemparameter uncertainties, r is the speed of themotor, and r is the angle of the motor.3.2 Nonlinear Speed Controller DesignThe block diagram of the speed controlsystem is shown in Fig. 2. To design anadaptive backstepping speed controller, thespeed tracking error can be defined as(3)Now, take the derivative of both sides to obtain30
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.org(4)Define the stability function as(13)From (13), one can obtain that the Lyapunovfunction is a non-increasing function because itsderivate is equal to or less than zero. Byintegrating both sides of (13), one can obtain(5)where Cw1 is a positive constant, and w1 is thestability function. The Lyapunov function canbe selected as(6)Furthermore, define e2 and, substituting (1) intoit, we can obtainThe values of e1(0), d1(0) are bounded. Inaddition, Vw2 0, therefore e1( ) are alsobounded. We can observe that the integration ofVw2 is bounded. It is easy to understand that theintegration of the e12 is bounded as well. Afterderiving that e1(t) is uniformly continuous, onecan use Bardalat’s lemma and obtain thefollowing :(7)Then, by taking the differential of Vw1 andsubstituting (4), (5), and (7) into it, we canobtain(8)To design the controller, which has the ability totrack commands and toe reject load disturbance,we add the related terms of d into Vw1 to obtaina new Lyapunov function as [12](15)To design the controller, which has the ability totrack commands and to reject load disturbance,we add the related terms of d into Vo1 to obtaina new Lyapunov function as [12](9)where d is the error between the value of d andis a positiveits estimated value d, andconstant. After that, by taking the differential ofV 2 and substituting (7) and (8) into thedifferential result, it is not difficult to obtainControl inputgenerationestimationof d Fig. 2.(10)According to (10), the control input u can bedesigned aswhere c23.3 Nonlinear Position Controller DesignThe position tracking error of the positioncontrol system can be defined as(11)is a positive constant. Then, the(16)Now, by taking the derivative of both sides, onecan obtain estimated value, d , can be obtained by solvingthe following:(12)Substituting (11) and (12) into (10), Vdetermined as(17)Define a stability function as2can be(18)31
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.orgwhere Cwhere C 2 is positive, and ay1 is a stabilityfunction. The Lyapunov function is selected as[12]is a positive constant. The derivative of the estimated value, d , can be expressed as(19)Furthermore, defined Z2 as(26)Substituting (25) and (26) into (24), Vy2 isfound to be negative and can be expressed as(20)Then, taking the differential of Vy1 andsubstituting (17), (18) and (20) into it, we canobtain(27)Figure 3 shows the block diagram of theproposed position control drive system. Byusing a similar method to that for the speedcontrol system mentioned previously, one canderive(21)Taking the differential of Z2 and substituting(1), (18), and (20) into it, we can derive(28)Generally speaking, the selection of the positiveconstants is similar for both speed and positioncontrols. A large value of C 1 and C 1 canincrease the transient response. On the otherhand, a small value ofcan reduce theconvergence time of the estimator. Theselection depends on the designer’s experience.If the positive constants are beyond theirlimitations, oscillations occur.(22)To design the position controller including theabilities of tracking input command andrejecting external load disturbance, we add the related terms of Z2 and d into Vy1 to obtain anew Lyapunov function as(23)where d is the estimated error between the value3.4 PI Speed Controller DesignThe closed-loop speed control system with a PIcontroller is easily obtained. The transferfunction of the closed-loop system at anexternal load equal to zero is d and its estimated value d , andis apositive constant. Finally, by using (21)–(23),and taking the differential of Vy2, it is notdifficult to derive(29)(24)According to (24), the control input u can beobtained as(30)(25)32
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.orgC ontr ol inputgeneration(31)estimationof d(32)Fig. 3 Block diagram of proposed positioncontrol drive system3.5 PI Position Controller DesignThe transfer function of the closed-loop positionsystem at(33)On the other hand, when the external load TL isadded, the speed dip of the drive system is(41)After suitably reducing the order, the transferfunction of the closed-loop position controlsystem can be approximated to(34)(35)(36)(37)(38)Substituting (30)-(33) into (25))-(38), one canobtain:(42)Where(39)and(40)In this paper, we select the parameters of thespeed-loop controller as Kp1 0.2 and KI1 0.3.(43)(44)(45)33
