Design of Discrete-Time Backstepping Sliding- Mode Control for LCL-Type Grid-Connected Inverter

This study designs a discrete-time backstepping sliding-mode control (DTBSMC) method for an LCL-type grid-connected inverter. Firstly, the dynamic model of a discrete-time three-order system is derived, and a discrete-time backstepping control method cascading with the sliding-mode control theory is designed via Lyapunov stability verification. Moreover, the system state equation is transformed into a special form by using a time-varying mapping for overcoming the difficulty of a non-causal problem. Besides, through the recursively subsystem design for the high-order LCL-type inverter, the asymptotic system stability can be ensured by step-by-step virtual control designs. In addition, the proposed method can combine both the advantages of the backstepping control method and sliding-mode control theory. Therefore, the inverter system can have strong robustness under the condition of a power grid with varied grid impedances. Finally, the effectiveness of the proposed DTBSMC method is verified by numerical simulations and experimental results in comparison with a traditional proportional-resonant (PR) method and a backstepping control method.


I. INTRODUCTION
Recently, with the increasing penetration of distributed generations (DGs), three-phase grid-connected inverters have been widely used as the interfaces between DGs and the power grid [1]. The important role of a grid-connected inverter is to provide high quality power to the power grid. Commonly, a filter is needed to reduce the switching harmonics for satisfying the requirement of low total harmonic distortion (THD). Compared with L-type filters, LCL-type filters have high harmonic attenuation ability at the switching harmonics, which permits systems to use smaller filters and increases input power quality [2], [3].
Due to the advantages of the harmonic attenuation ability, the LCL filter has attracted more and more attention, and a lot of linear control schemes are investigated in previous literatures. Without the limitation of a proportional-integral (PI) controller on the ac signals tracking [4], [5], linear control schemes, e.g., proportional-resonant (PR) or multi-PR The associate editor coordinating the review of this manuscript and approving it for publication was Zhilei Yao . controllers, are commonly used in LCL-type grid-connected inverters. Owing to the third-order system and resonance problem of an LCL filter, there exists inherent instability problem within the system. One strategy is to adopt the passive damping method, where the passive damping losses can be reduced in the certain degree by using additional reactive components in the filter circuit [6], [7]. On the other hand, many researchers have proposed active damping (AD) methods [8]- [11] without additional passive components. Liu et al. [8] presented a comparison of various AD methods, which were designed based on different feedbacks of four commonly used filter variables. Consequently, the gridside current feedback AD method is selected to realize a virtual resistor in parallel with the filter capacitor. Guan et al. [9] proposed a dual-current active damping control strategy based on the feedback of inverter-side and grid-connected currents. Nonetheless, the equivalent virtual impedance to be simulated in series with a filter inductance will decrease the stability margin. For the stabilization and dynamic characterization, the output admittance shaping method was investigated in [10], [11], where the output admittance or impedance can be revised to satisfy the system requirements. However, the increasing penetration of DGs and the long transmission cable will make the grid impedance to be inevitable, which will change the resonant frequency of an LCL filter and may cause the system instability. For dealing with the grid impedance variation, some improved methods have been proposed to expand the stability region [12], [13]. In [12], an optimal design of the capacitor-current-feedback coefficient was presented by evaluating the effect of grid impedance on the gain margins, and the optimal coefficient was selected to increase the critical frequency of the stability constrain from f s /6 to f s /2. However, the optimal design needs to consider the prior information of the accurate value of the grid impedance. Li et al. [13] expressed a new method with the converter-side current feedback and the capacitor voltagefeedforward. The control method in [13] can keep the lowfrequency system characteristic to be independent of the grid impedance. However, the synchronizing performance is less precise than the grid-side current feedback, and the complex compensation is necessary for the control delay. The active damping method will be helpful for damping the resonance of the LCL filter, but it is usually sensitive to the control delay or the parameters of the LCL filter and the grid impedance. Fortunately, some state-of-the-art improvements have been proposed to enhance the active damping performance [14]- [19]. In [14] and [15], modified sampling methods have been proposed for reducing the time of the control delay. Moreover, filter-based compensation methods also can be used to reduce the impact of the control delay, e.g., the infinite impulse response filter in [16] and the highpass filter in [17]. He et al. [18] presented a capacitorcurrent proportional-integral positive feedback active damping method for addressing the problem of the negative equivalent damping resistance, and achieving high robustness to the gird impedance. Chen et al. [19] adopted a secondorder high-pass filter on the feedback path of the capacitor current, and the improvement of the robustness against the grid impedance variation was achieved. In these methods, the damping strategy still requires to be carefully designed or an elaborate compensation strategy is necessary for enhancing the system stability performance. By considering the LCLtype grid-connected inverter to be a three-order nonlinear system, the nonlinear control strategies, such as the backstepping control method and the sliding-mode control (SMC) theory, are worthy to investigate.
