Analysis and Implementation of a Modified Robust Model Reference Adaptive

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Analysis and Implementation of a Modified Robust Model Reference AdaptiveControl with Repetitive Controller for UPS ApplicationsHilton Abilio Griindling , ES Mem ber, Emerson Giovani Carati, Jose Renes Pinheiro, IES Member

Federal University of Santa Maria - Santa Maria -RS [email protected] - Usually, the design for control systems beginstaking in account the nominal models. If there is any changein characteristics of these models, the system performance isdeteriored. Therefore, it is expected that the control systemsmust have robust stability. In this work, a modified RobustModel Reference Adaptive Control with a Repetitivecontroller (RMRAC-RP) for uninterruptible power supply(UPS) applications is presented. In the proposed scheme, avoltage source PWM inverter and its corresponding RLCload-filter are controlled by RMRAC-RP in order to obtain anearly sinusoidal output voltage for a single-phase UPS. Th emodified scheme guarantees that the closed-loop plant isglobally stable in the presence of unmodeled dynamics andcyclic fluctuating loads. The proposed theoreticalformulations are verified by system analysis and bothsimulated and experimental results are presented. In thiswork is also included an approach for the three phase scheme .

I. INTRODUCTIONRecent developments in closed-loop regulation of PWMinverters have been carried out to achieve good dynamicresponse and most of them were focused on the transientresponse improvement by using instantaneous feedbackcontrol [1]-[4]. In [ I ] a new adaptive repetitive controlscheme employs an auxiliary compensator to stabilize theclosed-loop system. Its parameters are tuned by an adaptivecontroller which recursively on-line identifies the plantdynamics. In order to guarantee the closed-loop stabilityunder plant variations and at the same time to eliminate

periodical errors, resulting by all frequency modes belowthe closed-loop bandwidth, this algorithm needs asophisticated plant model. This plant model includes thedynamics of the inductor, the capacitor-resistance and thesensor low-pass filters. Simulation and experimental resultsshow that the proposed control scheme can effectivelyeliminate periodic waveform distortions which are resultedby unknown periodic disturbances. However, theexperimental verification is carried out on a PWM inverterswitching @ 45 kHz. In [2 ] is presented a robust modelreference adaptive control algorithm including a repetitivecontrol for UP S applications. Such as in [ I ] , the proposedcontrol scheme can effectively eliminate periodicwaveform distortion, and moreover, it is globally stable inthe presence of unmodeled dynamics. This algorithm canbe designed for a reduced-order plant, without a prioriknowledge of the exact plant model of the PWM inverterand RLC load-filter system. This work proposes a modifiedRobust Model Reference Adaptive Control with aRepetitive controller (W R A C -R p) for uninterruptiblepower supply (UPS) applications. Unlikely of the previousalgorithms, this scheme includes the effect of the RPcontroller which is analyzed along with the RMRACcontrol law.

Following, a new modified error equation is developedand both the identifier stability and the closed-loop systemrobustness analysis are presented. In this work is alsoincluded an approach for the three phase scheme.11. DESCRIPTION OF THE PROPOSED SYSTEM

The conventional single-phase PWM inverter is shownin Fig. 1, where the inverter, LC filter, and resistive load Rare considered as the plant of the system. The state spaceequations of the plant arex:=A x t B u ,

where x=[v, V. a,, I / z , ';,, = ( 1 / 2 R ) m .

D1.+UR815Fig. 1. PW M inverter system

The power switches are turned on and offonce duringeach interv al T, such that Vi, is a voltage puls e ofmagnitude (E, 0, -E) and width AT centered in the intervalT . A sampled-data equation of the system at timet = ( k + l ) . T is

and the normalized output voltage y(k)& v, (k) /E is-y(k.) =[l/E 03. x(k) (2.3)

From (2.2) and (2. 3) a dlifference equation is obtained a sy( k +1)+a , y ( k )+a,y(k. - 1) =b,u(k) +b,u(k - 1) (2.4)

where the parameters are defined as( 2 . 5 )a1 = - ( ' P I 1 f (P22)b, =g J / E

6'2 = % I ' 0 2 2 -012 ' 0 21b2 =(gz ' 0 1 2 -g , ' 0 2 2 )T/E

U( lk)A, AT( k) /Tand the normalized input variable is

( 2 . 6 )

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where G(s) is strictly proper, G,(s) is the transfer functionof the modeled part of the plant, p. , (s) an d p. a s)are multiplicative and additive plant perturbations. Z,(s)and R,,(s) are monic polynomials of degree m an d n,respectively.For the modeled part of the plant Go(s) are made thefollowing assumptions:S1 - Zo(s) s monic Hurw itz polynomial of degree m ( 2 -1).S2 - b ( s ) is a monic polynomial of degree n.S3 - The sign of kp and the values of m and n are known.For the unmodeled p art of the plant are assum ed that:S4 - A ,(s) is a strictly proper stable transfer function.S5 - A,(s) is a stable transfer function.S6 - A lower bound p o 0 for which the poles of

The adaptive control objective can be stated as follows.A a (s-p) and A , (s-p) are stable is known.

