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Third-order Sliding Mode Controller for Buck Inverter Fei Xu, Hao Ma, Bin Liu, Liang Dong College of Electrical Engineering, Zhejiang University Email: [email protected] Abstract-A novel third-order sliding mode control strategy for Buck inverter is proposed based on Lie derivative method in this paper. The geometry associated with the sliding mode control is presented for the analysis and design of the control scheme. The inverter is composed of two symmetrical current bi-directional Buck converters controlled by two independent sliding mode controllers. Base on the equivalent circuit of Buck inverter, the design procedures of the third-order sliding mode controller are given in detail. The sliding domain and sliding motion are described by phase trajectory vividly. To simplify the analysis, three-dimensional trajectory transforms into the twodimensional one through the mapping transformation. Finally, experimental results are included to validate the proposed control strategy can ensure the system with improved tracking precision and well dynamic performances.

I.

INTRODUCTION

Sliding mode controller (SMC) was introduced initially for variable structure systems (VSS) which is well-known for its several advantages such as wide stability range, robustness for disturbance of system, and good dynamic response [1-2]. Characterized by switching, power electronics systems are inherently variable structured. Consequently, it is appropriate to apply SMC to power electronics systems [3-4]. As an alternative to PWM control strategies in dc-dc converters, the SMC has been widely implemented, which makes these systems highly robust to perturbations and greatly dynamic response [5-9]. For this reason, tracking control schemes based on SMC have also been applied to the dc-ac inverters [10-19]. And a lot of improvements have been made based on the traditional SMC. In [12], an autonomous (time independent) sliding surface is adopted to the SMC, that brings the system to a zeroth-order dynamics in sliding domain. In order to get a fix switching frequency, a hybrid controller using both sliding mode control and peak current control is presented [13]. In [14], the SMC is analyzed and designed in the frequency domain, and Tsypkin’s method together with the describing function is adopted to provide good estimation of the switching frequency. A SMC is used to a photovoltaic (PV) generation system, which allows to use much smaller, more reliable non-electrolytic capacitors and achieves stable closed loop performance [15-16]. The implementation of SMC is given in new voltage source inverters based on symmetrically bidirectional Buck converters. The equivalent control method and inverse function are used to analysis the reaching and existence conditions, and some design steps and equations are proposed

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[18]. The sliding motion and domain are researched by dividing the sliding surface into reference sliding surface and state variables surface, then the steady-state error of output voltage and the influence of ESR are analyzed by applying the new sliding surface [19]. So far, these traditional sliding mode control strategies are almost designed and analyzed based on the single input second-order controllers. And the existence condition must be obeyed, which is derived from Lyapunov’s second method to determine asymptotic stability to ensure that the sliding trajectory is maintained on the sliding surface. Infinite switching frequency should be achieved at ideal operation. This challenges the feasibility of applying SMC to power converters because of the excessive switching losses, core losses, and electromagnetic interference (EMI) noise issues. For limiting the frequency of SMC into a practical range, the previous proposed SMC is hysteresis modulation (HM) [18-20]. This method can reduce the chatting greatly, however, at the cost of a finite steady-state error. Therefore, integral action is introduced into SMC to achieve the zero steady-state error [21-23]. In this paper, a third-order sliding mode control strategy based on the Lie derivative method is presented, which is used to the Buck inverter. The geometric approach is adopted, associated with the sliding mode control, to analyze and design the control scheme which is easy to understand. The proposed third-order SMC is discussed in detail which can guarantee the system with improved steady-state precision and well dynamic performance. The sliding domain and sliding motion are described by phase trajectory vividly. Threedimensional trajectory transforms into the two-dimensional one through mapping method to simplify the analysis. To verify the proposed control strategy, experimental results show the good performances of this proposed control strategy. II. ANALYSIS OF SINGLE INPUT N-ORDER SMC A.

The Traditional Analysis of SMC Considering the dc-ac inverter as a single input n-order system, its state space equation is as follows, x = f ( x ) + g ( x ) ⋅ u (1)

where x ∈ R n , f ( x ) ∈ R n , g ( x ) ∈ R n , u ∈ {0,1} . While f ( x ) and g ( x ) are smooth vector fields defined on R n , and x is a n-dimensional vector made up of the error dynamics of system state variables.



