Complementary PID Controller to Passivity-Based Nonlinear Control of Boost Converters With Inductor Resistance

826 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 3, MAY 2012

Complementary PID Controller to Passivity-Based Nonlinear Control of BoostConverters With Inductor Resistance

Young Ik Son, Member, IEEE, and In Hyuk Kim, Student Member, IEEE

Abstract—Since the DC-DC boost converter exhibits highly non-linear and non-minimum phase properties, it is not an easy taskto design a controller that is robust against load perturbations.This paper presents a dynamic output feedback controller for aDC-DC boost converter that has a practical inductor and a seriesresistance. In order to maintain its robust output voltage regula-tion, the proposed controller adopts a simplified parallel-dampedpassivity-based controller (PD-PBC). A complementary pro-portional-integral-differential (PID) controller to the PD-PBChas been designed for removing the steady state error owing tothe parasitic resistance. We present sufficient conditions for theasymptotic stability of the augmented system with an additionaldynamic system. Computer simulations and experimental testsunder reference step changes and load perturbations confirm theimproved performance of the proposed approach.

Index Terms—Boost converter, load variation, output feedbackcontrol, parasitic resistance, passivity-based control.

I. INTRODUCTION

B ECAUSE the DC voltage generated by fuel cells or pho-tovoltaic systems varies widely in magnitude and unex-

pected transient states often result from uncertain load varia-tions, a reliable DC-DC boost (step-up) conversion stage is es-sential to provide a highly regulated DC voltage. This moti-vates the development of various control algorithms for boostconverters, along with the study of renewable energy sources[1]–[5].

Since the DC-DC boost converter has nonlinear character-istics and it is modeled as a non-minimum phase system witha right-half-plane (RHP) zero, the output voltage regulationproblem of the boost converter has attracted the attention ofmany control system researchers as well as power electronicsengineers. Although the controllers based on classical linearcontrol techniques are simple to implement, it is difficult to dealwith the variation of system parameters. Hence, there have beencontinuous efforts to design control strategies for improving theperformance of the power converter (see [6]–[18] and therein).

For example, Sira-Ramirez and Rios-Bolivar [6] presented anew sliding mode controller (SMC) to obtain a self-scheduling

Manuscript received December 28, 2010; accepted March 13, 2011. Dateof publication April 21, 2011; date of current version April 11, 2012. Thiswork was supported in part by the 2nd Brain Korea 21 Project and by the Ad-vanced Human Resource Development Program of Ministry of Knowledge andEconomy (MKE) through the Research Center for Intelligent Microgrid, My-ongji University.

Y. I. Son is with the Department of Electrical Engineering, Myongji Univer-sity, Kyunggi-do 449-728, Korea (e-mail: [email protected]).

I. H. Kim is with the Electric Development Team, Mitsubishi Elevator KoreaCo., Ltd, Incheon 404-812, Korea (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCST.2011.2134099

property under the change in operating conditions. In [7], a mod-ified current-mode control (CMC) algorithm was proposed toreduce the sensitivity to resistive loads, particularly the constantpower load. control was applied to overcome restrictions ofthe frequency-domain approach in [8]. The bifurcation behaviorof the boost converter using CMC was studied in [9]. The au-thors of [10] tested the performance of several different controlschemes with experimental comparisons. In [11], the operationof interleaved boost converters under SMC was analyzed to re-duce the ripple voltage amplitude. The effect of RHP zero onthe stability of the system was investigated in [12]. A synergeticcontrol approach was attempted in [13] to improve the previouscontrollers, e.g., SMC. In [14], the need of the inductor currentsensor was removed by a generalized PI SMC. An adaptive con-troller was proposed in [15] to reduce the output voltage ripplecaused by harmonic disturbances in the input voltage. In [16], anonlinear control of the exponential form was proposed to addan additional tuning parameter to the existing multi-loop CMC.A tri-state boost converter with an additional switch to removethe RHP zero was studied in [17]. A different type of a boostconverter with synchronous rectification was dealt with in [18]using a linear-to-nonlinear translator. Most of the above results,with the exception of the results of [14] and [18], require boththe input current and the output voltage measurements to imple-ment the control algorithms.

A passivity-based controller (PBC) as one of the robust con-trol algorithms has been applied to many practical control ap-plications including DC-DC power converters [19]. Several au-thors have considered PBCs to solve the regulation problemof the boost converter [19]–[23]. Among them, the parallel-damped PBC (PD-PBC) presented in [21] and [22] has achievedoutstanding performance under reference step changes as wellas load variations. The advantage of the parallel damping ap-proach over the series damping injection is that it is insensitiveto varying loads and uses only voltage measurements [21], [22].