The SMC method is one of attractive nonlinear control approaches in the power converter field due to its fast-dynamic response and strong robustness to parameter uncertainties and external disturbances [20]. In [21], a SMC method was presented for an LCL-type grid-connected inverter, where a sliding-surface function was formed by using the grid-current error, the capacitor-voltage error and its derivative. Wang and Wai [22] proposed a total SMC (TSMC) method for a single-phase grid-connected converter to enhance the power decoupling ability with respect to varied power coupling conditions. Because the control strategy in [22] has no reaching phase, it can ensure the robustness in the whole sliding-mode control system. However, the SMC method is an inversion control technique that requires internal dynamic stability [23]. The backstepping control method can guarantee the internal dynamic stability because of the stepby-step recursively algorithm with strictly stability condition. Each subsystem will be introduced with a virtual control law, and the error of each subsystem can be used to form the corresponding Lyapunov functions. The errors of each subsystem will achieve asymptotic convergence by each virtual control, and the nonlinear controller with global asymptotic stability can be eventually obtained [24], [25]. Although the idea of decomposed subsystems is very suitable for a threeorder LCL-type grid-connected inverter, parameter variations and unpredictable disturbances still have obvious effect on the system performance. Hence, the combination of the SMC and backstepping control methods is adopted in this study. The backstepping sliding-mode control (BSMC) scheme can guarantee the global stability and achieve strong robustness [26], [27]. In [26], a BSMC strategy was investigated to improve the stability of the DC bus voltage for a boost converter in a DC microgrid, and the Lyapunov theory was used to prove the global asymptotic stability of the overall closedloop system. Davila [27] presented a controller, which was designed by using the backstepping procedure, and combined the feedback linearization technique with high-order slidingmodes. In [27], the matched and unmatched disturbances can be compensated by the injection of a continuous term generated by the robust exact high-order sliding-modes differentiator. However, these methods in [26], [27] were considered under the continuous-time situation. It is common knowledge that discrete-time systems rather than the continuoustime systems are the closest for describing a real controlled system [28]. Although the linear discrete-time control system has been extensively studied in many literatures, a nonlinear discrete-time control method for an LCL-type grid-connected inverter has not been investigated to the same degree. What is more, there exists a non-causal problem in the design of nonlinear discrete-time control methods in [29], [30], which will make the virtual control law containing future information. The non-causal problem may lead the control law to be more complicated because it will be introduced recursively in each controller design step via the backstepping procedure, as will be discussed in Section III.
The research topic is motivated by the aforementioned discussion, and a discrete-time backstepping sliding-mode control (DTBSMC) strategy is proposed for an LCL-type grid-connected inverter in this study. The major contributions of the proposed DTBSMC method can be emphasized as follows: 1) the dynamic model of a discrete-time three-order LCL-type grid-connected inverter system is derived, and the discrete-time backstepping control method cascading with the SMC theory is designed with the discrete-time Lyapunov stability theorem. The proposed DTBSMC strategy can provide strong robustness to the grid impedance, and it is easily implemented in a digital controller; 2) the difficulty of the non-causal problem in the design of the proposed DTBSMC method can be overcome; 3) the proposed DTBSMC framework combines both of the merits of the backstepping control method and the SMC theory, which can guarantee global asymptotic stability without the specific design of damping methods, and 4) As our knowledge goes, there are fewer relevant researches with the proposed DTBSMC method for an LCL-type grid-connected inverter. The proposed DTBSMC strategy can achieve satisfactory performance and excellent control, and it can provide strong robustness under considering the grid impedance. The recursively subsystem design of the DTBSMC is also very suitable for a higher-order LCLtype grid-connected inverter, and it can ensure the system asymptotic stability by step-by-step virtual control designs.
This study is mainly organized into six sections. Following the introduction, the structure and the mathematical model of the LCL-type grid-connected inverter are described in Section II. In Section III, a discrete-time backstepping sliding-mode control method is proposed without the non-causal problem, which can achieve strong robustness and system stability. The corresponding numerical simulations and experimental results are presented in Sections IV and V, respectively. Finally, some conclusions are given in Section VI.

II. SYSTEM DESCRIPTION AND MODELING
The block diagram for a three-phase LCL-type gridconnected inverter is depicted in Fig. 1. As can be seen from Fig. 1, a three-phase voltage-source inverter (VSI) is connected to the power grid through an LCL filter. In Fig. 1, L 1 is the converter-side inductor, C f is the filter capacitor, and L 2 is the grid-side inductor. r 1 and r 2 are equivalent series resistances (ESR) of L 1 and L 2 , respectively. T a+ , T a− , T b+ , T b+ , T c+ , and T c− represent trigger signals for six power switches. The symbol of N is the neutral point of the filter capacitors. u dc and (u aN , u bN , u cN ) denote the dc-bus voltage and the inverter AC-side terminal voltage, respectively.
(i 1a , i 1b , i 1c ) and (u CaN , u CbN , u CcN ) are output three-phase currents of the inverter and three-phase voltages of filter capacitors, respectively. The grid currents (i 2a , i 2b , i 2c ) are controlled to be synchronized with the voltages at the point of common coupling (PCC). The weak power grid consisting of the equivalent grid inductor (L g ), the grid resistance (r g ), and ideal grid voltages (e ga , e gb , e gc ) can be modeled by the Thevenin's theorem. The model of the VSI is implemented in the stationary αβ frame, and all the state variables in the abc frame are transformed to this basis. From Fig. 1, the dynamic model can be expressed as T is the output current vector in αaxis and β-axis, which is transformed by (i 1a , i 1b , i 1c ) in the abc-frame; i 2 = [ i 2α i 2β ] T is the grid current vector in α-axis and β-axis, which is transformed by (i 2a , i 2b , i 2c ) in the abc-frame; u C = [ u Cα u Cβ ] T is the filter capacitor voltage vector in α-axis and β-axis, which is transformed by (u CaN , u CbN , u CcN ) in the abc-frame; e g = [ e gα e gβ ] T represents the power grid voltage vector in α-axis and βaxis, which is transformed by (e ga , e gb , e gc ) in the abc-frame.
