pwm controller design based on singular perturbation technique: a case study of buck-boost dc-dc...

PWM Controller Design based on Singular Perturbation Technique: A Case Study of Buck-Boost DC-DC Converter

Valery D. Yurkevich

Automation Department, Novosibirsk State Technical University, Novosibirsk, Russia, (e-mail: [email protected])

Abstract: The problem of DC voltage regulation for buck-boost DC-DC converter is discussed in presence of resistor load variations and voltage source variations as well. This paper addresses PI control with an additional low-pass filtering and a high-frequency pulse-width modulation (PWM) in control loop. The problem of controller design is reduced to the continuous-time controller design based on Filippov's average model of buck-boost DC-DC converter. The design methodology of cascaded PI controller via singular perturbation technique is used where two-time-scale motions are artificially induced in the closed-loop system. The method of singular perturbations is used throughout the paper. Numerical simulations for buck-boost DC-DC converter are presented.

Keywords: converters, pulse width modulation, PI controllers, singular perturbation method.

1. INTRODUCTION

The switching buck, boost, or buck-boost converters are widely used in various industrial applications such as power storage systems, power transmission systems, renewable energy systems, and motor control systems. The huge set of references is devoted to analysis and design of switching power converters. Some references are devoted to derivation of the converter circuit topology (Maksimovic and Cuk 1991; Mohan et al., 1995; Bryant and Kazimierezuk, 2002) while in the other ones the development of the switching controller design methods is discussed, for instance, there are various examples of controllers design based on LQR approach (Leung et al., 1993), H techniques (Kugi et al., 1999), sliding modes (Shtessel et al., 1998; Shtessel et al., 2002; Sira-Ramirez and Rios-Bolivar, 1994; Cunha and Pagano, 2002), etc. Due to the improvement of power electronic switching device characteristics, a high frequency switching mode for power converters can be easily provided, consequently, that gives a possibility to used the averaging techniques for analysis and design of switching power converters (see e.g. Kassakian et al., 1991; Sira-Ramirez et al., 1998). The switching control strategy for power converters can be provided based on pulse-width modulation (PWM) in feedback loop (Holtz, 1992; Mohan et al., 1995; Malesani and Tomasin, 1993; Gelig and Churilov, 1996). The existence of equivalence between sliding modes of variable structure control and PWM control responses under the high frequency sampling was discussed by Sira-Ramirez (1989) and by Sira-Ramirez and Lischinsky-Arenas (1990), where it was shown, if PWM controller is not saturated and the sampling frequency tends to infinity, then the response of discontinuously controlled system coincides with average * This work was supported by Russian Foundation for Basic Research (RFBR) under Grant no. 08-08-00982.

model (Filippov, 1964) where control variable is represented by continuous-time duty ratio function. In this paper, the parameters of proportional-integral (PI) controller are selected based on the Filippov's average model for buck-boost power converter via singular perturbation technique (Yurkevich, 2004) where two-time-scale motions are artificially induced in the closed-loop system. The analysis of the closed-loop system properties is provided via the method of singular perturbations (Tikhonov, 1948, 1952; Kokotovic et al., 1999; Naidu, 2002). The paper is organized as follows. First, the simplified circuit of buck-boost DC-DC converter, the output voltage regulation problem, and the cascaded control system structure with pulse-width modulator in control loop are discussed. Second, the Filippov's average model of the converter is introduced and PI current controller with an additional low-pass filtering is designed via singular perturbation technique. Such questions as the implementation of the current controller and high-frequency chattering caused by the pulse-width modulation are highlighted as well. Third, the outer voltage controller is designed based on the discussed two-time-scale design methodology. Finally, simulation results of the designed control system are included.

2. BUCK-BOOST DC-DC CONVERTER

Let us consider the simplified circuit of the DC-DC buck-boost converter (BBC) with inverting topology (see, e.g., Xu (1991), Mohan et al. (1995)) shown in Fig.1 where the ideal switched model of the converter is given by the following equations:

2

2 1

1

2

(1 )

(1 )

x

x

xE u uL L

x x uRC C

(1)

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

9739

where 1 Lx I is the current through the inductor with inductance L , 2 Cx V is the voltage drop across a capacitor with capacitance C , CI is the current through the capacitor, E is the value of the DC voltage source, RI is the current through the resistor (load) with resistance R , and out CV V . Assume that if the switch 1S is turned on (off), then

1u ( 0u ). If the switch 1S is turned on, then the switch

2S is turned off and vice versa.

