# Feedback Linearization and Sliding Mode Control for VIENNA Rectifier Based on Differential Geometry Theory

Authors: Xiang Lu; Yunxiang Xie; Li Chen
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Aiming at the nonlinear characteristics of VIENNA rectifier and using differential geometry theory, a dual closed-loop control strategy is proposed, that is, outer voltage loop using sliding mode control strategy and inner current loop using feedback linearization control strategy. On the basis of establishing the nonlinear mathematical model of VIENNA rectifier ind-q synchronous rotating coordinate system, an affine nonlinear model of VIENNA rectifier is established. The theory of feedback linearization is utilized to linearize the inner current loop so as to realize the d-q axis variable decoupling. The control law of outer voltage loop is deduced by utilizing sliding mode control and index reaching law. In order to verify the feasibility of the proposed control strategy, simulation model is built in simulation platform of Matlab/Simulink. Simulation results verify the validity of the proposed control strategy, and the controller has a strong robustness in the case of parameter variations or load disturbances.


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## Body

## 1. Introduction

With the development of power electronic technology, three-level pulse width modulation (PWM) rectifiers are widely used in high or medium power converters because of their excellent performance: low switch voltage stress, low input current harmonic distortion, high efficiency, input power factor is unit, and so on [1–7]. Three-phase/switch/level VIENNA rectifier (abbreviated as VIENNA rectifier) is one of the best three-level rectifier, which is proposed in 1994 by Kolar and Zach. Compared with traditional three-level rectifier, such as diode clamping three-level rectifier, VIENNA rectifier has lots advantages, such as small number of power switch tube, simple control circuit, and low design costs, without output voltage bridge arm shoot-through problems. So, more and more scholars and engineers focus their attention on the study of VIENNA rectifier and its control strategy [3–14].Since VIENNA rectifier is a typical strong coupling nonlinear system, it leads to difficulty in designing the controller. In [8, 9], the mathematical models of large and small signals topology are analyzed in detail, and the controller is designed by using proportion and integral (PI) algorithm. A control method of input/output accurate linearization is proposed in [10]. State-space average model is established, and PI control algorithm is used in the outer voltage loop; hysteresis control is used in the inner current loop in [11–13]. The control methods described above improved the performance of VIENNA rectifier to some extent. However, there are some disadvantages, such as system excessive dependence on the accurate mathematical model, inconvenience of parameter setting, complicated of control algorithm, and poor dynamic. In order to overcome the above drawbacks, this paper proposes a control strategy which combines feedback linearization control and sliding mode control.In recent decades, nonlinear control theory has made great progress, especially feedback linearization theory based on differential geometry. In this method, nonlinear system can achieve status or input/output linearization by using a certain nonlinear state transformation or feedback transformation. Feedback linearization control has been applied to three-phase voltage PWM rectifier [14–16], which is a multivariable and strong coupling nonlinear system, and achieved well control effect. All these methods can solve the problem of decoupling for original nonlinear system and obviously improve static and dynamic performance for three phase rectifier. Yet, this control method depends on an accurate mathematical model and is sensitive to system parameters. Sliding mode control is different from feedback linearization control method. Sliding mode control shows great robustness and stays out of parameter changes when the system is running in the sliding surface. In [17–20], sliding mode control has been applied to the three-phase PWM rectifier and achieved a good result.The three-phase PWM rectifier is a two-level rectifier, while the VIENNA rectifier is a three-level rectifier. Although their structure is not the same, there are some similarity in control strategy. Learning from the applications of feedback linearization control and sliding mode control for three-phase PWM rectifier, this paper integrates feedback linearization control method and sliding mode control method and eventually establishes a new type of VIENNA rectifier nonlinear control system. That is, sliding mode control is used in the outer voltage loop and state feedback linearization control is used in the inner current loop. At the same time, space vector pulse width modulation (SVPWM) technology is introduced to modulate the output signal of inner current loop [21]. In order to reduce the chattering phenomenon produced by sliding mode control, index reaching law is adopted to improve the whole approaching process. In order to verify the correctness and superiority of the proposed control strategy, numerical simulation is done.