T is the duty-cycle components vector in αaxis and β-axis, which is transformed by (D a , D b , D c ) in the abc-frame and can be expressed as where (D a , D b , D c ) represent the average of the switching state function (m a , m b , m c ) [31], and (m a , m b , m c ) can be expressed as [32] where c k represents the switching function which denotes the ON/OFF status of the devices in IGBT bridge, and it can be expressed as The terms of L G and r G in (1a) are defined as r G = r 2 + r g (5b) According to (1), the system dynamic equation without considering parameter uncertainties and unpredictable disturbance can be obtained as [24] ...
where K pwm is the pulse-width-modulation (PWM) gain, and u f = [ u f α u f β ] T is the control input; ω r is the resonance angular frequency of the LCL filter, which can be expressed as By defining three new state variables as x 1 = i 2 , x 2 =i 2 and x 3 =ï 2 , (6) can be rewritten as By applying the forward Euler discretization method [33], the discrete-time dynamic system model can be derived as where k is the sampling instant; T s is the sampling period; d r (k) = − r 1 L G T s +r G L 1 T s is the vector including the terms of ESR and grid resistances; d g (k) = − is the grid voltage disturbance vector; a n = diag(T s ω 2 r , T s ω 2 r ) and b n = diag K pwm T s /(L 1 C f L G ) , K pwm T s /(L 1 C f L G ) .
In practical applications, parameters of the LCL filter could be slightly different from nominal values or grid impedances could also be different from each other. Besides, considering the system with unpredictable disturbance and control delay (e.g., computation and PWM delays [13], which may be 1.5 sampling periods delay or another depended on the sampling method in the digital control), d u and d d will be introduced for representing the unpredictable disturbance and deviation of the control law caused by the control delay, respectively. Therefore, the system state equation in (9) can be rewritten as − (a n + a n )x 2 (k) where a n = diag( a nα , a nβ ) and b n = diag( b nα , b nβ ) represent system parameter variations; d p (k) is the lumped uncertainty vector, which can be expressed as As for a real circuit, the ESR is generally small, and the parameters deviations of the LCL filter are bounded. Moreover, the output control efforts are always limited by hardware digital/analog (D/A) ports, even for the divergence of the designed control inputs ( u f ). Therefore, lumped-uncertainty components in (11) can be reasonably assumed to be bounded by positive constants for a real system. Here, the lumped uncertainty vector is assumed to be bounded by d p (k) < ρ, in which · is the Euclidian norm, and ρ is a given positive constant.
By adopting the active damping method and considering the control delay, the active damping can be modeled as a virtual impedance, rather than as a pure resistor. However, it will shift the resonant frequency, and affect the damping performance and the system stability [12]. Note that, the proposed discrete-time backstepping sliding-mode control (DTBSMC) strategy does not need active damping mechanism, and the system stability is directly designed by the Lyapunov stability theorem. The influence of the control delay had been considered as the bounded lumped-uncertainty component ( d p ) in (10). Thus, the corresponding control law should be designed to satisfy the stability requirement in practical applications. Because the control delay will increase the value of d p in a certain extent, it may enlarge the chattering phenomenon in the traditional sliding-mode control (SMC) method. Fortunately, the element in d p caused by the control delay is limited and bounded, and the chattering phenomena can be alleviated by the design of an additional term in the proposed control framework.

III. DISCRETE-TIME BACKSTEPPING SLIDING-MODE CONTROL METHOD
In this section, the proposed discrete-time backstepping sliding-mode control (DTBSMC) method is designed to ensure the system stability and control the grid current vector (x 1 ) to track the reference signal vector (x d ), which is a purely sinusoidal signal synchronizing with the grid voltage. The control block diagram of the proposed DTBSMC strategy is depicted in Fig. 2. The three-order system model will be divided into several subsystems by the backstepping control method (in Step 1 and Step 2), and each virtual control law (α 1 and α 2 ) for the corresponding subsystem will be designed via the discrete-time Lyapunov stability theorem in every step. Because of the step-by-step recursively algorithm with strictly Lyapunov stability functions, this backstepping control method can guarantee the internal dynamic stability of the proposed DTBSMC strategy. The sliding-mode control method is designed in Step 3, which will ensure that the system state trajectory reaches the sliding surface, and the tracking error of the grid current can converge to zero. The final Lyapunov stability function in Step 3 is selected for guaranteeing the system to be global stability without using the active damping methods, even for the migration of the LCL filter resonance frequency or the existence of the grid impedance. The major reason of the proposed DTBSMC method without the utilization of extra damping methods is the overall control system to be directly designed via the Lyapunov stability theorem [34]. The control law of the proposed method is derived as a whole by considering system dynamic models and strictly Lyapunov stability functions, which satisfy both the requirement of the Lyapunov stability theorem in subsystems for internal dynamic stability and also the final Lyapunov stability function in Step 3 to achieve the objective of global stability. Hence, the proposed DTBSMC strategy does not need extra damping methods for compensating the system stability requirements. The detailed design process is step by step presented as follows.
Step 1: Define the tracking error vector between the controlled grid current vector ( x 1 ) and the reference signal vector ( x d ) as where e 1 = [ e 1α e 1β ] T and x d = [ x dα x dβ ] T . Then, the forward step of (12) can be obtained according to (10) as Choose the first Lyapunov function candidate [34] as where γ 1 is a positive constant. Hence, the difference of V 1 (k) can be derived as where T is the first virtual control law vector, and e 2 (k) = x 2 (k) − α 1 (k) is the second error vector between x 2 (k) and α 1 (k). The first virtual control law vector (α 1 (k)) in the step 1 can be given as By substituting (16) into (15), one can obtain As can be seen from (17), V 1 (k) will be negative definite if the condition of e 2 (k) = 0 can be ensured. Therefore, the next step is carried out to ensure the vector of e 2 for converging to zero.
