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410 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 4, APRIL 1991

Nonlinear P-I Co ntroller Designfor Switchmode dc-to-dcPower Co nvertersHebertt Sira-Ramirez, Senior Member, IEEE

Abstract --In this article, extende d linearizatio n techniqu es are pro-posed for the design of n onlinear proportional-integral P-I) controllersstabilizing, to a constant value, the average output voltage of pulse-widthmodulation (PWM) switch-reg ulated dc-to-dc converters. The Ziegler-Nichols method is employed for the P-I controller specification, asapplied to a family of parametrized transfer function models of thelinearized average converter behavior around a constant operating equi-librium point of the average PWM controlled circuit. The boost and thebuck-boost converters are specifically treated and the regulated perfor-mance i s illustrated through com puter simulation experiments.Keywords: dc-to-dc power con verters, extended Linearization, n onlinea rP-I controllers, pulse-width modulation.

I. INTRODUCTIONULSE-width modulation (PWM) control schemes usuallyP egulate dc-to-dc power converters in a variety of differ-ent arrangements, involving a high sampling frequency of therequired feedback signals. The available regulator designtechniques are based on approximate linear incrementalmodels of discrete-time nature (see Severns and Bloom [l],Middlebrook and Cuk [2], Csaki et al . [3], etc.). Nonlinearcontrol schemes have been recently proposed for this class ofcircuits, which do not necessarily resort to the discrete-timeapproximation scheme. Instead, the properties of suitablenonlinear continuous average models, obtained by imposingan infinite sampling frequency assumption, are convenientlyexploited. The proposed design schemes are, among others:Sliding mode control strategies, with switching surfaces pre-scribed on the basis of properties associated to the idealsliding dynamics (see Venkataramanan et al. [4] and Sira-Ramirez L.51); discontinuous control strategies, such as PWM,based on singular perturbation considerations related totime-scale separation properties of the converters averageresponses and their associated slow manifolds (see Sira-Ramirez and Ilic [6] and Sira-Ramirez [71); regulationschemes based on exact linearization of the average PWMcircuit model (Sira-Ramirez and Ilic [8]); and pseudolin-earization techniques applied on the nonlinear continuous

Manus cript received August 22, 1989; revised December 3, 1990. Thiswork was supported by the Consejo de Desarrollo Cientifico, Humanis-tico y Tecnologico of the Universidad de Los Andes under Grant1-32.5-90. This pa per w as recomm ended by Associate Editor R. K.Hester.The author is with the Control Systems Department of the SystemsEngineering School of the Universidad de Los Andes, Merida,Venezuela.IEEE Log Number 9042411.

average PWM converter model (see Sira-Ramirez [9]). Acomparison of the various proposed designs methods, con-tained in [5]-[9], is carried out in Section 3.3 of this paper .A new nonlinear regulator design scheme is proposed inthis article fo r the stabilization of output variables in dc-to-dcpower converters. The extended linearization control tech-nique, developed by Rugh and his co-workers (see [10]-[12]),is applied to nonlinear infinite frequency average models ofthe PWM controlled converters. Feedback controller designby means of the extended linearization approach constitutesa highly attractive nonlinear design technique with potentialfor many practical applications. The method is based on thespecification of a linear regulator inducing desirable stabilitycharacteristics on the behavior of an entire family of lin-earized plant models. The family of linear models isparametrized by constant operating points of a smooth sur-face of equilibrium points defined in the input-output spaceof the system. The linear design constitutes the basis for(nonuniquely) prescribing a nonlinear regulator that exhibitsthe fundamental property that its linearized model, com-puted about the same generic operating point, coincides withthe specified stabilizing controller. The obtained nonlinearregulator is known to exhibit the advantageous property ofself-scheduling with respect to constant reference opera t-ing equilibria which may be subject to sudden (and purpose-ful) changes in their nominal values.