## 2. Physical and Modeling Considerations

In this section, the physical system and the mathematical model of VIENNA rectifier are presented. The main circuit of VIENNA rectifier and its simplified model are shown in Figures1(a) and 1(b), respectively. The main circuit includes six fast-recovery diodes (D1–D6), three boost inductors, three bidirection power switching tubes (Sa, Sb, and Sc as shown in the dashed box), and two groups of output capacitances. Among them, ea, eb, ec are the AC input power of VIENNA rectifier; C1 and C2 are DC side output voltage filter capacitor, and their voltage across, respectively, are VC1, VC2; Rl is load resistance and the voltage across is Vdc, and il is the output current; L is the boosting inductor and R is defined as an equivalent resistance of inductor. In order to simplify the system, all the power-switching devices are seen as ideal and switching frequency is much higher than the grid frequency.Figure 1
Main circuit of VIENNA rectifier and its simplified model.
(a)
(b)The mathematical model of VIENNA rectifier inabc coordinates can be expressed as follows:(1)ZdXdt=AX+BE,where(2)Z=LLLC1C2,X=iaibicVC1VC2T,B=11100,E=eaebec00T,A=-R00-Sap-Sap+Sbp+Scp3San-San+Sbn+Scn30-R0-Sbp-Sap+Sbp+Scp3Sbn-San+Sbn+Scn300-R-Scp-Sap+Sbp+Scp3Scp-Sap+Sbp+Scp3SapSbpScp-1Rl-1Rl-San-Sbn-Scp-1Rl-1Rl.With Park’s transformation, the mathematical model of VIENNA rectifier ind-q coordinates system is given as follows:(3)Ldiddt=-Rid+ωLiq-SdpVC1+SdnVC2+ed,(4)Ldiqdt=-ωLid-Riq-SqpVC1+SqnVC2+eq,(5)CdVC1dt=32Sdpid+32Sqpiq-il,(6)CdVC2dt=-32Sdnid-32Sqniq-il,where ed and eq are the grid voltage variable ind-q coordinate system; id and iq are the grid current variable ind-q coordinate system. As for Sdp, Sdn, Sqp, and Sqn, they are the switching function ind-q coordinate system for Sxx=a,b,c. The equivalent circuit of VIENNA rectifier ind-q coordinate system is shown in Figure 2.Figure 2
Equivalent circuit of VIENNA rectifier ind-q coordinates.For three-phase balance power grid, because ofC1=C2=C, yield VC1=VC2=(1/2)Vdc. Assume that(7)Sd=Sdp-Sdn,Sq=Sqp-Sqn.Thus, (3) and (4) can be rewriten as follows:(8)Ldiddt=-Rid+ωLiq-Sd2Vdc+ed,Ldiqdt=-Riq-ωLid-Sq2Vdc+eq.With (5) plus (6), we can obtain(9)CdVdcdt=32(Sdid+Sqiq)-2il.


## 3. Control Goal and Control Strategies

The main control goal for VIENNA rectifier is to make sure that input current waveform is sinusoid and track the input voltage waveform, the power factor is unity, and DC side output voltage stabilized at the given reference voltage. Equations (8)-(9) show that VIENNA rectifier is a strong decoupling nonlinear system. In order to realize the variables decoupling ind-q axis, ensure that input current is sinusoid and tracks input voltage; the feedback linearization technology is used in the inner current loop. At the same time, sliding mode control based on index reaching law is used in the voltage loop to stabilize the output voltage and provide the reference directive current id* for inner current loop.


### 3.1. Inner Current Loop Controller Design Based on Feedback Linearization

The role of inner current loop is makingid and iq keep track of reference directive current id* and iq*, respectively, and realizes system unity power factor operation.


#### 3.1.1. Affine Nonlinear Models of VIENNA Rectifier

Select state variables as(10)x=x1x2=idiq.Select input variables as(11)u=u1u2=SdSq.Select output variables as(12)y1=h1xt=id,y2=h2xt=iq.According to (8), we get the affine nonlinear equation for two-input and two-output system as follows:(13)x˙=fx+g1xu1+g2xu2,y1=h1xt,y2=h2xt,where(14)fx=-RLid+ωiq+1Led-RLiq-ωid+1Leq,g1x=-Vdc2L0,g2x=0-Vdc2L.For the system which is described in (13), it is nonlinear to state variable X(t) while linear to controlled variable u.


#### 3.1.2. Linearization Processing of VIENNA Rectifier

Lemma 1 (see [22]).
Let one assumeBx=b1,…,bm as a nonsingular matrix and define the following sets:(15)G0x={b1,…,bm},G1x={b1,…,bm,adfb1,…,adfbm},⋮Gix={adfk:0≤k≤i,1≤k≤m},i=0,1,…,n-1.So, the necessary and sufficient conditions for these linearization solutions above are as follows:(1)
for eachi=0,1,…,n-1, the distribution Gix has constant dimension near x;(2)
for eachi=0,1,…,n-1, the distribution Gix is involution;(3)
the dimension ofGn-ix is n.According to the requirements of exact linearization for nonlinear system, verify the two-input and two-output affine nonlinear system described in (13) as follows: (16)adfg1x=∂g1x∂xfx-∂fx∂xg1x=0-ωVdc2L,adfg2x=∂g2x∂xfx-∂fx∂xg2x=ωVdc2L0.From (16), we can obtain this conclusion: the total relative degree r=1+1=2, which just right equals the system’s dimension n. Meanwhile, when n=2, the vector D=g1Xg2Xadfg1Xadfg2X is involution. Hence, it meets the requirements of linearization.Taking Lie derivative for (13), we can obtain(17)x˙1=Lfh1(x)+Lg1h1(x)u1+Lg2h1(x)u2,x˙2=Lfh2(x)+Lg1h2(x)u1+Lg2h2(x)u2,where(18)Lg1h1(x)=Lg2h2(x)=-Vdc2L,Lg2h1(x)=Lg1h2(x)=0,Lfh1(x)=-RLid+ωiq+edL,Lfh2(x)=-RLiq-ωid+eqL.


