Feedback Linearization Control of a Shunt Active Power Filter Using a Fuzzy Controller

Abstract

In this paper, a novel feedback linearization based sliding mode controlled parallel active power filter using a fuzzy controller is presented in a three-phase three-wire grid. A feedback linearization control with fuzzy parameter self-tuning is used to implement the DC side voltage regulation while a novel integral sliding mode controller is applied to reduce the total harmonic distortion of the supply current. Since traditional unit synchronous sinusoidal signal calculation methods are not applicable when the supply voltage contains harmonics, a novel unit synchronous sinusoidal signal computing method based on synchronous frame transforming theory is presented to overcome this disadvantage. The simulation results verify that the DC side voltage is very stable for the given value and responds quickly to the external disturbance. A comparison is also made to show the advantages of the novel unit sinusoidal signal calculating method and the super harmonic treatment property of the designed active power filter.

Keywords

Active Power Filter Feedback Linearization Technique Sliding Mode Control Fuzzy Control

1. Introduction

With the rapid development of the power electronics technology, more and more nonlinear and time-varying devices, such as inverters, rectifiers and switching power supplies, are used in grid generating power quality problems. A shunt active power filter (SAPF) is an effective device for implementing the harmonic current in the grid and is attracting increasing attention in modern society. Saetieo et al.^[1] designed a three-phase, three-line parallel active power filter, establishing the basic structure of an active filter. Researchers have studied the control theory and the application technique of APF from different aspects and made numerous achievements. Singh et al.^[2] proposed a three-phase three-wire parallel active power filter and compare the harmonic compensation effect of the direct and indirect current control techniques. Stanciu et al.^[3] and Fei et al._[4] investigated novel sliding mode controllers separately to implement reference current tracking control and reduce the total harmonic distortion (THD).

The control method can be calculated based on another type of modelling, such as the state space representation in the (d, q) reference frame, which allows the use of a feedback linearization control technique and the development of an adaptive control. Feedback linearization control is an effective method for establishing the model of an active power filter. Mendalek et al.^[5–6] proposed an exact feedback linearization control for both loops and used the pole placement strategy to synthesize the closed loop error dynamics of current tracking and DC bus voltage regulation. Matas et al.^[7] implemented a feedback linearization based sliding mode control method and demonstrated the superior harmonic compensation capability of the proposed active power filter. In the actual power system, the supply voltage may contain harmonics and the parameters of the system model may be different from the actual value. Adaptive control approaches have been successfully applied in the active power filter. Shyu et al.^[8] presented a novel model reference adaptive control (MRAC) and designed a single-phase active power filter to improve the line power factor and reduce line current harmonics. Ribeiro et al._[9] presented a robust adaptive control strategy for power factor correction, harmonic compensation and the balancing of nonlinear loads in APFs. Lee et al._[10] discussed the DC bus voltage control of three-phase AC/DC PWM converters using feedback linearization, while Tan et al.^[11] proposed adaptive feedforward and feedback control schemes for sliding mode controlled power converters. Other harmonic suppression approaches have been investigated in the literature ^[12–15]. Fuzzy control is very robust and capable of handling nonlinear systems. Fei et al.^[17] designed a direct adaptive fuzzy control using a supervisory compensator to implement the robust tracking of the MEMS gyroscope sensor. Ha et al._[18] proposed a novel tuning fuzzy vector field orientation feedback control method for a four track wheel, skid-steered mobile robot using flexible fuzzy logic control. As the model of the active power filter is nonlinear, fuzzy control is highly suitable for use in the control of an active power filter. Tsengenes et al.^[19] designed a fuzzy logic algorithm for tuning PI controllers and improving the performance of a shunt active power filter under non-ideal grid voltages. Colak et al.^[15] designed a single phase shunt active power filter and implemented the regulation of the DC side voltage using a fuzzy logic controller.

In this paper, the input-output feedback linearization is applied to implement the DC side voltage regulation in an indirect current-controlled active power filter, and a fuzzy parameter self-tuning method is designed to adjust the parameters of the feedback control law according to the error of the voltage regulation. Moreover, a novel sliding mode controller is incorporated into the reference current tracking control. The contribution of this paper can be summarized as follows:

A novel sliding mode control is proposed for reference currents tracking to reduce the error. The designed APF has superior harmonic eliminating performance and minimizes the harmonics for a wide range of variations of load currents, and therefore an improved THD performance can be achieved.

A fuzzy parameter self-tuning feedback linearization control method is presented to implement the indirect current control of a three-phase three-wire active power filter. The parameters of the feedback control law involve tuning according to the error between the DC side voltage and its reference value.

A novel unit signals calculating method is developed to implement the supply reference currents evaluation when the grid voltage contains harmonics. Thus, the designed active power filter could be used in a complex grid environment.

The paper is organized as follows. In section 2, a mathematical model of APF is derived from the principle of the average power balance. In section 3, the designed fuzzy parameter self-tuning feedback linearization control technique is presented and a novel integral sliding mode control is described as well. Detailed simulations and a comparison are reported in section 4. In section 5, the conclusion is presented.

