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Published in IET Power ElectronicsReceived on 13th August 2008Revised on 11th November 2008doi: 10.1049/iet-pel.2008.0271

ISSN 1755-4535

LMI robust control design for boostPWM convertersC. Olalla1 R. Leyva1 A. El Aroudi1 P. Garces1 I. Queinnec2

1Departament d’Enginyeria Electronica, Electrica i Automatica, Escola Tecnica Superior d’Enginyeria, Universitat Rovira iVirgili, Campus Sescelades 43007, Tarragona, Spain2CNRS, LAAS, University of Toulouse, 7 Avenue du Colonel Roche, Toulouse F-31077, FranceE-mail: [email protected]

Abstract: This work presents an analytical study and an experimental verification of a robust control design basedon a linear matrix inequalities (LMI) framework for boost regulators. With the proposed LMI method, non-linearities and uncertainties are modelled as a convex polytope. Thus, the LMI constraints permit to robustlyguarantee a certain perturbation rejection level and a region of pole location. With this approach, themultiobjective robust controller is computed automatically by a standard optimisation algorithm. Theproposed method results in a state-feedback law efficiently implementable by operational amplifiers. PSIMsimulations and experimental results obtained from a prototype are used to validate this approach. Theresults obtained are compared with a conventional PID controller.

1 IntroductionDc–dc switched-mode converters are power efficient devicesused to match the voltage level of an energy source to thespecifications of the load. The dynamics of such powerconverters is described by non-linear models. Despite thenon-linearities, dc–dc converters are usually driven bymeans of linear (state or output) feedback controllers, thusreducing the complexity and cost of the control system.The control objective of such devices is (i) to maintainregulation of the output voltage at the desired value, (ii) tomaximise the bandwidth of the closed-loop system in orderto reject disturbances and (iii) to satisfy certain transientcharacteristics (as e.g. to minimise output overshoot). Suchlinear controllers are usually designed considering alinearisation of the model at a certain operation point. Inthis case, large-signal transients may deteriorate the outputsignal or even make the system diverge from the desiredoperation point.

Owing to the intrinsic non-linear nature of the switchedregulators, several authors have proposed non-linearcontrollers in order to maintain stability over a certain rangeof operating conditions. Some of the first works on non-linear control for power converters can be found in [1, 2],

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where the authors propose non-linear strategies based onquadratic Lyapunov functions. More recently, Cortes et al.[3], He and Luo [4] and Leyva et al. [5], derive robust non-linear controllers for power converters. These last works seeklarge-signal stability of the regulator dynamics when acomplex control law is applied. The main disadvantages ofthe previous non-linear controllers are the difficulty topredict transient performances and the complexity of theimplementation.

Besides of non-linear control, other authors have adaptedlinear robust control techniques to power converters with theaim to assure stability under different operating conditions.Linear control laws are, a priori, easily implementable as, forexample, PID controllers [6–8]. Furthermore, the robustlinear control techniques, unlike conventional and non-linear control, allow to take into account parametricuncertainty. A correct treatment of uncertainties is of majorimportance in power converters, since some of the regulatorparameters such as the storage elements or the load areusually time dependent or partially unknown. Some of therobust methods successfully adapted to power electronics areH1 [9–11], m-synthesis [12], quantitative feedback theory(QFT) [13, 14] and approaches based on linear matrixinequalities (LMI) [15–18]. It is worth to point out some of

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their characteristics in order to compare such techniques:H1

and m-synthesis methods [9–12] require that the designerarbitrarily choose weighting functions to specify the requiredperformances and to approximate the transient properties.On the contrary, LMI-based control does not necessarilyrequire weighting functions and it can also deal withtransient requirements with pole placement constraints.Also, it can be pointed out that the controller expression isoptimised manually in the QFT technique [13, 14], whereasLMI control permits to optimise the controller parametersautomatically.

Since LMI provides the above advantages, the goal of thiswork is to adapt LMI robust control concepts to dc/dcregulator design. The fact that LMIs can be solvedautomatically by efficient standard numerical algorithms[19–21] has prompted a great number of researchers todescribe different control problems in terms of LMIs.Therefore this technique can be considered as mature inthe field of Control Theory.