#### 3.1.3. Coordinate Transformation and the Control Law

Select output vector as(19)y=y1y2=idiq.According to the definition of relative degree [22], the derivative of two input/output system output functions is(20)y˙1y˙2=Lfh1Lfh2+A(x)u1u2=αx+Axu,Where(21)Ax=-Vdc2L00-Vdc2L.In the meanwhile, since the matrixA(x) is invertible, the state feedback control law is given by(22)u1u2=-A-1(x)Lfh1Lfh2+v1v2.In order to track the desired value, a new control law is given by [14](23)v1=-k10y1*-y1v2=-k20y2*-y2.From (20), (22), and (23), the new state feedback control law of the original nonlinear inner current loop is obtained:(24)uduq=-2VdcRid-ωiq-edωid+Riq-eq+-k10y1*-y1-k20y2*-y2.So far, the original nonlinear system is linearization. Meanwhile, the control for inner current loop can be achieved by setting feedback coefficientsk10 and k20.


#### 3.1.4. Analysis of System Dynamic Stability

Because of the relative degree of (8) being 1 and according to the literature [14], we can get the error dynamic closed-loop system as follows:(25)e˙1+k10e1=0,(26)e˙2+k20e2=0,(27)z˙=az,e+y*,ξ,where ξ=y˙1y˙2T, ei=yi*-yi (i=1,2).Lemma 2.
For internal dynamic formula (27), when tracking error e vanishes, that is(28)z˙=az,y*,0,it is called the zero dynamics [14]. And if a nonlinear system’s zero dynamics is asymptotically stable, it is said to be minimum phase.Because the power supply is three-phase symmetrical voltage and the rectifier is operated with unity power factor and considered in the steady state, we haveed=Em, Vdc=Vdc*, id=Id* and iq=0. Therefore, (8) and (9) can be simplified as follows:(29)RId*+Sd′Vdc*2=Em,ωLId*+Sq′Vdc*2=0,32Sd′Id*=2Il,where Sd′, Sq′, Id*, and Il represent the steady-state values of the switching functions Sd and Sq, d-axis reference current id*, and output load current il, respectively.By contrast, we find that (29) are the same as (21)–(23) in [14], so the dynamic analysis of VIENNA rectifier can learn from the literature [14].Transform the original nonlinear system (13) into the linear form of (25)–(27) with z=x3=Vdc. The internal dynamics (27) becomes(30)x˙3=-3L2CidVdck10Id*-id+RLid+ωiq-EmL-3L2CiqVdc-k20iq+RLiq-ωid-1Cil.According to the energy balance and because of the tracking error vectore approaching zero, that is, id→Id* and iq→0, we can get the following zero dynamics equation:(31)x˙3=1RlCx3Vdc*2-x32.From (31), we can know the following. For a positive initial dc side output voltage, the steady-state value of x3 will eventually equal the desired value of Vdc*. So, when using cascaded current mode, the control system is a minimum phase system, and the dynamic is stability. Therefore, we can design a superior performance control system.


