An adaptive controller for a boost converter with harmonic reduction

G. Escohar*, A. Valdez, J. Leyva-Ramos and P.R. Martinez-Rodriguez

Absfracf- An adaptive controller for the compensation of harmonics in the input voltage is proposed for a Pulse Width Modulated (PWM) boost converter. Following the Lyapunov approach we designed an adaptive law to cope with uncertaintie in the disturbance signals. Complexity of the proposed controller is reduced by rotations which transform the adaptive terms into a sum of resonant filters having as input the output voltage error. The resonant filters are tuned at the frequencies of the harmonics under consideration. To facilitate the implementation we have tried tn preserve the structure of the proposed controller as close as possible to the conventional controller. The latter is usually composed by a voltage outer loop (basically a Proportional plus Integral (PI) control on the output voltage error) and an inner control loop (basically a Proportional control plus a Feedforward term). Thus, in the proposed controller, the hank of resonant filters appears as a refinement term which is added to the inner control loop. The proposed controller turns ont to he robust with respect to parameter uncertainties. Experimental results on a boost converter hoard, using a poorly regulated voltage source, are presented to assess the performance of our approach.

I. INTRODUCTION The main role of a dc-dc boost converter is to keep the

output voltage as close as possible to a desired constant reference. Although this task may be fulfilled by a simple open loop controller, it is usual to aggregate control terms to alleviate certain drawbacks. For instance, it is well known that open loop control is not able to cope for steady state errors due to changes in the input voltage and load variations. Usually, Proportional plus Integral (PI) controllers have provided a good answer to the regulation task in dc-dc boost converters. Due to the nonminimum phase nature of this converter [l], the designer is forced to control the output voltage indirectly by directly controlling the inductor current, this technique is referred as current or indirect control in the power electronics literature. Moreover, to facilitate the design, the designer usually appeals to the decoupling assumption, out of which the control design is split in two loops, namely, the. inner current loop and the outer voltage loop. The former is aimed to guarantee fast regulation of the inductor current towards its reference. usually a proportional term on the inductor current error plus either, a feedforward term of the input voltage, or a simple offset. The purpose of the outer voltage loop is to simply provide the inductor current reference to the inner

'Corresponding author G. Escobar. A. Valder, J. Leyva-Ramos, and P.R. Martinez-Rodriguez are

with Dept of Aplied Malhemathicr and Computer Systems-IPLCYT - AV. Venusliano Carranza 2425 A - Col. Bellas Lomas - San Luis Pomsl, SLP 78210 - Mexico - Tel +52 444 833 541 I - FAX +52 444 833 5412 - Tel +52 444 833 5411 - FAX +52 444 833 5412.E-mail - [gercobar,avaldezjleyva,pm~i"~~]@ipicyt.edu.rm

current loop, in this case a PI controller on the capacitor voltage error is the most commonly used.

Although most of the controllers designed so far have taken into account disturbances such as step load changes and slow input voltage variations, and they may be robust with respect to system parameters uncertainties, there are few results regarding the compensation of periodic disturbances on the input line voltage at frequencies in the audible range [3], [4], [51. This issue arises in applications, such as power factor correctors (PFC). where the delivered voltage varies over a wide rage. In these cases the input voltage is mainly polluted by a Znd harmonic component of the line voltage which is due to the rectification process in PFC.

In this paper we proposo an adaptive controller aimed to reduce the effects of harmonic disturbances present in the input voltage. Specifically, the proposed controller is aimed to reduce selected harmonics of the output capacitor voltage, hence, improving the audio-susceptibility chart, while main- taining an acceptable dynamical performance. We follow the Lyapunov approach to generate adaptation laws to estimate certain harmonic components of the disturbance to be compen- sated. The adaptations are later reduced, by means of rotations, into a hank of resonant filters tuned at the frequencies of the harmonics to be compensated. We also appeal to the decoupling assumption, hence, the final expression of the proposed controller includes an inner current loop and an outer voltage loop. In our case, the former is composed by a proportional term on the inductor current error, a feedforward term in function of the input voltage and the bank of resonant filters. The outer voltage loop is formed by a Low Pass Filter (LPF) term plus an integral term, both operating on the capacitor voltage error. We remark that, in our proposal, the usual proportional term has been substituted by a LPF to prevent the reinjection of further harmonics into the control loop due to the remanent harmonic content in the capacitor voltage. Our controller turns out to be very similar to the conventional one, where the main difference is the introduction of the bank of resonant filters acting as a refinement to the final control signal. It could be observed that the conventional controller and the feedforward control presented in [3] are particular cases of the proposed controller.

