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are discussed. I. INTRODUCTION. High frequency switching dc-dc converters are important power electronics devices widely used in a variety of appli- cations.
Guadalajara, MEXICO October20-24

Robust Sliding Mode Control for the Boost Converter

Doming0 Cortes, Jaime Alvarez

Joaquin Alvarez

Centro de Investigacidn y Estudios Avanzados del I.P.N., Departamento de lngenieria ElCctrica Secci6n de Mecatrdnica, P.O.Box 14-740, Mkxico, D.F ,07000 { dcortes/jalvarez } @mail.cinvestav.mx

Centro de Investigacidn Cientifica y de Educacidn Superior de Ensenada, MCxico. Departamento de Electr6nica y Telecomunicaciones P. 0. Box 434944, San Diego CA. 92 143-4944 [email protected]

Abstract-The sliding mode control of the boost converter is revisited. Several sliding surfaces are presented and analyzed. Some of the surfaces presented do not depend on the circuit load, eliminating the necessity to measure the current measurement. The analysis presented is based on the switching model of the circuit instead of the average model typically used. Practical aspects for implementing sliding mode controller with the proposed surfaces are discussed.

I. INTRODUCTION

'

High frequency switching dc-dc converters are important power electronics devices widely used in a variety of applications. They are also the circuits upon which more complex applications are based. From system theory point of view, dc-de power converters exhibit interesting dynamical properties. Their average models, which are the most common models used for analysis, are nonlinear, nonminimum phase systems, with boundedcontrol and a highly varying parameter (the load). Such properties have attracted the interest of many control specialists [I], [ 2 ] , [3], [4]. The average method yields a continuous-time model by assuming that the switching frequency is high. Continuous-time design techniques are applied to obtain a continuous-time control. The continuous-time control is then passed through a PWM block, which produces the commutation of the switching element. This way of implementing a commutation policy from a continuous-time control is known as PWM control. It has been shown that PWM control is related to sliding mode control (SMC). Indeed, PWM control makes the system evolve on a sliding surface. Equivalent control of such sliding surface is the same that the one driving the PWM block [ 2 ] . SMC is considered a powerful method to render a closedloop system robust against plant uncertainty and external disturbaces [ 5 ] , [6]. However, the chattering fenomena caused by the discontinuities of the SMC has limited its use in real applications. The elimination of chattering has been the main objective of a lot of works on SMC [6]. Most of the power electronics systems are inherently variable structure systems which are specially suitable to be controlled with SMC. Switching control elements are fundamental parts of power electronics systems, unlike most of the systems where the SMC have been applied.

0-7803-7640-4/02/$17.00 02002 IEEE

Due to the presence of switching elements, chattering is inherent to power electronics devices. Nevertheless, in these devices the chattering problem is compensated by the efficiency, simplicity, reliability and low cost gained by the switching. In this paper, we use the SMC to control the boost dc-dc converter, Several existing sliding surfaces are revisited and new ones are proposed and analyzed. One of the sliding surface proposed leads to is very robust controller and eliminates the necessity of inductor current measurement. Furthermore, the resulting SMC can be implemented with low cost components and without the requirement of a PWM block. Analysis, control design, and an implementation proposal are made by using the switching converter model instead of the traditional average model. The switched model is the natural model of the circuit and results in more robust controllers to load variations than its similar continuous controller. Such approach has been used in some works 1I], [3] but it is still far from being broadly accepted in the power electronics engineering community. Implementation of SMC yields to a variable switching frequency, which is unacceptable in most applications. In this document, a hysteresis loop is used instead of the sign function normally associated to SMC. Extensively simulations show that the hysteresis loop keeps the switching frequency constant. The paper is organized as follows. The boost switching model is disccused in Section 11. Section III shows the basis of SMC. Several sliding surfaces are analyzed in Section IV. In Section V, simulation results are presented, and implementation details are discussed. Conclusion and comments for further research are given in the last section. 11. BOOSTCONVERTER MODEL AND PROBLEM STATEMENT

Fig. 1 shows a diagram of the boost converter. A switched model of this circuit is given by

c-z d ---++z,, z2 dt 2 -

RL

where z,, z2 are the inductor current and the capacitor voltage respectively, and U E ( 0 , l ) defines the switch position that.