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vO

L

D1

R

R L1

v1

+

L2

S3

S1

C1

C2 D1

S2

Vin

v2

+ C

D2

S2

D4

+

vC

v2

− iC

D3 Vin

D2

+

S1

S4

S1 S2

k2 x 2 S(x)

iC1

S1

S3

iC2

v1

S2

S4

v2

k1 x 1

k3 x 3

k2

−

d dt

k1

x1

Vref

k 3 ∫ dt

Fig. 1. The buck inverter with sliding mode control

Fig. 2. The equivalent circuit of Buck inverter

In general design, the sliding mode surface is composed of a linear combination of the error dynamics of system state variables, which is given as S ( x) = K T x = 0 (2) where K = [k1 , k2 , …，kn ]T ∈ R n denotes the sliding surface coefficient. And the control function is given as ⎧0 when S < 0 1 + sgn( S ) u=⎨ or u = (3) 1 when S > 0 2 ⎩ In order to force the system moving on the sliding mode surface, the existence condition must be obeyed. S ( x ) S ( x ) < 0 (4) Substituting (1), (2) and (3) into (4), the inequality becomes T ⎪⎧ S ( x ) = K [ f ( x ) + g ( x )] < 0 when S ( x ) > 0 (5) ⎨ T when S ( x ) < 0 ⎪⎩ S ( x ) = K f ( x ) > 0 where K should be appropriately selected to obtain the stability, robustness and good dynamic response.

In order to maintain the sliding motion on the sliding surface, the equations bellow must be achieved. S = 0, S = 0 ⇒ S = 0, L f + ueq g S = 0 (9)

B. The Analysis Based on Lie Derivative In this paper, the Lie derivative is introduced to design and analyze the sliding mode controller. For the system of (1), the Lie derivative is expressed as follows, ⎡ ∂S ∂S ∂S ⎤ , ," , (6) L f S = 〈∇S , f 〉 = ⎢ ⎥f x x xn ⎦ ∂ ∂ ∂ 2 ⎣ 1 It is easy to see that the Lie derivative is the directional derivative of the scalar function S with respect to the vector field f . As a result of the control strategy (3), the existence and reaching condition (4) can be described as follows, ⎧⎪ L f + g S = 〈∇S , f + g 〉 < 0 when S > 0 SS < 0 ⇒ ⎨ (7) when S < 0 ⎪⎩ L f S = 〈∇S , f 〉 > 0 Then, it is obligatory to gratify the necessary condition which is also called the transversality condition of the vector field g on the sliding surface. (8) Lg S = 〈∇S , g〉 < 0

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Thus it can be seen ueq = −

Lf S 〈∇S , f 〉 =− 〈∇S , g 〉 Lg S

(10)

Combining (7), (8) and (10), a necessary and sufficient condition for the local existence of a sliding mode on the sliding surface is obtained such that 0 < ueq < 1 (11) III. DESIGN OF THIRD-ORDER SMC FOR BUCK INVERTER The basic structure of the Buck dc-ac inverter is shown in Fig. 1. The inverter is configured on two symmetrically current bi-directional buck converters. These two converters produce a dc-biased sine wave output respectively. The dcbias appeared at each end of the load have same values and the sine wave of each converter is 180° out of phase with each other. Then, the sinusoidal output on the load is obtained. It is assumed that all the components are ideal and the inverter operates in continuous conduction mode. According to the analysis above, the output voltage vO can be written as ⎧v1 = vm sinωt + vbias ⎪ (12) ⎨v2 = vm sin(ωt + π ) + vbias ⎪v = v − v = 2v sinωt 1 2 m ⎩ O To obtain good gain characteristic, it is better that dc-bias is equal to half value of input voltage, namely vbias = Vin 2 [18]. A. Dynamic Modeling of Buck Inverter The operation of the Buck inverter is better understood through the equivalent circuit shown in Fig. 2. Where L = L1 , C = C1 , vc = v1 , and the right-side Buck converter output voltage v2 is regarded as ideal voltage source.