The controller design shown in this work consists ofdescribing in terms of LMIs the following restrictions: first,the converter stability (in Lyapunov’s sense); second, aminimum level of perturbation rejection; and third, severalconstraints in pole location. In all the cases, the design takesinto account uncertainty in the operation point and loadvalue. Finally, the design is completed when the restrictionsare numerically solved. The result is an efficient statefeedback controller. The main contribution of this work isthe experimental implementation of a prototype whichvalidates the feasibility of an LMI approach, successfullyapplied in many engineering domains, in the area of powerelectronics.

It is worth to mention that LMI control has been previouslyreported in [15–17] to design a state-feedback controller for adc/dc converter. The approach shown here differs from theprevious works since we consider a step-up converter, whereasthe previous references consider a buck converter. Thereforeour work takes into account a non-linear system, whoselinearisation yields a non-minimum phase plant, whereas[15–17] deal with a minimum phase linear case. In addition,

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[15–17] do not describe any experimental prototype, whereaswe present accurate experimental measures.

In order to evaluate the performance achieved by the LMIcontroller, we have compared the results with a conventionaloutput-feedback PID law. Both controls can be easilyimplemented by operational amplifiers, a PWM circuit andstandard analogue elements.

The remaining sections are organised as follows. Section 2shows the uncertain model of the non-minimum phaseconverter. Section 3 introduces the LMIs and thecontroller constraints taken into account. The validity ofthe design is verified with experimental results from a boostprototype in Section 4, where we compare the performanceof the LMI controller with the conventional PID casenumerically using PSIM package. Section 5 summarises thekey aspects of the design method and presents someconclusions.

2 Boost converter uncertaintymodelFig. 1 shows the schematic circuit diagram of a dc–dc step-up(boost) converter and the relevant control signals. In Fig. 1, vo

is the output voltage, vg the line voltage and iload the outputdisturbance. The output voltage must be kept at a givenconstant value Vref . R models the converter load, while Cand L represent, respectively, capacitor and inductor values.

The binary signal (ub) that turns on and off the switches iscontrolled by means of a fixed-frequency pulse widthmodulation (PWM) circuit (see Fig. 1b). The constantswitching frequency is 1=Ts, where Ts is the switchingperiod equal to the sum of Ton (when ub ¼ 1) and Toff

(when ub ¼ 0) where the ratio Ton=(Ton þ Toff ) is the dutycycle dd. The duty cycle is compared with a sawtooth signalvs of amplitude VM. We assume that the converter operatesin continuous conduction mode (CCM) and that theinductor current is not saturated.

The following expressions show the state-space averagedand linearised model of the boost converter. Averaged

Figure 1 Schematic diagram of a boost converter and its control circuit

a Electric circuitb PWM waveforms

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models of dc–dc converters neglect the high-frequencydynamics because of the switching action, since we considerTs much smaller than the converter time constants. Themodel is then valid for a frequency range up to half theswitching frequency [22] in the vicinity of the operationpoint, since it has been linearised. The state-spacerepresentation is written as

_x(t) ¼ Ax(t)þ Bww(t)þ Buu(t)þ Bref Vref

z(t) ¼ Czx(t)þDzww(t)þDzuu(t)

�(1)

where A [ Rn�n, Bw [ Rn�r , Bu [ Rn�m, Cz [ Rp�n,Dzw [ Rp�r , Dzu [ Rp�m and

x(t) ¼

iL(t)

vo(t)

x3(t)

264

375, w(t) ¼ iload(t)

� �, u(t) ¼ dd(t)

� �,

z(t) ¼ vo(t)� �

The state variable x3 is the integral of the error signal obtainedfrom the difference between the reference Vref and the outputvoltage vo. At the equilibrium state, the voltage error is zero.Therefore x3 is constant. The line voltage is considered a dcvalue vg ¼ Vg. The disturbance vector w has been defined asan output current source in order to characterise the outputimpedance of the converter. Such output impedancedescribes the output voltage vo behaviour in presence ofchanges in the output current iload. The output z is theoutput voltage vo and represents the controlled outputwhose response has to fulfil the control requirements. Thematrices of the state-space representation are as follows

A ¼

0 �D0dL

0

D0dC�

1

RC0

0 �1 0

266664

377775, Bw ¼

0

�1

C0

264

375,

Bu ¼

V 0g

D0dL

�Vg

(D02d R)C

0

2666664

3777775, Bref ¼

0

0

1

264

375 (2)

where Dd is the operating point duty cycle and itscomplementary corresponds to D0d ¼ 1�Dd. The remainingstate-space matrices are

