# Optimal Sliding Mode Preview Repetitive Control for Three-Phase Z-Source Inverters

Authors: Jian-De Yan; Fan Peng; Wei-Cai Xie
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Aiming at high output voltage and obvious current fluctuation of the off grids inverters with unbalanced and nonlinear loads, a compound control strategy of sliding mode controller and optimal preview controller based on Z-source inverter (ZSI) is proposed. ZSI can boost the pressure and improve the system energy conversion rate through by combining the state with the linear quadratic design method of optimal control. The preview controller is introduced in the feedforward compensation link. With the help of difference operator, an extended state error system covering the target value signal and lag link feedback is designed. The optimal preview repetitive control (PRC) is transformed into a linear quadratic regulation matter of the discrete systems. Furthermore, a preview repetitive controller is obtained by using the Lyapunov method, linear matrix inequality, and the design method of optimal controller, which can realize sliding mode control, state feedback, repetitive control, and preview compensation. Finally, a 10 kV A prototype is built to verify the effectiveness of the proposed strategy.


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## Body

## 1. Introduction

With the aggravated environmental problems and energy crisis caused by traditional energy, renewable energy (wind energy, solar energy, etc.) has been applied in microgrids composed of distributed generation systems [1, 2]. When microgrids are off-grids, the control strategy of inverter affects the output power quality and system stability greatly [3].Z-source inverter (ZSI) is a single-stage power converter [4]. Compared with the traditional voltage source inverters (VSI), ZSI can prolong the output voltage (buck boost operation), prevent voltage sag, reduce inrush current and harmonic current from diode rectifier, enhance the reliability, and avoid gating errors of switching devices [5, 6].Sliding mode control (SMC) is a comprehensive mode of control systems. It is advanced with simple design and strong robustness to the influence of model uncertainty, parameter disturbance, and external disturbance on interference factors [7, 8]. It can deal with the nonlinear characteristics of inverter systems well. Therefore, SMC has been studied by many scholars in recent years [9, 10]. Preview control (PC) is an extended feedforward control method. The performance of the closed-loop systems can be improved by full use of the known future reference signals or interference signals. PC has received extensive attention since it was launched by Tomizuka in the 1970s and has developed different types such as adaptive PC [11], fault-tolerant PC [12], observer-based PC [13], and distributed PC [14]. Repetitive control (RC) can effectively track or suppress the periodic signals [15]. A positive feedback time-delay link is incorporated into the control system to improve the tracking accuracy through its own learning mechanism, thus tracking or suppressing the periodic reference signals without steady-state errors. Repetitive controllers have been improved and perfected continually due to lag link and the poor dynamic performance [16].Reference [17] adopts the control method combining repetition and PI to improve the dynamic performance of inverter and effectively reduce inverter harmonics, but the inverter performance is greatly affected by PI parameters. Reference [18] studies the design of sliding mode repetitive controller and its application in inverter. A proportional RC scheme is proposed [19] to enhance the fast transient response of the system, but the structure is complex. Incorporating the preview controller can improve the tracking ability of the system and significantly optimize the performance of the controller [20, 21]. The finite time bounded tracking control for linear continuous time-delay systems is analyzed in a reference [22]. In reference [23], the Lomberg state observer of linear discrete-time system with preview information is applied to the preview control system. In reference [24], a robust guaranteed cost preview repetitive controller is designed for polynomial uncertain linear discrete systems. In reference [25], the current situation of sliding mode control for permanent magnet synchronous motor (PMSM) is studied. By improving the traditional sliding mode surface, a sliding mode control method based on fractional order sliding mode surface is proposed. This method effectively suppresses the chattering phenomenon existing in the sliding mode surface sliding towards the equilibrium point and improves the tracking performance of the system. In reference [26], the sliding mode control of nonlinear systems with time-varying delay and external disturbances is studied. An integral sliding surface with time-varying delay is proposed. Lyapunov stability theory and improved interactive convex inequality are used, but the system is greatly affected by parameters. In reference [27], sliding mode control is applied to networked control systems affected by time delay, data packet loss, quantification, and uncertainty/interference, which improves robustness of the system and points out potential research challenges of sliding mode control for networked control systems.Although many achievements have been made in the SMC, RC, and PC of discrete systems, few studies have been done to combine the three methods and apply them to the control of microgrid systems. This observation inspires our current research. In this paper, the three-phase off-grid Z-source inverters are modeled, and the system is studied using a composite control strategy, which combines three control methods. The main contributions are summarized as follows: 1) the three-phase ZSI is mathematically modeled, transformed into a state space model, and further discretized. 2) The state feedback: RC, PC, and SMC are considered for design of a controller. The extended state error system is established by adopting the discrete lifting technology, and the optimal control is transformed into the regulation of system. 3) Stability of the system is proved, and the experimental analysis shows that this method contributes greatly to the off-grid inverter with unbalanced and nonlinear loads.