Finally, experimental results have been carried out in a boost converter board to asses the performance of the proposed controller. The converter is fed by a poorly regulated voltage source polluted by the second harmonic, i.e., 12OHz. For the sake of space, we present only the results of the modified version of the proposed controller, i.e., without feedforward term. For implementation purposes the resonant filters (which

0-7803-7906-3/03/$17.00 02003 IEEE. 568

have infinite gain at the resonant frequency) are replaced by Band Pass Filters (BPF) (which have limited gain at the tuned frequency) to guarantee a safer operation. Notice that the environment might be polluted by harmonics of frequency close to the resonance frequency, which might be introduced into the control loop with a huge gain. In our implementation only a single BPF tuned at 120Hz was included. Several tests have been proposed, such as the response to a step change in the load resistance, and connectioddisconnection of the resonant filter contribution, etc.

11. PROBLEM FORMULATION A circuit of the boost converter is shown in Fig. 1. We

have neglected, without loos of generality, the equivalent series resistances (ESR) of inductor, capacitor and Mosfet, as well as the voltage drop in the diode.

L

I I I I

where p, represents a unitary vector rotating at a frequency mw in counterclockwise direction, q,,, and V & , are the real and imaginary parts of the phasor Vs,,,,. A4 is the set of index of the harmonic components contained in uin.

The control objective consists in regulating the output capacitor voltage 2 2 towards a constant reference v d despite of the harmonic distortion in the input voltage. That is, the con- troller should be able to reject harmonic voltage disturbances present in the power supply. It is well known that, due to the nonminimum phase nature of this converter, it is preferable to indirectly control the capacitor voltage by directly regulating the inductor current towards a constant reference (this scheme is referred in literature as current or indirect control . As it will become clear later, a solution to our problem treated here is obtained by forcing the inductor current to track a harmonic distorted reference instead of the usual constant signal. The idea behind this approach is that, by distorting the inductor current reference, we incorporate a degree of freedom that allows compensation of harmonics in the capacitor voltage side.

Thus, we propose the following reference for the inductor current:

Z;(t) = I d + P:Ih,k (4) k E H

I 1 where Id is a constant reference, usually obtained (in a conventional controller) from a proportional plus integrative P I ) controller; and I h , k = [ I i , k , l A , k ] T a phasor representing

Fig. I . Boost converter circuit.

the harmonic components intraduceh to be reconstructed in an outer loop as well. H c M is the set of index of the harmonic components to be compensated.

The system dynamics of the boost converter shown in Fig. 1 are described by the following expressions:

LX1 = -ux2+uan (1) (2) CX2 =

where x1 is the inductor current, x2 is the capacitor voltage, vZn represents the voltage source (this signal is addressed indistinctly as input voltage or voltage source all along the paper), L is the inductance, C is the capacitance and R is the load resistance. We assume that parameters L,C and R are unknown positive constants. In the discontinuous model, i.e., U E {0, l}, the value u.= 0 corresponds to the situation where the transistor is conducting, while 'U = 1 corresponds to the case where the transistor is disconnected and thus the diode is conducting. In the average model used along the paper, it is assumed a sufficiently large switching frequency, hence, U represents the slew rate of a PWM signal feeding the gate of

x2 U 5 1 ~ -

R

Its time derivative i s given by

where we used the fact that bk = k w J p , . For the sake of simplicity, we assume that the inductor cur-

rent dynamics are faster than the capacitor voltage dynamics. This is a usual time scale separation principle advocated in many converter circuits to facilitate their control design. That is, the converter can be treated as two decoupled subsystems, a fast inductor current subsystem and a slow capacitor voltage subsystem. Therefore, dividing the control design in a inner current control loop and an outer voltage control loop.