Cz ¼ 0 1 0� �

, Dzw ¼ [0], Dzu ¼ [0] (3)

We consider that the load R and the duty-cycle D0d at theoperating point are uncertain or time-varying parameters. Wealso consider that all other parameters are well known. It isworth to point out that the same procedure can be used totake into account more uncertain terms. Nevertheless, themore uncertainty is considered in the converter, the lowerperformance level can be assured. Thus, in order to deal with

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the changes in parameters D0d and R, we express the systemmatrices (1) as function of these parameters

_x(t) ¼ A(p)x(t)þ Bww(t)þ Bu(p)u(t)þ Bref Vref

z(t) ¼ Czx(t)þDzww(t)þDzuu(t)

�(4)

Only matrices A and Bu depend on the uncertain terms, whichhave been grouped in a vector p. In a general case, the vector pconsists of N uncertain parameters p ¼ (p1, . . . , pN ), whereeach uncertain parameter pi is bounded between a minimumand a maximum value pi and pi

pi [ pi, pi

h i(5)

The admissible values of vector p are constrained in anhyperrectangle in the parameter space RN with L ¼ 2N

vertices {v1, . . . , vL}. The images of the matrix [A(p), Bu(p)]for each vertex vi corresponds to a set {G1, . . . , GL}. Thecomponents of the set {G1, . . . , GL} are the extrema of aconvex polytope, noted Co{G1, . . . , GL}, which contains theimages for all admissible values of p if the matrix[A(p), Bu(p)] depends linearly on p, that is

[A(p), Bu(p)] [ Co{G1, . . . , GL}

:¼XL

i¼1

liGi, li � 0,XL

i¼1

li ¼ 1

( )(6)

For an in-deep explanation of polytopic models of uncertaintysee [19 (Ch. 2), 23, 24 (Ch. 4)].

Since the boost converter matrices A and Bu do not dependlinearly on the uncertain parameters D0d and R, we define twonew uncertain variables d ¼ 1=D0d and b ¼ 1=(D02d R) inorder to meet with a linear dependence. Thus, theparameter vector is defined as p ¼ R, D0d, d, b

� �. By using

this parameter vector we can bound the uncertainty inside aconvex polytope.

Based on the uncertainty model described above, thesynthesis objective is to find a state-feedback gain K(u ¼ Kx), where uncertainty is restricted inside thefollowing intervals

R [ [Rmin, Rmax]

D0d [ [D0dmin, D0dmax]

d [ [1=D0dmax, 1=D0dmin]

b [ [1=(D02dmaxRmax), 1=(D02dminRmin)]

(7)

Note that the uncertain model is inside a polytopic domainformed by L ¼ 24 vertices.

The introduction of these new parameters d and b impliesa relaxation in the uncertainty restrictions, since we assumethe independence between uncertain parameters in order tohave linear relations. Consequently, the new relaxed model

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considers dynamic responses that do not correspond to anyreal case. Therefore we will obtain a potentially conservativesolution.

The objectives of the design are to guarantee the stability,to assure a minimum level of perturbation rejection and toconstraint the transient performance, for all the possiblecases [A( p), Bu( p)]. In next section, we will establish theLMI conditions which satisfy these objectives.

3 LMI design constraintsThis section introduces the concept of LMI and presents theconstraints used in the controller synthesis problem.

3.1 Lyapunov-based stability

The use of matrix inequalities to demonstrate certainproperties of dynamical systems can already be found inabout 1890, when Lyapunov establishes his well-knownstability method, whose linear case has been reproducedhere for completeness [24].