## 2. Modeling and Problem Analysis of Three-Phase Off-Grid ZSI

### 2.1. Principle of ZSI

Topology of the ZSI is shown in Figure1. A Z-source network is added between the direct current (DC) power supply and the invert bridge, which composes a symmetrical impedance source network (capacitance C1, C2, inductance L1, L2).Figure 1
Topology of the ZSI.The switches above and below the same bridge arm of a ZSI can be turned on simultaneously, which is a special conduction state of the ZSI (i.e., through zero vector). ZSI can realize step-down and step-up by combining the straight through zero vector and traditional vector.The Z-source network can be divided into through state and non-through state based on whether the switches above and below the same bridge arm are turned on simultaneously. The equivalent circuit topologies of ZSIs in nondirect and direct are shown in Figures2 and 3, respectively.Figure 2
Equivalent circuit topology for nondirect ZSIs.Figure 3
Equivalent circuit topology for direct-pass ZSIs.When the switches above and below the same bridge arm are turned on simultaneously, the ZSI is in the through state, and its DC link voltage is zero, which meansuin  = 0.In the case of nondirect connection, the DC link voltageuin can be written as follows:(1)uin=11−2D0udc=Budc,where B is boost factor of the Z-source network, D0 is the duty cycle of straight through zero vector, and the following equation can be obtained:(2)B=11−2D0≥1.Therefore, if the total gainG satisfies G=mB, the equation can be written as follows:(3)G=mB=m1−2D0.Mainstream pulse modulation methods for ZSIs [25] include the Simple Boost Control (SBC), Maximum Boost Control (MBC), Maximum Constant Boost Control (MCBC), and Modified Space Vector Pulse Width Modulation (MSVPWM) control. Among them, the constraint relationship between m and D0 is similar. SBC is selected as an example in this paper, and the constraint relationship between m and D0 satisfies the following:(4)m+D0≤1.The DC power supply voltage is assumed to be constant, and there is no fluctuation, andm = 1−D0 can be obtained by substituting (3).(5)G=m2m−1.SinceG cannot be negative, m must be greater than 0.5. A smaller m results in a larger D0 at the same voltage gain. This indicates that if D0 is fixed at a higher value all the time, the efficiency of the inverter will undoubtedly be reduced because the output voltage is small and m will be small. (Figure 4)Figure 4
Relationship betweenG and m.As shown in Figure 4, under this constraint, the total gain modulation of the ZSI is proportional with the same direct duty cycleD0. Similarly, the larger the duty cycle of the through zero vector is, the larger the total gain of the ZSI is when the modulation m is the same.


### 2.2. Modeling and Problem Analysis of ZSIs

The topology of three-phase off grid ZSIs is shown in Figure5. In the figure, udc is the input voltage of the DC bus, uin is the voltage of the DC bus after changing through the Z-source network, u is the input voltage of the controller, and D1 is the diode. The Z-source network includes is composed of capacitors C1 and C2, and inductors L1 and L2. RLf is the equivalent internal resistance of inductance Lf, Lf is the output filter inductance, and Cf is the output filter capacitance.Figure 5
Three phase off-grid Z-source inverter topology.In this paper, the state space equation of three-phase ZSIs is established by decoupling capacitance and inductance. Taking the decoupled d-axis as an example, the filtered inductance current and filtered capacitance voltageiL and filtered capacitance voltage vC of the d-axis are selected as state variables, and the controller output is undertaken as the input of the inverter. Then, a state space model can be deduced.(6)x˙=Acx+Bcu,y=Ccx,where(7)Ac=−RLfLf−1Lf−1Cf0,Bc=KPWM0,x=iLtvCt,C=01,y=vCt,where vectors iLt (t) and vCt represent the filtered inductance current and the filtered capacitance voltage, respectively; ut is the input voltage controlled by pulse width modulation (PWM), R is the amplification factor of the PWM power amplifier, Lf is the filter inductance, and Cf refers to the filter capacitance.Due to the digital control, a beat delay occurs during the generation of PWM, so the (6) is discretized based on external interference of the system and addition of a first-order lag link to the input.(8)xk+1=Axk+Buk−1+Edk,yk=Cxk.Remark 1.
A=eAcTs,B=∫0TseAcτdτBc,C=Cc, Ts is the discrete sampling period.
In equation (8), A,B,C, and E represent the constant matrixes. External disturbance of the dk three-phase ZSI control system indicates the uncertainty of the system.
In this paper, an appropriate controlleruk is planned to be designed for the discretized inverted system (7), so that the controller output u can match the DC link voltage uin under any initial conditions and uncertain disturbances, thus improving the stability of the system.