### 3.2. Voltage Loop Controller Design Based on Sliding Mode Control

The role of voltage loop controller is to guarantee output voltageVdc tracking the given reference voltage Vdc* and stable. In the meantime, voltage loop controller provides d-axis reference current id* for the current loop.For (24), id and iq can be acquired by taking Park’s transformation to input currents ia, ib, and ic. For dual closed loop control system which adopts inner current loop and outer voltage loop, id* is always provided by the outer loop. For three-level PWM rectifier, PI algorithm [23], fractional control algorithm [24], and sliding mode control algorithm [25] are used in the outer voltage loop. Since inner current loop uses feedback linearization control, it brings a shortcoming for this control system; that is, the control system is overreliance on accurate mathematical model. In order to compensate this shortcoming, sliding mode control algorithm is used in the outer voltage loop. One of the biggest advantages for sliding mode control is insensitive to the change of system parameters and having less demand for control system model. In order to avoid chattering, improve the approaching performance, index reaching law is introduced based on sliding mode control which is described in [25].Let us assume the error between dc output voltage and the given reference voltage as follows:(32)Ve=Vdc*-Vdc.Based on the principle of sliding mode control [26, 27], the sliding surfaces can be defined as(33)Sdc=Ve+k3∫0tVedt,k3>0.WhenSdc=0, the system runs in sliding mode surface. Take a derivate on both sides of (33), and because Vdc* is constant, yield(34)S˙dc=-dVdcdt+k3Ve.According to (9), (34) can be rewritten as(35)S˙dc=-32CSdid+Sqiq+2Cil+k3Ve.In order to guarantee the system having a good quality in the transition process, especially improving the quality of arrival stage, and eliminating the chattering phenomenon, index reaching law is applied [26]:(36)S˙=-εsgnS-λSε>0,λ>0.According to the literature [26], yield(37)St=ελ+S0-ελe-kt,S>0;St=-ελ+S0+ελe-kt,S<0.From (37), we can obtain(38)t*=1λln⁡S+ελ-ln⁡ελ.From (38), it shows that reducing ε and increasing λ can accelerate the approaching process.From (35) and (36), we can obtain(39)λSdc+εsgnSdc-32C(Sdid+Sqiq)+2Cil+k3Ve=0.Further, (39) can be transformed as follows:(40)λSdc+εsgnSdc-32CSqiq+2Cil+k3Ve·2C3Sd-id=0.According to the above assumption, power grid is symmetrical three-phase voltage. When the system is in steady state, yieldiq=0, diq/dt=0, ed=3URMS, eq=0, Vdc=Vdc*. At the same time, the processing speed of inner loop is faster than the outer loop. Thus, id can be seen as constant; yield did/dt=0. From (8), the following approximation algorithm can be obtained:(41)Sd≈2ed-RidVdc,Sq≈-2ωLiqVdc.From (40) and (41), we can obtain(42)id=CVdc33URMS-Rid·λVe+k3∫0tVedt+εsgnVe+k3∫0tVedtmmm+2ilC+k3VeVe+k3∫0tVedt.When the system is in steady state,id=id*. So, (42) is rewritten as(43)id*=CVdc33URMS-Rid·λVe+k3∫0tVedt+εsgnVe+k3∫0tVedtmmm+2ilC+k3Veeu+k3∫0tVedt.As a result, the output of outer voltage loop just is the current instructionsid* for the inner loop, which is relevant to the output voltage, output current, valid value of phase voltage, d-axis current id, and so on. Besides, id* is irrelevant to the switching function variables.


### 3.3. Control System Block Diagram

According to the above analysis, we can get the control system block diagram as shown in Figure3.Figure 3
Control system block diagram.


## 3.1. Inner Current Loop Controller Design Based on Feedback Linearization

The role of inner current loop is makingid and iq keep track of reference directive current id* and iq*, respectively, and realizes system unity power factor operation.


### 3.1.1. Affine Nonlinear Models of VIENNA Rectifier

Select state variables as(10)x=x1x2=idiq.Select input variables as(11)u=u1u2=SdSq.Select output variables as(12)y1=h1xt=id,y2=h2xt=iq.According to (8), we get the affine nonlinear equation for two-input and two-output system as follows:(13)x˙=fx+g1xu1+g2xu2,y1=h1xt,y2=h2xt,where(14)fx=-RLid+ωiq+1Led-RLiq-ωid+1Leq,g1x=-Vdc2L0,g2x=0-Vdc2L.For the system which is described in (13), it is nonlinear to state variable X(t) while linear to controlled variable u.


### 3.1.2. Linearization Processing of VIENNA Rectifier

Lemma 1 (see [22]).
Let one assumeBx=b1,…,bm as a nonsingular matrix and define the following sets:(15)G0x={b1,…,bm},G1x={b1,…,bm,adfb1,…,adfbm},⋮Gix={adfk:0≤k≤i,1≤k≤m},i=0,1,…,n-1.So, the necessary and sufficient conditions for these linearization solutions above are as follows:(1)
for eachi=0,1,…,n-1, the distribution Gix has constant dimension near x;(2)
for eachi=0,1,…,n-1, the distribution Gix is involution;(3)
the dimension ofGn-ix is n.According to the requirements of exact linearization for nonlinear system, verify the two-input and two-output affine nonlinear system described in (13) as follows: (16)adfg1x=∂g1x∂xfx-∂fx∂xg1x=0-ωVdc2L,adfg2x=∂g2x∂xfx-∂fx∂xg2x=ωVdc2L0.From (16), we can obtain this conclusion: the total relative degree r=1+1=2, which just right equals the system’s dimension n. Meanwhile, when n=2, the vector D=g1Xg2Xadfg1Xadfg2X is involution. Hence, it meets the requirements of linearization.Taking Lie derivative for (13), we can obtain(17)x˙1=Lfh1(x)+Lg1h1(x)u1+Lg2h1(x)u2,x˙2=Lfh2(x)+Lg1h2(x)u1+Lg2h2(x)u2,where(18)Lg1h1(x)=Lg2h2(x)=-Vdc2L,Lg2h1(x)=Lg1h2(x)=0,Lfh1(x)=-RLid+ωiq+edL,Lfh2(x)=-RLiq-ωid+eqL.