Nonlinear P-I controllers are proposed here for the regu-lation of the output voltage in dc-to-dc power converterssuch as the boost and the buck-boost power supplies. Thefrequency domain version of the Ziegler-Nichols prescrip-tion [13] is used for the determination of the linearized P-Iregulator gains, which result in a stable family of closed loopparametrized transfer functions relating the incremental out-put voltage to the incremental duty ratio function of theconverter. The nonlinear P-I controller is directly specifiedfrom the linear design in a manner entirely similar to thatproposed in [18]. In contrast to sliding mode control tech-niques, it should be remarked that constant output voltageregulation cannot be successfully accomplished by means ofsliding mode behavior induced on surfaces representing zerooutput errors. Within such a control technique, constantoutput voltage control may be achieved only when a combi-nation of the state variables is formed in a sliding line or,alternatively, indirectly through constant input current regu-lation [5]. Similarly, it is easy to verify that direct use ofPWM controllers, processing the same error signals, doperform constant output voltage regulation for a limited (but

0098-4094/91/0400-0410 01.00 991 IEEE

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SIRA-RAMIREZ: NONLINEAR P-I CONTROLLER 411

unnaturally restricted) range of desirable set points. Quiteon the contrary, the nonlinear P-I controller here proposedefficiently handles the output voltage regulation problem, ina direct manner, without noticeable instability effects, validat least in a local sense. Since the designed regulator is to beused in combination with an actual PWM actuator, produc-ing highly discontinuous signals, a proportional-integral-derivative (P-I-D) controller does not seem feasible due tolarge controller output values produced by the derivativeaction performed on the discontinuous error voltages.

A brief review of nonlinear P-I controller design, by meansof the extended linearization technique, is presented in Sec-tion 11. The class of systems treated there corresponds togeneral time-invariant bilinear systems. In Section I11 wepresent the procedure for synthesizing nonlinear P-I regula-tors for dc-to-dc converters of the boost and buck-boost type.These nonlinear P-I controllers are designed on the basis ofthe average models of the PWM controlled converters. Here,the form in which the designed nonlinear P-I controllers areto be used in the actual discontinuous PWM feedback schemeis also indicated. This section also presents some computersimulation experiments illustrating the performance of thenonlinear P-I controller. The last section contains someconclusions and suggestions for fu rther work.11. BACKGROUNDESULTS

The extended linearization technique is reviewed in thissection and applied to time invariant discontinuously con-trolled bilinear systems of the formh / d t = Ax +U( g ) + h

y = cx (2 .1)with x R , g and h are constant n-dimensional vectors,while A , B, and c are matrices of appropriate dimensions.The variable represents a switch position function actingas the control signal and taking values on the binary set{0,1}. The output y is assumed to be a scalar function.The feedback control strategy regulating the system isassumed to be of the discontinuous type and specified on thebasis of a sampled closed loop PWM control scheme of theform

between the actual PWM controlled responses and those ofthe average model, under identical initial conditions, remainuniformly arbitrarily close to each other. Conversely, foreach prespecified degree of error tolerance, a sufficientlyhigh sampling frequency may be found such that the actualand the average trajectories differ by less than such a giventolerance bound. The error can be made even smaller if thesampling frequency is suitably increased. Moreover, from apurely geometric viewpoint, in those regions of nonsatura-tion of the duty ratio function p , integral manifolds contain-ing families of s tate responses of the average model consti-tute actual sliding surfaces about which the discontinuousPWM controlled trajectories may exhibit sliding regimes [14].Outside the region of nonsaturation, the trajectories of boththe actual and the average PWM models entirely coincide.The average model dynamics plays then the role of the idealsliding dynamics (see Utkin [16] and Sira-Ramirez [17]) in thecorresponding variable structure control reformulation of thePWM control strategy [14].One formally obtains the average model of system(2.1)-(2.2) by simply substituting the duty ratio feedbackfunction p in place of the actual switch control function[14]. In order to differentiate the actual state vector x fromthe average states, we shall denote the average of the actualstate vector x by means of the vector 2:

d z / d t =Az p B z g hy = c z . (2 .3)

By means of Z , we denote an equilibrium state vector forthe average system (2.3). If such an equilibrium state existsthen it must, necessarily, correspond to a constant value ofthe duty ratio feedback function p . We could express such avalue by p Z ) . We prefer, however, to denote such a con-stant input value as U . It is also easy to see that for a givenU , he equilibrium state vector can, in turn, be expressed as afunction of U by means of a function Z ( U ) . This value Z ( U )coincides with the previously given equilibrium state Z if andonly if the matrix A UBI is invertible. In such a case weobtainZ U ) = (A UB) - Ug h) .