Given the reference modelY,(S) =W,(s) .r(s) =(K,/D")r(s) (3.3)

?here D,(s) is a monic Hurwitz polynomial of degreen = n - m and r(t) is uniformly bounded, design anadaptive controller so that for some p* 0 and any pE [0 ,p ) the resulting closed-loop plant is stable and theplant output y tracks the reference model output y m asclosely as possible for all plant perturbations A a ( s ) an dA, (s) satisfying S4 - S6.

The input U and the output y are used to generate an-1 dimensional auxiliary vector as

o (k ) =(z I - )-' . . u ( k )o (k ) =(zI - )-' . y(k) (3.4)

where F is a stable matrix and (F,q) is a controllable pair.The input of the plant is taken asu ( k ) = Q T ( k ) o ( k ) + u 2 ( k ) + c o r ( k )

where +=8- e * , 0: 1 are the desiredparameters of the controller, with q(k) =A(z)u(k) ,where A(z) is a strictly proper transfer function.

=10;'

Proof: Let us define =[e;T,e;T,8:], where 0;'0;' are vectors of dimension (n-1). From (3.4) andnd

(3.5) it follows that:u(kI1- f l z)- f ,(z) .G(z)]- u2(k ) =$T (k)w(k)+ r (3.7)

wheref , (z ) =f 2 ( z ) = 0 , ( z I - F ) - ' q + Q , *

(z I -F)-' q*T (3 .8)Due the controllability of the modeled part of the plant,

there exists a vector 0* such that[l- f , (z)- f 2 z ). G o z)] =W,-] (z).G (z) (3.9)

Using (3.7) and (3.9), (3.1) can be rewritten a s

where 8'(k) =18:(k),IT(k),6,(k)] is a (2n-1)dimensional control parameter vector, co(t) is a scalarfeedforward parameter, mT k) = (k), 05 (k), y(k)], andcp is a scalar parameter.can be expressed asor

Lemma 3.1: Combining (3.1)-(3.5) the tracking errorel (k) =Y(k)- Y ref (k)

e l (k )= w m z h T k)o(k)-wm ( z b 2 ( k )+ P T ( ~ ) (3.6)

IV. PARAMETER ADAPTATION ALGORITHMThere are a number of well-known parameter estimationtechniques which have been successfully applied to theidentification problems. In this scheme is considered thefollowing modified least-squares adaptation algorithm:

(4.2)where P =PT is such that

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0 ', 6, 2 1 , (4.4)6,where a , , 6 , , 6 , , h , cl an d R2 are positive constantsand 6, satisfies 6, + 6 , - < m i n [ p , , q , ] , q , E% + is suchthat the poles of W,(z-qo) and the eigenvalues of F +q , Iare stable and ?j2 is a positive constant. po >0 is definedin S6 and CI in (4.1) is given by

if ((0([ 2 M ,where M O> ~ ~ B * ~ ~nd oo> 2 p 2 / R 2 E '93 are design

I ei I rocess 7Fig 2 RMRAC - UPS

Lemma 4.1: The parameter adaptation algorithm in(4.1)-(4.5) and (3.14) subject to assumptions S1-S6 has thefollowing properties: a) V =+TP-'+5 v andb) 11$([* I K + a h R 2 V

Pro03 The proof of this lemma which guarantees theidentifier robustness follows the lines of the proof of thelemma 4. 2 in [6] and it will be om itted.Theorem 4.1: Consider the plant in (3.1), (3.2), subjectto assumptions SI&, the adaptive control structure defined

in (3.3)-(3.6) and (3.14) together with the parameteradaptation algorithm in (4.1)-(4.5). Then a j T>O can becomputed so that for each p E 0, k all the signals inthe closed-loop system are bounded for any bounded initialconditions. Furthermore, the tracking errore I =y - belongs to the residual set,

1 J

Prooj! The stability analysis follows the lines of theproof of the theorem 5.1 in [SI and it will be omitted.V. EXPERIMENTAL RESULTS

A based-on IGBT prototype of the inverter has been built inthe laborato ry to verify the referen ce tracking capability of theRMR AC repetitive control scheme.TABLE IPARAMETERS OF PWM INVERTER AND LOAD

Filter inductance 1 L = 5 . 4 m HC =80pFI E = 4 0 VVref= 30V, 60Hz1 Load cauacitance I ;;I3;&[ Sampling time [ t, =(1/18OO)sThe detailed schem atic of the converter and load is shown inFig. 1, while the comp onent values are reported in Table I.The controller was implemented using a micro-controllerplatform where the following tasks are performed.

1) Measure o f capacitor voltage v, from the A/Dconverter (12 bits, 100 IC%).2)3) Charge the programmable counterhime for pulsegeneration (three-level F'WM pattem).