### 3.1.3. Coordinate Transformation and the Control Law

Select output vector as(19)y=y1y2=idiq.According to the definition of relative degree [22], the derivative of two input/output system output functions is(20)y˙1y˙2=Lfh1Lfh2+A(x)u1u2=αx+Axu,Where(21)Ax=-Vdc2L00-Vdc2L.In the meanwhile, since the matrixA(x) is invertible, the state feedback control law is given by(22)u1u2=-A-1(x)Lfh1Lfh2+v1v2.In order to track the desired value, a new control law is given by [14](23)v1=-k10y1*-y1v2=-k20y2*-y2.From (20), (22), and (23), the new state feedback control law of the original nonlinear inner current loop is obtained:(24)uduq=-2VdcRid-ωiq-edωid+Riq-eq+-k10y1*-y1-k20y2*-y2.So far, the original nonlinear system is linearization. Meanwhile, the control for inner current loop can be achieved by setting feedback coefficientsk10 and k20.


### 3.1.4. Analysis of System Dynamic Stability

Because of the relative degree of (8) being 1 and according to the literature [14], we can get the error dynamic closed-loop system as follows:(25)e˙1+k10e1=0,(26)e˙2+k20e2=0,(27)z˙=az,e+y*,ξ,where ξ=y˙1y˙2T, ei=yi*-yi (i=1,2).Lemma 2.
For internal dynamic formula (27), when tracking error e vanishes, that is(28)z˙=az,y*,0,it is called the zero dynamics [14]. And if a nonlinear system’s zero dynamics is asymptotically stable, it is said to be minimum phase.Because the power supply is three-phase symmetrical voltage and the rectifier is operated with unity power factor and considered in the steady state, we haveed=Em, Vdc=Vdc*, id=Id* and iq=0. Therefore, (8) and (9) can be simplified as follows:(29)RId*+Sd′Vdc*2=Em,ωLId*+Sq′Vdc*2=0,32Sd′Id*=2Il,where Sd′, Sq′, Id*, and Il represent the steady-state values of the switching functions Sd and Sq, d-axis reference current id*, and output load current il, respectively.By contrast, we find that (29) are the same as (21)–(23) in [14], so the dynamic analysis of VIENNA rectifier can learn from the literature [14].Transform the original nonlinear system (13) into the linear form of (25)–(27) with z=x3=Vdc. The internal dynamics (27) becomes(30)x˙3=-3L2CidVdck10Id*-id+RLid+ωiq-EmL-3L2CiqVdc-k20iq+RLiq-ωid-1Cil.According to the energy balance and because of the tracking error vectore approaching zero, that is, id→Id* and iq→0, we can get the following zero dynamics equation:(31)x˙3=1RlCx3Vdc*2-x32.From (31), we can know the following. For a positive initial dc side output voltage, the steady-state value of x3 will eventually equal the desired value of Vdc*. So, when using cascaded current mode, the control system is a minimum phase system, and the dynamic is stability. Therefore, we can design a superior performance control system.


## 3.1.1. Affine Nonlinear Models of VIENNA Rectifier

Select state variables as(10)x=x1x2=idiq.Select input variables as(11)u=u1u2=SdSq.Select output variables as(12)y1=h1xt=id,y2=h2xt=iq.According to (8), we get the affine nonlinear equation for two-input and two-output system as follows:(13)x˙=fx+g1xu1+g2xu2,y1=h1xt,y2=h2xt,where(14)fx=-RLid+ωiq+1Led-RLiq-ωid+1Leq,g1x=-Vdc2L0,g2x=0-Vdc2L.For the system which is described in (13), it is nonlinear to state variable X(t) while linear to controlled variable u.