Given a linear time-invariant (LTI) system

_x ¼ Ax (8)

the existence of a quadratic function of the form

V (x) ¼ x0Px . 0, 8x = 0 (9)

that satisfies _V (x) , 0 is a necessary and sufficient conditionto assure that the system is stable (i.e. all trajectories convergeto zero) [24]. Since V (x) has quadratic form, this condition isreferred as quadratic stability. In this case, the condition_V (x) , 0 can be rewritten as follows

_V (x) ¼ x0(A0P þ PA)x , 0, 8x = 0 (10)

Thus, the system is stable if and only if there exists a symmetricmatrix P that is positive definite (in the following, the notationP . 0 means that the matrix P is positive definite) for which_V (x) , 0. Inequality (10) is satisfied if and only if the term(A0P þ PA) is negative definite, that is

9P . 0 s.t. A0P þ PA , 0 (11)

In this case, P is the matrix variable to be found to prove thestability. Ref. [25] showed that these inequalities thatpresented linear dependence on the variables, afterwardscalled LMIs, can be solved by convex optimisation methods.We will take advantage of such convex optimisationmethods, that have been implemented in computeralgorithms [19–21], in order to solve the LMIs that arise inthe control of the boost converter. In the followingsubsections we introduce the LMIs to ensure the quadraticstability, to deal appropriately with disturbances and toconstraint the pole placement of the closed-loop system.

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3.2 Quadratic stability LMIs

The following theorem [23] adapts the quadratic stabilityinequality (11) for a closed-loop system with a state feedbacku ¼ Kx.

Theorem 3.1: System (1) is stabilisable by state feedbacku ¼ K x if and only if there exists a symmetric matrixW [ Rn�n and a matrix Y [ Rm�n such that

W . 0AW þWA0 þ BuY þ Y 0B0u , 0

�(12)

a controller for such state feedback is given by K ¼ YW �1.

The decomposition of K in matrix variables Y and Wallows to satisfy the linearity condition of these inequalities.A detailed proof can be obtained in [23]. The case withpolytopic uncertainty in matrices A and Bu directly extendsby computing (12) at all the vertices {G1, . . . , GL} of theconvex polytope Co{G1, . . . , GL} [24, Ch. 5]. Thisextension allows us to assure the quadratic stability of auncertain plant.

3.3 H1 control LMIs

The H1 norm of a stable scalar transfer function f (s) is thepeak value of jf (jv)j as a function of frequency [26]. It isused as a measure of the performance of a system, forexample, to evaluate the minimum attenuation level of anexternal disturbance. Considering the transfer function H(s)from disturbances w to outputs z, the H1 norm of suchsystem is equal to

kH (s)k1 W supv(t)=0

kzk2kwk2

(13)

where k � k1 and k � k2 stand for the infinity and the Euclidiannorms, respectively.

The following theorem, adapted from [27], guarantees amaximum H1 norm g (i.e. a minimum level of disturbanceattenuation).

Theorem 3.2: System (1) is stabilisable by state-feedbacku ¼ K x and kzk2=kwk2 , g if and only if there exists asymmetric definite positive matrix W [ Rn�n and a matrixY [ Rm�n such that the following inequality holds

AW þWA0 þ BuY þ Y 0B0u Bw W C 0y þ Y 0D0yu

B0w �g1 0

CyW þDyuY 0 �g1

264

375 , 0

(14)

a controller for such state feedback is given by K ¼ YW21.


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A proof is given in [27]. Note that satisfaction of inequality(14) implies satisfaction of (12) and therefore this theoremalso ensures quadratic stability. The polytopic case isdirectly applicable by satisfying (14) for all the verticesG1, . . . , GL

� �of the polytopic domain.

3.4 Pole placement LMIs

It is a desirable property of the closed-loop system that itspoles are located in a certain region of the complex plane,in order to assure some dynamical properties like overshootand settling time. In [28], a region of the complex planeS(a, r, u), depicted in Fig. 2, is defined such that the(complex) poles of the system in the form x + jy satisfy

x , �a , 0, jx + jyj , r, y , cot (u)x (15)

In such a case the poles of the system x + jy ¼ �zvn + jvd

ensure a certain damping at the desired rate (see [29]). Thepresented region is equivalent to a minimum decay rate a,a minimum damping ratio z . sin u and a maximumdamped natural frequency vd , r cos u, wherevd ¼ vn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� z2

p.

The following theorem allows to constraint the location ofthe closed-loop poles in the region S(a, r, u).

Theorem 3.3: The closed-loop poles of the system (1) witha state-feedback u ¼ Kx are inside the region S(a, r, u) ifthere exists a symmetric definite positive matrix W and amatrix Y such that

AW þWA0 þ BuY þ Y 0B0u þ 2aW , 0 (16)

�rW WA0 þ Y 0B0u

AW þ BuY �rW

�, 0 (17)

cos u(AW þWA0 þ BuY þ Y 0B0u)

sin u(�AW þWA0 � BuY þ Y 0B0u)

sin u(AW �WA0 þ BuY � Y 0B0u)

cos u(AW þWA0 þ BuY þ Y 0B0u)

�, 0 (18)

and K ¼ YW 21 is the state feedback gain.