208

~

111. SLIDING MODE CONTROL OF SWITCHING SYSTEMS In this section, the main ideas behind the SMC and necessary conditions for sliding modes to exist in a class of systems containing the boost converter model [ 11, are revisited. Let us consider the system,

1

1

I

.-

4

It = f(x) 4 g(x)u -

Fig. I . Schematic diagram of the boost converter.

plays the role of the control input. The inductor L, the capacitor C , and the source voltage v,, are constants and they are supposed to be known. The resistance R, is unknown but is considered to be constant for analysis purposes. Although the average model will not be used, it is important to mention that it has the same structure that the model (I); however, in the average model, the control input U denotes the duty cycle of one switching period with values in the interval [0, I], and the states z are also average values. The average model is valid at high switching frequencies and away from zero inductor currents 121. For an easy analysis, a normalized model can be obtained from ( 1 ) via the following transformations

and a time scaling t =

m~ yielding , the model x, = 1 - ux2,

(3a)

where the upper dot denotes the time derivative with respect to the new time z, which we will denote by t in the sequel. Since the normalizing transformation is linear, any analysis result for (3) can be translated into a result for (1). The control objective is to regulate the (normalized) capacitor voltage x2 to a desired value x ~That ~ is, . to make x2 -+ X2d

(5)

-

(4)

with a fast response and robust stability against load variations while keeping overall system stability. Fast response is an essential characteristic required from the control law. It is what makes to control the boost converter a difficult task. Without this requirement, the control problem would be trivial since the system is open-loop stable regardless the load value. When U is switched at high frequency such that it has a constant average value C, then we can find an average equilibrium point (2,&) = ( 1 /(Rii&), 1/ I e qIn ) . the open-loop case, we can make I,, = 1 /x2d and obtain the equilibrium point ( ~ { , / R , X So, ~ ~ when ) . xz =xZd,then x, =x;,jR. Furthermore, since Zeq can only take values between 0 and 1 , then xu 2 1.

where the control U E (0,l). This class of systems includes the most common switching dc-dc converters such as the buck, the boost, the buck boost and the Cuk converter. The underlying idea of the SMC is to make the system evolves on a surface containing the desired equilibrium point. Such an equilibrium point must be an asymptotically stable point of the system’s motion restricted to the surface. Sliding regimes in system (5)are trajectory movements on a surface where the system can not evolve with a sole value of the control U. So the motion through this surface is achieved by appropriate switching of U between its two possible values. Let us express the surface in which we are interested the system evolves, by (T(x) = 0 (4) For a sliding mode to exist the state x should be driven to reach the surface (6) and afterwards be constrained to this surface. That is, the surface must be rendered attractive and invariant by an appropriate switching of the control U . Furthermore, the desired operation point must be an asymptotically stable equilibrium point of the dyamics of the system constrained to evolve in the surface (6). Attractiveness and invariance of the surface (6) can be assured by making the trajectories of the system evolve towards the surface. This can be achieved for example by choosing U such that ir > 0 when c s 0 If the system is constrained to evolve on the surface (6)then 6 ( x )=0

(8)

should be satisfied. By solving (8) for u we get an expression for the control addressed as the equivalent control ueq. It is known that the physical meaning of the equivalent control is the average value of U when the system evolves on the sliding surface [5]. The description of the dynamics when the system is constrained to the surface (6) is given by (see [ 5 ] )

Iv. SLIDING MODE CONTROL O F TNE BOOST CONVERTER Several sliding surfaces are presented in the first part of this section. Attractiveness and stability of the surfaces are proved in the second part. Along the sliding surface presentation and analysis, some practical aspects for the selection of the controller parameters are discussed.