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The state space modeling equation of the equivalent circuit can be obtained as ⎡ v ⎤ ⎡ 0 1 0 ⎤ ⎡ vc ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎢ c ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ v ⎥ ⎥ dv V 1 d ⎢ dvc ⎥ ⎢ 1 ⎥ ⎢ c ⎥ + ⎢ in ⎥ u + ⎢ 2 ⎥ 0 = − − ⎢ RC ⎥ dt ⎢ dt ⎥ ⎢ LC RC ⎥ ⎢ dt ⎥ ⎢ LC ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ 0 0 ⎥⎦ ⎢ vc dt ⎥ ⎣⎢ 0 ⎦⎥ ⎢ ∫ vc dt ⎥ ⎣ 1 ⎣ ⎦ ⎣ ⎦ ⎣∫ ⎦ (13) where variable u is the switches status, taking values in the discrete set {1,0}. A new state variables matrix of the system is defined which consists of the output voltage error x1 , the output voltage error dynamics x2 , and the integral of output voltage error x3 . Then, a new state space equation of the equivalent circuit can be obtained as x = Ax + Bu + D (14) ⎡ Vref − vc ⎤ ⎥ ⎡ x1 ⎤ ⎢ ⎢ d (Vref − vc ) ⎥ ⎢ ⎥ where x = ⎢ x2 ⎥ = ⎢ ⎥, dt ⎥ ⎢⎣ x3 ⎥⎦ ⎢ ⎢ (Vref − vc ) dt ⎥ ⎣∫ ⎦ 1 0⎤ ⎡ 0 ⎢ 1 ⎥ 1 A = ⎢− − 0⎥ , ⎢ LC RC ⎥ ⎢ 1 ⎥ 0 0 ⎣ ⎦ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎢ V ⎥ in ⎥ , D = ⎢ F ( t )⎥ , B = ⎢− ⎢ ⎥ ⎢ LC ⎥ ⎢ ⎥⎦ 0 ⎢ 0 ⎥ ⎣ ⎣ ⎦ Vref Vref v F (t ) = + + Vref − 2 . LC RC RC And from (12), we can get v2 = −Vref . Therefore, Vref 2Vref F (t ) = + + Vref . LC RC Then we define that f ( x ) = Ax + D , g ( x ) = B . B. Analysis of Third-Order SMC Based on Lie Derivative In this paper, a third-order SMC is presented, which is joined with the integral of the output voltage dynamic error. The third-order SMC eliminates the steady-state error essentially. The sliding surface function is selected as S ( x ) = k1 x1 + k2 x2 + k3 x3 (15) = k (V − v ) + k (V − v ) + k (V − v ) dt 1

ref

c

2

ref

c

3

∫

ref

c

where k1 > 0 , k2 > 0 and k3 > 0 . Substituting (15) into (8), we will have ⎡ ∂S ∂S ∂S ⎤ k2Vin , , Lg S = 〈∇S , g 〉 = ⎢ <0 ⎥B =− ∂ x ∂ x ∂ x LC 2 3⎦ ⎣ 1

which is satisfied with the transversality condition. Then substituting (15) into (10), we will have

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(16)

S new = 0 l1 = 0

S new = −k3 x3 (t1 )

x2

l2 = 0

S new = − k3 x3 (t3 )

S new = − k3 x3 (t2 )

Vref − Vin 1 − k3 LC

0

Vref

x1

1 − k3 LC

Fig. 3. The sliding domain and sliding motion of third-order SMC

ueq = −

Lf S Lg S

=

⎤ ⎛ k1 1 ⎞ 1 ⎞ LC ⎡⎛ k3 ⎢⎜ − ⎟ x1 + ⎜ − ⎟ x2 + F ( t ) ⎥ (17) Vin ⎣⎢⎝ k2 LC ⎠ ⎝ k2 RC ⎠ ⎦⎥

when k1 k2 = 1 RC , x1 = Vref − vc , it will yield ueq =

⎤ k LC ⎡ vc 2Vref  + + Vref + 3 (Vref − vc ) ⎥ ⎢ Vin ⎣⎢ LC RC k2 ⎦⎥

(18)