## 2.1. Principle of ZSI

Topology of the ZSI is shown in Figure1. A Z-source network is added between the direct current (DC) power supply and the invert bridge, which composes a symmetrical impedance source network (capacitance C1, C2, inductance L1, L2).Figure 1
Topology of the ZSI.The switches above and below the same bridge arm of a ZSI can be turned on simultaneously, which is a special conduction state of the ZSI (i.e., through zero vector). ZSI can realize step-down and step-up by combining the straight through zero vector and traditional vector.The Z-source network can be divided into through state and non-through state based on whether the switches above and below the same bridge arm are turned on simultaneously. The equivalent circuit topologies of ZSIs in nondirect and direct are shown in Figures2 and 3, respectively.Figure 2
Equivalent circuit topology for nondirect ZSIs.Figure 3
Equivalent circuit topology for direct-pass ZSIs.When the switches above and below the same bridge arm are turned on simultaneously, the ZSI is in the through state, and its DC link voltage is zero, which meansuin  = 0.In the case of nondirect connection, the DC link voltageuin can be written as follows:(1)uin=11−2D0udc=Budc,where B is boost factor of the Z-source network, D0 is the duty cycle of straight through zero vector, and the following equation can be obtained:(2)B=11−2D0≥1.Therefore, if the total gainG satisfies G=mB, the equation can be written as follows:(3)G=mB=m1−2D0.Mainstream pulse modulation methods for ZSIs [25] include the Simple Boost Control (SBC), Maximum Boost Control (MBC), Maximum Constant Boost Control (MCBC), and Modified Space Vector Pulse Width Modulation (MSVPWM) control. Among them, the constraint relationship between m and D0 is similar. SBC is selected as an example in this paper, and the constraint relationship between m and D0 satisfies the following:(4)m+D0≤1.The DC power supply voltage is assumed to be constant, and there is no fluctuation, andm = 1−D0 can be obtained by substituting (3).(5)G=m2m−1.SinceG cannot be negative, m must be greater than 0.5. A smaller m results in a larger D0 at the same voltage gain. This indicates that if D0 is fixed at a higher value all the time, the efficiency of the inverter will undoubtedly be reduced because the output voltage is small and m will be small. (Figure 4)Figure 4
Relationship betweenG and m.As shown in Figure 4, under this constraint, the total gain modulation of the ZSI is proportional with the same direct duty cycleD0. Similarly, the larger the duty cycle of the through zero vector is, the larger the total gain of the ZSI is when the modulation m is the same.


## 2.2. Modeling and Problem Analysis of ZSIs

The topology of three-phase off grid ZSIs is shown in Figure5. In the figure, udc is the input voltage of the DC bus, uin is the voltage of the DC bus after changing through the Z-source network, u is the input voltage of the controller, and D1 is the diode. The Z-source network includes is composed of capacitors C1 and C2, and inductors L1 and L2. RLf is the equivalent internal resistance of inductance Lf, Lf is the output filter inductance, and Cf is the output filter capacitance.Figure 5
Three phase off-grid Z-source inverter topology.In this paper, the state space equation of three-phase ZSIs is established by decoupling capacitance and inductance. Taking the decoupled d-axis as an example, the filtered inductance current and filtered capacitance voltageiL and filtered capacitance voltage vC of the d-axis are selected as state variables, and the controller output is undertaken as the input of the inverter. Then, a state space model can be deduced.(6)x˙=Acx+Bcu,y=Ccx,where(7)Ac=−RLfLf−1Lf−1Cf0,Bc=KPWM0,x=iLtvCt,C=01,y=vCt,where vectors iLt (t) and vCt represent the filtered inductance current and the filtered capacitance voltage, respectively; ut is the input voltage controlled by pulse width modulation (PWM), R is the amplification factor of the PWM power amplifier, Lf is the filter inductance, and Cf refers to the filter capacitance.Due to the digital control, a beat delay occurs during the generation of PWM, so the (6) is discretized based on external interference of the system and addition of a first-order lag link to the input.(8)xk+1=Axk+Buk−1+Edk,yk=Cxk.Remark 1.
A=eAcTs,B=∫0TseAcτdτBc,C=Cc, Ts is the discrete sampling period.
In equation (8), A,B,C, and E represent the constant matrixes. External disturbance of the dk three-phase ZSI control system indicates the uncertainty of the system.
In this paper, an appropriate controlleruk is planned to be designed for the discretized inverted system (7), so that the controller output u can match the DC link voltage uin under any initial conditions and uncertain disturbances, thus improving the stability of the system.