209

A. Sliding surjaces for rhe boost converter

Taking into account the control goal (4), a straightforward surface is given by ~ ( xz) x2 -xu. It is known that such sliding surface can be made attractive only at the expense of the permanent increase of xI,that is, such sliding surface is unstable [I]. ) x, Other natural sliding surface is (6) with ~ ( x E which has been proved to be attractive and asymptotically stable [1]. Unfortunately, for implementing such sliding surface it is necessary to know the load R, which is an important practical drawback. Futhermore, the response is somewhat slow. The following expression for (T combines the two previous ones.

9

(

(T(x) E

X]

2)+

--

k, (x2- xZd)

(IO)

where k , > 0. The sliding surface (IO) has already been used in real circuits [7], however, no stability analysis exists. In [8],it is shown that the performance and the robustness of a sliding mode controlled system can be improved by adding feedback to the sliding control algorithm. Applying this idea, we propose the following expression for (T derived from (10)

B. Attractiveness and stability of the sliding su$aces Qualitative analysis and a priori knowledge of the circuit is used in this section to obtain conditions for the sliding surfaces (IO) and ( 1 3) to be atractive and stable. At the same time a commutation policy is derived. In view of the circuit operation, we restrict the analysis to positive values of xi and x2. Analysis of the other sliding surfaces can be made along similar lines that those presented here. Proposition I : Let us consider the system (3). The sliding surface ( ~ ( x )= 0, with CT(X) given by (IO), is attractive if the control policy 0 if Q(X) < 0 U = 1 if Q(X) > 0

{

is applied. The region of attraction is the first quadrant of the (xl,x2)-plane. Furthermore, if k, < R/x2,, the point ( x $ ~ / R ! xis~a~local ) asymptotically stable point of the system’s motion restricted to evolve on the sliding surface. Proo$ The sliding surface is depicted in Fig. 2. Let us suppose that at certain time, o(x)< 0. In this case U = 0 and the system equations become

x2

x2=-- R K2

where ki > 0. We have used x1 instead of x1 - -fbecause the

4

effect of dropping out the constant term can be compensated by the integral term. Note that this sliding surface does not depend on the load, so, the resulting SMC will be robust to load variations. The x , term appearing in (IO) and (1 I ) can be avoided by using the integral reconstructors proposed in [9]. From (3a), we can write

x,(f)=l(l

(12)

-U2)dffX,(O)

Using expression (12), new sliding surfaces can be obtained from the previous ones. For example, combining (1 I) and (1 2) we obtain a sliding surface that does not depend on the load . neither on the current: O(X) E

I’

(1 - ux2)dt

thus, x, increases and x2 tends to zero. Therefore, the system eventually crosses the sliding surface (see Fig. 2 ). On the contrary, if Q(X) > 0 then U = 1 and the system equations turn into

x2 +x, i2 = --

R

In these conditions the system evolves towards the point ( x 2 / R , 1 ) . This point is “below” the sliding surface, so the system crosses the sliding surface again (Fig. 2). Previous arguments show that control (15) makes the sliding surface attractive and the region of attraction is the first quadrant of the (xl,x2)-plane. For the stability analysis, we need a description of the system motion on the sliding surface, so, it is necessary to find the equivalent control. Solving &(x) = 0 for U ,results

(13)

+ k,

(x2 - xu) + k,

b 2 -X2d)

df

ucq=

b X l

where we have ridded of the constant x,(O) appearing in (12) because it can be compensated by the integral terms. Remark I: Using integral reconstructors and slightly different ideas to the ones exposed here, in [9], the following expression for o(x) is proposed CT (x) =

/(‘

0

1

laz) dt

-x l d

+ ki /;

(x2 - x

~ dt~ ) ( 14)

4, - 1

R2 - xz

From (1 8) and (1 0) we can write (3b) as

Equation (1 9) entirely describes the system on the sliding surface.