However, the nonlinear controller does not take into consid-eration the parasitic resistance such as the resistance in the in-ductor. Although the parasitic resistance is relatively very small,it cannot be ignored in the practical DC-DC boost converterbecause it increases the model uncertainty. Owing to the par-asitic resistance, the PD-PBC cannot maintain robust perfor-mance under load variations.

The objective of this paper is to provide a robust output feed-back controller for the DC-DC boost converter that has two par-asitic resistors including the inductor resistance. Based on thefact that the PD-PBC exhibits robust performances for the idealboost converter, this paper extends the application of the ap-proach to the boost converter with a practical inductor. A newcontrol algorithm that combines a proportional-integral-differ-ential (PID) controller with the PD-PBC is proposed to achieve

1063-6536/$26.00 © 2011 IEEE

SON AND KIM: COMPLEMENTARY PID CONTROLLER TO PASSIVITY-BASED NONLINEAR CONTROL OF BOOST CONVERTERS 827

Fig. 1. DC-DC boost converter having parasitic resistance.

robust output regulation against load uncertainties. Sufficientconditions for the asymptotic stability of the augmented systemwith the additional system are presented. In the proposed ap-proach, the PID controller uses an additional system state vari-able instead of a constant reference value for generating an errorsignal. This is the main difference between the proposed con-troller and conventional PID controllers.

This paper is organized as follows. Section II introduces themathematical model of the DC-DC boost converter. Section IIIsummarizes the simplified PD-PBC and describes the design ofthe proposed controller for the system considered in this paper.In Section IV, computer simulations and experimental resultsshow the performance of the proposed controller in the presenceof reference step changes and load perturbations. In the simula-tions, we have compared the closed-loop performance with fourdifferent controllers, which were designed using: 1) a general-ized PI sliding mode controller (GPI SMC) [14]; 2) a simplifiedPD-PBC [22]; 3) a conventional PI controller; and 4) the pro-posed controller. Finally, Section V presents the conclusions.

II. DC-DC BOOST CONVERTER MODEL

This paper deals with the output voltage regulation problemof the DC-DC boost converter shown in Fig. 1. Unlike the in-ductor considered in [12]–[14] and [22], we consider a practicalinductor with a parasitic resistance . This model also includesa resistor to represent unavoidable voltage drops and a cur-rent-sensing resistor [24]. The nominal values of the resistorsare assumed to be known.

By using an average switching method, the mathematicalmodel of Fig. 1 is described by

(1)

where is the inductor current; is the output voltage;is the DC source voltage; the control input is the duty ratio

; and , , and denote the inductance, thecapacitance, and the load resistance, respectively.

Let be the desired output voltage. Then, the equilibriumvalues , , andsatisfy the following equations:

(2)

(3)

Replacing in (2) with (3) gives

(4)

If , the value of can then be obtainedas shown in [22]

(5)

The control input does not depend on the variation of loadresistance .

On the other hand, the solutions of (4) are given by

(6)

(7)

In the above equations, owing to and , the variation ofmakes deviations from the nominal equilibrium values. It is

noted that and when .Section III first reviews a simplified PD-PBC using (5). A

complementary PID controller and its stability analysis are pro-vided.

III. PID CONTROLLER COMBINED WITH SIMPLIFIED PBC

A. Drawback of a Previous PD-PBC

Recently, a simplified PD-PBC has been proposed in [22] forthe case, where

(8)

(9)

Replacing in (5) with yields a controller, describedin (8), that is independent of the circuit parameters. Hence, thecontroller exhibits robust performances under reference stepchanges and load variations without a steady state error [22].However, when or , (5) is no longer valid, andthe controller described in (8) leads to a steady-state error.

Instead of using (5) to deal with the non-ideal case, we firstconsider the equilibrium value shown in (6). By replacing in(4) and (6) with , we can obtain

(10)

(11)

The rest of this section proves the feasibility of the controllerin (11) when and the stability of as in [22].

The time derivative of (10) yields

(12)


Fig. 2. Stability of the new input for a boost converter.

Using (11) and letting coincide with its desired value in(12) and (9), we can obtain

(13)

In addition, from (10), we can obtain

(14)

Since the denominator of (14) is positive, the phase-plane dia-gram (see Fig. 2) can be described by (13) and (14) with positiveconstants , and . This figure proves the fea-sibility of the controller described by (11) and the asymptoticstability of [22].