2. Mathematical Model of the Active Power Filter

Suppose the supply voltage is

(V_{s p} \cos (w t), V_{s p} \cos (w t - 2 π / 3), V_{s p} \cos (w t - 4 π / 3)),

and define (R_ca, R_cb, R_cc) and (L_ca, L_cb, L_cc) as resistances and inductances in compensation circuits. The DC side capacitor is named C_dc and its voltage is V_dc. The supply currents, load currents and compensating currents are defined as (i_Sa, i_Sb, i_Sc), (i_La, i_Lb, i_Lc) and (i_ca, i_cb, i_cc) separately. As such, the schematic block diagram of the designed shunt active power filter is shown as in Figure 1.

Figure 1.

Overall diagram of the proposed SAPF system

The transformation formula from the abc frame to the dq frame is given as (2.2) [5]. For the three-phase sinusoidal signals, the component in phase abc is a constant while the quadrature component is zero. Thus, in order to compensate the harmonic in the load currents and make the sinusoidal supply currents, compensating currents should eliminate the quadrature component of the load currents. That is:

j I_{c q} = - j I_{L q}

(2.1)

[\begin{array}{l} I_{d} \\ I_{q} \end{array}] = \sqrt{2} [\begin{matrix} \begin{array}{l} \cos (θ - π / 6) \\ - \sin (θ - π / 6) \end{array} & \begin{array}{l} \sin θ \\ \cos θ \end{array} \end{matrix}] [\begin{matrix} I_{a} \\ I_{b} \end{matrix}]

(2.2)

The principle of average power balance is used to determine the approximate model of the compensator. The mathematical model is derived based on the following assumptions.

The supply voltages are balanced and contain small harmonics, such that the quadrature component of the supply voltage can be ignored.

Only the fundamental components of the currents are considered, as the harmonic components do not affect the average power balance expressions.

The supply impedance has been ignored, and all losses of the system are lumped and represented by an equivalent resistance R_c.

IGBT is an ideal switch inverter.

The load current in the dq frame can be described as I_L = I_Ld + j^I_Lq, while the compensating current in the dq frame is I_c = I_cd + j^I_cq. According to formula (2.1), the supply current is:

I_{S} = I_{L d} + j I_{L q} + I_{c d} + j I_{c q} = I_{L d} + I_{c d}

(2.3)

As shown in figure 1, the voltage at the SAPF connection points is equal to the supply voltage. That is:

{\begin{cases} v_{a} = V_{S p} \cos (w t) \\ v_{b} = V_{S p} \cos (w t - 2 π / 3) \\ v_{c} = V_{S p} \cos (w t - 4 π / 3) \end{cases}

(2.4)

By synchronous frame transforming, formula (2.5) can be derived:

{\begin{cases} v_{c o m d} = \sqrt{\frac{3}{2}} V_{S p} \\ v_{c o m q} = 0 \end{cases}

(2.5)

The compensator power is:

P_{c o m} = v_{c o m d} I_{c d} + v_{c o m q} I_{c q} = v_{c o m d} I_{c d}

(2.6)

If, for a particular operating point, I_cq is constant, then the inductance power is:

P_{L} = n \frac{d}{d t} [\frac{1}{2} L_{c} I_{c d}^{2}] n = 3

(2.7)

The power loss in the resistor is given by formula (2.8):

P_{R} = n R_{c} (I_{c d}^{2} + I_{c q}^{2}) n = 3

(2.8)

The capacitor average power is:

P_{C} = C_{d c} V_{d c} \frac{d V_{d c}}{d t}

(2.9)

where V_dc is the instantaneous DC side voltage.

According to the power balance principle of the compensator, the following equation can be established:

v_{c o m d} I_{c d} - n R_{c} (I_{c d}^{2} + I_{c q}^{2}) - n \frac{d}{d t} [\frac{1}{2} L_{c} I_{c d}^{2}] = C_{d c} V_{d c} \frac{d V_{d c}}{d t}

(2.10)

Then, the mathematical model of the active power filter can be derived:

\begin{array}{c} v_{c o n d} (I_{S} - I_{L d}) - n R_{c} [I_{S}^{2} - 2 I_{S} I_{L d} + I_{L d}^{2} + I_{c q}^{2}] - n L_{c} (I_{S} - I_{L d}) \frac{d I_{S}}{d t} \\ + n L_{c} (I_{S} - I_{L d}) \frac{d I_{L d}}{d t} = C_{d c} V_{d c} \frac{d V_{d c}}{d t} \end{array}

(2.11)