Obviously, Vref  vc , Vref  vc and Vref ≈ vc when the system is in steady state, it will yield that 0 < ueq ≈ vc Vin < 1 . Hence, it is clear that the third-order SMC satisfies the necessary and sufficient condition. C. The Sliding Domain and Sliding Motion The equation (7) forces the state trajectories moving on the sliding surface, and the robustness of inverter will be guaranteed. Then substituting (14) and (15) into (7), the sliding domain is obtained as ⎛k ⎛ k1 Vin 1 ⎞ 1 ⎞ (19) 0≤⎜ 3 − ⎟ x1 + ⎜ − ⎟ x2 + F ( t ) ≤ LC ⎝ k2 LC ⎠ ⎝ k2 RC ⎠ The sliding domain is a three-dimensional space which is between two parallel planes. Convenient for analysis, the sliding domain is mapped into the two-dimensional plane. The relationship among the sliding surface, sliding motion and sliding domain is shown in Fig. 3. The boundaries of sliding domain are two defined lines as follows. ⎧ ⎛ k3 ⎛ k1 1 ⎞ 1 ⎞ ⎪l1: ⎜ − ⎟ x1 + ⎜ − ⎟ x2 + F ( t ) = 0 k LC k RC ⎪ ⎝ 2 ⎠ ⎝ 2 ⎠ (20) ⎨ ⎛ k3 ⎛ k1 Vin 1 ⎞ 1 ⎞ ⎪ ⎪l2 : ⎜ k − LC ⎟ x1 + ⎜ k − RC ⎟ x2 + F ( t ) = LC ⎝ 2 ⎠ ⎝ 2 ⎠ ⎩ When the system enters into the sliding domain, its equivalent trajectory can be ideally described as S ( x ) = 0 . Then using mapping method, a new sliding surface function can be obtained as



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Snew = k1 x1 + k2 x2 = −k3 x3 = −k3 ∫ x1dt

(21)

Let k2 = 1 and k1 = 1 RC , the sliding domain and sliding motion can be analyzed into three aspects. a) k3 = 1 LC In this situation, the whole plane is the sliding domain. b) k3 < 1 LC The boundaries of sliding domain have to cross the points ⎛ Vref ⎞ ⎛ Vref − Vin ⎞ , 0 ⎟ and ⎜ , 0 ⎟ respectively. of ⎜ ⎝ 1 − k3 LC ⎠ ⎝ 1 − k3 LC ⎠ The system starts up at the point of (Vref , 0 ) which is in the

sliding domain. After starting up, the system state trajectory tends to the sliding surface. Here, due to x1 > 0 and the effect of −k3 ∫ x1dt , the sliding surface will move left parallel away from the system state trajectory. Then a proper k3 has to be selected to prevent that the system state trajectory can not reach the sliding surface. The system state trajectory reach the sliding surface at the time t1 . The system enters into the sliding motion. The sliding surface will keep moving left parallel until x1 = 0 at the time t2 . Then x1 < 0 , the sliding surface will move right parallel. If the system state trajectory can not reach the steady state at the time t3 when x1 = 0 , the sliding surface will move left parallel again. Finally, the system state trajectory convolutes into the steady state. c) k3 > 1 LC The two lines l1 and l2 interchange the position. And the analysis is as same as the situation of k3 < 1 LC .

For this Buck inverter, we usually define that k2 = 1 and k1 = 1 RC . And the bandwidth of the controller’s response f BW is equal to the transition frequency of system f P , which yields 1 k3 1 f BW = = fP = (25) 2π k2 2π LC Hence, we can get k3 = 1 LC , which is matched the foregoing discussions. IV. IMPLEMENTATION AND EXPERIMENTAL RESULTS A prototype of the Buck DC-AC inverter (Fig. 1) has been built in the laboratory, which is in order to verify the proposed control scheme experimentally. And the rated output signals are v1 = 24 + 16sin100πt , and v2 = 24 − 16sin100πt , so vo = 32sin100πt . The parameters of this system are listed in TABLE I. Fig. 4 shows the experimental results of steady state output voltage and output current waveforms under third-order SMC. These results are explicit sinusoid with very little distortion. Table II shows the steady-state performance of the third-order SMC. The experimental results are obtained when the desired output voltage is set at different amplitudes. It is clear that the third-order SMC has excellent tracking performance in the steady-state operation. The experimental results, for the case where no load suddenly turns to full load under the third-order SMC, are illustrated in Fig. 5. And Fig. 6 shows the opposite case. These waveforms of the output voltage show the robustness and dynamic response of the control scheme.