## 3. Design of the Optimal Sliding Mode Preview Repetitive Controller

### 3.1. Design of the Preview Repetitive Controller

The basic discrete repeat controller is shown in Figure6.Figure 6
Basic repetitive controller.According to the discrete nature of the z-transform, the time domain form of the basic repetitive control in Figure6 is expressed as follows:(9)vk=ek,0≤k<L,vk−L+ek,k≥L.ek=rk−yk.In the above equations,ek is the error signal of the system; L is the delay parameter of the delay link, which is equal to the period of the reference input signal rk; and vk is the output of the repetitive controller.The interference signal is assumed to bedk=0, and the nominal system form of the discrete system (7) is given as follows:(10)xk+1=Axk+Buk−1,yk=Cxk.The preview repetitive control system designed in this paper is shown in Figure7. K1,K2 and K3 are the gain matrix of repetitive controller, state feedback controller, and preview controller, respectively, and CRz is the basic repetitive controller. Therefore, the preview repetitive controller designed in this paper is as follows:(11)uk=∑i=0L−1K1ivk−i+1+K2xk+1+∑i=0MrK3irk+i+1.Figure 7
Structure of preview repetitive controller.The above controller consists of three parts: a basic repetitive controller to strengthen the error adjustment ability of the periodic signal, a state feedback controller to enhance the system stability in the periodic signal, and a preview controller to improve the overall tracking performance by compensating the system with future target values.For the inverter discrete system (10), the performance index is introduced as follows:(12)J=∑k=0∞eTkQeek+ΔuTkHΔuk,where Qe and H are given symmetric positive definite weighted moments.The objective of this paper is to design a preview repetitive controller (11), such that the closed-loop system can satisfy the following performance:(13)limk⟶∞ek=limk⟶∞rk−yk=0.The following hypotheses are made for the target signalrk of the system:The number of predictable steps for the target value signalrk of cycle L is set to MrMr<L; that is, the current value of k, rk and the future value of Mr step rk,rk+1,…,rk+Mr are known, and a constant after Mr step can be taken.This hypothesis is about signal predictability of the target value. A signal value, which is far away from the current moment, has no significant impact on system performance.A variable of the system is defined as follows:(14)Δαk=αk−αk−L.αk in the above equation is an arbitrary vector.Based on the hypothesis, the future information of the target signal is introduced into the linear discrete system, and the result is given as follows:(15)xrk+1=Arxrk,where(16)xrk=ΔrkΔrk+1⋮Δrk+Mr,Ar=0I0⋯000I⋯0⋮⋮⋮⋮000⋯I000⋯0.Based on the (15) and parallel (10), the system error can be expressed with the following equation:(17)Δxk+1=AΔxk+BΔuk−1,ek+1=ek+1−L−CAΔxk−CBΔuk−1+Δr(k+1,The equation above can be transformed into the below one by introducing a state vector:(18)x¯1k+1=A¯x1k+GrΔrk+1+GΔuk−1,Δyk=C1x¯1k,where(19)A¯=0⋯0I−CAI⋯000⋮⋱⋮⋮⋮0⋯I000⋯00AGr=10⋯00T,G=−CB0⋯0BTC1=0⋯0C.Combined with the error system (17), Δuk−1=Δuk is added to the state vector by using the discrete lifting technique, and the extended state error equation of the system is obtained as follows:(20)X¯k+1=ΦxX¯k+GxΔuk,yk=C¯X¯k,where(21)Φx=A¯GxuGxr00000Ar;X¯k=x¯1kΔuk−1Xrk,Gx=G00TC¯=C100,Gxr=0Gr0⋯0.For the extended state error system, the performance index (12) can be rewritten as equation (19):(22)J=∑k=0∞X¯TkQX¯k+ΔuTkHΔuk,where(23)Q=diagQ100.With the discrete lifting technology, the control of the linear discrete system (10) under the performance index (12) can be transformed into stability adjustment of the extended state error system (18) under the performance index (21).Based on equations (8) and (11), (24) can be written as follows:(24)Δuk=∑i=0L−1K1iek−i+K2Δxk+∑i=0MrK3iΔrk+i=K1K2x¯k+K3Xrk=K1K2K3X¯k=KX¯k.Obviously, if the extended state error system (18) is equipped with a feedback control law Δuk=KX¯k, the linear discrete system (7) can be provided with an optimal preview repetitive controller in the form of equation (11).Lemma 1 (Schur complement [28]).
For a given symmetric matrixS11∈Rr×r where the following three conditions are equivalent:(1)
S<0(2)
S11<0,S22−S121S11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2.
The hypothesis is true if and only if there are positive symmetric definite matricesX and Y, and if the following condition holds:(25)−XΦxX+GxY00∗−XXQ12YTH12∗∗−I0∗∗∗−I<0.
the extended state error system (18) obeys an optimal feedback control law Δuk=KX¯k, and K=YX−1.Proof.
WhenΔuk=KX¯k is the feedback control law of the extended state error system (18), the corresponding closed-loop system is known by combining equations (18) and (20).(26)X¯k+1=ΦxX¯k+GxΔuk=Φx+GrKX¯k.
The Lyapunov function is selected:(27)VX¯k=X¯kTPX¯k,where(28)P>0.
The difference ofVX¯k is calculated along any trajectory of the closed-loop system (22).(29)VX¯k+1−VX¯k=X¯Tk+1PX¯k+1−X¯TkPX¯k=X¯TkΦx+GxKTPΦx+GxKX¯k−X¯TkPX¯k.(30)Φx+GxKTPΦx+GxK−P+Q+KTHK<0.
If (30) is true, (26) can be written as follows:(31)VX¯k+1−VX¯k<−X¯TkQ+KTHKX¯k<0.
According to Lyapunov stability theory, the closed-loop system (22) is asymptotically stable. Further, the following equation can be obtained from equation (26):(32)X¯TkQX¯k+ΔuTkHΔuk<−VX¯k+1−VX¯k.
Sum fromk=0 to k=∞ on both sides of the inequality(33)∑k=0∞X¯TkQX¯k+ΔuTkHΔuk≤X¯0TPX¯0,where X0 is the initial state of the extended state error system.
The (27) can be transformed into the below one after application of lemma 1.(34)−P−1Φx+GxK∗−P+Q+KTHK<0.
After both sides of the above equation are multiplied bydiagI,P−1 concurrently, the below equation can be obtained:(35)−P−1Φx+GxKP−1∗−P−1+P−1Q+KTHKP−1<0.
IfX=P−1(36)−XΦx+GxKX∗−X+XQ+KTHKX<0.K=Y−1X is defined, and lemma 1 is adopted to obtain (20).
To sum up, the extended state error system (18) is subjective to a feedback control law Δuk=KX¯k, so that the linear discrete system (7) has to be provided with an optimal preview repetitive controller uk in the form of equation (11).