Step 2: For guaranteeing the vector of e 2 to converge to zero, the second Lyapunov function can be chosen as where γ 2 is a positive constant to satisfy the condition of γ 1 T 2 s < γ 2 < 2/T 2 s . The ratio factor γ 1 and γ 2 here are used to introduce the physical units for the sum of Lyapunov functions to have a unified physical unit. According to (10), one can obtain the error vector of e 2 as The difference of V 2 (k) can be derived as where α 2 (k) = [ α 2α (k) α 2β (k) ] T is the second virtual control law vector, and e 3 (k) = x 3 (k) − α 2 (k) is the third error vector between x 3 (k) and α 2 (k). The second virtual control law vector ( α 2 (k)) can be designed as By substituting (21) into (20), V 2 (k) can be derived as The second virtual control law vector (α 2 (k)) contains the future information in the right side of (21), which will make the control law more complex as the virtual control law to be introduced recursively in each step by the backstepping design procedure. This is called a non-causal problem to be mentioned in [29], [30]. The non-causal problem will emerge when one constructs a controller for a general strict-feedback nonlinear system via the backstepping design in the discretetime domain. This drawback can be solved by transforming the system state equation into a special form, which is suitable for the backstepping design via a time-varying mapping technique [30]. The basic idea is that if one considers the dynamic system model in (10) to be a one-step ahead predictor, and then one can transform the one-step ahead predictor into an equivalent maximum three-step ahead predictor, which can predict the future states [35]. By the time-varying mapping, the dynamic system model in (10) can be transformed as By moving one more step for x 1 and x 2 , that is moving the state k + 1 to k + 2 of the first two equations in (10), the state variable x 1 (k+1) and x 2 (k+1) will become equivalent as Similarly, after further step, the state variable x 1 of the first equation in (10) will be transformed into x 1 (k+3), and it can be expressed as Thus, the state variables of x 1 (k+1), x 2 (k+1), and x 3 (k+1) in (10) will be transformed into x 1 (k+3), x 2 (k+2), and x 3 (k+1) as shown in (23). As a result, the non-causal problem in the derivation of the discrete-time backstepping design can be avoided when the control law is constructed based on the maximum three-step ahead predictor form. Fortunately, one can redesign the virtual control law vectors just as the previous steps, but without the causality contradiction. Theorem 1: Consider the discrete-time dynamic system model of a three-phase LCL-type grid-connected inverter with a three-step ahead predictor represented by (23), if the control law vector of the proposed DTBSMC method is designed as (26), then the system stability of the designed DTBSMC method can be guaranteed.
/T s ; sgn(·) represents the sign function; k 1 and k 2 are positive constants; and the term of F(k) can be expressed as For satisfying the stability requirement, the coefficients of 0 < k 1 < (γ 2 − γ 1 T 2 s )/2 and d p (k) < ρ should be satisfied, and the ratio factors (γ 1 and γ 2 ) should meet the condition of γ 1 T 2 s < γ 2 < 2/T 2 s . Note that γ 1 and γ 2 adopted in V 1 and V 2 are just used to introduce the physical units so that the sum of Lyapunov functions can have a unified physical unit. Thus, both the values of γ 1 and γ 2 can be directly set to be 1. The detail stability proof is presented in the Appendix A.
In this study, a discrete-time control strategy without using the damping method has been designed. By divided into several subsystems, each virtual control laws (α 1 and α 2 ) have been designed with strictly discrete-time Lyapunov stability functions (V 1 and V 2 ) in Step 1 and Step 2 without the noncausal problem. Because of the design of strictly Lyapunov stability functions in each subsystem, the internal dynamic stability can be ensured for the discrete-time sliding-mode control in Step 3. The final control law ( u f ) of the proposed DTBSMC method has been designed in (26) without any requirement of the damping method. As can be seen the stability analysis in the Appendix A, the final control law ( u f ) can guarantee the difference of final Lyapunov function candidate ( V n 3 ) in (A15) to be a negative-definite function, which means the system stability can be ensured even without the complex damping method. Moreover, the variation of L G mainly affects the value of b n in (10), which is namely a part of d p as shown in (11). As can be seen in (A15), the negative-definite function can be guaranteed all the time if the condition of d p (k) < ρ can be satisfied. By considering the value of L G and the margin for other parts in d p to be bounded in practice, the system stability can be ensured if a relatively larger value of ρ is selected. In order to alleviate the chattering phenomena in (26), the parameter of k 2 will be also helpful to dominate the fact that V n 3 < 0, which also increase the robustness to the grid impedance. As our knowledge goes, no research papers investigate the combination of backstepping control and SMC to deal with the non-causal problem for the application of the discrete-time control strategy without using additional damping methods to an LCL-type grid-connected inverter under a power grid with varied grid impedances.