Fig.1 The DC-DC buck-boost converter circuit The DC voltage regulator is being designed for the discussed buck-boost converter so that to maintain the desired output DC voltage in the presence of varying load R and varying amplitude E that is

lim ( ) dC Ct

V t V

(2)

where dCV is the reference value (reference input) of voltage

drop CV across a capacitor. Moreover, the controlled transients of CV should have desired transient performance indices. These performance indices should be insensitive to parameter variations of the buck-boost converter and external disturbance represented by varying value of the resistor ( )R t and the value of the DC voltage source ( )E t .

Fig.2 Block diagram of the cascaded control system with the inner current controller, the outer voltage controller, and the pulse-width modulator in control loop.

In this paper a cascaded control law structure is used as shown in Fig.2 where, firstly, an inner fast continuous PI controller ( 1C ) of the current LI through the inductor is designed such that the condition

lim ( ) dL Lt

I t I

(3)

holds where dLI is the reference value (reference input) of the

current through the inductor and, secondly, an outer slow continuous PI controller ( 2C ) is constructed in order to meet

the requirement (2) by the tuning of the reference value dLI .

A discontinuous control strategy is provided by the pulse-width modulator shown in Fig.2 where the input signal of the pulse-width modulator is defined as the scalar variable

which takes values in the interval [0,1] . The output signal of the pulse-width modulator is defined as the switching function ( )u t given by

1 ( )0 ( )

s

s s

for t t t t Tu

for t t T t t T

(4)

where sT is the sampling period of the pulse-width modulation, ( )t is the value of the duty ratio function when t t , st T , and 0,1,2, .

3. PIF CONTROL WITH PWM FEEDBACK LOOP

3.1 Filippov's Average Model

The system (1) can be rewritten, for short, as the nonlinear system given by

( , )x f x u (5)

where 1 2[ , ]Tx x x . Denote ( ) ( , 1)f x f x u and

( ) ( , 0)f x f x u , then the system (4)-(5) can be rewritten as

( ) (1 ) ( )x uf x u f x (6) Assumption 1: The pulse-width modulator given by (4) is not saturated, that is the following inequality 0 1 holds. Assumption 2: The sampling period sT is assumed to be sufficiently small in compare with time constants associated with the dynamics of the system (5) or, in other words, the sampling frequency 1/s sf T tends to infinity.

From Assumptions 1 and 2, by following to the Filippov's approach (Filippov, 1964), the geometric approach to PWM control (Sira-Ramirez, 1989; Sira-Ramirez and Lischinsky-Arenas, 1990), the response of discontinuously controlled system given by (4)-(5) coincides with Filippov's average model

( ) (1 ) ( )x f x f x (7) where (7) can be rewritten as

( ) [ ( ) ( )]x f x f x f x (8) and the following inequality 0 1 holds. Finally, in accordance with (8), the Filippov's average model of the discussed buck-boost converter is given by

21

2

2

2 1 1

x

x

x E xL Lx x x

RC C C

(9)

where 0 1 .

3.2 Current PIF Controller

Consider the inner current continuous-time controller ( 1C ) given by the following differential equation:

21 1 1 1 1 1 1 1[( ) / ]d k r x T x (10)


9740

where 1 is a small positive parameter of the controller,

1 0, 1 0d , 1dLr I , and 1 0T . The control law (10)

can be expressed in terms of transfer functions, that is the structure of the PI controller with an additional low-pass filtering (PIF controller) given by

11 1 1 1

1 1 1 1

1( ) [ ( ) ( )] ( )( )

ks r s x s x ss d sT

.

Consider the average closed-loop system given by (9)-(10). Denote 1 2 1, .The replacement of 1x in (10) by the right member of (9) yields the average closed-loop system in the form

1

2

1 1 2

1 11 2 1 1 1

1

2 21

2 1 11

2 21 2

xx

x

r x

E xL Lx x x

RC C

k d k

C

E xL T

xL

(11)

Since 1 is the small parameter, the above equations (11) are the singularly perturbed differential equations. If 1 0 , then fast and slow modes are artificially forced in the closed-loop system, where the time-scale separation between these modes depends on the parameter 1 . From (11), the average fast-motion subsystem (FMS)

2 21

1 1 2

1 11 22 1 1 1

1

r xE x xL L

k d kT

(12)

results where 1x and 2x are treated as the frozen variables during the transients in (12). Remark 1: The stability of FMS transients is provided by selection of the gain 1k such that the condition

21 0k E x

L

holds given that 1 0 and 1 0d .

