A Multi-Slope Sliding-Mode Control Approach for Single-Phase Inverters under Different Loads

In this paper, a new approach to the sliding-mode control of single-phase inverters under linear and non-linear loads is introduced. The main idea behind this approach is to utilize a non-linear, flexible and multi-slope function in controller structure. This non-linear function makes the controller possible to control the inverter by a non-linear multi-slope sliding surface. In general, this sliding surface has two parts with different slopes in each part and the flexibility of the sliding surface makes the multi-slope sliding-mode controller (MSSMC) possible to reduce the total harmonic distortion, to improve the tracking accuracy, and to prevent overshoots leading to undesirable transient-states in output voltage that occur when the load current sharply rises. In order to improve the tracking accuracy and to reduce the steady-state error, an integral term of the multi-slope function is also added to the sliding surface. The improved performance of the proposed controller is confirmed by simulations and finally, the results of the proposed approach are compared with a conventional sliding-mode controller (SMC) and a synchronous reference frame PI (SRFPI) controller.


Introduction
The single-phase inverters are widely used in various applications, such as uninterruptable power supplies [1,2], power filter [3,4], motor derives [5,6], renewable energy conversion [7,8], and etc.The major requirement of its control system is to achieve a proper ac voltage regulation with fast dynamics response for sudden change at loads, good disturbance rejection and a stringent frequency regulation during transients, all this while retaining almost zero steady-state error under linear and non-linear loads [9].
In order to achieve these requirements, numerous control methods have been proposed in the literature [10,11].Owing to availability and desired process of implementation, prevalent techniques based on repetitive control [12,13], deadbeat control [14,15], multi-loop regulation strategies [16], neural network methods [17,18], iterative learning control [19], adaptive control method [20], have been introduced recently.Repetitive control has acceptable ability to reject periodic disturbances but slow dynamics, poor tracking accuracy, and inappropriate performance to non-periodic disturbances are the weaknesses of this method.
Deadbeat controller provides fast dynamics performance in direct control of inverters.This method is easy to implementation and appropriate design of this method leads to prevent overshoot and ringing, but suffers from the drawbacks, such as sensitivity to system parameters, uncertainties and loading conditions.Continuous-time control strategies based on sliding-mode control technique have been proposed in [21][22][23][24][25], a discontinuous method has been proposed in [21] and a Lyapunov-Function-Based control strategy is also proposed in [22].The approach in [23] has proper performance but not desired due to the high inductor's current of the output filter which is used as state variable, and moreover, implementation of its switching function is based on the computation of separate sliding surface for each leg with the associated hardware complexity.The sliding mode controller with linear surface is simple to achieve but due to the single-slope and inflexible sliding surface of this method for all values of error, the proper tracking accuracy and transient-states for different values of the error cannot be achieved.In this control structure, for small slopes of the surface, the tracking accuracy is decreased and for large slope of the sliding surface, poor transient-states and large overshoots are generated in output voltage in moments of loading.
It has been shown that the inability of the conventional sliding-mode controller and fast SRFPI controller [24] to achieve high-speed operation in reducing the tracking error and desired transientstates in moments of loading at the same time can be overcome by using the proposed control method without the need for complex algorithms.It is worth mentioning that most of the aforementioned drawbacks occur due to the constant slope of the sliding surface or inflexibility of the control method.
In this paper, a new approach to the sliding mode-control of single-phase inverters is introduced.In this approach, by using a non-linear multi-slope (MS) function, a non-linear sliding surface function is generated.In general, this surface consists of two parts with different slopes and slope of the surface for low and large values of the error are different.Therefore, by utilizing this multi-slope function, a considerable improvement can be achieved in transient-states, tracking accuracy and speed of error reduction.The important consequences of this sliding-mode controller are low total harmonic distortion (THD) under different loading conditions, high-speed performance with desired transientstates with a significant reduction in overshoots in loading moments.