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights reserved.www.jatit.orgThe specification of the position control systemis shown as follows. The steady-state positionerror ess 0, the risetime tr 0.15s, themaximum overshoot Mpo15%, the maximumposition dip at an external load o7.8%. Then,according to the required specification, one canderive the following:(46)Then, it is possible to rearrange (42) into twoparts as follows:(47)(55)(48)(56)(57)(49)(58)Substituting (48)–(51) into (55)–(58), one canderive9.5 Kp2 13.2(59)and17.3 KI2 24.8(60)In this paper, we select the position controller asKp2 10 and KI2 20. The relationshipbetween PI speed controller and PI positioncontroller cannot be derived because the speedcontrol system and the position control systemhave different specifications.Regarding the controller design, a goodtransient response is required for the IM drivesystem. In addition, the load disturbanceresponse should perform a low dip and a smallvalue of the integral square error (ISE). Byusing the proposed adaptive backsteppingcontroller, the specification of the risetime forspeed controller and position controller are setas 0.2 and 0.15s, respectively. Therefore, thereference model in Figs. 2 and 3 can be derivedas Md(s) 167.96/(s2 25.92s 167.96) andMd (s) 672.36/(s2 51.86s 672.36)individually. As for the PI controller design, inorder to decide the parameters of Kp and KI, thespecifications of the drive system are required.For example, the risetime of the transientresponse, the maximum overshoot and themaximum dip of the external load disturbanceshould be clearly determined. For a faircomparison, the risetime of the speed drive isset as 0.3s and the risetime of the position(50)(51)On the other hand, the transfer function betweenthe position dip and the external load is(52)After suitably reducing the order of (52), onecan obtain(53)and(54)34
Journal of Theoretical and Applied Information Technology 2007 JATIT. All rights ,fromexperimental results, the adaptive backsteppingcontroller performs better when the controllertracks a time-varying command, such as asinusoidal command or a triangular command.The major reason is that a PI controller maycause serious steady-state error and phase-lagproblem. As we know, a PI controller can tracka step input but cannot track a time varyingcommand well owing to its phase-lag problem.The details of the experimental results will beshown and discussed in Section 4. It is possibleto choose new parameters of the PI controllerand the adaptive backstepping controller for thedrive system carefully to reduce the risetime.According to the experimental results, for thebest case, the risetime of the speed controllerand position controller can be set as 0.13 and0.1 s, respectively. The results are also shown inSection 4. According to the experimentalresults, the performance of the adaptivebackstepping controller is again better than thatof the PI controller.control system is set as 0.15s. In this paper, tosimplify the design procedures, Kp isdetermined first. Then, KI is carefully chosen.By decreasing the value of KI, the overshoot ofthe transient response can be reduced. However,reducing KI could cause a large speed dip or aposition dip when the external l
The AC/AC matrix converter has several advantages. For example, the matrix converter is a single-stage converter. It does not require any DC-link energy storage component. In addition, it has a high-power-factor sinusoidal input current with a bidirectional pow
nonlinear control methods that have been applied to linear induction motor is the backstepping design [8, 9, 10]. Backstepping is a systematic and recursive design methodology for nonlinear feedback control. This approach is based upon a systematic procedure for the design of feedback control strategies suitable for the
In the approach of attitude controller design, dynamic inversion[5], feed linearization and sliding mode control[6], model reference adaptive[7] have been widely used. Backstepping control design[2-4], due to its simple, has become an effective approach for controller design. The rotational dynamics of the hexarotor UAV satisfies the strict
where a neural network adaptive controller is used for the transition from horizontal ight to hover, and [16], where a nonlinear dynamic inversion approach is used for formation ight. Within this context, an autopilot con guration for longitudinal and latero-directional aircraft control based on nonlinear backstepping is pre-
controller design. Many of these techniques make use of some aspect of the nonlinear dynamic inversion principle while choosing the variables for feedback and the controller structure. Block backstepping (BBS) is a well known technique for the design of nonlinear controllers(2-7) based on nonlinear dynamic inversion principle.
focused on backstepping control and sliding mode control. In [25], a backstepping technique was proposed to control the un-deractuated USV under constant environmental disturbances. In [3], a siding mode control was proposed to address the underactuated USV control problem, and experiments were carried out to verify the effectiveness.
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Chapter Two first discusses the need for an adaptive filter. Next, it presents adap-tation laws, principles of adaptive linear FIR filters, and principles of adaptive IIR filters. Then, it conducts a survey of adaptive nonlinear filters and a survey of applica-tions of adaptive nonlinear filters. This chapter furnishes the reader with the necessary
Highlights A large thermal comfort database validated the ASHRAE 55-2017 adaptive model Adaptive comfort is driven more by exposure to indoor climate, than outdoors Air movement and clothing account for approximately 1/3 of the adaptive effect Analyses supports the applicability of adaptive standards to mixed-mode buildings Air conditioning practice should implement adaptive comfort in dynamic .