Assume that the control law parameters 1k , 1 , and 1d have been selected such that the FMS (12) is stable as well as time-scale decomposition is maintained in the closed-loop system (11). Take, for example,

1 ( )/ dCEL Vk (13)

then the FMS (12 ) characteristic polynomial is given by 2 2

1 1 1 1s d s (14)

when 2d

Cx V . Then, letting 1 0 in (12), we obtain the steady state (more precisely, quasi-steady state) of the FMS (12), where 1 1

id (that is inverse dynamics solution) and

21

1 1

2 1

id r x xT

LE x L

. (15)

Substitution of 1 1id into (9) yields the average slow-

motion subsystem (SMS) given by

2 1 1

1 11

1

2

2

1 12

1( )x x x L

r xxT

r xxT

xRC C C E x L

(16)

where the average behavior of 1x , that is the current through the inductor, is prescribed by the stable reference equation of the form 1 1 1 1( ) /x r x T and by that the requirement (3) is maintained. Remark 2: The parameter 1T is selected in accordance with the desired settling time SMSt for the current 1x such that 1 / 3SMST t . The time-scale decomposition is maintained in the system (11) by selection of the parameter 1 such that the condition 1 1 1/T holds where, for example, 1 10 . The desired damping of the FMS transients is provided by selection of the parameter 1d , for example, 1 3d .

The steady-state for the current 1x yields 1d

L Lx I I , then the system (16) degenerates into the following one:

21

22 1 1,

( )x E r

RC Cx x r cons

E xt

(17)

The system (17) has the unique asymptotically stable positive equilibrium point 2

sx given by

12

41 12

s r RExE

.

By linearization of (17) at the equilibrium point 2sx , we

obtain intz a z where int 0a and

1int 2

1

41

1 1 4 /

raRC CE Rr E

.

3.3 Implementation of PIF Current Controller

In order to practical implementation, the discussed PIF controller (10) can be rewritten in the form given by

1 1 1 0 1 0 1x x c r (18) where

1 1 1 11 1 0 02 2

1 1 10, , ,d k k a c

.

Then, from (18), we may get the equations of the discussed PIF controller in the state-space form without an ideal differentiation of 1x or 1r , that is

1 1 1 2 1 1 2 0 1 0 1 1, ,u u u x u x c r u . (19) In order to avoid excitation of transients caused by the initial condition mismatching between (9) and (19), the following expressions

1 2 1 1 1 1(0) (0), (0) (0) (0)u u u x (20)


9741

can be used for initial conditions selection of the controller (19) where 1 1(0) (0)x r .

3.4 High-Frequency Chattering Attenuation

The maximum impact of the pulse-width modulation on the amplitude of the high-frequency chattering in the behavior of the current 1x can be estimated by

1

22( )( )x ss

E xAL

(21)

when 0.5 where 2 /s sT . Accordingly, the maximum impact of the pulse-width modulation on the amplitude of the high-frequency chattering in the behavior of the duty ratio function can be estimated by

1

12

| |( ) ( )s x ss

kA A

(22)

Remark 3: The sampling period sT of the pulse-width modulator is selected by taking into account such additional requirements as

1 1 2( ) , ( )x s sA A (23)

where 1 and 2 are the amplitude of the admissible chattering in the behavior of the current 1x and the duty ratio function , 1 0 , and 2 0 .

4. OUTER VOLTAGE CONTROLLER DESIGN

By selection of 1T in (10), let us provide that the transients of the inner control loop for the current 1x are much faster in compare with the transients of the outer control loop for the output voltage 2 C outx V V . Then, assume that

11d

L Lx I I r and, by following to the discussed two-time-scale design methodology, the outer controller can be designed based on the model of the degenerated system (17), where 1r is treated as the new control variable as well as 2x is the output variable as shown on Fig.2. Consider the outer voltage controller ( 2C ) of the same structure as the continuous-time PIF controller given by (10), that is

22 1 2 2 1 2 2 2 2 2[( ) / ]r d r k r x T x (24)

where 2 is a small positive parameter of the controller,

2 0 , and 2 0T . Hence, the closed-loop system of the voltage control loop is given by (17) and (24). The replacement of 2x in (24) by the right member of (17) yields

2

2 2 22 1 2 2 1

2

21

21

22

2( )

