Cascaded Control System Design for a Cuk Converter via Singular Perturbation Approach

Aksenov E.A., Yurkevich V.D. Cascaded Control System Design for a Cuk Converter via Singular Perturbation Approach. Proc. of 16th Annual International Conference of Young Specialists on Micro/Nanotechnologies and Electron Devices (EDM 2015), 29 June - 3 July, 2015, pp. 534 -541. (http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7184600)

Cascaded Control System Design for a Cuk Converter via Singular Perturbation Approach Efim A.Aksenov, Valery D.Yurkevich, Member, IEEE Automation Department, Novosibirsk State Technical University, Novosibirsk, Russia Abstract –The problem of cascaded control system design for a Cuk converter is discussed where control system consists of three feedback loops. In the first one the input current control is provided by means of pulse-width modulated (PWM) proportional-integral (PI) controller. In the second one the voltage across capacitor for transferring energy is regulated by PI controller. Finally, in the third one the voltage across capacitor for storing energy is maintained by integral (I) controller. The both PI controllers are designed based on singular perturbation technique (time-scale separation technique) such that multi-time-scale motions are artificially induced in the closed-loop system. Numerical simulations are included in order to show the efficiency of the proposed design methodology.

low-pass filtering are used. In the last outer loop the integral controller is applied. Finally, numerical simulations are included. II. CUK CONVERTER MODEL Let us consider the simplified circuit of the DC-DC buck-boost converter shown in Fig.1, which is called as the Cuk converter [13-15].

Index Terms –DC-DC converters, pulse-width modulation, PI controller, singular perturbation method.

I. INTRODUCTION

C

UK CONVERTORS ARE WIDLY USED in various industrial applications, for example, such as photovoltaic cells [1], electrical vehicles [2], telecommunication systems [3], and motor drive control systems [4]. Complicated dynamics and slow-scale oscillations are inherent properties of Cuk converters. Therefore, the problem of controller design for Cuk converters gives an appropriate example of a high nonlinear dynamical system and allows demonstrating efficiency of different controller design methodology. Problem of controller design for Cuk converters is treated in numerous research works, for example, the application of the time-invariant multi-frequency modeling of PWM Cuk converter is shown in [5], the peak current control Cuk converter is discussed in [6], the state-space averaging method to design an output feedback neural controller for the Cuk converter in sliding-mode operation is used in [7], PI and sliding mode control of the Cuk converter is presented in [8], the sliding mode control approach for the Cuk converter is presented in [9]. Comprehensive analysis of different control techniques for power converters, including the Cuk converter, such as the sliding mode control, the Hamiltonian system approach, the feedback linearization, and passivity based control has been done in [10]. This paper is a continuation and further inquiry of the approach discussed in [11,12] for the Cuk converter where a distinctive feature is that two-time-scale motions are artificially forced in the closed-loop system. Stability conditions imposed on the fast and slow modes, and a sufficiently large mode separation rate, can ensure that the full-order closed-loop system achieves desired properties: the output transient performances are as desired, and they are insensitive to parameter variations and external disturbances. The paper is organized as follows. At the beginning, the Cuk converter mathematical model, the control problem statement, cascaded control system structure, and averaged model are presented. Then two control loops are designed based on singular perturbation technique where PI controllers with an additional

Fig. 1. The simplified Cuk converter circuit

The Cuk-converter contains two inductors L1 and L2 , two capacitors C1 and C2 , a load resistor R , an input voltage source E , and two switches S1 and S2 . These switches can be implemented based on, for example, MOSFET switch (VT) and diode (VD) which are treated in the paper as the ideal switches closing the circuit in opposite phase (Fig.2).

Fig. 2. The Cuk converter circuit with ideal switches

The Cuk converter has two operating modes assigned by switching function u . The circuit is in charging mode if the switch S1 is turned on and the switch S2 is turned off as shown in Fig.3(a), then u  1 . Accordingly, the circuit is in discharging mode if the switch S1 is turned off and the switch S2 is turned on as shown in Fig.3(b), then u  0 .