210

u = 1 and the system equations are given by (17), in this case the state tends to (1 / R , 1 ), therefore

22

t

p =0

(7

I

I

k

I

P

h

>

Fig. 2. Atractivetiess and stability of surface (10)

Stability of (19) is analized by Lyapunov method using the positive function V(x2) = (x, - ~ ~ ) ~ Taking / 2 . the time derivative along (1 9) yields

Since the numerator of (20) is always positive, then

+ k , (1 -xZd) + k i / (

1 -x,,)dt

consequently, o ( x ) eventually goes negative and the system crosses the sliding surface. So, we have a situation as the one Fig. 2 shows. For stability analysis, we can define

Thus, the system can be described as (3) together with i=(1-~~Z)-t-ki(xz-x~)

(26)

+

and the sliding surface is rewritten as Q s z k, (xz - x,(/) = 0. When the system (3), (26) evolves on the sliding surface (1 3), the dimension of the system becomes 2 because there i s an algebraic relationship beetwen z and x2. Solving Ir = 0 for U, the following equivalent control is obtained $x2- 1 - ki(x2 Ucy =

Ax,- x 2

must accomplish for stability of point (19). hequality (21) is true if

Inequality (22) specifies an attractive region of the point x, = x~~~in the differential equation (1 9). For assuring this region to include the point x2 = x~~ itself, the condition k,, < R / x , must be held. The previous analysis shows that if the systems reaches the sliding surface on a point where (22) is held then x2 -+ xZd and since cs = 0 then xl -+ & / R . Attractiveness of the sliding surface (13) can be proved along the same lines above, however, the stability analysis is more difficult because the sliding surface is dynamical in this case. Proposition.2: Let the system (3) and consider the sliding surface ~ ( x=) 0 with Q ( X ) given by (13). If the control policy ( I 5) is considered and k,x, < 1, then the sliding surface is attractive being the first quadrant of the (xl ,x2)-plane the region of attraction. Furthermore, if k, < R / x then the point 2d ( x & / R , x , ~ ~is) a local asymptotically stable point of the system motion restricted to evolve on the sliding surface. Pro05 If at a certain time o(x) < 0 then U = 0 and the system equations are given by (16), consequently x2 tends to zero, therefore 0

-+

r

] dt - k+,,

rf

- k, J0 xZddt

If kix2(, < 1 then ~ ( xeventually ) goes positive and the system crosses the sliding surface. On the contrary, if ~ ( x>) 0 then

Subtituting (27) in (3) and linearizing around the point ( X L / R , X ~ we ) , get the system

which has negative eigenvalues if conditions ki < i/xZd and k, < R / x Z Jhold. Remark 2: Unlike Proposition 1 the stability proof in Proposition 2 does not give us an estimation for the region of attraction of the equilibrium point when the system evolves on the sliding surface. V. SIMULATION RESULTS The analysis presented in the previous section supposes that the control U can switch arbitrarily fast. Of course, this is not true in practice. The control policy (15) has to be modified in order to achieve a bounded switching frequency. One solution could be to examine the sign of cs periodically at discrete times. However, this results in an erratic behavior of the states, that is, the state ripples have many frequency components which is unconvenient in many applications. Different approaches in order to keep a constant switching frequency and a regular behavior of the state have been tried; extensive simulations have shown that introducing hysteresis in the control policy (see Fig. 3) achieves these requirements. Moreover among the methods to bound the switching frequency, introduction of hysteresis is one of the easiest to implement. The width of the hysteresis loop is inversely proportional to the switching frequency.

211

m. r”

XZ

Fig. 3. Introduction of hysteresis achieves constant switching frequency 5 0. 4 5-

,---Open loop voltage (x, )

’,; -1.0

Fig. 5. Block diagram of the controller

,!‘I, 0 20

,

, 40

c l o h o o p current (x ,) ,

, . ,

60

80

,

,

100

,

,

120

,

b’

S t !