( )

x

r xr d r

x E rRC C E x

x E rRC C x

kT E

(25)

Denote 1 1 2 2 1,u r u r , then the system (25) may be rewritten as the system of the singularly perturbed differential equations, that is

2

2 1 2

2 22 2 2

21

2

2

22

21 2 2

( )

( )

x

u u

r x

x E uRC C E x

xE uC E

u k d u kx CT R

(26)

From (26), similar to the above, the standard technique for two-time-scale motions analysis yields the FMS given by

21

2 1 2

2 2 22 2 2 2

22

2( )E xu

u u

k r xu d u kTC E x RC

(27)

where 2x is treated as the frozen variable during the transients in (27), and the SMS given by 2 2 2 2( ) /x r x T . Note, the desired settling time for the voltage 2x is provided by selection of the controller parameter 2T . In order to provided the FMS (27) stability, take, for example,

2 ) /( dCE Vk C E (28)

then the FMS (27) characteristic polynomial is given by 2 2

2 2 2 1s d s (29)

when 2d

Cx V . Remark 4: The time-scale decomposition is maintained in the system (26) by selection of the parameter 2 such that the condition 2 2 2/T holds where, for example, 2 10 . The desired damping of the FMS transients in (27) is provided by selection of the parameter 2d , for example,

2 3d . Remark 5: The time-scale decomposition between control loop of the current LI and control loop of the voltage CV is maintained by selection of the parameters 1T and 2T such that the condition 1 2 0/T T holds where, for example,

0 10 . Similar to (19) and (20), we may get the state-space form of

(24), that is 1 1 1 2 1 2 2 0 2 0 2 1 1, ,z z z x z x c r r z (30)

where the initial conditions of the controller (30) are defined by the following expressions:

1 1 2 1 1 1 2(0) (0), (0) (0) (0)z x z z x (31)

5. SIMULATION RESULTS

Let the converter parameters are as the following ones: 15 ,E V 0.02 ,L H 180R . Take 2 (0) d

Cx V and

2 (0) 50x V , then the steady-state of (9) yields

2 2 21

2

(0)[ (0)] (0)(0) , (0)(0)

x E x xxRE E x

.


9742

In accordance with (13) and (28) we get 41 3.077 10k

and 2 0.0043k . Take 1 0.01T s , 2 0.1T s , 1 3d , and

2 3d . Accordingly, we get 1 0.001s , 2 0.01s . The sampling period of the pulse-width modulation is selected as

45 10sT s, accordingly, 41.2566 10s rad/s. Hence, by (21) and (22),

1( ) 0.1646x sA and ( ) 0.004sA

given that 2 50dCx V V . The initial conditions of the

controllers (19) and (30) are defined by (20) and (31), respectively. Simulation results for the system (1), (4) controlled by (19), (30) are displayed in Fig.3-10. The simulation results confirm the analytical calculations.

Fig. 3 Simulation results: Plot of 1 Lx I (A).

Fig. 4 Simulation results: Plot of 2 C outx V V (V).

Fig. 5 Simulation results: Plot of 1r (A).

Fig. 6 Simulation results: Plot of .

Fig. 7 Simulation results: Plot of R ( ).

Fig. 8 Simulation results: Plot of E (V).

Fig. 9 Simulation results: Plot of LV (V).


9743

Fig. 10 Simulation results: Plot of ( )u t where [0.4,0.41]t s.

6. CONCLUSIONS

The discussed PIF controller with the small parameter (high gain) in feedback produces slow-fast decomposition in the closed-loop system. It has been shown that if a sufficient time-scale separation between the fast and slow modes in the closed-loop system and stability of FMS transients are provided by selection of controller parameters, then SMS has the form of the prescribed reference equation, thus the desired transient performance indices of the current ( )LI t and the voltage ( )CV t in the closed-loop system are maintained.

The presented method of switching regulator design allows us to obtain the desired transient performances for buck-boost DC-DC converter under uncertainty in the model description and in the presence of load disturbances and voltage source variations.

REFERENCES

Bryant, B. and M.K. Kazimierezuk (2002). Derivation of the buck-boost PWM DC-DC converter circuit topology. Proc. of the IEEE Int. Symp. on Circuits and Systems, vol. 5, pp.V841-V844.

Cunha, F. B. and D. J. Pagano (2002). Limitations in the control of a DC-DC boost converter. Proc. of 15th IFAC World Congress, Barcelona, Spain.