Fig. 3. Charging (a) and discharging (b) modes of the Cuk converter circuit

In accordance with the assigned above operating modes, the dynamical behavior of the converter is given by the following equations [8,10]: E 1   I L1  L  L U C1 (1  u ) 1 1   1 1 U C1  U C1u  I L1 (1  u ) C1 C1  (1)   I   1 U u  1 U C1 C2  L2 L2 L2  U  1 I  1 U  C 2 C2 L 2 RC2 C 2 III. CONTROL PROBLEM STATEMENT The control system for the Cuk converter is being designed so that to maintain the desired output voltage in the presence of varying load R and varying voltage source E , that is lim U C 2 (t )  U Cd 2 (2) t 

where U Cd 2 is the reference value (reference input) of voltage drop U C 2 across the capacitor C2 . Moreover, the controlled transients of U C 2 should have desired transient performance indices. These performance indices should be insensitive to parameter variations of the converter and external disturbances represented by the varying value of the resistor R and the value of the voltage source E . In this paper a cascaded control law structure is used as shown in Fig.4 where, firstly, an inner controller ( CIL1 ) of the current I L1 through the inductor L1 is designed such that the condition

lim I L1 (t )  I Ld1 t 

(3)

holds where I Ld1 is the reference value of the current through the inductor L1 . Secondly, an outer controller ( CUC1 ) is constructed in order to meet the requirement lim U C1 (t )  U Cd1 (4) t 

by the tuning of the reference value I Ld1 where U Cd1 is the reference value of voltage drop U C1 across the capacitor C1 .. Thirdly, the next outer controller ( CUC 2 ) is constructed in order to meet the requirement (2) by the tuning of the reference value U Cd1 .

Fig.4 Block diagram of the cascaded control system with the pulse-width modulator in control loop.

A discontinuous control strategy is provided by the pulse-width modulator (PWM) shown in Fig.4 where the input signal of PWM is the duty ratio function d which takes values in the interval [0,1] . The duty ratio function d is the output signal of the controller C IL1 . The PWM output signal is defined as the switching function u (t ) given by

1 for t  t  t  d (t )Ts u (5)  0 for t  d (t )Ts  t  t  Ts where Ts is the sampling period of the pulse-width modulation,

d (t ) is the value of the duty ratio function when t  t , t   Ts , and   0,1, 2,  . IV. AVERAGED MODEL OF THE CUK CONVERTER The Cuk converter operation consists of a periodical sequence of the two modes which are defined by the switching functions u (t ) in accordance with (5).. Assumption 1: The pulse-width modulator is not saturated, that is the following inequality 0  d  1 holds. Assumption 2: The sampling period Ts is assumed to be sufficiently small in compare with time constants associated with the dynamics of the converter. From Assumptions 1 and 2, by following to the Filippov's approach [16], the geometric approach to PWM control [17,18], and by taking into account the definition of the switching function u (t ) , the response of discontinuously controlled system given by (1) coincides with Filippov's average model E 1   I L1  L  L U C1 (1  d ) 1 1   1 1 U C1  U C1d  I L1 (1  d ) C C  1 1 (6)  1  I   U d  1 U C1 C2  L2 L2 L2  U  1 I  1 U  C 2 C2 L 2 RC2 C 2 where 0  d  1 . The average model (6) is used below in order to control system design. V. EQUILIBRIUM STATE OF THE CUK CONVERTER From the average model (6), by taking IL1  0 , U C1  0 , IL 2  0 , and U C 2  0 , the following equilibrium state equations can be easily derived:

 Ed 2 I   L1 R(1  d ) 2   E U C1  1 d   I   Ed  L2 R(1  d )  U   Ed  C 2 1 d

(7)

It can be verified, for example, based on linearized approximation of (6) at the equilibrium state, that the equilibrium state defined by (7) of the system (6) is stable. From (7), the plots for components of the equilibrium state result as the functions of the duty cycle d (Fig.5).

E 1 IL1   U C1 (1  d1 ) L1 L1 1 1 U C1  U C1d1  I L1 (1  d1 ) C1 C1 1 1 IL 2   U C1d1  U C 2 L2 L2 U C 2

(11)

1 1  I L2  UC 2 C2 RC2

 I 1d1  d 2  I 1d2  

Fig. 5. Plots for components of the equilibrium state of the Cuk converter as the functions of d