System Description
The main structure and power stage of a single-phase voltage source inverter (VSI) which is used in this research, as shown in Fig. 1 consisting of a switching part constructed by insulated-gate bipolar transistor (IGBT) or power Mosfet followed by 2nd or more order LC filter.Load in this case is located in parallel with the capacitor of the filter .In practice, to reduce the ESR effect and current ripple, several low ESR capacitors can be connected in parallel structure for the LC filter and it is worth mentioning that the small size of the filter capacitor is necessary and throughout this paper, the dclink voltage is assumed to be constant.It can be realized by utilizing large capacitances at the dc-link of the inverter.The parameters of the inverter are given in Table 1.The equations describing the dynamics of the inverter can be obtained as: where m describes the control variable.inductor can be ignored.The state-space equation describing the operation of the system can be written in matrix form, as: where Based on (1) and ( 2), the output voltage error (e(t)) and its derivative (rate of change of the output voltage error) are defined as: where 1 x  denotes the time derivative of 1 x , , the time-varying term D(t) is considered as a disturbance, and is the reference for output voltage.The disturbance term is defined as: The behavior of the system can be expressed by the following state space controllable canonical form: Control of single-phase inverters under linear and non-linear loads using the sliding-mode method is well-improved research topic now.In order to achieve a desired control performance in different loading conditions, the controller should have the ability to ensure the low steady-state error in the presence of the inverter model uncertainties as well as other disturbances.In addition to reducing the tracking error in moments of heavy loading, the controller must provide desired transient-states with negligible overshoots in the output voltage during the load current change.Therefore, the previous and single-slope method of sliding-mode control suffers from many problems such as single-slope and inflexible sliding surface, undesirable and poor transient-states in loading moments for sliding-mode controller with large slope of the surface, inability to achieve high-speed response and tracking accuracy simultaneously because of a significant reduction in existence regions for large slope of the surface.

Suggested Control Structure
In this section, the problems of conventional SMC will be explained briefly first, then, the novel non-linear function using to produce the multi-slope SMC (MSSMC) will be introduced, after that, its theoretical basis is developed and its coefficients adjustment is explained briefly.Finally, based on a mathematical analysis, the improved performance of the proposed method will be confirmed.Fig. 2 illustrates the suggested control structure, which consists of a MSSMC to regulate the instantaneous output voltage, state generator to provide the system states in order to use in proposed controller and a gate driver part, which is employed to trigger the switches of the VSI.The conventional sliding-mode controller with single-slope sliding surface suffers from the following drawbacks:

Conventional SMC Drawbaks
-In the conventional sliding function, the integral term is not usually used to accumulate the existing steady-state errors.Even if the integral term is used, lack of flexibility in the integral term still exists for different values of the error.The principal reason behind this problem is the fixed-slope of the integral function for any values of error which can lead to wind-up the integral for different values of the tracking error.
-As mentioned before, one of the most important drawbacks of the conventional sliding-mode controllers is single-slope and inflexible sliding surface.On the other hand, these controllers does not have the ability to achieve both high speed and desired transient-states in loading moment's simultaneously.The surface with small slope leads to a slow response, low tracking accuracy and the surface with large slope leads to a poor transient-state during the load current change.A tradeoff between small and large slopes of the surface, cannot realize both fast response and low THD response without any overshoots in output voltage simultaneously in moments of loading.

SMC Using Non-Linear Function
In order to overcome the conventional SMC drawbacks, to achieve a desired controller with high speed in reducing the tracking error in moments of loading, and to alleviate steady-state error under different types of loads without loss of the desired transient response, a flexible sliding surface composed of different terms of a special non-linear function is introduced.It should be noted here that each of these terms is made by a multi-slope non-linear function.Proposed sliding surface with an acceptable approximation consists of two general parts with different slopes, which are shown in Fig. (3-a) with part A and part B. As can be obtained, when the error is generated by load current disturbance, for driving the system states to the sliding surface, a control low is applied to the system.In the sliding mode, the state trajectory reaches to part B of the surface earlier and align the surface goes to the part A with high speed.Therefore, when the error value increases and output voltage drops because of the high current spikes of the load capacitance, the states trajectory hits the part B of the sliding surface first which can lead to a faster reaching mode.It is important to know that in the sliding mode, the VSI will be robust and by changing the slope of the surface, the system dynamics will also change.