Filippov, A.F. (1964). Differential equations with discontinuous right hand sides. Am. Math. Soc. Transl., vol. 42, pp.199-231.

Gelig, A.Kh. and A. N. Churilov (1996). Pulse-Modulated Systems: a Review of Mathematical Approaches. Functional-Differential Equations, vol. 3, no. 3-4, pp.267-320.

Holtz, J. (1992). Pulsewidth modilation: a survey. IEEE Trans. Idustrial Electronics, vol. IE-39, no.5 pp.410-420.

Hoppensteadt, F. C. (1966). Singular perturbations on the infinite time interval. Trans. of the American Mathematical Society, vol. 123, pp.521-535.

Maksimovic, D. and S. Cuk (1991). Switching converters with wide DC conversion range. IEEE Trans. Power Electronics, vol. PE-16, no.1, pp.151-157.

Malesani, L. and P. Tomasin (1993). PWM current control techniques of voltage source converters - a survey. Proc. of the IECON'93 Int. Conf. on Industrial Electronics, Control, and Instrumentation, vol. 2, pp. 670-675.

Mohan, N., T.M. Undeland, and W.P. Robbins (1995). Power Electronics: Converters, Applications and Design, Wiley.

Kassakian, J.G., M.F. Schlecht, and G.C. Verghese (1991). Principles of Power Electronics. Addison-Wesley. Massachusetts.

Kokotovic, P.V., H. K. Khalil, and J. O'Reilly (1999). Singular Perturbation Methods in Control: Analysis and Design, Philadelphia, PA: SIAM.

Kugi, A. and K. Schlacher (1999). Nonlinear controller design for DC-to-DC power converter. IEEE Trans. on Contr. Syst. Tech., vol.7, no.2, March, pp.230-237.

Leung, F.H.F., P.K.S. Tam, and C.K. Li (1993). An improved LQR-based controller for switching DC-DC converters. IEEE Trans. on Ind. Elect., vol. 40, no.5, Oct., pp.521-528.

Naidu, D. S. (2002). Singular Perturbations and Time Scales in Control Theory and Applications: An Overview. Dynamics of Continuous, Discrete & Impulsive Systems (DCDIS), Series B: Applications & Algorithms, vol. 9, no. 2, pp. 233-278.

Shtessel, Y.B., A.S.I. Zinober, and I.A. Shkolnikov (2002). Boost and buck-boost power converters control via sliding modes using dynamic sliding manifold. Proc. of the 41st IEEE Conf. on Decision and Control, vol. 3, pp.2456-2461.

Shtessel, Y., O. Raznopolov, and L. Ozerov (1998). Control of multiple modular DC/DC power converters in conventional and dynamic sliding surfaces. IEEE Transactions on Circuits and Systems, vol. 45, no.10, pp.1091-1101.

Sira-Ramirez, H. (1989). A geometric approach to pulse-width-modulated control in nonlinear dynamical systems, IEEE Trans. Automatic Control, Vol. 34, no.2, pp.184-187.

Sira-Ramirez, H. and P. Lischinsky-Arenas (1990). Dynamical discontinuous feedback control of nonlinear systems. IEEE Trans. Automatic Control, Vol. 35, no.12, pp. 1373-1378.

Sira-Ramirez, H., R. Ortega, and M. Garcia-Esteban (1998). Adaptive Passivity Based Control of Average DC-to-DC Power Converter Models. Int. J. of Adaptive Control and Signal Processing, vol. 12, pp. 63-80.

Sira-Ramirez, H. and M. Rios-Bolivar (1994). Sliding mode control of DC/DC power converters via extended linearization. IEEE Transactions on Circuits and Systems. Vol. 41, no.10, pp. 652-661.

Tikhonov, A. N. (1948). On the dependence of the solutions of differential equations on a small parameter. Mathematical Sb. Moscow, vol.22, pp.193-204.

Tikhonov, A. N. (1952). Systems of differential equations containing a small parameter multiplying the derivative. Mathematical Sb. Moscow, vol. 31, no. 3, pp. 575-586.

Xu, Jianping (1991). An analytical technique for the analysis of switching DC-DC converters. Proc. of the IEEE Int. Symp. on Circuits and Systems, vol. 2, pp.1212-1215.

Yurkevich, V. D. (2004). Design of nonlinear control systems with the highest derivative in feedback. - World Scientific.


9744

pwm controller design based on singular perturbation technique: a case study of buck-boost dc-dc...

Documents