Figure 2 LMI region S(a, r, u)

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A detailed proof is reported in [28], where the poleplacement in generic regions of the complex plane isexplored. A different point of view on robust pole placementis given in [30]. Once again, the polytopic case directlyextends by satisfying (16)–(18) for all G1, . . . , GL

� �.

Therefore by taking into account these conditions in eachvertex of the convex polytope Co G1, . . . , GL

� �, we will

ensure a proper performance despite uncertainty.

3.5 Remarks

We summarise the LMI synthesis method as follows. Thedesign of a robust control for a boost converter consists ofsolving inequalities (14) and (16)–(18) to find the matrixvariables Y and W that minimise the H1 norm g and thatsatisfy the pole placement constraints of the region S(a, r, u)for all the extrema of the polytopic model G1, . . . , GL

� �minY ,W

g under conditions

(14), (16), (17) and (18)8 Gi

� �, i ¼ 1, . . . , L

(19)

It is necessary to remark that the stability and theH1 bound ofthe closed-loop system is guaranteed for arbitrarily fast changesin Dd

0 and R. However, the pole placement constraint is onlysatisfied if the time-dependent parameters changes are slowlyenough to recover the steady state of the system. For a surveyon how the rate change of parameters can be taken intoaccount, see [17].

In the next section, we develop in detail the givenprocedure for a specified boost converter.

4 Control implementation andexperimental resultsIn this section, we present the results of the LMI designmethod proposed above. First, we identify the parametervalues of the boost converter. Next, the state-feedback gainK is numerically found. The resulting controller has beensimulated with a switched model of the converter by usingPSIM software. We have compared the simulated responseof the proposed controller with a conventional PIDcontroller, in order to evaluate the robustness andperformance achieved with our approach. Finally, thesimulation results have been verified with an experimentalset-up that is explained in detail at the end of this section.

The parameters of the dc–dc converter, that have beentaken from [13], can be found in Table 1. The nominalvalues of the uncertain converter parameters are R ¼ 50 V

and D0d ¼ 0.5. Note that since the output voltage vo isconsidered constant, an uncertain operation point Dd

0 in theinterval [D0dmin

, D0dmax] introduces a range of possible line

voltages Vg [ [vo �D0dmin

, vo �D0dmax

] that are included in thepolytope Co G1, . . . , GL

� �defined in (6).

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Given the parameters of Table 1, the uncertain variables dand b, defined in (7), belong to the intervals [1, 3.33] and[0.02, 1.11], respectively.

4.1 Controller synthesis

As it has been stated before the synthesis objective is tominimise g while a pole placement constraint S(a, r, u) issatisfied. The values of (a, r, u) can be found in Table 2.In order to have the poles of the system inside the validfrequency range of the model, r has been set to 1/10 ofthe switching frequency. For a minimum damping ratio of

Table 1 Boost converter parameters

Parameter Value

R [10, 50] V

D0d [0.3, 1]

vo, Vref 24 V

C 600 mF

L 310 mH

Ts 5 ms

Table 2 Pole placement parameters

Parameter Value

a 130

r2p

10Ts

u 258


0.4, u has been set to 258. Finally, for a fast decay rate, wehave tested several different values of a. For the presentcase, the parameter a has been set to 130; a higher value ofa results in an unfeasible problem for this parameter set.

Solving problem (19) using Matlab’s standard LMItoolbox [19], which yields a state-feedback controller K

K ¼ [�1:95 �2:00 725:22] (20)

and a guaranteed H1 bound from disturbances to outputs ofg ¼ 2.89, which is equivalent to 9.21 dB. The correspondingcontrol law that yields the duty cycle is

dd(t) ¼ �1:95iL(t)� 2:00vo(t)þ 725:22x3(t) (21)

4.2 Simulations

The switched model of Fig. 1a with the controller K has beenimplemented using PSIM simulator [31], taking the nominalvalues of the converter.