111. PROPOSED CONTROLLER ASSUMING vin IS AVAILABLE the boost converter, i.e., U = (1 - d ) where d is the duty ratio.

We assume that the input voltage uin is polluted by higher order harmonics. This phenomenon appears in some applica- tion where the input voltage may vary on a wide range, for

A. Carrent control loop

in terms of its increments as follows:

'

Let us rewrite the inductor current subsystem dynamics (1)

instance, in PFC's, or when the converter is fed by a poorly Lbl = --U22 + vin - LX: regulated voltage source, most of them occurring in relatively high power applications. In these cases, we assume that the invut voltage can he reDresented as follows:

Assuming signal uin is available from measurements, then a control law can be proposed as:

mEM I

A where kz is a positive design constant, Z1 = x1 - xi, where xi is obtained later in the outer loop.

569

The closed loop dynamics yields

k 2 2 2 - ~2 -V, K! K!

LL1 = -- xi + ~ (Lx; -U

with equilibrium point given by

which is stable provided that all design parameters are chosen positive.

2) a c component: For the sake of clarity, let us define the following transformation

Recall that I h , k represents the kth control input for this subsystem, while v , , k represents the kth harmonic component of the perturbation.

The subsystem is now rewritten as

A - where &k = (*k - * k ) . Following the Lyapunov approach, we propose the follow-

ing storage function

whose time derivative given by

is made negative semidefinite by proposing the following adaptive laws

+ k = y k P k x Z h k E H

where ^/k are positive design constants representing the adap- tation gains, and we used the fact that 6 k = k'k since * k are constants, for all k E H. This yields the time derivative

, 0 ut of which X2h is bounded and goes to zero asymptotically. Moreover, following the Lassalle's invariance principle, X2h

C. Implementation discussion Using the descriptions of X I and Si (4)-(5), and the trans-

formations (13),then controller (6) can be rewritten in terms of the estimate * k as follows

o implies &k = 6 k = 0.

Notice that the controller above requires the generation of vectors pk. which might complicate its physical implemen- tation. To overcome this problem, we propose the following transformations

(14) T " c l , k = P:&k 3 5 2 , k P k J*k

which yields the following expression for the controller

57

with adaptive expressions given by

< l , k = "ikX2h - W k c 2 , k

52 ,k = W&,k

which expressed in the form of transfer functions are

Y k k W 52,k s2 + k 2 w 2 . x 2 h

for every k E H. Thus, the controller is rewritten as

I -b where k; = k and y k - E Remark 111.1 Notice that this controller is composed by a feedforward term 2 plus a proportional term kh(x1 - I d ) , which are the same terms present in the conventional approach. In contrast with the conventional approach, our proposed controller includes, as well, a sum of resonant filters to cope with the harmonic distortion. 0

It is clear that X2h and 2 2 0 are not available from mea- surements. Fortunately, thanks to the selective nature of the resonant filters we can assume

X2h E - v d

and leaning on the LPF capability of the proposed controller (12) we can assume

5 2 0 E 5 2 - Vd

In conclusion, the final expressions for the controller are

I d = -k,v - k;< = 2 2

v = -bu+a52

where ita = x 2 - Vd. A block diagram of controller (15) is shown in Fig. 2.

I v . EXPERIMENTAL RESULTS A boost,converter and controller (15) have been imple-

mented. The converter parameters are given in Table I. The inductor current is sensed via a precision resistor of 0.05R connected in series with the inductor. A typical circuit SG3524 is used to generate the PWM signal. A conventional non regulated power supply using a full bridge diode rectifier with a 4700pF capacitor filter is used as a voltage source. The voltage provided by this source is polluted mainly by a 2nd harmonic, i.e., at 120Hz. which, as expected, increases for a higher current demand. To guarantee a safer operation, we have preferred to use BPF's instead of resonant filters (ideally, resonant filters have infinite gain at the resonant frequency,

' 1

.........................................

+

Fig. 2. Block diagram of proposed controller measuring vin.

while BPF's have a limited gain at the resonant frequency) In our implementation only a single BPF tuned at 120Hz was included. This BPF has been implemented following the guidelines in [Z], whose transfer function is given by

7;s - U 0 v i _ -

s2 + 2 s + k2w2 where the design parameter A, > 0 is the desired gain of the BPF at the resonant frequency kw. Notice that, in the case of an ideal resonant filter A, a 03.