### 3.2. Design of the Sliding Mode Controller

Surface of the double power sliding mode controller is designed as(37)sk=CX¯k,where C∈R1×n+p+Mr, and CGx≠0Remark 2.
In order to ensure the accuracy of equation (34) Δu2≠0, so CGx≠0.
For the discrete system (7), the reaching law of the double power sliding mode controller is defined as follows:(38)sk+1−sk=−ε1Tskαsgnsk−ε2Tskβsgnsk−qTsk,where q>0,1−qT>0,ε1>0,0<α<1,ε2>0,β>1 and T is the sampling step.
According to (38), the double power reaching law consists of two parts. ε1Tskαsgnsk plays a major role when the system is far away from the sliding surface, and ε2Tskαsgnsk plays a major role when the system is close to the sliding surface. Therefore, the advantage of double power sliding mode controller is that it can select the corresponding gain according to the state and demand of the system, improve the speed reaching the sliding mode surface, and enhance the dynamic characteristics.Lemma 3.
The control law of double power sliding mode preview repetitive controller is composed of a linear part and a nonlinear part:(39)Δuk=Δu1k+Δu2k,where(40)Δu1k=∑i=0L−1K1iek−i+K2Δxk+∑i=0MrK3iΔrk+i,Δu2k=CGx−1ε1Tskαsgnsk+CGx−1ε2Tskβsgnsk−CGx−11−qTsk.Proof.
Reference Lyapunov function(41)vk=12s2k.
when the system meets the following conditions:(42)ΔVk=s2k+1−s2k<0,sk≠0.
According to Lyapunov theorem, the necessary and sufficient condition for the global asymptotic stability of the system is that the sliding mode surfacesk<0. Regardless of the initial value of the system state vector, its final value will approach the sliding mode surface sk. Therefore, the arrival condition of the sliding mode controller is as follows:(43)s2k+1<s2k.
When the sampling step of the system is small, the necessary and sufficient conditions for the existence and accessibility of the sliding mode controller are given as follows:(44)sk+1−sksgnsk<0,sk+1+sksgnsk>0.
The following equation can be obtained by substituting the approach law proposed by (38) into (44):(45)sk+1−sksgnsk=−qTsk−ε1Tskα−ε2Tskβ<0.
From (45) and (46), it can be obtained that the double power sliding mode surface meets the two conditions of (44); that is, the double power sliding mode controller meets the existence and accessibility, and the system shows a good dynamic performance.Theorem 1.
For a given matrixQ and H, if there are matrices X and Y with appropriate dimensions so that linear matrix equation (21) holds, the optimal sliding mode preview repetitive controller of discrete system equation (7) under performance index function equation (12) is as follows:(46)uk=u1k+u2k,where(47)u1k=∑i=0L−1K1ivk−i+K2xk+∑i=0MrK3irk+i,u2k=CGx−1qT−1vk+ε1Tϕk+ε2Tφk,where(48)vk=vk−1+sk,ϕk=ϕk−1+skαsgnsk,φk=φk−1+skβsgnsk.