1, fOrfk

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form p J s ) asG,( S := y,( s)/p, , S ) = e ( SI- A U S ] ) [ B Z( U ) g ] .

(2 .6)The description of the linearized system as a proper ratio-nal transfer function is only valid in the region of nonsatura-tion of the duty ratio function, i.e., for 0 10 for m ( t ) < 0.

This bounding scheme may cause saturation effects on thePWM actuator, especially for those initial conditions whichare far from the required equilibrium point. The use ofanti-reset windup schemes ([13], pp. 10-14) may be at-tempted in such cases. We do not give further considerationto this topic here since it implies only a minor modificationof the control scheme proposed here.The high-frequency discontinuous trajectories induced bythe PWM actuator on the system state and output variables(known as chattering) requires some additional processingto fur ther approximate the proposed control scheme to thatof the designed average system. The feedback design pre-sented above is based on the infinite frequency averaged

m ( t ) for 0 < m ( t ) < 1 (2.11)

C , ( s ) = p 8 ( s ) / e , ( s )= K , ( U ) K 2 ( U ) / s . 2.8) filter and its associated phase lag is made sufficiently small.The above gains are easily computed in terms of theultimate frequency and the ultimate gain as prescribed bythe Ziegler-Nichols design rules:

111. P-I CONTROLLERESIGNOR dc-To-dcPOWER ONVERTERS3.1. Boost ConverterK l ( U ) = 0 . 4 K o ( U ) , K 2 ( U )= K l ( U ) W O ( U ) / l . 6 r . ( 2 . 9 ) Consider the boost converter model shown in Fig. 1. Thisi181, a nonlinear p-I controller may then be converter is described by the following bilinear system ofspecified by considering the dynamical regulator: controlled differential equations:

d X 1 / d t = w o x 2 u w 0 x 2 bd r 2 / d t = 00x1 - ~ 1 x 2 U w o X l( t )= s t ) K , [ l ( t ) l e ( t )

(2.10)be taken as the specification of the duty ratio function, forwhere m ( t ) s the nonlinear controller output signal that is to Y = x2 (3 .1)the PWM actuator, only in the region where such an outputsignal m ( t ) does not violate the saturation limits naturallyimposed to the duty ratio function as 0 < p < 1. We refer tom as the computed duty ratio function.Linearization of the nonlinear state equations, describingthe dynamical controller represented by (2.10), around theoperating point e ( U )= 0, l ( U )=U , produces an incrementalmodel whose transfer function entirely coincides with (2.8).The operation of the average nonlinear controlled system(2.3) in the vicinity of the given equilibrium point ( U ,Z ( U ) )thus exhibits the same stability characteristics that the lin-earized P-I controller (2.8) imposes on the family of lin-earized plants represented by (2.6). By the results com-mented upon in Remark 1, the behavior of the actualdiscontinuous PWM controlled system (2.1)-(2.2) in conjunc-tion with the designed nonlinear P-I controller will exhibitthe same qualitative stability characteristics caused by thestabilizing design on the average PWM closed loop system,provided a sufficiently high sampling frequency is used in(2.2).

where x 1= ZJL, x 2 =VJC represent normalized input cur-rent and output voltage variables, respectively. The quantityb = E/JL is the normalized external input voltage, andwo= 1/JLC and w 1= 1/RC are, respectively, the L C (in-put) circuit natural oscillating frequency and the RC (output)circuit time constant. The variable U denotes the switchposition function, acting as a control input, and taking valuesin the discrete set {O l}. System (3.1) is of the same form as(2.1), with g =0 and h =[ b 01. We now summarize, ac-cording to the theory presented in the previous section, theformulas leading to a nonlinear P-I controller design for theaverage model of (3.1).

Average boost converter model:

y = 2 2

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SIRA-RAMIREZ: NONLINEAR P I CONTROLLER 413

I T{Fig. 1. Boost converter.

