Adaptive Repetitive Control of a PWM Inverter for AC Voltage

7/27/2019 Adaptive Repetitive Control of a PWM Inverter for AC Voltage

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Adaptive Repetitive Control of a PWM Inverter for AC VoltageRegulation with Low Harmonic DistortionShing-Chung Yeh and Ying-Yu Tzou, Member, IEEE

Power Electronics &Motion C ontrol Lab.,Institute of Control EngineeringNational Chiao Tung Univ., Taiwan, R.O.C.

Abstracf% adaptive repetitive control scheme is proposedand applied in control of a PWM inverter used in high-performance ac power supply. The proposed control scheme canadaptive eliminate periodic distortions resulted by unknownperiodic load disturbances in an ac power supply. The proposedadaptive repetitive controller consists of a voltage regulatorusing state feedback control, a repetitive controller with tuningparameters, and an adaptive controller with recursive least-squares estimator. This adaptive repetitive controller used forac voltage regulation has been realized by using a single-chipdigital signal processor @SP) TMS320C14 from TexasInstruments. Experimental verifications have been carried outon a 2 kVA PWM inverter. Simulation and experimental resultshow that the DSP-based adaptive repetitive controller canachieve both good dynamic response and low THD under largeload disturbances and plant uncertainties.

I. INTRODUCTIONClosed-loop regulated pulsewidth modulated (PWM)inverters have been widely applied in various type of acpower conditioning systems, such as unintermptible powersupply (UPS) systems, automatic voltage regulators (AVR),

and programmable ac source (PAS). One major requirementof these applications is that the system is required to m aintaina low-distortion waveform under transient or periodic loaddisturbances. Some research have been carried out on theclosed-loop regulation of PWM inverters to achieve gooddynamic response and most of them were focused on thetransient response improvement by using instantaneousfeedback control [1]-[7]. Although satisfactory results havebeen obtained for transient load disturbance, they leaveperiodic distortions in the output waveform when the loaddisturbance is periodic.Repetitive control theory [SI-[lo] originated from theinternal model principle [1 I ] provides a solution to eliminateperiodic error occurred in a dynamic system. A repetitivecontroller can be viewed as a periodic waveform generatoraugmented within the control loop of a co ntrol system, whichis closed-loop regulated by a feedback controller, so that theperiodic errors can be eliminated. A number of repetitivecontrol schemes have been developed and ap plied to variousindustrial a p p k ti o n s [12]-[14]. Nishida and Haneyoshi [15],and Hane, um. t a l . [161 had applied the repetitive controltechnique to elimina te periodic distortions resulted in a PW Minverter. In their approaches, the repetitive controller wasdesigned based on the exact model of the closed-loop0-7803-2730-61954.00 995 EEE

controlled system with one-step prediction. Therefore, theperformanc e and stability of such repe titive control systems ishighly dependent upon the robustness of the original feedbackcontrol system.Torelax the stringent stability requirement of a repetitivecontrol system, a modified repetitive control scheme withfinite frequency mode elimination has been developed [171.However, the conve rgence of such system will be deteriorateddue to plant uncertainties. For time-varying or system withlarge plant uncertainties, adaptive repetitive control schemeswere developed to eliminate periodic errors [181-[191.Although these methods can track the changing plantdynamics, they have the drawback that the number ofparameters to be estimated are proportional to the frequencymodes selected to be c anceled.In this paper, we have developed a new adaptive repetitivecontrol scheme that employs an auxiliary compensator tostabilize the closed-loop system and its p arameters are tunedby an adaptive tuning controller which recursively on-lineidentify the plant dynam ics. The ada ptive repetitive controllerwill guarantee the closed-loop stability under plant variationsand, at the same time, elim inates periodical errors resulted byall frequency modes below the closed-loop bandw idth.This paper is organized as following. In Sec. 11, we firstlyintroduce the operational principle, and then we derive thestability criterion and convergence of the periodic error of arepetitive control system. Sec. I11 describes the proposedadaptive repetitive control scheme. Sec. IV describes themodeling and the DSP-based repetitive control of a PWMinverter for ac voltage regu lation. Sec. V gives the simulationand experim ental verification of the proposed c ontrol schemeand Sec. VI is the conc lusion.