A main drawback of (11) is that load variations will cause asteady state error in the regulated output. This drawback comesfrom the change of (6) owing to and . Hence, the values of

and as well as should be precisely known to achievethe control objective.

As a way of making the best use of the robust performanceof the PD-PBC by removing the steady-state error, this paperpresents a new PID controller combined with the dynamics in(9).

B. Design of the Proposed Controller

The proposed controller is represented by

(15)

(16)

where . The main difference between the proposedand the conventional PID controller is that the additional statevariable is employed instead of a constant reference forgenerating the error signal .

In order to determine the parameters of (9) and (16), we firstdefine error variables as

(17)

The error dynamics are obtained by

(18)

The equilibrium values of (18) are and , where

(19)Let us define . Jacobian linearization of (18) at the

above equilibrium point yields

(20)

where is the linearized state and

The rest of the design procedure for gains , , , ,and consists of two steps.

In the first step, gains and are determined from theLaplace transform of . When and

, is represented by

(21)

where,

, and

.In the derivation , we used (21) and the following equation:

(22)

The stability conditions for (21) (or the matrix ) are that( 0, 1, 2) and . When and, and the are positive because

(23)


When , an additional condition is necessary to obtaina positive value for

(24)

It can also be shown that the inequality is sat-isfied when the values of and are positive. This impliesthat the system matrix is stable for all (positive) system pa-rameters , , , , , , and if the positive constants

and satisfy inequality (24).Remark 1: It is not difficult to determine and that

satisfy inequality (24)

(25)

Because the values of and are relatively small as com-pared to that of , the resulting value of (25) is typically verysmall, as in the example of Section IV. In the appendix, a neces-sary and sufficient condition is also provided to obtain a positivevalue of .

We determine the positive constants and such that thedenominator of (21) becomes

(26)

In the second step, the PID controller described in (16) isdesigned for system (20). Since the system is already stable withthe above values for and , the complementary controllercan be designed by various design techniques, including the rootlocus method. This paper adopts both the pole-zero cancellationand the root locus method.

If a dominant pole of system (20) is much smaller than theother two poles, the PI controller can be designed without using

(27)

Otherwise, the numerator of (16) is designed to remove twodominant poles of system (20). After the pole-zero cancella-tion, gain in (27) or in (16) is adjusted in order for theclosed-loop system to remain stable by using the root locus.

The next proposition summarizes the result of this section.Proposition 1: In the presence of or the error system

matrix is Hurwitz when the values of and in (9) sat-isfy the condition of (24) with positive values. In this case, theclosed-loop system [see (1), (9), and (15)] is locally asymp-totically stabilized and the regulation problem can be solvedwithout the steady state error by using the complementary con-troller described by (16). The approach results in a dynamic

TABLE INOMINAL PARAMETERS OF DC-DC BOOST CONVERTER

output feedback controller using only the output voltage mea-surement.

The performance of the proposed controller is tested by twoexamples in Section IV.

IV. DESIGN EXAMPLES

This section includes two examples for computer simulationsand experimental tests. All the computer simulations have beenperformed with Matlab/Simulink. The nominal system parame-ters are listed in Table I.

A. Example 1 [22]: Computer Simulations

We first test whether the proposed controller can be success-fully combined with the previous PD-PBC [22] to maintain theperformance without the steady-state error. Hence, the exampleprovided by [22] (except ) has been considered underthe same reference and load perturbations.

Through the pole-zero cancellation of , gains andare chosen as 0.4204 and 0.7645, respectively. The transfer

function of system (20) is then obtained by

(28)

Because the dominant pole is much smaller thanthe other poles, a PI controller is used to cancel the smallestpole. With the controller described by

(29)

gain is selected as 0.014 through the root locus method, andgain .

Comparative simulations were performed with the general-ized PI sliding mode controller (GPI SMC) [14], the simpli-fied PD-PBC [22], and a conventional PI controller. The de-sired voltage 5 V is changed to 3.5 V at ;it is then changed from 3.5 to 6 V at 5 ms and finallyfrom 6 to 5 V at 9 ms. The load resistance varies from

to at 12 ms. The initial conditionsare , and .

Because the SMC is well known for its robustness againstmodel uncertainties, several authors have presented SMCs forboost converters (see, e.g., [6], [10], [13], [14]). In [14] the needof the inductor current sensor was removed by a generalized PI


Fig. 3. Results with GPI SMC when� � � and� � ��. (a) Input current;(b) output voltage.