D. Selection of Sliding Coefficients From (17), the coefficient of x1 and x2 are shown the regulating characteristic of system steady state and dynamic state respectively. Namely, the larger of k3 k2 , the better of the steady characteristic, and the larger of k1 k2 , the better of dynamic characteristic. From (14) and (15), when the system enters into the sliding domain it yields S ( x ) = k1 x1 + k2 x1 + k3 ∫ x1 = 0 (22) Rearranging the time differential of S ( x ) = 0 , we can get a standard second-order system form,  (23) x1 + 2ζωn x1 + ωn2 x1 = 0 where ωn = k3 k2 is the undamped natural frequency and

ζ = k1 2 k2 k3 is the dumping ratio [9]. In a criticallydamped system, the bandwidth of the controller’s response f BW is f BW =

ωn 1 = 2π 2π

k3 k2

k,(((

(24)

TABLE I PARAMETERS OF THE BUCK INVERTER Item

Symbol

Value

Unit

DC link voltage

Vin

48

V

Filter capacitor

C1 / C2

100

μF

Filter inductor

L 1 / L2

200

μH

Nominal load

R

3

Ω

TABLE II STEADY-STATE PERFORMANCE OF THE THIRD-ORDER SMC (MEASURED BY DIGITAL POWER METER WT2030) References of Output Voltage

Experimental Results of Output Voltage RMS (V) Error (V)

Peak (V)

RMS (V)

32

22.63

22.62

0.01

28

19.80

19.78

0.02

24

16.97

16.95

0.02

20

14.14

14.15

-0.01

16

11.31

11.31

0.00

12

8.49

8.47

0.02

8

5.66

5.67

-0.01

5

3.54

3.55

-0.01



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Fig. 4. Steady state outputs under third-order SMC

V. CONCLUSIONS In this paper, a novel third-order sliding mode control strategy is applied to the Buck inverter. The proposed control strategy is analyzed and designed detailedly based on the Lie derivative method. The geometry associated with the sliding mode control is easy to understand. The Buck inverter with the proposed third-order SMC is discussed in detail which guarantees the steady-state and dynamic properties. The sliding domain and sliding motion are described by phase trajectory vividly. Three-dimensional trajectory transforms into the two-dimensional one through mapping method to simplify the analysis. The proposed control strategy can improve tracking precision and maintain dynamic performances.

Fig. 5. Transient responses from no load to full load

ACKNOWLEDGMENT This work is supported by the National Nature Science Foundation of China (50777056). REFERENCES [1] V.I. Utkin, “Variable structure systems with Sliding Modes,” IEEE Trans. Automat. Contr., Vol. ac-22, No. 2, pp. 212–222, Apr. 1977. [2] J.Y. Hung, W.B. Gao, and J.C. Hung, “Variable structure control: a survey,” IEEE Trans. Ind. Electron., Vol. 40, No. 1, pp. 2–22, Feb. 1993. [3] M. Carpita and M. Marchesoni, “Experimental study of a power conditioning system using sliding mode control,” IEEE Trans. Power Electron., vol. 11, pp. 731–741, Sept. 1996. [4] G. Spiazzi and P. Mattavalli, “Sliding-mode control of switched-mode power supplies,” in The Power Electronics Handbook. Boca Raton, FL: CRC Press LLC, ch. 8, 2002. [5] V.M. Nguyen and C.Q. Lee, “Indirect implementations of sliding-mode control law in buck-type converters,” in Proc. IEEE Applied Power Electronics Conf. and Expo. (APEC), Vol. 1, pp. 111–115, Mar. 1996. [6] S.A. Bock, J.R. Pinheriro, H. Grundling, and H.L. Hey, “Existence and stability of sliding modes in bi-directional DC-DC converters,” in Proc. IEEE Power Electronics Specialists Conf. (PESC), pp. 1277–1282, Jun. 2001. [7] S.C. Tan, Y.M. Lai, M.K.H. Cheung, and C.K. Tse, “On the Practical Design of a Sliding Mode Voltage Controlled Buck Converter,” IEEE Trans. Power Electron., Vol. 20, No. 2, pp. 425–437, Mar. 2005. [8] M. Castilla, L. García de Vicuña, J. M. Guerrero, J. Matas, and J. Miret, “Design of voltage-mode hysteretic controllers for synchronous buck converters supplying microprocessor loads,” IEE Proc. Electr. Power Appl., Vol. 152, No. 5, Sept. 2005. Fig. 6. Transient responses from full load to no load