Com pute the pulsewidth At using (2.6).

r I

(DI.~,C,R),.OADime scale 2.5ms/div

t Fig 4 Output v,( 10 V/div) and filter input v,,(20 V/div) with(DI .~ ,C ,R)LOADim e scale 2 Sms/divFi g3 show s the output voltage v, and the input current

of the rectifier. Fig.4 shows the output voltage v, and thefilter input voltage v,, waveform with a load (DI-4,CL, RL).The experimental results, as shown in figures 3 and 4,demonstrate the robustness of th e W R A C p ro po se dcontroller. It is verified that the output voltage v, is asinusoidal waveform even in the presence of the powersupply variations (E ) and unmodeled dynamics.Fig. 6 shows the FFT of the output voltage which ispresented in Fig. 5. Experimentally, the TH D of the outputvoltage for a load (rectifier -RC) was equal to 3,25%.

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432I0I23

Fig 5 . 0 u t p u t v,(10 V/div) and iLoAD(1 N d i v ) with(D - ~ ,C & ) LO A D35 I30252015105" 0 5 I O I5 20 25 30 35 40 45 50

Fig 6 FFT output voltage TH D =3 25 'YOVI. THREE PH ASE CASE

Fig. 7shows the reference model block diagram for a idealthree phase U P S. The basic circuit configuration of theproposed reference model consists of a three-phase RL C load-filter, which is connected in w ye (Y) configuration.

Fig. 7. Reference model for three phase scheme

At this point it is important to note that ymd and yms areFig.8 shows the control function block diagram of the three-independent signals.phase UPS system.

J" gFig. 8. Block diagram for the proposed three phase system.

The output of a three-phase PWM inverter is connected tothe LC filter-load (Plant) in delta (A) configuration and itsinput is supplied by a battery. In the Fig. 8, two control laws u dand uqare calculated using the algorithm presented in Fig. 2.

Fig. 9 presents the reference signals used in the Matlabsimulation. The simulation results of the proposed three-phaseU P S system for a resistive load is presented in the Fig. 10.The purpose of this simulation is to verify the controlstrategy, to design the controller parameters and to study thestatic and dynamic performance of the system before buildingthe laboratory prototype.

In Fig. 10, when the initials conditions are far fiom thedesired values can be seen a small transient and afterwards theconvergence improves con siderably.

- 4 4 1 . , . , , , , , , , , I44

I Q

2Q

iQ

Q

- i o

-2a

.> e

-4 0

I Q 20 10 40 5Q 6Q 70 80 9Q 100 110 120

Fig. 9.Reference Signals

i a 21 >a 4~ sa 6a 70 ao QQ i o 0 i i c I L QFig. 10. Output voltage waveforms.

VII. CONCLUSIONSThis work describes a modified robust model referenceadaptive control with a repetitive controller for UPS

applications. A new proposed control law yields new errorequations, which are employed in the adaptationparameters algorithm of the controller. Experimentalresults show that the new proposed control scheme caneffectively eliminate periodic waveform distortion resultingby unknown periodic d i sturbances and m o d e l e ddynamics. Furthermore, this scheme also can be designedfor a reduced-order plant, without a priori knowledge ofthe exact plant model of the PWM inverter system. Theanalysis and the approach for the three-phase UPS systemis not different from the conventional methods. The

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simulations results have demonstrated that the proposedscheme is feasible and applicable for future developments.It was observed that the proposed scheme presents a goodperformance either for balanced as unbalanced loads. Thistype of W R A C -R P controll er and PW M inverter systemis particularly suitable for high-performance ac high-powersupply systems.

VIII. REFERENCESS. C . Yeh and Y. Y . Tzou, Adaptive repetitivecontrol of a PWM inverter for AC voltage regulationwith low harmonic distortion, IEEE Power Electron.Specialists Con$, pp. 157-163, 1995.H. A. Griindling, E. G. Carati, J. R. Pinheiro, Arobust model reference adaptive controller for UPSapplications, 23rd Intern. Co ns on IndustrialElectron. Control and Instbum . vol. 2, pp. 90 1-905,Nov. 1997.L. G. Bames I1 1 and R. Krishman, An adaptive three-phase UPS inverter controller, IEEE Power Electron.Specialists ConJ, pp. 473-479, 1995.H. B. Mohr and G. E. Mondardo, PWM voltageinverter implementation using deadbeat digitalcontrol, IEEE Power Electron. Specialists ConJ, pp .P. Ioannou and K. Tsakalis, A Robust DirectAdaptive Controller, IEEE Transactions onAutomatic Control, Vol. AC-31, no 11, pp. 1033-1043, 1986.R. Lozano-Leal, J. Collado and S. Mondie, ModelReference Robust Adaptive Control Without A PrioriKnowledge of the High Frequency Gain, IEEETransactions on Automatic Control, Vol. AC-35, no

2 17-221 , 1996.

3, pp. 71-78, 1990.

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Analysis and Implementation of a Modified Robust Model Reference Adaptive