IV. NUMERICAL SIMULATIONS
In order to verify the effectiveness of the proposed discrete-time backstepping sliding-mode control (DTBSMC) strategy, numerical simulations are carried out by the MAT-LAB/Simulink software. The circuit parameters and operational conditions are given in Table 1. The values of the converter-side and grid-side inductances (L 1 and L 2 ) in the three-phase inverter via the SVPWM are determined according to the design criteria in [7] and [36]. According to [7], the value of L 1 is designed as the requirement of the ripple current on L 1 , and, the expression of L 1 can be expressed as where f sw is the switching frequency; m r is the modulation index; I 2 is the ripple current and can be chosen as 25% of the rated current. When the values of U dc = 350V, I 2 = 25%×10.6A, and f sw = 10kHz are selected, one can obtain that L 1 ≥ 978µH. Thus, the value of L 1 is selected as 1mH in this study. The value of L 2 is designed according to the guideline in [36], where the converter harmonic voltages by Bessel functions is applied to design L 2 for reducing the gridcurrent harmonics and meeting the harmonic distortion limits according to the IEEE-519 standard [37]. According to [36], the expression of L 2 can be expressed as where |V aN (jω h )| is the output voltage of h-th order harmonic on phase-a; ω h is the h-th order angular frequency; λ h is the ratio of the value of filtered current at ω h to the rated grid current. By considering the largest converter voltage harmonics occurred at the switching side-band frequency as h = 10k − 120 = 9.88kHz, the value of ω h = 19760π rad/s can be determined. Because the order of h is higher than 33th according to IEEE-519 standard, the harmonic should be less than 0.3%, and then λ h = 0.15% is taken. When the values of L 1 = 1mH, C f = 20µF, ω h = 19760π rad/s, λ h = 0.15%, and I 2 = 10.6A are selected, one can obtain that L 2 ≥0.41mH. By considering the harmonic frequency in comparative experiment to be higher than onesixth of the sampling frequency without damping, the value of L 2 is selected as 0.45mH in this study. Note that, filtering inductors in numerical simulations have already considered their parasitic resistances, where the corresponding values are determined according to the physical inductors used in experiments. The values of the equivalent series resistances (ESR) of the filtering inductors (L 1 and L 2 ) are 0.044 and 0.028 , respectively, which are measured by the LCR meter (LCR-819, Gwinstek Company). The system stability of the proposed DTBSMC strategy is directly designed by the Lyapunov stability theorem, and the influence of the ESR ( d r (k)) had been considered as the bounded lumped-uncertainty component in (11). Thus, the corresponding control law can satisfy the stability requirement in practical applications. Moreover, the control coefficients of the proposed DTBSMC method are chosen according to the following guidelines. The value of k 1 is decided by considering the convergent time of the tracking error vectors, and they should satisfy the conditions of 0 < k 1 < (γ 2 − γ 1 T 2 s )/2 to ensure the Lyapunov stability requirement. The value of k 2 is selected to ensure the system stability, and it is helpful to reduce the chattering phenomena introduced by the sign function. Increasing the value of k 2 properly can dominate the fact that V n 3 < 0 even the worst case d p (k) > ρ happens, which means that the parameter ρ could be conservatively selected to avoid increasing the chattering phenomena caused by the term of ρsgn(s(k)). The parameter ρ can be roughly determined by the limited range of the possible occurrence of parameter variations and external disturbance. By considering the value of the LCL filter deviating 20% and the grid impedance to be 10mH, the estimated value of ||d p || can be calculated according to (11). By using the parameters in Table 1, one can roughly obtain the theoretical bound value of ||d p || to be about 1.47 × 10 9 . However, the selection of a large value ρ will cause the control effort with serious chattering phenomena and deteriorate the corresponding control performance. Fortunately, the selection of a large value k 2 is helpful for reducing the value of ρ in the proposed methodology because the termk 2 s(t) in (26) will dominate the fact of V n 3 < 0. By considering a better transient control performance as well as the requirement of stability, and combining the possible occurrence of uncertainties through numerical simulations, the values of k 1 = 0.1, k 2 = 50, and ρ = 30 via the trialand-error procedure are selected in the proposed DTBSMC method.
In order to illustrate the superiority of the proposed DTB-SMC method, numerical simulations of a traditional quasi proportional-resonant (PR) method in [38], and a backstepping control method in [24] are compared in this study. The intention of this study is not to draw the conclusion that the non-linear methods are definitely better than linear control methods, but to adopt a non-linear control structure, which combines the merits of the backstepping control method and the SMC theory, to design a discrete-time control strategy without using additional damping methods. The linear control structure, which adopts the PR controller commonly used in the control strategy without using additional damping methods, is taken as a contrast to show the robustness of the proposed DTBSMC strategy with respect to the grid impedance. Similarly, another non-linear control structure, which adopts the backstepping structure, is also compared with the proposed DTBSMC strategy. The PR controller can be expressed as where s is the Laplace operator; k rp and k ri are the proportional gain and the resonance gain, respectively; ω 0 and ω b are the resonance frequency and the bandwidth of the resonance controller, respectively. The discrete form of (26) can be expressed as [38] G PR (z) = k rp + 2k ri ω b T s (z − 1) where z is the z-transform operator. ω 0 = 120π rad/s is the fundamental angular frequency, and ω b = 1.2π rad/s is selected to deal with a typical ±1% variation of the grid fundamental frequency. k rp = 4 and k ri = 80 are chosen by the root locus method as the design guideline in [38].
Ge et al. [24] adopted the backstepping control method for an LCL-type converter to be used as an active power filter. The control structure of the backstepping control method in [24] is implemented into three steps, and each step has a corresponding Lyapunov function for ensure the stability of each control law in every step. Finally, the tracking errors in each step will asymptotically converge to zero, and the grid current can be ensured to track the reference signal. Hence, the idea of the backstepping control method in [24] is adopted in this study as a comparative algorithm. By following the design steps in [24], the control structure is also divided into three steps to eventually ensure the grid current to be controlled. In addition, the backstepping control method is also discretized in this study for a fair comparison with the proposed DTBSMC strategy. The design procedure of the backstepping control method is similar to the design steps in the Appendix A, but without the SMC method in Step A3. The control law (u fb ) of the backstepping control method can be expressed as The detail derivation of (30) can be referred to Appendix B. In this study, the focus is to design a control strategy without using the damping method, and keep the strong robustness to the grid impedance. A common proportional-resonant (PR) controller without the damping method is adopted for comparisons in this study, and it belongs to a linear control structure. The PR controller also can ensure the system stability without using the damping method, but only in the case that the initial value of the LCL resonance frequency is higher than the critical frequency (f s /6) for the grid-side current feedback [39]. More importantly, the proposed DTBSMC method does not use extra damping method, and can ensure the system stability without the limit of the critical frequency, whatever the value of the LCL resonance frequency changes. The proposed DTBSMC method can guarantee the system stability without using the complex active damping strategy. Although the PR controller can also ensure stability without damping if the LCL filter resonance frequency is higher than the critical resonance frequency, which is one-sixth of the sampling frequency (f s ), the strong robustness to the grid impedance will be hardly kept. In Fig. 3, three control methods are carried out under an ideal power grid, where the grid impedance is zero. The resonance frequency of the LCL filter in Table 1 is 2kHz, which is larger than f s /6 (1.6kHz). It can be seen from Fig. 3(a)-(c) that all these control methods can keep the grid current with a very good sinusoidal waveform under the ideal power grid. In Fig. 3(c), the total-harmonicdistortion (THD) value of the proposed DTBSMC method is 1.21% to be the lower one. The proposed DTBSMC strategy can provide a high-quality current to the power grid, and the system stability can be guaranteed with the control design by the Lyapunov stability theorem in Section III.