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[9] S.C. Tan, Y.M. Lai, C.K. Tse, and M.K.H. Cheung, “A Fixed-Frequency Pulsewidth Modulation Based Quasi-Sliding-Mode Controller for Buck Converters,” IEEE Trans. Power Electron., Vol. 20, No. 6, pp. 1379– 1392, Nov. 2005. [10] H. Pinheiro, A.S. Martins, and J.R. Pinheiro, “A sliding mode controller in single phase voltage source inverters,” in Proc. IEEE Ind. Electron. Conf. (IECON), Vol. 1, pp. 394–398, Sept. 1994. [11] L. Malesani, L. Rossetto, G. Spiazzi, and Z. Zuccato, “An AC power supply with sliding-mode control,” IEEE Ind. Applicat. Mag., Vol. 2, pp. 32–38, Sept./Oct. 1996. [12] D. Biel, E. Fossas, F. Guinjoan, E. Alarcón, and A. Poveda, “Application of sliding-mode control to the design of a buck-based sinusoidal generator,” IEEE Trans. Ind. Electron., Vol. 48, No. 3, pp. 563–571, Jun. 2001. [13] S. Pierfederici, J.P. Martin, F. Meibody-Tabar, and B. Davat, “Fixed frequency AC current controller for a three phases voltage source inverter,” in Proc. IEEE Power Electronics Specialists Conf. (PESC), Vol. 2, pp. 490–494, Jun. 2003. [14] R. Gupta and A. Ghosh, “Frequency-domain characterization of sliding mode control of an inverter used in DSTATCOM application,” IEEE Trans. Circuits and Syst. I, Vol. 53, No. 3, pp. 662–676, Mar. 2006. [15] J. Knight, S. Shirsavar, and W. Holderbaum, “An improved reliability Cuk based solar inverter with sliding mode control,” IEEE Trans. Power Electron., Vol. 21, No. 4, pp. 1107–1115, Jul. 2006. [16] R.J. Wai and W.H. Wang, “Design of Grid-Connected Photovoltaic Generation System with High Step-Up Converter and Sliding-Mode Inverter Control,” in Proc. IEEE International Symposium on Intelligent Contr. (ISIC), Web02.2, pp. 1179–1184, Oct. 2007.

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[17] P. Sanchis, A. Ursæa, E. Gubía, and L. Marroyo, “Boost DC-AC inverter: a new control strategy,” IEEE Trans. Power Electron., Vol. 20, No. 2, pp. 343–353, Mar. 2005. [18] H. Ma and S. Han, “Buck Inverter With Sliding Mode Control Strategy”, in Proc. IEEE Ind. Electron. Conf. (IECON), Vol. 1, pp. 592–597, Nov. 2004. [19] H. Ma and T. Zhang, “Analysis and design for sliding mode controlled buck inverter based on reference sliding surface and state variables surface,” in Proc. IEEE Applied Power Electronics Conf. and Expo. (APEC), pp. 1472–1477, Mar. 2006. [20] H. Ma and T. Zhang, “Sliding mode control for AC signal power amplifier”, in Proc. IEEE Ind. Electron. Conf. (IECON), pp. 1012–1017, Nov. 2005. [21] I.-C. Baik, K.-H. Kim, and M.-J. Youn, “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique,” IEEE Trans. Contr. Syst. Technology, Vol. 8, No. 1, pp. 47–54, Jan. 2000. [22] S. Seshagiri and H.K.Khalil, “On introducing integral action in sliding mode control,” in Proc. IEEE Decision and Control, Vol. 2, pp. 1473– 1478, Dec. 2002. [23] S. Mahieddine-Mahmoud, R. Ramanou, L. Kefsi, and L. Chrifi-Alaoui, “Sliding mode control: a new contribution using an integral action in the boundary layer,” in Proc. IEEE Ind. Technology, pp. 681–686, Dec. 2006.



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