Fig. 2. A nonlinear P-I control scheme for the output voltage regula-tion of the boost converter.

Constant operating equilibrium points:p =U ; Z l ( U ) = bw / [ m i 1 U,]

Z ( U ) = b / b o ( l - U ) ] . (3 .3 )Parame trized fam ily of linearized systems about the constantoperating p oints:

(3 . 4 )withz j s ( t ) = z j ( t ) - Z i ( U ) ; i = 1 , 2Y s ( t ) = Y ( t1 Y U ) = z2 t ) - Z,( U ) Pa( ) = p ) U .Family of parametrized transfer functio ns:

(3 . 5 )

(3 . 6 )Crossover requency:

WO ) = J2 W O 1- U ) .

P 0 ( U ) :=2 T / W0 U)=J 2 T / [ w o (l U ) ]K o ( U ) = w o ( l - U ) / b . (3 .7 )

Ultimate period and ultimate gain:

Ziegler Nichols P-I controller gains fo r the linearized fami lyof converters:K , ( U ) = 0 . 4 w 0 ( l - U ) / b ; K 2 ( U ) = ~ ; ( l - U ) ~ / ( 2 J 2 ~ b ) .

(3 .8 )Nonlinear P-I controller:

d [ ( t ) / d t =[ -( t ) ) 3 / ( 2 J 2 T b ) ~ e ( t )~ ( t )[ ( t ) + [ 0 . 4 ~ 0 1 - ( t ) ) / b ] e ( t )e ( t ) = Y d ( U )- Y ( t )= Z , ( U ) - Z(t) . (3 .9 )

214 2-8.2 e e z 8. 4 8 . 6 8 . 8Fig. 3. State trajectories of the ideal average boost converter modelcontrolled by a nonlinear P-I regulator.

Fig. 4. Average controlled state response of the boost converter sub-

In this case, upon integration of the differential equationfor [ ( t ) , an explicit expression can be obtained for thenonlinear P-I controller in terms of the error signal integral.

ject to a sudden change in the set point value.

with[ O ) = l o nd k = a p be ( t ) = Y d ( W - Y ( t )= Z , ( W Z (t ).

Low-Pa ss filter: A first-order low-pass filter may be usedto yield an approximation to the ideal average output func-tion z 2 required by the nonlinear P-I controller. Such a filteris characterized by a sufficiently small time constant of value( l / T J , and a state f .d f ( t ) / d t = - ( l / ~ c ) ( f ( t ) - ~ ( t ) ) ?2 (t ) = f ( t ) * ( 3.1 0)

The proposed regulation scheme corresponding to theboost converter is shown in Fig. 2.3.4. A Simulation Example

A boost converter circuit with parameter values R = 30 Cn,C = 20 pF, L = 20 mH, and E = 15 V was considered fornonlinear P-I controller design. The constant operating valueof p was chosen to be U = 0.6 while the correspondingdesirable normalized constant output voltage turned out tobe Z2(0.6)= 0.1677. Fig. 3 shows several state trajectoriescorresponding to different initial conditions set on the ideal

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9.15.

9.6,

:Z(t)l i l t w e d o u t p u t )

9.4.

9 e Z t ) f i l t m d o ut pu t) t i n t [rsle 29 39 4e 59

Fig. 5. State response of the actual PW M-controlled boost converter.

/2B U ) 69 89

Fig. 6. Filtered output response of the actual PWM boost convertercontrolled by a nonlinear P-I regulator.

Fig. 7. Buck-boost converter.

average boost converter model controlled by the nonlinearP-I regulator of the form (3.9). The figure represents theprojection of the closed loop system three-dimensional stateonto the z 1- 2 average state plane. The average controlledstate variables, z1 and z 2 ,are shown to converge toward thedesirable equilibrium point represented by Zl U ) = 0.4419and Z 2 ( U )= 0.1677. Fig. 4 shows the average controlledvariables evolution when subject to a 100% step change inthe output set point value, from 0.1677 to 0.3354 (the corre-sponding change in the operating point of the duty ratio wasfrom 0.6 to 0.8). Figs. 5and 6 show, respectively, the stateresponse of the actual (i.e., discontinuous) PWM controlledcircuit and the filtered output response. The sampling fre-quency for the PWM actuator was chosen as 1 kHz and theoutput low-pass filter cut-off frequency was set at 0.1 rad/s.3.3. Buck -Boost Converter