11. REPETITIVEONT ROLYSTEMA . Principle of Repetitive Control

A servomechanism system is required to regulate thecontrolled variables to reference commands without steady-state error against unknown and unmeasurable disturbanceinputs. In servo mecha nism system design, the internal modelprinciple proposed by Francis and Wonham [11] plays animportant role. The internal model principle states that thecontrolled output tracks a set of reference inputs withoutsteady-state error if the model which generates thesereferences is included in the stable closed-loop system. The157

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IIGepebtive I Tracking I

L3asic Servo Plant F'( 21 )Repebt ive

Controller

Pig. 1. Conligurations ofrepztitivc control s ys te m

.............. I I

................................ IFig 2 Block d i a q a m of a disci-etc. im e repetitive control system.

internal model principle therefore reveals that the highaccuracy asymptotic tracking properties for periodicexogenous inputs can be achieved by locating the modelwhich generates a set of periodic signals. Repetitive control isa control scheme, in which the frequency modes of theperiodic error to be eliminated arc included in the stablecontrol loops. Repetitivc control can be easily realized usingmicroprocessor-based digital controller and with the advanceof high-performance microprocessor and digital signalprocessor, more frequency modes can be included in thecontrol loops. This reveals the feasibilit! of implementationof superprecision servomechanism system.

In a repetitive control systcm. a rcpetitivc controller isinsertcd in thc control loop i n addition to the conventionaltracking controller. There are various control configurationsfor the repetitive control systems. Fig. 1 shows two basiccontrol structures of the repetitive control system. Fig. l(a)sho\vs a cascaded type repetitive controller, in which it iscascaded as an outer loop controller. Fig. l(b) provides afeedforward path to the repetitive controller. The majorpurpose of the tracking controller is to improve the systemtransient response and make i t insensitive to esternaldisturbance. u l~ il e he repetitive controller is to reduceperiodic error resulted by periodic reference or disturbance s.Fig. 2 shows the block diagram of the proposed repetitivecontrol system. where P(z-') represents the closed-loop

transfer function of the plan t i n which it is closed-loopregulated by a tracking controller, S(z- ') an d e(.-')are theauxiliary compensator of the repetitive controller, r (k ) is thereference signal, v(k) s system output. e ( k ) is tracking error.an d r ,(k) is compensated reference co mm and.

The transfer function from the disturbance input d(k) o thetracking error e ( k ) in Fig 2is

where E @ ' ) an d D( z- ' )are z-transforms of e ( k ) an d t l (k) . Ifd(k) is a periodic disturbance with period of N , then theFourier series representation of d ( k ) can be expressed as

N-l

,=Owhere { c , , } denotes the Fourier coefficients and itscorresponding frequency response in s-domain is

f f ( j m )=fI(z-' )Iz . ! l l l l (31I n a special case, if Q(z- ' )= an d P(z") is stable. we can find

l f I ( j o ) ) l = O a t m = 2 n n / N , n = O , . . N - I . (4)This reveals that these frequency modes of the periodic errorhave been eliniinated by the repetitive controller, thus perfecttracking is achieved in this condition. However, such aperfect tracking imposes a stringent stability criterion in thesynthesis of S'(z-'), In practical situation, we can relax thisstability criterion by choosing Q(z- ' ) a low-pass filter or aconstant less than unit such that

( 5 )where pow) specifies the attenuation of the frequency modesof the periodic distu rbance

IH( j m < p ( j w ) at w =2 n n l N , n =O,..N -- I

B. Stability Awa&sisFrom Fig. 2 . we can find

an d

Eliminating I ( z ) from (6) an d (7), we can getE (ZL1=E(z I )z " (Q(z ' ) -P (z~)S(z ' ) )

+ ( I - P ( z '))(IQ (z '12 N)R(Z ). (8)Fig 3 shows the block d iagram representation of (8 ) and wecan observe that if

(9)an d P(z- ' ) I S stable, then e ( k ) is bounded which means thesystem is stable