Diode MBRICJ45 Power Mosfet IRF540 Inductor 350mH Capacitor 47pF Load resislor 18/36 R

TABLE I PARAMETERS OF THE BOOST CONVERTER

The tests performed include: 1) Enabling and disabling the harmonic compensation. That

is, connecting and disconnecting the BPF contribution, respectively, while keeping a constant load resistance R = 180.

2 ) Step changes in load resistance between 18R and 36R are presented to show the robustness of the proposed controller against load variations.

Fig. 5 shows the responses of capacitor voltage 2 2 , inductor current x1 and the dc component of the inductor current reference I d (from top to bottom). In this figure the harmonic compensation is enabled after a given period of time. We observed that after a relatively short transient, the distortion in the output voltage capacitor is considerably reduced.

. . . . . , . , . . , , , . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . . . . . . . . . . . .

I ! . . 1 ' .

... . . . . . . -. . . . . . . . . . I . .

Fig. 3. Transient responses after enabling the harmonic compensation, with R = 1812. (From top to bottom) capacitor

voltage x2. inductor current i : ~ and dc component of the inductor current. reference Id.

Fig. 4 shows the frequency spectrum of 2 2 without and during compensation (from top to bottom). We observed that the 2"d harmonic component (the one under compensation) decreases almost 30dB, while the rest of harmonics are main- tained almost unchanged.

WE I I O . O d 8 6 2 . 5 H Z , Fig. 4. Frequency spectrum of capacitor voltage 52, with

R = 18R: (Top) without harmonic compensation, and (Bottom) under harmonic compensation.

Fig. 5 shows the responses of capacitor voltage x2, inductor current x1 and dc component of the inductor current reference Id (from top to bottom), when the compensation is disable after a certain period of time.

Fig. 6 shows the frequency spectrum of the inductor current x1 without and under compensation (from top to bottom). As predicted by theory, the harmonic content of the inductor current increases, roughly s,peaking, it is necessary to distort the inductor current in such a way to allow compensation in the capacitor voltage 22.

Once the system is operal.ing under compensation, i.e., with the BPF connected, we proceed to change the load from 36R to 18R. Fig. I shows the transient response of voltage x2 and inductor current x1 (from top to bottom). We observed that after 3 small transient the voltage recuperates its desired value

.

512

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . i..? ' . . . .

r

Fig. 5 . Transient responses after disabling the harmonic compensation, with R = 180: (From top to bottom) capacitor

voltage 5 2 , inductor current ZI and dc component of the inductor current reference I d .

f r i I j . . . i . : . . . . . . . . .

. .

I Zo.od8 Sz . IHz1 Fig. 6. Frequency spectrum of inductor current XI. with

R = 18R: (Top) without harmonic compensation, and (Bottom) under harmonic compensation.

24V. in average. In Fig. 8 the inverse process is performed, that is, we switch the load resistance from 18R to 36R.

V. CONCLUSIONS We have presented a controller for the boost converter

whose structure is very close to the conventional one. The main difference consists in the introduction of a bank of resonant filters aimed to compensate for a selected group of harmonic components (in the audible range) contained in the output capacitor voltage. This type of disturbance is mainly due to a voltage source polluted by harmonics in the audible range. The idea behind the proposed approach is that,, by distorting the inductor current reference, we incorporate a degree of freedom that allows compensation of harmonics in the capacitor voltage side. Implementation of the controller requires the measurement of the inductor current, capacitor voltage and input voltage. A set of tests have been carried out in an experimental prototype to assess the performance of the proposed controller. To guarantee a safer operation in the real implementation we have preferred to use BPF's instead of pure resonant filters. In the experimental results we compare

. . . . . . . . . . . . . . . . . f ' " " " ,

. . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7. Transient response for a load step change from R = 36R to R = 18R: (From top to bottom) capacitor voltage XI. inductor current X I and dc component of the inductor current reference I d . p.----- . . . . . .c. .

. . . . . . . . . . . . . . . . . . 1 ' :

. . . . . . . . * . . . . . . .

Fig. 8. Transient response for a load step change from R = 180 to R = 36R: (From top to bottom) capacitor voltage XI, inductor current XI and dc component of the inductor current reference I d .

the responses obtained with and without the aforementioned harmonic compensation. Transient responses to step changes in the load are also presented to exhibit the robustness of the proposed controller against load variations.

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