For verifying the robustness of the proposed DTBSMC strategy under the existence of the grid impedance, numerical simulations with different grid impedances (2mH and 3mH) are depicted in Figs. 4 and 5, respectively. In Fig. 4, the grid impedance is increased to 2mH, and the resonance frequency of the LCL filter is declined to 1.3kHz, which is smaller than f s /6 (1.6kHz). As can be seen from Fig. 4(a), the grid current caused by the PR method is hardly to keep stability owing to the increased grid impedance. Although the backstepping control method still can guarantee the system stability in Fig. 4(b), the THD value of the grid current is increased to be 2.74%. When the grid impedance continues to increase to 3mH, the THD value of the grid current is further deteriorated to 3.04%, which is increased more than 125% comparing with the one under ideal power grid in Fig. 3(b). In contrast, because of the strong robustness of the proposed DTBSMC strategy, the grid current still can keep desirable quality when the grid impedance exists, and its THD values in Fig. 4(c) and Fig. 5(c) are only 1.86% and 1.97%, respectively. Compared with Fig. 5(c) and the one under ideal power grid in Fig. 3(c), the increment of the THD value is reduced almost 55% than the backstepping control method in Fig. 5(b). According to the analysis of the Lyapunov stability theorem in Section III, the proposed DTBSMC strategy can guarantee the strong robustness, and the stability is independent of the varied grid impedance. Therefore, the system performance under wide range grid impedance variation from 1mH to 10mH is depicted in Fig. 6. Comparing with the performance of the backstepping control method in 10mH grid impedance, the increment of the THD value of the proposed strategy can be reduced more than 64%. Even though the grid impedance has large variations, the proposed DTBSMC method still can guarantee the global asymptotic stability as well as the desirable current quality.
In order to examine the influence of the values of ρ and k 2 on the control chattering phenomena, numerical simulations    of the three-phase LCL-type grid-connected inverter system operated from Case 1 to Case 4 are given in Fig. 7. In Fig. 7, the values of ρ and k 2 are respectively selected as ρ = 1×10 6 and k 2 = 0 for Case 1; ρ = 8 × 10 6 and k 2 = 0 for Case 2; ρ = 2 × 10 7 and k 2 = 0 for Case 3; ρ = 30 and k 2 = 50 for Case 4. As can be seen from Fig. 7(a), the system cannot keep stability because only the value of ρ is considered and its value is too small. By gradually increasing the value of ρ for Case 2 and Case 3 in Fig. 7(b), the system can achieve the stable control objective. Unfortunately, the control chattering phenomena are accordingly augmented with the increasing value of ρ, and the value of THD is increased to 18.54%. As for Case 4 in Fig. 7(b), owing to the parameter of k 2 is considered, a smaller value of ρ can be selected and the corresponding chattering phenomena can be alleviated.
Remark 1: In general, the PR controller via the selection of appropriate control parameters also can ensure the system stability without using the damping method if the LCL filter resonance frequency is higher than the critical resonance frequency, which is one-sixth of the sampling frequency (f s ). However, the strong robustness to the grid impedance will be hardly kept by the PR controller under the variation of the resonance frequency to be lower than this critical resonance frequency. When the LCL filter resonance frequency is smaller than f s /6 owing to the increased grid impedance,  the PR controller without the damping method cannot keep the system stability as shown in Fig. 4(a). By comparing Fig. 4(a) with Fig. 4(c), the proposed DTBSMC method without any extra damping strategy can ensure the system stability no matter where the resonance frequency locates.

V. EXPERIMENTAL RESULTS
An experimental prototype has been constructed as shown in Fig. 8 to verify the effectiveness of the proposed discretetime backstepping sliding-mode control (DTBSMC) scheme. The control algorithm is implemented by a digital signal processor (DSP) and a field-programmable gate array (FPGA) digital microcontroller, where the DSP (TMS320F28335) is a main algorithm controller, and the FPGA (XC3S400)   is an assistant controller for implementing the pulse-widthmodulation (PWM), sampling signal processing, and fault protection, etc. The experimental circuit parameters and operational conditions are the same as Table 1. By considering the practical situation, the actual parameters of the LCL filter could be slightly different from nominal values in Table 1.
Although the derivation of the proposed DTBSMC strategy has many equations in Section III, most of them are intermediate variables or the derivations of the stability for each step. The overall control law of the proposed DTBSMC strategy is u f (in (26)), which can be easily implemented in a DSP. Only grid-current sensors are required for sampling the grid currents ( x 1 ) and send them to the DSP. Then, the final DTBSMC law ( u f ) in (26) can be calculated, where the virtual control law (α 1 and α 2 ) and sliding-surface ( s) are just intermediate variables, and the actual calculation of u f in the DSP depends only on x 1 and the current reference signal (x d ). Moreover, the control law will be sent to the program of space-vector pulse-width-modulation (SVPWM) in a FPGA digital microcontroller for generating corresponding drive signals. VOLUME 8, 2020   [38], the backstepping control method in [24], and the proposed DTBSMC method under an ideal power grid, where the experimental conditions are the same as the ones in Fig. 3. In Fig. 9-11, all of the three control methods can ensure the system stability under the ideal power grid, and all the grid currents can be kept in good quality. The tracking error index can be defined as e s = e 2 1α + e 2 1β , where e 1α and e 1β are the tracking errors between the grid currents and its references in the stationary αβ frame, respectively. The maximum value of e s for the proposed DTBSMC method is 0.22A, which is 15% and 12% smaller than the PR method and backstepping control method in Fig. 9(a) and 10(a), respectively. As can be seen from Fig. 11, the proposed DTBSMC strategy can provide a high quality current to the power grid, where the corresponding total-harmonic-distortion (THD) value is 1.816% to be the lowest one of the three control methods.