VI. CONTROL SYSTEM DESIGN

maintain the condition (3). Let us consider the controller CIL1 in the form of the following differential equation:  I21 d (2)  d I 1  I 1 d (1)  k I 1[( I Ld1  I L1 ) / TI 1  I L(1)1 ] (8) where  I 1 is a small positive parameter of the controller,  I 1  0, d I 1  0 , and TI 1  0 . The control law (8) can be expressed in terms of the Laplace transform that is the structure of the PI controller with an additional low-pass filtering given by  1  kI 1 d ( s)  [ I Ld1 (s )  I L1 ( s)]  I L1 ( s)  .   I 1 (  I 1s  d I 1 )  sTI 1  Denote d1  d , d 2   I 1 d (1) , then the controller (8) can be rewritten in the form  I 1d1(1)  d 2 (9)  I 1d 2(1)  d I 1d 2  k I 1[( I Ld1  I L1 ) / TI 1  I L(1)1 ] Finally, the closed-loop system analysis can be provided based on the consideration of the system (6) with controller (9), that is E 1 IL1   U C1 (1  d1 ) L1 L1 1 1 U C1  U C1d1  I L1 (1  d1 ) C1 C1

(10)

1 1 U C 2  I L2  UC 2 C2 RC2  d  d I1 1

Since  I 1 is the small parameter, the above equations (11) are the singularly perturbed differential equations [19-23]. Hence, fast and slow modes are artificially forced in the closed-loop system (11) as  I 1  0 . The degree of time-scale separation between these modes depends on the parameter  I 1 . From (11), the fast-motion subsystem (FMS)  I 1d1  d 2

 I 1d2  

A. Controller of the Current I L1 The controller CIL1 of the current I L1 is designed so that to

1 1 IL 2   U C1d1  U C 2 L2 L2

Id  I  k I 1U C1 E 1 d1  d I 1d 2  k I 1  L1 L1   U C1  L1 L1 L1  TI 1 

2

 I 1d2  d I 1d 2  k I 1[( I Ld1  I L1 ) / TI 1  IL1 ] The replacement of I L(1)1 in the last equation of (10) by the right member of the first equation of (10) yields the closed-loop system in the form

Id  I  k I 1U C1 E 1 d1  d I 1d 2  k I 1  L1 L1   U C1  L1 L1 L1  TI 1 

(12)

results where I L1 ,U C1 , I Ld1 and E are treated as the frozen variables during the transients in (12). Remark 1: The stability of FMS transients of (12) is provided by selection of the gain k I 1 such that the condition k I 1U C1 / L1  0 holds given that  I 1  0 and d I 1  0 . Assume that the control law parameters k I 1 ,  I 1 , and d I 1 have been selected such that time-scale decomposition is maintained in the system (11) as well as the FMS (12) is stable. Take k I 1  L1 / (U C1   ) (13) where  is a positive value, for example,   1 in order to avoid division by zero. Then the characteristic polynomial of the FMS (12) is given by  I21 s 2  d I 1 I 1 s  1 Letting  I 1  0 in (11), we obtain the steady state of the FMS (12), where d1  d1s , that is  L1  I Ld1  I L1 E 1   U C1  .  U C1  TI 1 L1 L1  From (11), the steady state of the FMS (12) yields the following reduced order system: E 1 IL1   U C1 (1  d1s ) L1 L1 d1s 

1 1 U C1  U C1d1s  I L1 (1  d1s ) C1 C1 1 1 IL 2   U C1d1s  U C 2 L2 L2 U C 2

1 1  I L2  UC 2 C2 RC2

0  d 2s 0

Id  I  k I 1U C1 s E 1 d1  d I 1d 2s  k I 1  L1 L1   U C1  L1 L1 L1  TI 1 

(14)

By eliminating d1s and d2s from (14), we obtain the slowmotion subsystem (SMS) equations given by Id  I IL1  L1 L1 TI 1 1 E  E U C1    I L1  IL2  C1U C1  C1 C1U C1  E 1 1 IL 2   U C1  U C 2 L2 L2 L2

(15)

(19)

U 1 z2  dU 1 z2  kU 1[(U Cd1  U C1 ) / TU 1  U C(1)1 ]

B. Controller of the Voltage U C1 In accordance with the discussed multi-time-scale design methodology, we assume that the transients in the inner control loop for the current I L1 are much faster as the transients in the outer control loop for the voltage U C1 . Hence, take the steadystate of the inner control loop, that is I Ld1  I L1 , then from (15), we get the reduced order system given by 1 E  E U C1    I Ld1  IL2  C1U C1  C1 C1U C1  E 1 1 IL 2   U C1  U C 2 L2 L2 L2

(16)

1 1 U C 2  IL2  UC 2 C2 RC2

where I Ld1 is treated as the new control variable. Note that the system (16) has two equilibrium points, where the first one is unstable and defined as

I Ld1 E . R

The second equilibrium point is stable and defined as

Id E U C1  E  RI Ld1 E , U C 2   RI Ld1 E , IL 2   L1 . R The controller CUC1 of the voltage U C1 is designed so that to maintain the condition (4) by the tuning of the reference value I Ld1 . Take the controller CUC1 in the same form as (8), that is d C1

1 1 U C 2  IL2  UC2 C2 RC2

U 1 z1  z 2

1 1 U C 2  IL2  UC 2 C2 RC2 where from the first equation of the SMS (15) it follows that the current I L1 through the inductor L1 exhibits the stable prescribed behavior such that the requirement (3) is maintained.