Consider the buck-boost converter model shown in Fig. 7 .This device is described by the following constant bilinearstate equation model:

du1 / d t = ~ 0 x 2 u w 0 x 2 ubd x , / d t = - w ~ x ~ - w ~ x ~ + u w ~ x ~

Y = x 2 (3 .11)

where x 1= iJL, x 2 = V J C represent normalized input cur-rent and output voltage variables, respectively, b = E / JL isthe normalized external input voltage (here assumed to be anegative quantity, i.e., reversed polarity) while w o=1/ JLCand w 1= 1 / R C are, respectively, the L C (input) circuitnatural oscillating frequency and the R C (output) circuittime constant. The switch position function, acting as acontrol input, is denoted by U and takes values in thediscrete set ( 0 , l ) . System (3.11) is of the same form as (2.1),with h = 0 and g = b 01. We now summarize the formulasleading to a nonlinear P-I controller design for the averagemodel of (3.11).Average buck - boost converter model:

d z / d t = W O Z ~ C L W ~ Z ~p bd z2 / d t = O Z ~- w l z q L LWO Z ~

y = z2. (3 .12)Constant equilibrium points:

p = U ; Z l ( U ) = b U w l / [ w t ( l - U ) 2 ] ;Z , ( U ) = - b U / [ w o ( l - U ) ] . ( 3 . 1 3 )

Parameterized family of linearized systems about the con-stant operating points:d [ 0 wo 1 - ' 1 [Z l l ( l ) ]dt 228 - w o l - U ) - 0 1 z 2 s ( t )[ w l I j ; ; ( ~ ~ l ~ U ) ' ] ] P aY d t >= Z28( t ) (3 .14)

withz i s ( t ) = z i ( t ) - Z i ( U ) ; i = 1 , 2y s ( t > =y ( t ) - Y ( U ) : = z 2 ( t ) - Z Z Z ( U ) ; p g ( t ) = C L ( t ) - U .Family of parametrized transfer functions:

s - b [Z l ( U ) ]- . (3.15)Crossover frequency:

Wo U)= w o l - U ) 1 + 1 / U ) l 2 (3 .16)Ultimate period and ultimate gain:P o ( U ) =2Tr/ WO(U)= 2 4 o l - U ) 1 + 1 / u ) 1 '2]KO( U )= W O 1- U ) ' ] /( bW. ( 3 .17)Ziegler Nichols P-I controller gains fo r the linearized familyof converters:

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IFig. 8. A nonlinear P-I control scheme for the output voltage regula-

-9.1ion of the buck-boost converter.

21-8.1-8A -9.1 -8.2 9 9.2

-0.

Fig. 11. State response of the actual PWM controlled buck-boostconverter.

O l IFig. 9. State trajectories of the ideal average buck-boost convertermodel controlled by a nonlinear P-I regulator.

-1 tine Is10.15 9.2 8.25 9 , 3

Fig. 10. Average controlled state response of the buck-boost con-verter subject to a sudden change in the set point value.

(in this case an explicit closed-form expression for l ( t ) isimpossible to obtain).Low-Pass filter: A simple first-order low-pass filter may beproposed to yield an approximation to the ideal averageoutput function z 2 required by the nonlinear P-I controller.Such a filter is characterized by a sufficiently small timeconstant of value (1/ T c)and a state variable f .d f ( t ) / d t = -(l/TC)( f ( t ) - ~ 2 ( t ) ) ;2 t ) =f t ) . (3.40)

The proposed regulation scheme corresponding to thebuck-boost converter is shown in Fig. 8.3.4. A Simulation Example