In order to illustrate the strong robustness of the proposed DTBSMC strategy, experimental current responses and spectrums with 1.92mH grid impedances for all three methods are depicted in Fig. 12-14. Because the varied resonance frequency is lower than the critical resonance frequency (f s /6) attributed to the grid impedance, the PR method in Fig. 12 is hard to keep stability any more. Although the total inductance of the LCL filter is only about 1.5mH, the resonance frequency will migrate to about 1.342kHz (lower than the f s /6) by considering the 20µF filter capacitor and the 1.92mH grid impedance. As can be seen from Fig. 12, the resonance still occurs, even the practical situation contains resistive components (e.g., ESR). In Fig. 12(b), the resonance frequency is the same about 1.3kHz, which is accordance with the theoretical analysis. The similar result also can be seen from the simulation in Fig. 4(a), where the grid impedance is 2mH. As can be seen from Fig. 14, the proposed DTBSMC method still can guarantee the system stability, even though the grid impedance is increased to be 1.92mH. Moreover, the grid current caused by the proposed DTBSMC method can be kept in desirable performance, where the THD value is 2.458%. Although the backstepping control method also can keep the system stability as shown in Fig. 13, the grid impedance has more serious impact on it. The corresponding THD value in Fig. 13(b) is 3.279%, where the increment  is 48% larger than the proposed strategy, and the ratio will continue to grow as the gird impedances increase. Similar to numerical simulations in Figs. 4-6, owing to the advantages of the backstepping control structure and the slidingmode control theory, the proposed DTBSMC scheme can have strong robustness to the variations of grid impedances. As can be seen from Fig. 6, the THD value of the proposed DTBSMC strategy is 1.86% as the grid impedance is 2mH. Although the value of the grid impedance is increased to be 10mH, the increase of the THD value is relatively limited, and its THD value is only 2.48%. Note that, inductors in numerical simulations are ideal devices and their values are fixed. However, inductors in the experiments are non-ideal and varied, and the inductance permeability will be decreased as the current is increased. By comparing Fig. 14 with Fig. 6, the THD value of the grid current in the experimental result is larger than the one in the numerical simulation. Figure 15 shows the experimental results under an unbalanced power grid for all the three methods, where the grid voltage drops 10% and 20% in phase-B and phase-C, respectively. The definition of m u is the unbalanced current index, and it can be represented as where Max(·), Min(·) and Avg(·) represent the maximum, minimum and average root-mean-square (RMS) current of the three-phase grid currents (i 2a , i 2b , i 2c ), respectively. As can be seen from Fig. 15(b), the unbalanced current index (m u ) of the backstepping control method and the PR method is 8.3% and 3.7%, respectively. Owing to the characteristic of strong robustness, the grid current of the proposed  [24] and PR method in [38].
DTBSMC method still can maintain desirable sinusoidal waveforms in despite of the unbalanced grid voltage. The unbalanced current index (m u ) of the proposed DTBSMC strategy in Fig. 15(a) is 2.7%, which is 27% and 67% smaller than the PR method and the backstepping control method shown in Fig. 15(b), respectively. Figure 16 shows the experimental results of the proposed DTBSMC method, the PR method in [38] and the backstepping control method in [24] as the grid voltage to be suddenly dropped 15%. As for the occurrence of the grid voltage variation, the maximum fluctuation of the tracking error index (e s ) by the PR method is about 1A in Fig. 16(b); the one by the proposed DTBSMC method in Fig. 16(c) is reduced almost 50%. As can be seen from Fig. 16, the grid voltage dip has very slight impact on the grid current of the proposed DTBSMC method. Moreover, the grid current fluctuation of the proposed DTBSMC method can be stabilized within about 1.6ms, which is 16% and 36% faster than the PR method and the backstepping control method in Fig. 16(b) and 16(c), respectively.
In order to verify the dynamic response, the experimental results of the proposed DTBSMC method, the PR method in [38] and the backstepping control method in [24] under the variations of current references are depicted in Fig. 17. The step change of the current reference is from 6 A to 4 A. By comparing with the PR method and the backstep-  [38]. (c) Backstepping control method in [24]. ping control method in Fig. 17(b) and 17(c), the proposed DTBSMC strategy can have a very fast dynamic performance in Fig. 17(a), and the grid current can track the changed reference rapidly in almost 1.7ms, which is 19% and 57% faster than the PR method and backstepping control method, respectively. Owing to the step-by-step recursively algorithm with strictly stability condition, the backstepping control architecture can ensure all the state variables of the SMC method to be internal dynamic stability. Therefore, the characteristic of fast-dynamic response in SMC still can be retained because the error of state variables will converge exponentially to zero once the system state trajectory reaches the sliding surface. The experimental performance comparisons of three control methods are summarized in Table 2. As can be seen from Table 2, the proposed DBSMC strategy indeed yields superior performance than the PR method in [38] and the backstepping control method in [24].