SMC. The GPI SMC described in [14] has been chosen for com-parison because it can be implemented using the voltage mea-surement only. Fig. 3 shows the effect of the parasitic resistance

on the GPI SMC represented by

forfor

(30)

(31)

(32)

where , , , andas in [14].

The simulation results with the PD-PBC [see (8)] describedby [22] when are shown in Fig. 4, where

Fig. 4. Results with the simplified PD-PBC when � � � and � � ��. (a)Input current; (b) output voltage.

and . The PD-PBC showed better performancesthan the GPI SMC when ; however, the two controllersyielded the steady-state error for the small value of .

Fig. 5 shows that the proposed algorithm achieves the controlobjective without the steady state error. In Fig. 6, the proposedcontroller is compared with a conventional PI controller, where

and . Although the PI controllerhas been widely used in practical systems because of its simplestructure and robustness, it is very difficult to find one that leadsto a desirable transient response for the system parameters of[21] and [22]. In Figs. 5 and 6, the real value of is assumedto be larger than its nominal value by 50% to test the robustperformance against the parameter uncertainty.

B. Example 2 [24]: Experimental Tests

This section presents the experimental results obtained byusing a laboratory boost converter [24] (see the parameters


Fig. 5. Proposed controller vs. PD-PBC when � � ��. (a) Input current;(b) output voltage.

in Table I). The objective is to check whether the proposedcontroller can deal with unmodeled dynamics of the convertersystem.

By using the zero of , we have chosen gainsand . With these gains, (24) is satisfied.

The transfer function is given by

(33)

As in the first example the smallest pole is cancelled out witha PI controller and the gains are given as and

via the root locus method.Fig. 7 shows the simulation results comparing the perfor-

mances with a conventional PI controller. It is not difficult tostabilize this system using a PI controller. The gains of the con-ventional PI controller are and .

Fig. 6. Proposed versus PI controller when � � ��. (a) Input current; (b)output voltage.

The load resistance varies from toat 5 ms by connecting two 40 resistors in par-

allel with the nominal load. The desired voltage 18 V ischanged to 15 V at 10 ms. In the simulation 53.25 mand are used as opposed to the nominal values of

35.5 m and seen in Table I.The experimental results are shown in Fig. 8 (proposed

controller) and Fig. 9 (PI controller). Since the system doesnot have the inductor current sensor, the current responsecould not be included. The divisions of the axes are 1 ms/divfor the -axis and 5 V/div for the -axis. The initial values are

and . The discrete-time controllersare implemented using a TI DSP TMS320F28335 and thesampling frequency is 20 kHz. The results ensure that theproposed controller can deal with unmodeled system dynamicsand parameter uncertainties.


Fig. 7. Simulations with uncertain � and � . (a) Input current; (b) outputvoltage.

V. CONCLUSION

Here, the output feedback control problem of the DC-DCboost converter with an inductor parasitic resistance is studied,along with variations in the load. Because the simplifiedPD-PBC [22] has successfully achieved the control objectivefor the converter with an ideal inductor, this paper has tested amodification of the previous controller to check whether it canmaintain robust performances for the system with a practicalinductor under load variations. In order to remove the steadystate error owing to the parasitic resistance, this paper designsa complementary PID controller to the PD-PBC. The proposedapproach resulted in a dynamic output feedback controllerusing only the output voltage measurement. The robust asymp-totic stability of the augmented system has been proved underthe positive gain conditions. Through comparative computersimulations and experimental tests, the proposed controller hasbeen shown to have an improved performance and a robuststability.

Fig. 8. Experimental results with proposed controller. (a) Voltage step up; (b)load variation; (c) reference change.

APPENDIX

A. Necessary and Sufficient Condition for

Coefficient can be represented by

(34)


Fig. 9. Experimental results with PI controller. (a) Voltage step up; (b) loadvariation; (c) reference change.

If we define and such that

(35)

then the following condition should be satisfied for to bepositive:

(36)

Let us define such that

(37)

then, (19) and (23) and the new parameters and arerewritten as

Some manipulations yield

Hence, the condition in (36) is equivalent to the next inequality

(38)

For the example in Section IV, parameter is also positive, andthe above condition is satisfied for all positive constants and

. It should be noted that, unlike (24), the condition in (38) isa necessary and sufficient one for .

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments to improve this manuscript.

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Complementary PID Controller to Passivity-Based Nonlinear Control of Boost Converters With Inductor Resistance