## 3.1. Design of the Preview Repetitive Controller

The basic discrete repeat controller is shown in Figure6.Figure 6
Basic repetitive controller.According to the discrete nature of the z-transform, the time domain form of the basic repetitive control in Figure6 is expressed as follows:(9)vk=ek,0≤k<L,vk−L+ek,k≥L.ek=rk−yk.In the above equations,ek is the error signal of the system; L is the delay parameter of the delay link, which is equal to the period of the reference input signal rk; and vk is the output of the repetitive controller.The interference signal is assumed to bedk=0, and the nominal system form of the discrete system (7) is given as follows:(10)xk+1=Axk+Buk−1,yk=Cxk.The preview repetitive control system designed in this paper is shown in Figure7. K1,K2 and K3 are the gain matrix of repetitive controller, state feedback controller, and preview controller, respectively, and CRz is the basic repetitive controller. Therefore, the preview repetitive controller designed in this paper is as follows:(11)uk=∑i=0L−1K1ivk−i+1+K2xk+1+∑i=0MrK3irk+i+1.Figure 7
Structure of preview repetitive controller.The above controller consists of three parts: a basic repetitive controller to strengthen the error adjustment ability of the periodic signal, a state feedback controller to enhance the system stability in the periodic signal, and a preview controller to improve the overall tracking performance by compensating the system with future target values.For the inverter discrete system (10), the performance index is introduced as follows:(12)J=∑k=0∞eTkQeek+ΔuTkHΔuk,where Qe and H are given symmetric positive definite weighted moments.The objective of this paper is to design a preview repetitive controller (11), such that the closed-loop system can satisfy the following performance:(13)limk⟶∞ek=limk⟶∞rk−yk=0.The following hypotheses are made for the target signalrk of the system:The number of predictable steps for the target value signalrk of cycle L is set to MrMr<L; that is, the current value of k, rk and the future value of Mr step rk,rk+1,…,rk+Mr are known, and a constant after Mr step can be taken.This hypothesis is about signal predictability of the target value. A signal value, which is far away from the current moment, has no significant impact on system performance.A variable of the system is defined as follows:(14)Δαk=αk−αk−L.αk in the above equation is an arbitrary vector.Based on the hypothesis, the future information of the target signal is introduced into the linear discrete system, and the result is given as follows:(15)xrk+1=Arxrk,where(16)xrk=ΔrkΔrk+1⋮Δrk+Mr,Ar=0I0⋯000I⋯0⋮⋮⋮⋮000⋯I000⋯0.Based on the (15) and parallel (10), the system error can be expressed with the following equation:(17)Δxk+1=AΔxk+BΔuk−1,ek+1=ek+1−L−CAΔxk−CBΔuk−1+Δr(k+1,The equation above can be transformed into the below one by introducing a state vector:(18)x¯1k+1=A¯x1k+GrΔrk+1+GΔuk−1,Δyk=C1x¯1k,where(19)A¯=0⋯0I−CAI⋯000⋮⋱⋮⋮⋮0⋯I000⋯00AGr=10⋯00T,G=−CB0⋯0BTC1=0⋯0C.Combined with the error system (17), Δuk−1=Δuk is added to the state vector by using the discrete lifting technique, and the extended state error equation of the system is obtained as follows:(20)X¯k+1=ΦxX¯k+GxΔuk,yk=C¯X¯k,where(21)Φx=A¯GxuGxr00000Ar;X¯k=x¯1kΔuk−1Xrk,Gx=G00TC¯=C100,Gxr=0Gr0⋯0.For the extended state error system, the performance index (12) can be rewritten as equation (19):(22)J=∑k=0∞X¯TkQX¯k+ΔuTkHΔuk,where(23)Q=diagQ100.With the discrete lifting technology, the control of the linear discrete system (10) under the performance index (12) can be transformed into stability adjustment of the extended state error system (18) under the performance index (21).Based on equations (8) and (11), (24) can be written as follows:(24)Δuk=∑i=0L−1K1iek−i+K2Δxk+∑i=0MrK3iΔrk+i=K1K2x¯k+K3Xrk=K1K2K3X¯k=KX¯k.Obviously, if the extended state error system (18) is equipped with a feedback control law Δuk=KX¯k, the linear discrete system (7) can be provided with an optimal preview repetitive controller in the form of equation (11).Lemma 1 (Schur complement [28]).
For a given symmetric matrixS11∈Rr×r where the following three conditions are equivalent:(1)
S<0(2)
S11<0,S22−S121S11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2.
The hypothesis is true if and only if there are positive symmetric definite matricesX and Y, and if the following condition holds:(25)−XΦxX+GxY00∗−XXQ12YTH12∗∗−I0∗∗∗−I<0.
the extended state error system (18) obeys an optimal feedback control law Δuk=KX¯k, and K=YX−1.Proof.
WhenΔuk=KX¯k is the feedback control law of the extended state error system (18), the corresponding closed-loop system is known by combining equations (18) and (20).(26)X¯k+1=ΦxX¯k+GxΔuk=Φx+GrKX¯k.
The Lyapunov function is selected:(27)VX¯k=X¯kTPX¯k,where(28)P>0.
The difference ofVX¯k is calculated along any trajectory of the closed-loop system (22).(29)VX¯k+1−VX¯k=X¯Tk+1PX¯k+1−X¯TkPX¯k=X¯TkΦx+GxKTPΦx+GxKX¯k−X¯TkPX¯k.(30)Φx+GxKTPΦx+GxK−P+Q+KTHK<0.
If (30) is true, (26) can be written as follows:(31)VX¯k+1−VX¯k<−X¯TkQ+KTHKX¯k<0.
According to Lyapunov stability theory, the closed-loop system (22) is asymptotically stable. Further, the following equation can be obtained from equation (26):(32)X¯TkQX¯k+ΔuTkHΔuk<−VX¯k+1−VX¯k.
Sum fromk=0 to k=∞ on both sides of the inequality(33)∑k=0∞X¯TkQX¯k+ΔuTkHΔuk≤X¯0TPX¯0,where X0 is the initial state of the extended state error system.
The (27) can be transformed into the below one after application of lemma 1.(34)−P−1Φx+GxK∗−P+Q+KTHK<0.
After both sides of the above equation are multiplied bydiagI,P−1 concurrently, the below equation can be obtained:(35)−P−1Φx+GxKP−1∗−P−1+P−1Q+KTHKP−1<0.
IfX=P−1(36)−XΦx+GxKX∗−X+XQ+KTHKX<0.K=Y−1X is defined, and lemma 1 is adopted to obtain (20).
To sum up, the extended state error system (18) is subjective to a feedback control law Δuk=KX¯k, so that the linear discrete system (7) has to be provided with an optimal preview repetitive controller uk in the form of equation (11).