## 3.1.2. Linearization Processing of VIENNA Rectifier

Lemma 1 (see [22]).
Let one assumeBx=b1,…,bm as a nonsingular matrix and define the following sets:(15)G0x={b1,…,bm},G1x={b1,…,bm,adfb1,…,adfbm},⋮Gix={adfk:0≤k≤i,1≤k≤m},i=0,1,…,n-1.So, the necessary and sufficient conditions for these linearization solutions above are as follows:(1)
for eachi=0,1,…,n-1, the distribution Gix has constant dimension near x;(2)
for eachi=0,1,…,n-1, the distribution Gix is involution;(3)
the dimension ofGn-ix is n.According to the requirements of exact linearization for nonlinear system, verify the two-input and two-output affine nonlinear system described in (13) as follows: (16)adfg1x=∂g1x∂xfx-∂fx∂xg1x=0-ωVdc2L,adfg2x=∂g2x∂xfx-∂fx∂xg2x=ωVdc2L0.From (16), we can obtain this conclusion: the total relative degree r=1+1=2, which just right equals the system’s dimension n. Meanwhile, when n=2, the vector D=g1Xg2Xadfg1Xadfg2X is involution. Hence, it meets the requirements of linearization.Taking Lie derivative for (13), we can obtain(17)x˙1=Lfh1(x)+Lg1h1(x)u1+Lg2h1(x)u2,x˙2=Lfh2(x)+Lg1h2(x)u1+Lg2h2(x)u2,where(18)Lg1h1(x)=Lg2h2(x)=-Vdc2L,Lg2h1(x)=Lg1h2(x)=0,Lfh1(x)=-RLid+ωiq+edL,Lfh2(x)=-RLiq-ωid+eqL.


## 3.1.3. Coordinate Transformation and the Control Law

Select output vector as(19)y=y1y2=idiq.According to the definition of relative degree [22], the derivative of two input/output system output functions is(20)y˙1y˙2=Lfh1Lfh2+A(x)u1u2=αx+Axu,Where(21)Ax=-Vdc2L00-Vdc2L.In the meanwhile, since the matrixA(x) is invertible, the state feedback control law is given by(22)u1u2=-A-1(x)Lfh1Lfh2+v1v2.In order to track the desired value, a new control law is given by [14](23)v1=-k10y1*-y1v2=-k20y2*-y2.From (20), (22), and (23), the new state feedback control law of the original nonlinear inner current loop is obtained:(24)uduq=-2VdcRid-ωiq-edωid+Riq-eq+-k10y1*-y1-k20y2*-y2.So far, the original nonlinear system is linearization. Meanwhile, the control for inner current loop can be achieved by setting feedback coefficientsk10 and k20.


## 3.1.4. Analysis of System Dynamic Stability

Because of the relative degree of (8) being 1 and according to the literature [14], we can get the error dynamic closed-loop system as follows:(25)e˙1+k10e1=0,(26)e˙2+k20e2=0,(27)z˙=az,e+y*,ξ,where ξ=y˙1y˙2T, ei=yi*-yi (i=1,2).Lemma 2.
For internal dynamic formula (27), when tracking error e vanishes, that is(28)z˙=az,y*,0,it is called the zero dynamics [14]. And if a nonlinear system’s zero dynamics is asymptotically stable, it is said to be minimum phase.Because the power supply is three-phase symmetrical voltage and the rectifier is operated with unity power factor and considered in the steady state, we haveed=Em, Vdc=Vdc*, id=Id* and iq=0. Therefore, (8) and (9) can be simplified as follows:(29)RId*+Sd′Vdc*2=Em,ωLId*+Sq′Vdc*2=0,32Sd′Id*=2Il,where Sd′, Sq′, Id*, and Il represent the steady-state values of the switching functions Sd and Sq, d-axis reference current id*, and output load current il, respectively.By contrast, we find that (29) are the same as (21)–(23) in [14], so the dynamic analysis of VIENNA rectifier can learn from the literature [14].Transform the original nonlinear system (13) into the linear form of (25)–(27) with z=x3=Vdc. The internal dynamics (27) becomes(30)x˙3=-3L2CidVdck10Id*-id+RLid+ωiq-EmL-3L2CiqVdc-k20iq+RLiq-ωid-1Cil.According to the energy balance and because of the tracking error vectore approaching zero, that is, id→Id* and iq→0, we can get the following zero dynamics equation:(31)x˙3=1RlCx3Vdc*2-x32.From (31), we can know the following. For a positive initial dc side output voltage, the steady-state value of x3 will eventually equal the desired value of Vdc*. So, when using cascaded current mode, the control system is a minimum phase system, and the dynamic is stability. Therefore, we can design a superior performance control system.