\i, , .

VI. COMMENTS AND CONCLUSIONS Sliding mode control of dc-dc converters was revisited in this paper. Some sliding surfaces have been analyzed. One of them results in a high performance, robust, low cost and easy (although not simple) to implement controller. Introducing an hysteresis loop in the control policy, constant switching frequency was achieved. Some insights about the region of stability and the valid parameters values for several controllers have been obtained from analysis. Further research can be focused on clarifying the relationship between parameters values and the performance or the relationship between the width of hysteresis loop and the switching frequency. Different sliding surfaces could also be proposed.

L‘

,

140

160

~

180

,

I

200

Fig. 4. Simulation results for the sliding surface (13)

Let us consider the application of the control law given by (15) with an hysteresis loop and 0 given by (13) to the boost converter model with parameter values: R , = 48Q, C = 28.2pF, L = 0.36mH, and vin = 50V, which represent practical values. The desired capacitor voltage i s taken as zZd = 15OV. With these values, the parameters of the normalized model are given by R = 13.4343, xzd = 3. Fig. 4 shows the high performance of this controller. To show the controller robustness, the load was suddenly changed from R to 0.5R at t = 77. At t = 137 the load was changed from 0.5R to 3R. The open-loop response is also shown in the figure for comparison. The controller’s parameters were set to k, = 0.1 and k, = 0.1. The width of the hysteresis used was 0.2. Simulations results depicted in Fig. 4 were performed in Matlab 0using the switched model of the boost circuit. Fig. 5 shows the block diagram of the program developed for simulation, This block diagram resembles the power electronics devices that would be necessary for implementing the controller. Note that only one integrator, some comparators and some gains are needed.

REFERENCES [ I ] H. Sira-Ramirez, “Sliding motions in bilinear switched networks”, IEEE Trans. on Circrtirs and Systents, Vol. 34,No. 8 , August, pp. 919-933, 1987 121 Sira-Ramirez, H., Pbrez-Moreno, M., Ortega, R., and Garcia-Esteban, M., “Passivity based controllers for the stabilization of DC-DC power converters,” Aufomatica, Vol. 39, No. 4., pp. 499-513, 1997. 131 G. Escobar, R. Ortega, H. Sira-Ramirez, J-P. Villiain, I. Zein “An expenmental comparison of several nonlinear controllers for power converters” ZEEE Control Systeni Magazine, pp. 66-82, feb. 1999. [4J A. Kugi, K. Schlaeher. “Nonlinear H , controller design for DC-to-DC power converter”, IEEE Truns. on Cont. Syst. Tech., pp. 230-237, vol. 7. no. 2, mar. 1999. [5] V. I. Utkin, “Variable structure control systems with sliding mode,” E E E Trans. on Aut. Control, vol. 22. no. 2, pp. 210-22, 1977. [6] J. Y. Hung, W. Gao and I. C. Hung, “Variable structure control: a survey”. IEEE Trans. on Ind. Electron., vol. 40, no. 1, pp. 2-21, 1993. [71 N. VAzquez, J. Almazan, J. Alvarez, C. Aquilar, J, Arau, “Analysis and experimental study of the buck boost, and huck-boost inverters”, IEEE Power Electronics Specialist Con$ PESC ’99,pp. 801-806, 1999. [8] J. Alvarez-Gallegos, R. Castro-Linares, M. Velasco-Villa “Sliding modes techniques for nonlinear model matching” Proc. of N lutinoamerican congress ofautomatic control, F’uebla, Mexico, Nov. 1990, pp. 872-877. (91 H. Sira-Ramirez, R. Mirquez and M. Fliess “Generalized PI sliding mode control of dc-dc power converters”, IFAC syinposirtm on sysfem structure and control, Prague, Czech Republic, August 29-31, 200 I .

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