Multi-Slope (MS) Function
The Multi-Slope (MS) function can be described as: (10) where α1, α2 and δ are adjustable coefficients of the function, which define the general form of the function and it is clear that MS function is a continuous and differentiable function.The variation of slopes in A and B parts and extent of these parts for different values of the multi-slope function coefficients (α1,α2,δ) are shown in Fig. (3-b).This clearly shows that the MS function includes two slopes and for small x values, the slope of the part A and for large value of x, the slope of the part B is more effective in this function.
In order to confirm the effectiveness of the MS function on the controller operation and to demonstrate the performance of the controller, the effectiveness of the proposed MSSMC has been described on a VSI loading behavior.

MS Function Coefficients Setting
The coefficients of the MS are α1, α2 and δ and to achieve a desired and fast dynamic response with almost negligible overshoot in the output voltage, these coefficients must be adjusted wisely.The task of these coefficients in the MS function structure are as follows: 1. δ coefficient: This coefficient adjusts the slope of the part A and should be positive (δ ˃0).On the other hand, by changing this coefficient, the gain of the function will be changed for small error values.
2. α1 coefficient: This coefficient is utilized in MS function to adjust the slope of the part B and its effect on slope of the part A is negligible.In principle, this coefficient is utilized to adjust the gain of the MS function for large values of the error.
3. α2 coefficient: This coefficient adjusts the height of the part A. Therefore, by increasing this coefficient, the height of the part A will increase.It is worth mentioning that by increasing the value of this coefficient, low values of error in the input of the function creates more effect in the output of the function.
In general, the MS function is a non-linear function and use of this function in sliding function leads to yield a fast response with a desired transient-state without the need for any complex algorithm.On the other hand, if the coefficients of the MS function are adjusted wisely, a desired sliding surface will be generated which leads to achieve a fast error convergence with a desired transient-state without any overshoots simultaneously.

Sliding Surface and Stability Considerations
The sliding surface in MSSMC is composed of three MS functions and can be defined as: In the sliding function, the integral term is added in order to reduce the steady-state error in output voltage and p, i, and d indexes represent the MS function in one of these three terms.When the sliding mode is reached (S=0), we have: As can be seen, due to the structure of the MS function, the sliding surface is non-linear.It is worth mentioning that for proper performance of the controller, α m1,2 must be positive where m can be p, i, and d.
The existence condition must be satisfied for stability of reaching mode as: We consider the Lyapunov function as , and it is clear that for global stability V  must be negative definite [25].If this condition is satisfied, the state trajectory will be moved toward the surface and sliding can be preseved until the trajectory reached to the origin.By taking the time derivative of S, substituting ( 6), ( 7) into the resulting equation gives: The above equation can be simplified as: The time derivative of the Lyapunov function is obtained as: In (17), ε 1 (x 1 ) and ε 2 (x 2 ) are defined as: Substitution of m=-sign(S) into (17) yields: The following inequalities can be obtained from (22): Fig. (3-c) illustrates the regions of existence of the sliding mode.It is clear from ( 23) and ( 24) that A 1 =0 and A 2 =0 are two lines in the x 1 -x 2 plane for the practical values of the inverter parameters.Note that the slopes of these lines are equal which means that these lines are parallel with each other.This slope can be obtained as: The intersection points of these lines with x 1 and x 2 axes can be obtained from ( 26)- (29).
If the following condition holds: Equations ( 28) and ( 29) can be approximated by: where k 1 (t) and k 2 (t) are defined as: Due to stability considerations, the existence regions should be kept as large as possible.As can be obtained from ( 25 can be adjusted wisely so that the high speed response is yielded and the overshoot is eliminated from the output voltage.Therefore, the distance between the intersection points of the x 2-axis and origin will be increased.In most cases, the following condition holds: It is clear that the system become unstable with very large value.In most cases, α i1 and α i2 are chosen so that the following condition holds: According to (36) and combination ( 23) and ( 24), we obtain: can be determined by: Clearly, the upper bound does not depend on one parameter and has a flexible and adjustable structure which can lead to a fast response without any overshoots.