## 3.2. Voltage Loop Controller Design Based on Sliding Mode Control

The role of voltage loop controller is to guarantee output voltageVdc tracking the given reference voltage Vdc* and stable. In the meantime, voltage loop controller provides d-axis reference current id* for the current loop.For (24), id and iq can be acquired by taking Park’s transformation to input currents ia, ib, and ic. For dual closed loop control system which adopts inner current loop and outer voltage loop, id* is always provided by the outer loop. For three-level PWM rectifier, PI algorithm [23], fractional control algorithm [24], and sliding mode control algorithm [25] are used in the outer voltage loop. Since inner current loop uses feedback linearization control, it brings a shortcoming for this control system; that is, the control system is overreliance on accurate mathematical model. In order to compensate this shortcoming, sliding mode control algorithm is used in the outer voltage loop. One of the biggest advantages for sliding mode control is insensitive to the change of system parameters and having less demand for control system model. In order to avoid chattering, improve the approaching performance, index reaching law is introduced based on sliding mode control which is described in [25].Let us assume the error between dc output voltage and the given reference voltage as follows:(32)Ve=Vdc*-Vdc.Based on the principle of sliding mode control [26, 27], the sliding surfaces can be defined as(33)Sdc=Ve+k3∫0tVedt,k3>0.WhenSdc=0, the system runs in sliding mode surface. Take a derivate on both sides of (33), and because Vdc* is constant, yield(34)S˙dc=-dVdcdt+k3Ve.According to (9), (34) can be rewritten as(35)S˙dc=-32CSdid+Sqiq+2Cil+k3Ve.In order to guarantee the system having a good quality in the transition process, especially improving the quality of arrival stage, and eliminating the chattering phenomenon, index reaching law is applied [26]:(36)S˙=-εsgnS-λSε>0,λ>0.According to the literature [26], yield(37)St=ελ+S0-ελe-kt,S>0;St=-ελ+S0+ελe-kt,S<0.From (37), we can obtain(38)t*=1λln⁡S+ελ-ln⁡ελ.From (38), it shows that reducing ε and increasing λ can accelerate the approaching process.From (35) and (36), we can obtain(39)λSdc+εsgnSdc-32C(Sdid+Sqiq)+2Cil+k3Ve=0.Further, (39) can be transformed as follows:(40)λSdc+εsgnSdc-32CSqiq+2Cil+k3Ve·2C3Sd-id=0.According to the above assumption, power grid is symmetrical three-phase voltage. When the system is in steady state, yieldiq=0, diq/dt=0, ed=3URMS, eq=0, Vdc=Vdc*. At the same time, the processing speed of inner loop is faster than the outer loop. Thus, id can be seen as constant; yield did/dt=0. From (8), the following approximation algorithm can be obtained:(41)Sd≈2ed-RidVdc,Sq≈-2ωLiqVdc.From (40) and (41), we can obtain(42)id=CVdc33URMS-Rid·λVe+k3∫0tVedt+εsgnVe+k3∫0tVedtmmm+2ilC+k3VeVe+k3∫0tVedt.When the system is in steady state,id=id*. So, (42) is rewritten as(43)id*=CVdc33URMS-Rid·λVe+k3∫0tVedt+εsgnVe+k3∫0tVedtmmm+2ilC+k3Veeu+k3∫0tVedt.As a result, the output of outer voltage loop just is the current instructionsid* for the inner loop, which is relevant to the output voltage, output current, valid value of phase voltage, d-axis current id, and so on. Besides, id* is irrelevant to the switching function variables.


## 3.3. Control System Block Diagram

According to the above analysis, we can get the control system block diagram as shown in Figure3.Figure 3
Control system block diagram.


## 4. Simulation Results

In order to verify the correctness and superiority of the proposed control method, the simulation model is built in the simulation platform of Matlab/Simulink. The main simulation parameters are given in Table1. SVPWM algorithm based on two-level space vector is used in modulation methods of VIENNA rectifier. Small positive vector is set as the first vector, by judging the direction of phase load current which is connected to the output neutral point and adjusting relative action time according to the imbalance between small positive and negative vectors. Voltage regulating factor r (0<r<1) is introduced. The adjustment to vector action time is done so as to realize the midpoint potential balance control.Table 1
Simulation parameters.
Input phase voltage
e
(
a
,
b
,
c
)
=
110 V
Output reference voltage
V
d
c
*
=
400 V
Boost inductance
L
=
2.8 mH
Line equivalent impedance
R
=
0.1 Ω
Output filtering capacitor
C
1
=
C
2
=
680 μF
Nominal load resistance
R
l
=
100 Ω
Power rating
P
=
1.6 KW
Switching frequency
f
=
10 KHz


### 4.1. System Startup Responses

When the system is started, DC side output voltage rapidly rises from 0 V and stabilizes at reference voltageVdc* approximately at 0.0035 s, the response waveforms of output voltage are shown in Figure 4. At the same time, it shows that the system has a rapid response, without overshoot and static error. Also, it verifies that using sliding mode control method could force the system running path moving fast to sliding surfaces, thereby accelerating the convergence process of the system.Figure 4
Response waveforms of output voltage when system starts up.