I Q( z ' 1- P ( z ' ) S ( Z ' )I ~ i a r l 1 f a al l w

158

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..Fig. 3 . Block diagram representation ofth e error convergence. p x p..The design of S(z") an d Q(z- ' ) s a comprom ise between thedegree of relative stability and the convergence rate of theperiodic error elimination. For simplicity, if we choose Q(z-')to be a constant less than and close to unit. we can arbitrarilychoose an , ~ ( ~ - 1 ) vev gain tocriterion of (9). However. the periodic error may be still large. There are a number of well-known parameter estimationTo satis@ both requirements (5) an d (9), we can choose Q(z-') techniques that have been successfully applied to theto be a constant less than and close to unit and S(Z-') to be a identification problem [20]. The recursive-least squaresdigital filter with phase-lead characteristics, An optim al estimator (LSE) has the advantages Of unbias, fastchoice of S ( z - ' ) n terms of large relative stability and fast convergence. and its aPPlicabilif3' to a wide range ofconvergence rate can be achieved when s(z ) ~ ( ~ - ' ) ossesses application in whi ch other statisticabestimation theo ries Inaya nearly zero-phase-shift characteristics [191. Th is can be be diffi cult to apply [211. In the aPPl*cation of an adaptiveaccomplished by choositlg ~ ( ~ - 1 )s the inverse of ~ ( ~ - 1 )nd control system, the most imp ortant considerati on factor is itsaugmented with a realizable low-pass filter with appropria te feasibility to be realized with an acceptable sanding *ate.cut-off frequency. The recursive LSE (RLSE) parameter identificationalgorithm used in this paper is:

Fig. 4. The proposed adaptive repetitive control scheme.

the stability where the parameters a,@) an d a,@) are left to be identified.

C . Convergence .4nalysi.v

also be viewed as a perform ance index for the convergence ofP ( k - )&k -1)8(k)=&k - 1) +The i e ( z ~ ' ) - S ( z - ' ) P ( z - ' ) lin the stability criterion of (9 ) ca n u + 4 ( k - I ) ' P ( k - - 2 ) 4 ( k - l ) '

( y , )- c (k- ) - ( k- )'&$ -1)) (13)periodic error. A smaller IQ(z - ' ) -S ( z~ ' )P( - - ' ) ( results a fastererror convergence. The convergence index is defined as 1. 14 )P ( k - 2)qi(k - ) # ( k- ) 'P(k - )P (k - ) =- ( k - 2) -a a +$( k - )'P(k- )+(k- 1)

B ( k - l ) = [ a , ( k ) a,(k)l' , (13l [h =( Q ( z- ' ) P ( z - ' ) s ( z - ' ) ~_ (10)

If h=O, the periodic error can be eliminated after one cycle.However, such a condition requires a perfect match of thesuch situation, we define

I

plant model P(2.l) and this is obviouslv unrealistic. To avoid @(k - ) =[ y(k - ) - y ( k -2] , (16)where &0) is the initial guess of the parameters to be

g Q@ ' > identified, P ( k ) is a positive definite measure of theestimation error and its elements tend to decrease as theS(z-1) =--T + 1 P ( z - ' ) (''I identification reaches its steady state, and CL is a forgettingis a 'Onstant between zero an d unit' and ' s a factor to weight new data more heavily than old data. Whentx=l , all data are weighted equally. For O


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P V I M I NM RTE R j LCFILTER j L O A 0 /SENSOR FILTERI

E

Fig. 7. Hardware configuration ofth e digital controlled PW M inverter

Fig. 5. Parameter identification of a PW M inverter connected with a bridge-rectifier RC load, (a) output voltage and current waveforms, @) estimatedparameterusmg the RLSE, (c) estimated paramcter using the modified RLSE, d)estimatedparameters ofthe averaging model using modified RLSE.vonags sensor and filter

Fig. 8. State feedback control block diagram of PWlM inverter with nominalload R =40 ohm.

an dFig. 6. Diserete adaptively repetitive controller.