U C1  E  RI Ld1 E , U C 2  RI Ld1 E , I L 2 

1 E  E U C1    z2  IL2  C C U C U  1 1 C1  1 C1 E 1 1 IL 2   U C1  U C 2 L2 L2 L2

U  U C1  U2 1 ( I Ld1 )(2)  dU 1 U 1 ( I Ld1 )(1)  kU 1   (U C1 )(1)  . (17) T U1   Denote z1  I Ld1 , z2  U 1 I Ld1(1) , then the controller (17) can be rewritten in the form U 1 z1  z2 (18) U 1 z2   dU 1 z2  kU 1[(U Cd1  U C1 ) / TU 1  (U C1 )(1) ] Hence, the outer controller (17) is being designed based on analysis of the reduced closed-loop system given by

The replacement of U C(1)1 in the last equation of (19) by the right member of the first equation of (19) yields the closed-loop system in the form 1 E  E U C1    z1  IL2  C C U C U  1 1 C1  1 C1 E 1 1 IL 2   U C1  U C 2 L2 L2 L2 1 1 U C 2  IL2  UC2 C2 RC2

(20)

U 1 z1  z2 U 1 z2  

kU 1 E z1  dU 1 z2 C1U C1

U d  U C1  1 E    kU 1  C1    I L2   C1 C1U C1    TU 1 Since U 1 is the small positive parameter, the above equations (20) are the singularly perturbed differential equations [19-23] where fast and slow modes are forced in (20) as U 1  0 . From (20), the FMS U 1 z1  z2

U 1 z2  

kU 1 E z1  dU 1 z2 C1U C1

(21)

U d  U C1  1 E    kU 1  C1    I L2   C1 C1U C1    TU 1 results where I L 2 , U C1 , U Cd1 and E are treated as the frozen variables during the transients in (21). Remark 2: The stability of FMS transients of (21) is provided by selection of the gain kU 1 such that the condition kU 1 E / C1U C1  0 holds given that U 1  0 and dU 1  0 .

Assume that the control law parameters kU 1 , U 1 , and dU 1 have been selected such that time-scale decomposition is maintained in the closed-loop system (20) as well as the FMS (21) is stable. Take, for example, kU 1  C1U C1 / E , (22) then the characteristic polynomial of the FMS (21) is given by U2 1 s 2  dU 1U 1 s  1 . Letting U 1  0 in (21), we obtain the steady state of the FMS (21) where z1  z1s and C1U C1  U Cd1  U C1  1 E       I L2  . E  TU 1 C C U  1 1 C1   From (20), the steady state of the FMS (21) yields the following reduced order system: z1s 

1 E  E U C1    z1s  IL2  C C U C U  1 1 C1  1 C1 E 1 1 IL 2   U C1  U C 2 L2 L2 L2 1 1 U C 2  I L2  UC2 C2 RC2 0z



1  kU 2  0 . RC2

(29)

holds, then the system (27) is stable and the requirement (2) is maintained for the steady state of (27).. The system (27) has the stable equilibrium point which is defined as (23)

s 2

I L002  U Cd 2 / R, U C002  U Cd 2 , U Cd1  E  U Cd 2 . Remark 3: The time-scale separation between the transients in the control loops can be maintained by selection of controller parameters in accordance with the following conditions: (30) 0   I 1  TI 1  U 1  TU 1  1 / min Re( si )

k E 0   U 1 z1s  dU 1 z2s C1U C1

i 1, 2 ,3

U d  U C1  1 E    kU 1  C1    IL2   C1 C1U C1    TU 1

By eliminating z1s and z2s from (23), we obtain the SMS equations given by U d  U C1 U C1  C1 TU 1 E 1 1 IL 2   U C1  U C 2 (24) L2 L2 L2 1 1 U C 2  I L2  UC 2 C2 RC2 where from the first equation of the SMS (24) it follows that the voltage drop U C1 on capacitor C1 exhibits the stable prescribed behavior such that the requirement (4) is maintained. The system (24) has one stable equilibrium point which is defined as