In order to show the robustness of the proposed controller,we have simulated the transient behaviour in presence of anabrupt load change under nominal conditions and out ofnominal conditions. Fig. 3a depicts the output voltagewhen the converter operates at the nominal equilibriumpoint and reacts in front of a load step change of 0.5 A. Itcan be noted that the output voltage response presents atime constant of approximately 10 ms, that corresponds toa decay rate of 400, which as expected is larger than theminimum guaranteed decay rate (a ¼ 130). Furthermore,the output voltage exhibits a small overshoot that agreeswith the damping ratio restriction. Fig. 3b shows thesimulation out of the nominal condition, without anychange in the controller parameters. The duty cycle at thenew equilibrium point is D0d ¼ 0.3, which corresponds to aVg equal to 7.2 V. Again, we simulate the regulator

Figure 3 PSIM simulated response for a loading step of 0.5 A with the nominal load R ¼ 50 V

a Under nominal duty cycle D0d ¼ 0.5b Out of nominal duty cycle D0d ¼ 0.3


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Figure 4 PSIM simulated response for a loading step of 0.5 A with the nonnominal load R ¼ 10 V

a Under nominal duty cycle D0d ¼ 0.5b Out of nominal duty cycle D0d ¼ 0.3

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response in front of a step load perturbation of 0.5 A. It canbe observed that despite the new operation point, theproposed controller maintains a small overshoot of theoutput voltage. Although this overshoot is larger than inthe nominal case, it is yet inside of the specified dampingratio restriction. Besides, the decay rate is approximately200 which is smaller than in the nominal case but largerthan the specified a. Fig. 4 shows the same simulations ofFig. 3 with a different non-nominal load of R ¼ 10 V. Thesimulations show that the LMI controller satisfies thedamping ratio specification and also the decay rate is betterthan the specified a.

In order to contrast the performance and robustness of theproposed regulator, we have also simulated the boostconverter with a conventional PID controller, whose controllaw at low frequencies is given in (22). This comparison hasbeen made with the aim to show the benefits of using a robustcontrol method compared with a conventional approach.Obviously, this comparison could have been done withanother state-feedback or current-mode controller.Nevertheless, the robust method would keep, to a greatextent, its advantages over a design disregarding uncertainty.

dd(t) ¼ Ki

ð(Vref � vo(t))þ Kp(Vref � vo(t))

þ Kd

d(Vref � vo(t))

dt(22)

Equation (23) shows the control-to-output transfer function ofthe boost converter, whereas (24) is the transfer function of thePID controller, which corresponds with a PID controller withKi ¼ 31.25, Kp ¼ 0.0209432186 and Kd ¼ 0.0000447531.This PID controller has been designed at the nominal modelof the converter. We impose as PID specifications a phasemargin greater than 608 and a gain margin greater than20 dB, which are common specifications in power electronicscontrol. We also impose that the closed-loop poles present adamping ratio greater than 0.4 and a decay rate greater than

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130, which are similar to those of the LMI controller. Toselect such PID controller we have used sisotool of Matlab.

vo(s)

u(s)¼

Vg(R � (L=D02d )s)

CLRs2 þ Ls þD02d R(23)

PID(s) ¼0:0000447531s2

þ 0:0209432186s þ 31:25

s((s=1� 105)þ 1)((s=1� 105)þ 1)(24)

Fig. 3a depicts the transient simulation of the converter with thePID controller for a load step change of 0.5 A in nominalconditions. When compared with the LMI simulation, it canbe observed that the response of the PID controllers is betterin terms of decay rate and perturbation rejection, but it isslightly worse in terms of damping ratio, since they have beenbuilt using the concepts of phase and gain margin. However itis important to remark that all the controllers presentadequate damping properties at the nominal operation point.