Fig. 5(d) shows the estimated parameters of the averagingmodel using modified RLSE method. The estimatedparameters of the averaging model are derived using thefollowing algorithm

Since there are switching ripples in the capacitor voltageand inductor current, they are sensed through low-pass filters.Considering the dynamics in these filters, we can get

where N is the number of samples during one half period ofthe output waveform. T he auxiliary compe nsator S(z-l)of therepetitive controller is adjusted by rising the estimatedparameters of a,(??).Z 2 ( n )as shown in Fig. 6 .

From (19)-(22), the state equation and output equation ofthe plant isi ( t ) =k ( t )+Bu(t) , (23)

IV. MODELINGND CONTROL FPWM INVERTERSYSTEMThe hardware configuration of the proposed DSP-baseddigital controlled PWM inverter system is shown in Fig. 7, inwhich, the combination of H-bridge PWM inverter, LC filter,and Rectlfier-type RC load is conside red as plant.A . Plant Modeling

The capacitor voltage vc and the inductor current iL arechosen as the state variables and the system dynamicequations can be derived a s

u ( t )=v, ( t ) , (27)

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A =

an dB =[1/L 0 0 01,

RR + r , R + r ,

The corresponding discrete-time model can be derived asx ( k + l ) =G p x ( k ) + H p u ( k ) , (3 1)

G , =e A T , (33)H P=A-'(eAT )B , (34)

where Ti s the sampling period.B. State Feedback Control

Considering the state feedback control block diagram inFig. 8 . The control law can be derived as

u ( k ) =k , u , ( k ) - k , u , ( k ) - k , v , ( k ) , ( 3 7 )where uref(k) s a table of reference command stored inmemory of DSP, the state feedback gains k,,,k , , an d k, can bedetermined by method in [221. Combining (31), (32), and(37), we can obtain the state space equation of the digitalcontrolled system as

where

an dK =[0 0 k, k 2 ] . (42)

The discrete-time transfer function from reference commandto output voltage is

~

161

0-20-40

1 5d e o re e

-400 . . . . . .. . . . . . I I2 510 l o 3 rad/sec lo 4 10(a)

2 , I

1.5 1 A Q ( 2 - l ) =e(.-')- O.SP,(Z-~)P,~~(Z-')1

0. 5

E o-0.5

-1s tab le boundary-1 5 1

- 2 1 5 1 0. 5 0 0 5 1 1 5 2real(b)

Fig. 10. (a) Frequency responses of the state feedback controlled PWM invertersystem P, ( i ) and the approximate model P,(z ') . (b) Nyquist plot of Q(z')-0.5Pn(z1~ , (zL) ,hich show s stability of the repetitive control system when

1

S(Z-I)=llP*(i')

(43)It should be noted that the plant m odel P,(z-') is derived basedon a nominal op erating condition and in practical condition itwill encounter large model uncertainties due to large loadvariations.C. Design Example

TableI lists some of the key parameters of the constructedPWM inverter system for 60 Hz, lO V (RMS) ac voltageregulation. A a single-chip DSP TMS320C14 from TexasInstruments has been adopted to realize the proposed adaptiverepetitive controller. The sampling frequency of the digitalcontrolled is 15 kHz and there are 250 samples in each cycleof the sinusoidal output. The sampling rate of the adaptiveparameter tuner is 120 Hz and it adjust the controlparameters of the rep etitive control every half cycle.The state feedback gain k , an d k, are determined tominimize the output voltage distortion due to transient load

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TABLEI. PARAMBTERS OF TH E PW M JNVERTER SYSTEMI Item I Symbol INominal value I UnitSampling RateFilter Inductor 0.6

f,L

Filter Capacitor r 3 3 PFInductor ESR r L ( I 5 nCap serial resistor rc 1 0 R

~ Noininal load R 40 R--

, DC link voltage c 30 0 VoltsOutput voltage ' j 0 110 Volts (rms)

disturbance and the feedforward gain k,, is a scalingparameters to let the system have an unit gain at 60 Hz. Withthe given parameters as shown in Table I, the k l = -0.9 an dk,=-0.45 are selected as state-feedback gains and thecorresponding feedforward gain k , is 0.56. The nominaltransfer function Pn(z- ') of the digital state-feedbackcontrolled PW M inverter is