A buck-boost converter circuit with the same parametervalues as in the previous example was considered for nonlin-ear P-I controller design. The constant operating value ofwas again chosen to be U = 0.6 while the correspondingdesirable normalized constant output voltage turned out tobe 2J0.6) =0.1006. Fig. 9 shows several state trajectoriescorresponding to different initial conditions set on the idealaverage buck-boost converter model controlled by the non-linear P-I regulator of the form (3.19). Fig. 9 represents theprojection of the closed loop system three-dimensional stateonto the z1- 2 average state plane. The average controlled

tine la09 19 19 39 48Fig. 12. Filtered output response of the actual PWM buck-boostconverter controlled by a nonlinear P-I regulator.

state variables, z1 and z2, are shown to converge toward thedesirable equilibrium point represented by Z , ( U )= .2652and Z 2 ( U >= 0.1006. Fig. 10 shows the ideal average con-trolled state variables evolution when subject to a 100 stepchange in the output set point value, from 0.1006 to 0.2012(the corresponding change in the operating point of the dutyratio was from 0.6 to 0.75.) Figs. 11and 12 show, respec-tively, the state response of the actual (i.e., discontinuous)PWM controlled circuit and the filtered output response.The sampling frequency for the PWM actuator was chosenas 1 k z and the output low-pass filter cut-off frequency wasset at 0.1 rad/s.3.5. Discussion and Review of Som e C ontroller DesignTechniques for dc-to-dc Converters

A n overview and comparison of some recently publishedtechniques for regulating dc-to-dc power converters seems tobe in order at this point, especially those appearing in[5]-[9], published by the author of this paper.Regulation circuits for dc-to-dc power supplies have beentraditionally designed by a combination of approximate lin-earization exercised on discretized converter models, andPWM actuators used as simple proportional controllers[11-[3], [20]. The techniques are analytically cumbersome andvalid over a rather limited range of regulating conditions. In[5]-[9], nonlinear design techniques were proposed on thebasis of rigorous analytical developments with some discus-sion about practical implementation from two fundamentalviewpoints: sliding mode control and PWM. The slidingmode approach, extensively used in [5], explores the funda-mental advantages and limitations of control schemes basedon active switching of the converter about linear sliding

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416 IEEE TRANSACITONS O N CIRCUITS A ND SYSTEMS, VOL. 38, NO. 4, APRIL 1991

surfaces that guarantee either constant input current orconstant output voltage regulation for the different convert-ers. It was shown, by using Lyapunov stability analysis of theresulting nonlinear ideal sliding dynamics, that some of theseschemes result in stable sliding motions (particularly thosepursuing regulation of input current to a constant value) andsome do result in unstable sliding motions (basically thosepursuing regulation of output voltage to a constant value).The class of proposed stable sliding modes were particularlysimple to achieve since only one state variable of the con-verter need be measured. Sliding regimes, on the other hand,are known to be robust with respect to parameter variationsand are easy to implement from a hardware standpoint.However, only indirect constant output voltage control canbe stably achieved by constant input current regulation in allof the converter cases. The associated transients may besignificant and thus undesirable for certain applications. Inorder to overcome some of these limitations, sliding modecontrol performed on affine switching lines was proposed in[6] and [7]. The resulting controlled action naturally led t o adecoupling of the input current and output voltage dynamicswith predetermined exponential convergence of the statevariables toward the equilibrium point. In this respect, [6]and [7] showed that time scale separation properties could bemade inherent to the converter by appropriate circuit param-eter specifications. This could be advantageously exploited,in case of input source or parameter perturbations, fordiminishing undesirable effects of transients toward the sta-ble equilibrium point. However, the techniques in [6] and [7]call for the accurate measuring of all state variables of theconverter circuit and more elaborate hardware for theswitching line synthesis as compared to the methods in [51.The sliding mode techniques basically tried to achieve lin-earization of the circuit dynamics in the range of existence ofthe sliding motion. But, of course, this can also be achievedwithout use of sliding modes. With the advent of the geomet-ric theory of nonlinear systems and the spur of theoreticalresults dealing with the possibilities of feedback regulationfor nonlinear plants, a new avenue of nonlinear controllerdesign techniques was initiated in [SI, [9], and the presentpaper for dc-to-dc power supplies. Such techniques ar e basedon the possibilities of feedback regulation of the convertervariables by means of exact linearization [211, pseudolin-earization [221, and extended linearization [Ill, 1121 of theaverage PWM controlled converter model. Aside from show-ing in [81 and [91 that modern nonlinear feedback designmethods based in the geometric theory of nonlinear systemscould be easily handled from an analytical viewpoint for theconverters case, the simplicity of regulating a linear system inBrunovsky controllable canonical form is highly attractivefrom both the conceptual and practical standpoints. Ofcourse, the limitations of the approach are referred to thenon-globality of the state and input coordinate transforma-tions needed to achieve linearization, and the hardwarecomplexity for synthesizing the required state-space coordi-nate transformations. Even though such limitations are beingovercome, day by day, with the advances of modern electron-ics, the approach in [8] is still highly dependent upon theconstant operating point of the converter and its lack ofdesirable self-scheduling properties may be significant. In [9],a controller design based in pseudolinearization of the aver-age PWM model is proposed so as to free the linear designfrom the equilibrium point dependence present in the tech-