U C01  U Cd1 , U C0 2  E  U Cd1 , I L0 2  ( E  U Cd1 ) / R . C. Controller of the Voltage U C 2 The last outer control loop for the voltage U C 2 .is being designed under assumption of the steady state of the transients in the control loop for the voltage U C1 . So, assume that the condition U Cd1  U C1 holds and from (24), we get the reduced order system given by E 1 1 IL 2   U C 2  U Cd1 L2 L2 L2 (25) 1 1  UC 2  I L2  UC 2 C2 RC2

where s1 , s2 , and s3 are the roots of the characteristic polynomial (28). VII. SIMULATION OF CONTROL SYSTEM Let the Cuk converter parameters are as the following ones: E  15 V , L1  0.02 H , L2  0.02 H , C1  0.005 F , C2  0.005 F , R  20  . The sampling period of the pulse-width modulator is selected as Ts  0.0005 s. The gain kU 2 is chosen as kU 2  1 . Then, for the assigned above converter parameters, we get s1  1 and s2,3  5  j100 . In accordance with (13), (22), (29), and (30), the following controller parameters were chosen: TI 1  0.02 s,  I 1  0.00067 s, d I 1  2, TU 1  0.2 s, U 1  0.01 s, dU 1  2. The simulation has been done based on Matlab/Simulink Tools. The response of the current I L1 during simulation of the converter model (1) and (5) with the single current controllers (8) is shown in Fig.6 where I Ld1  10 A . Figs.7 and 8 show the simulation results of the converter model (1) and (5) with the current controller (8) and the voltage controller (17) where U Cd1  90V . Finally, the simulation results of the discussed converter based on the model (1) and (5) with controllers (8), (17) and (26) are displayed in Figs.9-13 where U Cd 2  60V .

where U Cd1 is treated as the new control variable. As far as the system (25) is the linear one and the sake of simplicity, take integral controller given by U Cd1  kU 2 (U Cd 2  U C 2 ) (26) From (25) and (26), we get closed-loop system in the form E 1 1 IL 2   U C 2  U Cd1 L2 L2 L2 1 1 U C 2  I L2  UC 2 (27) C2 RC2 U d  k (U d  U ) C1

U2

C2

C2

where the characteristic polynomial of (27) is given by k 1 2 1 s3  s  s  U2 . RC2 C2 L2 C2 L2 By the Routh–Hurwitz stability criterion, if the condition

(28)

Fig. 6. Simulation results of the system with single inner controller CIL1

Fig. 7. Simulation results of the system with controllers CIL1 and CUC 1

Fig. 10. Simulation results of the system with controllers CIL1 , CUC 1 , and CUC 2

Fig. 8. Simulation results of the system with controllers CIL1 and CUC 1

Fig. 11. Simulation results of the system with controllers CIL1 , CUC 1 , and CUC 2

Fig. 9. Simulation results of the system with controllers CIL1 , CUC 1 , and CUC 2

Fig. 12. Simulation results of the system with controllers CIL1 , CUC 1 , and CUC 2

Fig. 13. Simulation results of the system with controllers CIL1 , CUC 1 , and CUC 2

The external disturbances are represented by step-wise varying values of the resistor R and the voltage source E as shown in Fig.13. From Figs.9-12, it follows that the effective rejection of the external disturbances is provided in the discussed cascaded control systems. VIII. CONCLUSION The advantage of the presented singular perturbation technique of controller design for the discussed Cuk converter is that the desired performance indices of transients are maintained in the presence of varying values of the voltage source and the load current. The other advantage is that analytical expressions for selection of controller parameters are derived in accordance with the imposed performance indices. IX. ACKNOWLEDGEMENT This work was supported by Russian Foundation for Basic Research (RFBR) under grant no.14-08-01004-a.

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Efim A. Aksenov is a student of the Automation Department, Novosibirsk State Technical University. His areas of research are nonlinear control systems, switching controllers for power converters, voltage source inverters.

Valery D. Yurkevich is a professor in the Automation Department, Novosibirsk State Technical University. His areas of research are nonlinear control systems, digital control systems, flight control, distributed parameter control systems, robotics, switching controllers for power converters, singular perturbations in control