The proposed DTBSMC method can achieve the strong robustness to the grid impedance, and does not need the additional carefully design for the damping strategy. By considering the dynamic system model, the control system is directly designed as the requirement of the Lyapunov stability theorem [34]. Although the derivation contains many equations, VOLUME 8, 2020   [38]. (c) Backstepping control method in [24]. most of them are intermediate variables and the stability derivation by following the Lyapunov stability theorem. What is more, the control law is derived in the discrete-time, which will make the form of the equation complex, but it is more suitable for the digital signal processor. Eventually, only the control law in (26) is finally implemented to the control object. The execution times of each control method are also provided in Table 2. As can be seen from Table 2, the execu-  tion times of the PR method in [38], the backstepping control method in [24], and the proposed DTBSMC method are 13µs, 14.25µs and 16.25µs, respectively. Although the execution time of the proposed DTBSMC method is 25% larger than the one of the PR controller, the stronger robustness and better performance can be achieved by the proposed DTBSMC method. In addition, by considering the advanced computing power of the microprocessor and the 100µs sampling time, the execution time of the proposed DTBSMC method is still acceptable.
In order to test the performance of the proposed DTBSMC strategy with considering the grid impedance, experimental results with 1.92mH grid impedance under the grid voltage drip and the step change of current reference are depicted in Figs. 18 and 19, respectively. The grid voltage is dropped 15% in Fig. 18, and the reference current is changed from 6A to 4A in Fig. 19. Compared with the performances in Figs. 16(a) and 17(a), the existence of the grid impedance will increase the settling time. Fortunately, the corresponding impact is still slight owing to the strong robustness of the proposed DTBSMC strategy.

VI. CONCLUSIONS
This study has successfully designed a discrete-time backstepping sliding-mode control (DTBSMC) method for an LCL-type grid-connected inverter. Through the step-by-step virtual control designs, the proposed DTBSMC strategy can ensure the system asymptotic stability. Based on the discretetime Lyapunov stability theory, a suitable control law is derived to guarantee the accurate control and global stability of the overall closed-loop system. Moreover, a detailed design procedure is presented for overcoming the difficulty of a non-causal problem. What is more, the proposed DTBSMC method combines both advantages of the backstepping control method and the sliding-mode control theory. Therefore, the internal dynamic stability can be ensured by the several subsystems design. In addition, the proposed DTBSMC scheme can realize the features of good dynamic performance and strong robustness to the grid impedance without the specific design of damping strategy. Compared with the traditional proportional-resonant (PR) method and the backstepping control method in experiments, the total-harmonicdistortion (THD) value of the proposed DTBSMC method is 2.458%, where the increment of the THD value is reduced 48% when the grid impedance is 1.92mH. Furthermore, the settling time under the reference variation of the proposed DTBSMC strategy is 19% and 57% faster than the traditional PR method in [38] and the backstepping control method in [24], respectively. Numerical simulations and experimental results are provided to verify the effectiveness of the proposed DTBSMC strategy.

APPENDIX A
The design procedure and stability verification of the proposed DTBSMC method in Theorem 1 are represented as follows.
Step A1: The first Lyapunov function candidate is chosen as the same as (14). By considering the tracking error vector ( e 1 ) in (12), the difference of V 1 (k) can be derived as where e n 2 (k) = x 2 (k) − α n 1 (k − 2) is the new second error vector because of the two-step ahead predictor form of x 2 in (23). In order to achieve the negative-definite form of e 1 in (A1), a new first virtual control law vector (α n 1 ) in the step A1 can be given as which can be equivalently expressed as By substituting (A2) into (A1), one can obtain V n 1 (k) = γ 1 2 T 2 s e nT 2 (k)e n 2 (k) − γ 1 2 e T 1 (k)e 1 (k) (A4) The difference of V 1 (k) in (A4) is similar to the one in (17). In order to force (A4) to be a negative-definite function, the next step is to guarantee the vector of e n 2 for converging to zero.
Step A2: For ensuring the second error vector (e n 2 ) can converge to zero, one can choose a new second Lyapunov function candidate as V n 2 (k) = γ 2 2 e nT 2 (k)e n 2 (k) + V 1 (k) (A5) Therefore, the difference of V n 2 (k) can be derived as V n 2 (k) = γ 2 2 x 2 (k)+T s e n 3 (k)+T s α n 2 (k −1)−α n 1 (k −1) T × x 2 (k) + T s e n 3 (k) +T s α n 2 (k − 1) − α n 1 (k − 1) For guaranteeing the term of V n 2 to be negative definite, and ensuring the vector of e n 3 to converge to be zero, the next step must be constructed.
Step A3: In this step, a sliding-mode control method is designed after the two steps in the backstepping design procedure. Define a sliding-surface vector as s(k) = e n 3 (k) + k 1 e n 2 (k) (A10) VOLUME 8, 2020 where k 1 is a positive constant. The control law vector of the proposed DTBSMC method is assumed to take the following form: u f (k) = b −1 n [F(k) − ρsgn(s(k))−k 2 s(k)−e n 3 (k)+k 1 e n 2 (k)] (A11) where sgn(·) represents the sign function; k 2 is a positive constant; and the term of F(k) can be expressed as F(k) = −x 3 (k) + a n x 2 (k) − k 1 x 2 (k) − k 1 T s x 3 (k) + α n 2 (k) + k 1 α n 1 (k − 1) (A12) Define the third Lyapunov function candidate as If the condition of d p (k) < ρ is satisfied, and the coefficients of 0 < k 1 < (γ 2 − γ 1 T 2 s )/2 is selected, V n 3 (k) is a negative-definite function. According to the Lyapunov stability theorem [34], the system stability of the proposed DTBSMC method can be guaranteed, and the corresponding condition is independent of the grid impedance. In addition, the error vectors of e 1 , e n 2 and e n 3 will asymptotically converge to zero, and it implies that the grid current can track the current reference signal ( x d ).

APPENDIX B
The design procedure of the backstepping control method is also divided into three steps. Because the backstepping structure is the same as the part of the proposed DTBSMC strategy, the derivation of the first two steps of the backstepping control method makes no difference (Step A1 and Step A2). The difference of the derivation is in the third step without using the SMC method as shown as follows.