0 53 1Oz '+0 3h55z - 04852 +0 O O O ~ Z - ~P " ( Z ')=1 - 0 5 1 8 2 ~ + 0 4 7 8 9 z '-0127iz ' - 0 0 0 1 5 ~ - ~ 45)The recursive LSE method has been used to identifjr thetransfer function of the closed-loop controlled PWM inverterbased on a second-order approximate model of (12). Theidentified plant model Po(z-' ) s

2(46)Z1; ( z - ' =__-__1 - 0 3 1 1 ~+ O 2962

The frequency responses of P,(z-') and P,(z-') are shown i nFig. 9(a) and it can be observed that there are closeresemblance between P,(z") an d P,(z-'). The Nyquist plot of~(z~')-P,(z~')Un(z~')s shown in Fig. 9(b) an d it can beobserved that it is within the stabilit) b oundary and thisguarantee the stability of the repe titive control system.

v. SIMULATION AN D EXPERIMENTALERIFICATIONFig. 10 shows the simulation results of the adaptiverepetitive controlled PWM inverter for ac voltage regulation.Fig. l0(a) shows a 3-dimension plot of the error convergenceof the PW M inverter connected with bridge-rectifier RC load,Fig. 1001) shows the time responses of the estimatedparameter estimation of a second-order approximate averagemodel.The experimental verification of the proposed adaptiverepetitive control scheme is carried out on a 2 kV A PW Minverter connected with a rectifier-RC load with current crest-

~

162

samples( 8 )

0.5--'0.40.30.20.10

-0.1-0.2-0.3-0.4 t 1

-1 4 I I I 'ycles10 20 30 40 50 6 00.5 (b')Fig. 10. (a) Simulation results of th e m o r convergence using the proposedadaptive repetitive control scheme, (b) Time responses of the estimatedparameters oft he approximate average model.factor of 3. Fig. I I(a) shows the time responses of the outputvoltage and current of th e PWM inverter using the digitalstate feedback control and Fig. Il(b) shows the samecondition when the adaptive repetitive control is included.Fig. 12 shows the error convergence of the PW M inverterunder repetitive control and it can be observed that it elapsesabout 12 cycles for the settling of the periodic error. For ;I 60Hz output, it corresponds to 0.2 sec to elirniriate the periodicdisturbance resulted by a step changed rectifier-RC load.

VI. CONCI,IJSIONAn adaptive repetitive control is proposed and successfullyapplied for the closed-loop regulation of a PW M inverter usedin high-performance ac power supply. Simulation andexperimental results show that the proposed control schemecan effectively eliminate periodic waveform distortionresulted by unknown periodic disturbance. Compared withthe conventional repetitive control methods, thc proposedadaptive repetitive control scheme can not only achieve fasterconvergence rate. it also guarantees stability robustness underlarge load variations. The THD for the rectifier-RC load ofcurrent crest factor 3 is 8% by using the state feedbackcontrol and it can be reduced to about 1% within 0 .2 sec for

60 Hz rated output. An important merit of the proposedadaptive repetithe control scheme is that rt can be designedan d implemented independently without knowing the exact

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Volt Amp200150100500-50-100-150

0 5 10 15 20Time (msec)Volt (a)200150100500

-50-100-150-200 1 ----.~~..... ..-..-...-.. . . ....~---- ---~-...----.-.------.......................-200 5 10

(b)15 20Time (msec)

Fig. 11. (a) The 60 Hz output w aveforms for rectify load, Lvrrent crest fad or3 with only state feedback control, (b) output waveform when the adaptiverepetitive control is applied.

Vo l t510 I -1

-I___ ;2 0 4 0 6 0 8

Tiiriiilsarl

Fig. 12. Error convergence oft he PWM inverter under repetitive control.plant model of the PWM inverter system. This reveals thefeasibility of the construction of an adaptive repetitive controlmodule (ARCM) which can be inserted within the controlloop of a conventional analog controlled PWM inverter andsignificantly improves th e waveform quality for ac voltageregulation

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Adaptive Repetitive Control of a PWM Inverter for AC Voltage