## 3.2. Design of the Sliding Mode Controller

Surface of the double power sliding mode controller is designed as(37)sk=CX¯k,where C∈R1×n+p+Mr, and CGx≠0Remark 2.
In order to ensure the accuracy of equation (34) Δu2≠0, so CGx≠0.
For the discrete system (7), the reaching law of the double power sliding mode controller is defined as follows:(38)sk+1−sk=−ε1Tskαsgnsk−ε2Tskβsgnsk−qTsk,where q>0,1−qT>0,ε1>0,0<α<1,ε2>0,β>1 and T is the sampling step.
According to (38), the double power reaching law consists of two parts. ε1Tskαsgnsk plays a major role when the system is far away from the sliding surface, and ε2Tskαsgnsk plays a major role when the system is close to the sliding surface. Therefore, the advantage of double power sliding mode controller is that it can select the corresponding gain according to the state and demand of the system, improve the speed reaching the sliding mode surface, and enhance the dynamic characteristics.Lemma 3.
The control law of double power sliding mode preview repetitive controller is composed of a linear part and a nonlinear part:(39)Δuk=Δu1k+Δu2k,where(40)Δu1k=∑i=0L−1K1iek−i+K2Δxk+∑i=0MrK3iΔrk+i,Δu2k=CGx−1ε1Tskαsgnsk+CGx−1ε2Tskβsgnsk−CGx−11−qTsk.Proof.
Reference Lyapunov function(41)vk=12s2k.
when the system meets the following conditions:(42)ΔVk=s2k+1−s2k<0,sk≠0.
According to Lyapunov theorem, the necessary and sufficient condition for the global asymptotic stability of the system is that the sliding mode surfacesk<0. Regardless of the initial value of the system state vector, its final value will approach the sliding mode surface sk. Therefore, the arrival condition of the sliding mode controller is as follows:(43)s2k+1<s2k.
When the sampling step of the system is small, the necessary and sufficient conditions for the existence and accessibility of the sliding mode controller are given as follows:(44)sk+1−sksgnsk<0,sk+1+sksgnsk>0.
The following equation can be obtained by substituting the approach law proposed by (38) into (44):(45)sk+1−sksgnsk=−qTsk−ε1Tskα−ε2Tskβ<0.
From (45) and (46), it can be obtained that the double power sliding mode surface meets the two conditions of (44); that is, the double power sliding mode controller meets the existence and accessibility, and the system shows a good dynamic performance.Theorem 1.
For a given matrixQ and H, if there are matrices X and Y with appropriate dimensions so that linear matrix equation (21) holds, the optimal sliding mode preview repetitive controller of discrete system equation (7) under performance index function equation (12) is as follows:(46)uk=u1k+u2k,where(47)u1k=∑i=0L−1K1ivk−i+K2xk+∑i=0MrK3irk+i,u2k=CGx−1qT−1vk+ε1Tϕk+ε2Tφk,where(48)vk=vk−1+sk,ϕk=ϕk−1+skαsgnsk,φk=φk−1+skβsgnsk.