Simulation Results
In this section, the simulation results of an inverter controlled by proposed multi-slope sliding-mode controller under linear and non-linear loads have been evaluated.Due to limited switching frequency of an inverter, the direct implementation of m=-sign(s) is not possible for a VSI in practice.In order to prevent chattering, operating frequency has to be limited to a finite form by replacing a suitable hysteresis function instead of sign function.It is worth mentioning that the simulations are carried out using the Sim-Power-Systems toolbox of Simulink.As can be obtained, in both cases, the trajectories reach to each other at the point A, but as can be seen, the drop of the output voltage for the inverter which is controlled by proposed method is lower than the conventional

Figure 2 .
Figure 2. Block diagram of the sliding-mode-controlled inverter with the proposed method

( 1 ε
mode existing region which can lead to an overshoot in output voltage.It is clear that for very large value of ) 0 , instability can be occurred in the system because of the excessive reduction of the existence regions.In order to increase the sliding mode existing regions and to reduce the overshoots in output voltage for large value of )

×
Fig. (4-a) shows the output voltages for an inverter controlled by proposed MSSMC and a fast controller with high performance proposed in [26].As can be obtained, both of them are adjusted to generate high speed response with a very low voltage ripple in output voltage.It is clear that, MSSMC leads to an output voltage without any overshoot and Fig. (4-b), shows a larger view of this figure in the loading moment.The simulation results of the proposed controller are obtained when the controller coefficients are chosen as:By utilizing these coefficients, the system response is fast with a negligible overshoot.Fig.(5-a) illustrates the simulated waveforms of output voltage and load current for a resistive load (R=3Ω).As can be seen, the regulation of the output voltage is preserved and the output THD is 0.51%.Fig.(5-b) depicts the output voltage and load current waveforms for a nonlinear triac-controlled resistive load (R=3Ω).The load current changes at the peak of the output voltage and as can be observed, the output voltage regulation is fully preserved and the tracking error reduces with high speed and the output voltage THD is limited under 2.69 % for this load.Fig. (6-a) shows the output voltage and sliding function for a nonlinear triac-controlled resistive load case.Because of the structure of m, the sliding function varies between zero and negative or positive value in positive and negative cycles of the output voltage respectively.Fig. (6-b) illustrates that the output voltage remaining Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 19 July 2016 doi:10.20944/preprints201607.0059.v1Peer-reviewed version available at Electronics 2016, 5, 68; doi:10.3390/electronics5040068constant when a sudden change occurs in the load resistance (at t =7ms) and confirms that the MSSMC has good robustness against the load disturbances.Output voltages of the inverter controlled by fast controller and proposed MSSMC Output voltages of the inverter controlled by proposed MSSMC and a fast controller when loading is performed at the peak of the output voltage

Figure 4 .
Figure 4. Output voltage of the inverter

Fig. ( 7
Fig. (7-a) shows the output voltage and load current for a bridge rectifier having 10Ω in parallel with a 700µF at the output of the bridge.Output voltage THD for this case was obtained as 0.51%.The output voltage and the sliding function for this bridge rectifier load is shown in Fig. (7-b).Fig. (7-c) depicts the state trajectoriesof the inverter controlled with proposed and conventional sliding mode controller.It is obvious that the response of the output voltage can be made faster and the output voltage drop (x 1 value) can be reduced by using the proposed MSSMC method.When the current disturbance occurs and the trajectory moves toward the surface, the reaching mode can be reached faster than the conventional SMC because of the improved surface structure.

Fig. ( 7
Fig.(7-d)  shows a comparison between the proposed and conventional SMC with the same reaching time.

Preprints
(www.preprints.org)| NOT PEER-REVIEWED | Posted: 19 July 2016 doi:10.20944/preprints201607.0059.v1Peer-reviewed version available at Electronics 2016, 5, 68; doi:10.3390/electronics5040068SMC.Fig.8shows the output voltage spectrum obtained by simulation and it is clear that the magnitude of the output voltage fundamental component is very larger than other components under non-linear load.

Figure 5 .Figure 6 .
Figure 5. Simulation results for a resistive load

Table 1 .
Parameters for inverter model