Figure 5 PSIM simulated output impedance of the converterwith the LMI controller. The simulation was made with acurrent sink of 200 mA at nominal duty cycle D0d ¼ 0.5

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Out of these nominal conditions, in Fig. 3b, it is shown thetime-response simulation in presence of the sameperturbation of 0.5 A when the converter operates at a dutycycle equal to D0d ¼ 0.3. Again, we have maintained theprevious controller parameters. Since the PID controller wasnot designed taking into account the uncertainty of theconverter, the regulator with PID controller loses its damping


properties and exhibits poor response when the duty cycle D0ddecreases far from the nominal value. The simulations withthe non-nominal load R ¼ 10V with the PID controller areshown in Fig. 4. While in the nominal duty cycle D0d ¼ 0:5the behaviour is close to the case with the nominal loadR ¼ 50 V, when the duty cycle is out of its nominal valueD0d ¼ 0:3, the output voltage behaviour is worse than the

Figure 6 Implementation diagram of a boost converter with the proposed LMI state-feedback regulation

Figure 7 Detail of the circuit implementation of the LMI state-feedback controller and the PWM regulator


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LMI controller in terms of perturbation rejection, decay rate anddamping ratio.

Besides the transient response characteristics, we havesimulated the steady-state output impedance of the regulatorwith the LMI controller. The frequency response for aperturbation iload ¼ 200 mA around the nominal duty cycleD0d ¼ 0.5 has been depicted in Fig. 5 for the cases of thenominal load R ¼ 50 V and a non-nominal load R ¼ 10 V.The output impedance gain peak with the LMI controller isapproximately 7 dB, which agrees with the H1 normspecification bound g ¼ 9.21 dB found in the LMI synthesis.

4.3 Experimental verification

To demonstrate the advantages of the proposed controlscheme, an experimental prototype of a boost converter hasbeen implemented. The structure of the converter with thestate-feedback controller is shown in Fig. 6. The state-feedback controller, whose electric diagram is also given inFig. 7, only requires a small number of operationalamplifiers and discrete components.

Fig. 8 shows the prototype transient response under aloading step of 0.5 A, as it was carried out in thesimulations. The slew rate of the current step is greater than150 A/ms. For this figure, the converter was working at thenominal operation point. The transient closely resembles thesimulation in Fig. 3. The robustness of the controller isverified in Fig. 9, where the loading step is applied out ofthe nominal conditions (D0d ¼ 0.3). Once again, theexperimental waveform agrees with the simulation in Fig. 4.

Besides the transient experiments, the output impedance ofthe regulator has been measured with the help of a frequency

Figure 8 Experimental response for a loading step of 0.5 Aunder nominal duty cycle D0d ¼ 0.5. Upper traces are outputvoltage. Lower traces are output current


analyser connected to a voltage-controlled current sink of200 mA. The measurements have been carried out forR ¼ 10 and 50 V, at the nominal duty cycle D0d ¼ 0.5. Themeasurements, depicted in Fig. 10, show a perfect agreementwith the simulation results and demonstrate that despite ofuncertainty, the LMI controller fulfils the objectives markedin the synthesis procedure.

5 ConclusionsThis work has presented a robust controller design frameworkbased on LMIs for switched-mode dc–dc converters. The

Figure 9 Experimental response for a loading step of 0.5 Aout of nominal duty cycle D0d ¼ 0.3. Upper traces are outputvoltage. Lower traces are output current

Figure 10 Experimental output impedance of the converterwith the LMI controller. The measurement was made with acurrent sink of 200 mA at nominal duty cycle D0d ¼ 0.5

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LMI design method has been applied to a boost converter andthe results have been compared with a conventional PIDcontroller by means of both numerical simulations. Also theexperimental measurements from a laboratory prototype of aboost converter have verified the results of the LMIcontroller. The LMI control circuit implementation consistsof a standard current sensor, some OPAMPs and a PWMdevice. We have shown that the state-feedback LMIapproach is a valid synthesis method for non-minimumphase converters. With the proposed method, transient andsteady-state performances can be taken into account and theresulting design has, in this case, good performancecharacteristics, despite the conservatism in the uncertaintymodel. The main advantage of this approach is that thestate-feedback controller can be synthesised automatically,differently from other robust control methods, for which thecontroller must be synthesised manually or using CADtools. Since the synthesis is carried out by means of aMatlab’s LMI toolbox, the method can be readily extensiblein order to consider parasitic resistances in the inductor andcapacitor by modifying the state-space matrices of themodel. Finally, experimental verifications show a perfectagreement with the design constraints despite uncertainty.Future work will deal with the application of the method tomore complex power converters such as high-order circuits,multiphase and multilevel converters.

6 AcknowledgmentThis work was partially supported by the Spanish Ministeriode Educacion y Ciencia under grants TEC2004-05608-C02-02 and TEC2007-67988-C02-02.

7 References

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