### 4.2. System Steady-State Characteristics

When VIENNA rectifier is operating in steady state, input current waveform tracks input voltage waveform very well and shows sinusoid, as shown in Figure5(a); the active current id can well track reference current id* given by outer voltage loop, as shown in Figure 5(b); the reactive current iq is 0, which means the rectifier operates with unit power factor, as shown in Figure 5(c). The simulation waveforms indicate that the VIENNA rectifier reaches predetermined target.Figure 5
Steady-state simulation waveforms.
(a)
(b)
(c)


### 4.3. Transient Responses to Step Changes in Load

When load suddenly changes, the simulation waveforms are shown in Figure6. The value of load resistance Rl is changed from 100 Ω to 200 Ω at 0.05 s. As shown in Figure 6(a), DC side output voltage rises to about 6 V instantaneously and then reverts to a stable value (400 V) after 0.001 s. Figure 6(b) shows that input current also suddenly changes. However, input current waveform can correctly track the input voltage waveform and maintain sinusoid. Figure 6(c) shows that active current id can correctly track the given active current reference id*. At the same time, the system response time is short. In a word, it shows significant anti-interference capability to external disturbance by using the proposed strategy.Figure 6
Transient responses to step changes in load.
(a)
(b)
(c)


### 4.4. Transient Responses to Step Changes in the Given Output Reference Voltage

Assuming output voltage value instantaneously declines to 350 V at 0.06 s, the simulation waveforms are shown in Figure7. From Figure 7(a), output voltage begins to decrease at 0.06 s and stabilized at 350 V after 0.003 s. From Figure 7(b), a-phase input current value is 0 at 0.06 s and lasts about 0.003 s. Since then, it starts to increase rapidly and tracks input voltage and remains sinusoid.Figure 7
Transient responses to step changes in the given output reference voltage.
(a)
(b)


## 4.1. System Startup Responses

When the system is started, DC side output voltage rapidly rises from 0 V and stabilizes at reference voltageVdc* approximately at 0.0035 s, the response waveforms of output voltage are shown in Figure 4. At the same time, it shows that the system has a rapid response, without overshoot and static error. Also, it verifies that using sliding mode control method could force the system running path moving fast to sliding surfaces, thereby accelerating the convergence process of the system.Figure 4
Response waveforms of output voltage when system starts up.


## 4.2. System Steady-State Characteristics

When VIENNA rectifier is operating in steady state, input current waveform tracks input voltage waveform very well and shows sinusoid, as shown in Figure5(a); the active current id can well track reference current id* given by outer voltage loop, as shown in Figure 5(b); the reactive current iq is 0, which means the rectifier operates with unit power factor, as shown in Figure 5(c). The simulation waveforms indicate that the VIENNA rectifier reaches predetermined target.Figure 5
Steady-state simulation waveforms.
(a)
(b)
(c)


## 4.3. Transient Responses to Step Changes in Load

When load suddenly changes, the simulation waveforms are shown in Figure6. The value of load resistance Rl is changed from 100 Ω to 200 Ω at 0.05 s. As shown in Figure 6(a), DC side output voltage rises to about 6 V instantaneously and then reverts to a stable value (400 V) after 0.001 s. Figure 6(b) shows that input current also suddenly changes. However, input current waveform can correctly track the input voltage waveform and maintain sinusoid. Figure 6(c) shows that active current id can correctly track the given active current reference id*. At the same time, the system response time is short. In a word, it shows significant anti-interference capability to external disturbance by using the proposed strategy.Figure 6
Transient responses to step changes in load.
(a)
(b)
(c)


## 4.4. Transient Responses to Step Changes in the Given Output Reference Voltage

Assuming output voltage value instantaneously declines to 350 V at 0.06 s, the simulation waveforms are shown in Figure7. From Figure 7(a), output voltage begins to decrease at 0.06 s and stabilized at 350 V after 0.003 s. From Figure 7(b), a-phase input current value is 0 at 0.06 s and lasts about 0.003 s. Since then, it starts to increase rapidly and tracks input voltage and remains sinusoid.Figure 7
Transient responses to step changes in the given output reference voltage.
(a)
(b)


## 5. Conclusions

In this paper, a dual closed loop control method, that is, outer voltage loop based on sliding mode control and inner current loop based on feedback linearization, is proposed. Simulation results show that the proposed control strategy has a good control effect. The main contributions of this paper are as follows.(i)
It presents feedback linearization control strategy for VIENNA rectifier inner current loop such that it solves the linearization problems and realizesd-q axis variable decoupling.(ii)
It presents sliding mode control strategy for VIENNA rectifier outer voltage loop and introduces the index reaching law such that it solves stability of output voltage, system startup response, and dynamic characteristics.(iii)
The combination control strategy overcomes the disadvantage that the system is overreliance on the accurate mathematical models by using feedback linearization control strategy. Meanwhile, the proposed control strategy greatly improves the robustness of the system.



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*Source: 573016-2015-03-23.xml*