nique developed in [81. (For a discussion, see 19, p. 864,remark 31.1 The present paper gives a more concise andcomplete answer to the nonlinear feedback regulation ofstate variables in dc-to-dc power converters, yet with a tradi-tional flavor. Only averaged input-output data is needed,classical linear design techniques are exploited, and a conve-nient degree of self-scheduling, characteristic of extendedlinearization, is also achieved. This is highly desirable incases where the reference output voltage is subject to suddenchanges, a feature not considered before in [5]-[9]. Thisarticle thus constitutes a partial completion of the picture ofutilization of the available nonlinear feedback regulator de-sign techniques for dc-to-dc power converters. Each nonlin-ear controller design method proposed thus far implicitlybears the advantages and limitations of the underlying non-linear design technique.

IV. CONCLUSIONSND SUGGESTIONSFOR FURTHERESEARCH

Output voltage regulation in PWM controlled dc-to-dcpower supplies of the boost and the buck-boost types hasbeen demonstrated through use of nonlinear controllers de-signed on the basis of extended linearization techniques inconjunction with classical control methods. The stabilizingdesign considers nonlinear proportional-integral regulatorsderived from a linearized family of transfer functionsparametrized by constant equilibrium points of idealized(infinite-frequency) average PWM controlled converter mod-els. The nonlinear controller scheme is shown to comply withthe same qualitative stabilization features imposed on theaverage model, provided the output signal used for feedbackpurposes is properly low-pass filtered and a suitably highsampling frequency is used for the PWM actuator. As aresearch topic that deserves further work, one could investi-gate the local character of the stabilizing properties of theproposed nonlinear P-I regulation scheme.The proposed control scheme can be extended to the Cukconverter and to its various modifications. Application of themethod to higher order converters requires the use of sym-bolic algebraic manipulation packages such as REDUCE,MAPLE, or MATEMATICA. Other classical compensatordesign techniques can also be proposed. In particular, theuse of the frequency domain observer-controller regulator,or even the classical analytical design theory [19] and itsassociated integral-square error minimization, could be analternative to the Ziegler-Nichols design recipe for thenonlinear P-I controller specification. The feasibility of theseapproaches remains to be demonstrated. Actual laboratoryimplementation of the nonlinear controllers looks promising,but elaborate with present day technology. Efforts should beconducted in such a direction.REFERENCES

R. P. Severns and G. Bloom, Modem dc-to-dc Switch-ModePower Converter Circuits. New York: Van Nostrand-Reinhold,1985.R . D. Middlebrook and S. Cuk, Advances in Switched ModePower Conversion.F. Csaki, K. Ganszky, I. Ipsitz, and S. Marti, Power Electronics.Budapest , Hungary: Akademia Kiado, 1983.R. Venkataramanan, A. Sabanovic, and S. Ciik, Sliding modecontrol of dc-to-dc converters, in Proc. VZI nt. PCI Conf., pp.331-340, Apr. 1983.

P asad en a , C A T E S L A C o . , 1981.