## 4. Experimental Results and Analysis

A laboratory prototype is constructed to verify the correctness and feasibility of the control strategy proposed in this paper. The digital signal processor (DSP) of TMS320F28335 chip produced by TI company is undertaken as the digital control platform to build an experimental platform with a rated capacity of 10 kV·A. The experimental sample machine is shown in Figure8, and the specific experimental parameters are shown in Table 1. It includes a Z-source network, which is connected to a three-phase voltage source inverter (VSI), and the inverter output adopts LC filtering.Figure 8
Experiment prototype.Table 1
Parameters of three-phase ZSI.
SymbolQuantityValuevdcDC power supply voltageV300CZ-source capacitor (μ F)1000LZ-source inductormH5LfFilter inductancemH2CfFilter capacitor (μ F)1200RLfLine resistancemΩ30fsSwitching frequency of the inverter (kHz)10Parameters for the optimal sliding mode preview repetitive controller are set as follows:L=10, sampling time T=0.01s, the number of preview steps is Mr=2 to ensure system stability, the state weighting coefficient Qe  = 100, and the control weighting coefficient H=1, q=6,ε1=ε2=0.1,KPWM=1.2,α=0.1,β=1.01.According to theorem1, it is solved by using LMI toolbox of MATLAB:(49)A=0.86070.00670.00021B=0.012050.01200C=01,K1=−0.1590−0.01570.00240.0015−0.00010.00040.00260.01580.15742.4537,K2=−2.3457−5.2758,K3=0.16471.26470.0166.To verify the dynamic performance and stability of the system, the peak DC link voltage is adjusted to the reference value of 400V to verify the boost performance of ZSI (as shown in Figures9 and 10). Figures 11–13 set the load switching of the system from no-load to full load under linear load conditions.Figure 9
Z source side DC link voltage (gray) and peak value (green).Figure 10
Detail diagram of DC link voltage and peak value at Z source side.Figure 11
Repetitive controller under linear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)Figure 12
Preview repetitive controller under linear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)Figure 13
Optimal sliding mode preview repetitive control under linear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)To verify the performance of the proposed optimal preview repetitive controller, Figures11–16 compare the RC, PRC, and SPRC under linear and nonlinear loads and collect the waveform distortion of voltage at steady state. Figures 17 and 18 verify the change of three-phase output voltage and current waveform under unbalanced load, where ia and vaN are the output current and voltage of phase inverter, respectively.Figure 14
Repetitive controller under non-linear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)Figure 15
Preview repetitive controller under non-linear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)Figure 16
Optimal sliding mode preview repetitive control under nonlinear load. (a) Voltage and current waveforms. (b) Voltage waveform distortion rate.
(a)(b)Figure 17
Output voltage under unbalanced load.Figure 18
Output current under unbalanced load.The Z-source controller increases the input voltage through a specific topology and adjusts the peak DC link voltage to 400V, as shown in Figure9. To further demonstrate the performance of the Z-source controller, some switching cycles in Figure 9 are enlarged. The steady state waveform of the Z-source controller is shown in Figure 10.Figures11–13 show the voltage and current waveforms of RC, PRC, and SPRC under linear load conditions, respectively. The total harmonic distortion (THD) of output voltage is 3.12% for RC, 2.48% for PRC, and 0.78% for SPRC. In no-load to full-load switching experiments, when RC is used, the output voltage reaches steady state after 10 ms of linear load input. When PRC and SPRC are used, the output voltage can reach steady state after 5 ms of linear load input. The control strategy presented in this paper is superior to traditional repetitive control and preview repetitive control in the case of linear load. It is validated that the control strategy proposed in this paper can effectively improve the voltage quality and improve the tracking performance of the system.Figures14–16 are the voltage and current waveforms of RC, PRC, and SPRC under nonlinear load conditions, respectively. The THD of output voltage is 6.72% for RC, 4.68% for PC, and 0.92% for SPRC. When the system is under a nonlinear load, due to the presence of supporting capacitance, the adjustment time is longer than that under a linear load, and the dynamic response time is about one wave period. The proposed control strategy uses state feedback of inductance current and output voltage and compensates the first-order lag of input, which further improves the stability compared with the traditional RC.To verify the ability of the SPRC strategy proposed in this paper to suppress disturbances with unbalanced loads, 50% linear balanced loads are connected at the load side, and 20Ω resistance is applied on the AC phase to simulate unbalanced loads. The experimental results are shown in Figures 17 and 18. It shows the three-phase output voltage and current waveforms under unbalanced load condition respectively. It reveals that, under the unbalanced load conditions, the sliding mode preview repetitive strategy proposed in this paper can ensure the balanced three-phase output voltage of the inverter.To sum up, under the disturbance of unbalanced load, nonlinear load, and load switching, the control strategy proposed in this paper shows good dynamic and steady state performance. It can be explained that the proposed control strategy is better than the tracking and anti-interference performance of the system, which verifies the feasibility of the control strategy proposed.


## 5. Conclusion

When the disturbance loads (such as unbalanced load and nonlinear load) of the off-grid inverter are strong, the control ability of traditional RC is insufficient. To solve this deficiency, an optimal sliding mode preview repetitive control is proposed in this paper. The design of sliding mode preview repetitive controller is transformed into the stability of discrete system to improve the performance of the controller. Based on the stability theory of discrete systems and with the help of LMI processing techniques, the existence conditions of sliding mode preview repetitive controller and the solution method of controller parameters are obtained. The experimental results show that the SPRC shows better stable load tracking performance when the input voltage and output voltage are unbalanced. Therefore, the optimal sliding mode preview repetitive control strategy proposed in this paper shows better anti-interference ability and dynamic response performance. Of course, there are still many problems to be solved in the microgrid system. How to run multiple systems jointly or in parallel is a major and challenging problem. In addition, it should be noted that how to use the composite control method in the microgrid system and switch between the grid and the disconnected network is the focus of the next step.



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*Source: 9660028-2022-09-13.xml*