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H. Sira-Ramirez, Sliding motions in bilinear switched net-works, IEEE Trans. Circuits Syst., vol. CAS-34, pp. 919-933,Aug. 1987.H. Sira-Ramirez and M. Il ic, A geometric approach to th efeedback control of switch mode dc-to-dc power supplies,IEEE Trans. Circuits Syst., vol. 35, pp. 1291-1298, Oct. 1988.H. Sira-Ramirez, Sliding-mode control on slow manifolds ofdc-to-dc power converters, Int . J. Contr., vol. 47, pp.1323-1340, May 1988.H. Sira-Ramirez and M. Ilic-Spong, Exact linearization inswitched-mode dc-to-dc power converters, Int . J Contr., vol.H. Sira-Ramirez, Switched control of bilinear converters viapseudolinearization, IEEE Trans. Circuits Syst., vol. 36, pp.858-865, June 1989.W. Baumann and W. J. Rugh, Feedback control of nonlinearsystems by extended linearization, IEEE Trans. Automat.Contr., vol. AC-31, pp. 40-46, Jan. 1986.W. J. Rugh, The extended l inearization approach for nonlin-ear systems problems, in Algebraic and Geometric Methods inNonlinear Control Theory, M. Fliess and M. Hazewinkel, Eds.Dordr echt, Holland: D . Reidel Publishing, 1986, pp. 285-309.J. Wang and W. J. Rugh, Feedback l inearization families fornonlinear systems, IEEE Trans. Autom at. Contr., vol. 32, pp.K. J. Astrom and T. Haggliind, Automatic Tuning of PIDControllers. Research Triangle Park, NC: Instrument Societyof America, 1988.H. S ira-Ramirez, A geometric approach to pulse-width-mod-ulated control in nonlinear dynamical systems, IEEE Trans.Automat. Contr., vol. 34, pp. 184-187, Fe b. 1989.-, Design of nonlinear PWM controllers: Aerospaceapplications, in Variable Structure Control for Robotics andAerospace Applications, K. K. David Young, Ed. Amsterdam,Holland: Elsevier, to be published.V. I. Utkin, Sliding Modes and Their Applications in VariableStructure Systems. Moscow: MIR, 1978.H. Sira-Ramirez, Differential geometric methods in variablestructure control, Int. J . Contr . , vol. 48, pp. 1359-1390, Oct.1988.W. J. Rugh, Design of nonlinear PID controllers, Tech. Rep.J H U / E E 87-1, The Jo hns Hopkins University, Baltimore, M D,1987.

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[19] G. C. Newton, L. A. G ould, and J. F. Kaiser, Analytical Designof Linear Feedback Controls.[20] N. Mohan, T. M. Undeland, and W. P. Robbins, Power Elec-tronics. New York: Wiley, 1989.[21] L. R . Hu n t , R . Su, and G. Meyer, Global transformations ofnonlinear systems, IEEE Trans. Automat. Contr., vol. AC-28[22] C. Reboulet and C. Champetier, A new method for lineariz-ing non-linear systems: The pseudolinearization, Znt. J . Contr . ,

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Hebertt Sira-Ramirez M75-SM185) re-ceived the electrical engineering degree fromthe Universidad de Los Andes Mbrida-Venezuela), in 1970, and the M.S.E.E. andPh.D. degrees in electrical engineering in1974 and 1977, respectively, from MIT , Cam -bridge, MA.He is a full professor in the Control Sys-tems Department of the Systems EngineeringSchool of the Universidad de Los Andes. Hehas held the positions of Head of the ControlSystems Department and was elected Vice-president of the Univer-sity for the per iod 1980-1984. During 1986 he was a visiting profes-sor in the Department of Aeronautical and A stronautical Engineer-ing, the D epartm ent of Electrical Engineering, and the CoordinatedScience Laboratory of the University of Il l inois at Urbana-Cham paign. H e has also held brief visiting positions in the School ofElectrical Engineering of Purdue University, West Lafayette, IN.His interests include the theory and ap lications of discontinuousfeedback control for nonlinear dynamicaf) ystems.Dr . Sira-Ram irez was the recipient of the 1987 Venez uelanCollege of Engineers A ward f or Scientific Work. In 1990, he wasgranted a Scholarship Prize of the Censejo Nacional de Investiga-ciones Cientificas y Tecnol6gicas CONICIT) as senior researcherfor the period 1990-1994.

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