3. Fuzzy Parameter Self-tuning Feedback Linearization Control with a Novel Sliding Mode Control Method.

3.1 Feedback Linearization Control

Defining two state variables ${\begin{cases} x_{1} = I_{S} \\ x_{2} = V_{d c} \end{cases}$ and $u = \frac{d x_{1}}{d t}$ as input, then the mathematical model according to (2.3) can be written as:

\begin{array}{c} \overset{•}{x} = [\begin{array}{l} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} 0 \\ \frac{v_{c o n d} (x_{1} - I_{L d}) + n R_{c} (2 x_{1} I_{L d} - x_{1}^{2} - I_{L d}^{2} - I_{c q}^{2}) + n L_{c} (x_{1} - I_{L d}) \frac{d I_{L d}}{d t}}{C_{d c} x_{2}} \end{array}] + \\ [\begin{array}{l} {_{}}_{} {_{}}_{} {_{}}_{}_{} 1_{}_{} \\ \frac{n L_{c} (I_{L d} - x_{1})}{C_{d c} x_{2}} \end{array}] u \\ y = x_{2} \end{array}

(3.1)

The former system is a single input single output (SISO) system. It can be described as:

{\begin{cases} \overset{•}{x} = f (x) + g (x) u \\ y = h (x) \end{cases}

(3.2)

where the state variables are x ∊ R², f(x),g(x): R² →R², h(x): R² →R², and f(0) = 0 and h(0) = 0. Then:

\overset{•}{y} = \frac{\partial h}{\partial x} \overset{•}{x} = \frac{\partial h}{\partial x} f (x) + \frac{\partial h}{\partial x} g (x) u = f_{1} (x) + g_{1} (x) u

(3.3)

We design a feedback linearization control law:

u = \frac{R - f_{1} (x)}{g_{1} (x)}

(3.4)

Eq. (3.3) can be linearized as:

\overset{•}{y} = R

(3.5)

Supposing a position reference signal as y_d(t), and defining:

R = \overset{•}{y_{d}} - k_{1} (y - y_{d}) - k_{2} \int (y - y_{d}) d t,

(3.6)

where k₁ > 0 and k₂ > 0. Then, the Eq. (3.6) can be changed into:

\overset{•}{e} + k_{1} e + k_{2} \int e = 0, w h e r e e = y - y_{d}

(3.7)

The former formula (3.7) is the error dynamic equation and e(t_•) will exponentially converge on zero. If e(0) = e(0) = 0, e(t) will be always zero when t ≥ 0.

For the shunt active power filter, the reference value of the DC side voltage is constant, so:

{\overset{•}{y}}_{d} = 0 a n d R = - k_{1} (V_{d c} - V_{d c r e f}) - k_{2} \int (V_{d c} - V_{d c r e f})

(3.8)

We can deduce the feedback linearization control law as (3.9):

\begin{array}{c} u = \frac{v_{c o n d} (x_{1} - I_{L d}) - n R_{c} (2 x_{1} I_{L d} - x_{1}^{2} - I_{L d}^{2} - I_{c q}^{2}) + n L_{c} (x_{1} - I_{L d}) \frac{d I_{L d}}{d t}}{n L_{c} (x_{1} - I_{L d})} \\ - \frac{C_{d c} x_{2} [k_{1} (V_{d c r e f} - V_{d c}) + k_{2} \int (V_{d c r e f} - V_{d c}) d t]}{n L_{c} (x_{1} - I_{L d})} \end{array}

(3.9)

By integrating the control law $u = \frac{d x_{1}}{d t}$ , the amplitude of the supply reference currents I*_sp can be derived and applied to calculate the reference supply currents of the indirect current control method.

3.2 The Fuzzy Parameter Self-tuning Method

The feedback linearization control method implements an exact feedback control law. However, changes in the nonlinear load and system parameters in an actual power grid usually reduce the precision of the control and affect the harmonic compensating performance of the active power filter. A fuzzy parameter self-tuning method is applied to adjust the parameters of the feedback linearization control law.

The feedback linearization control method is used to implement the regulation of voltage in the DC side. The fuzzy parameter self-tuning method is designed to adjust the parameters of the feedback control law k₁ and k₂, according to the control error of the DC side voltage.

Define e as the control error of the DC side voltage and e_c as its velocity error. Choose the fuzzy set as {NL, NM, NS, ZO, PS, PM, PL}. The parameters of the feedback control law must be a positive number, so the fuzzy set of the output is defined as {S, L}. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform the mapping. The fuzzy rule can be expressed as:

IF e is A₁ and e_c is A₂, THEN k₁ is y₁ and k₂ is y₂.

The schematic block diagram of the fuzzy parameter self-tuning controller is shown in Figure 2 and the rule of the fuzzy parameter self-tuning control is described in Table 1.

Figure 2.

Schematic block diagram of the fuzzy parameter self-tuning controller

Table 1.

Table of fuzzy rules

e_c	e
NL	NM	NS	ZO	PS	PM	PL
NL	L/L	L/L	L/L	L/L	L/L	L/L	L/L
NM	S/L	L/L	L/S	L/S	L/S	L/L	S/L
NS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
ZO	S/L	S/L	S/L	L/S	S/L	S/L	S/L
PS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PM	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PL	S/L	L/L	L/S	L/S	L/S	L/L	S/L

3.3 Integral Position Tracking Sliding Mode Control

A novel sliding mode control method is proposed to reduce the THD of the supply currents and steady-state errors. In this paper, the integral sliding mode controller is applied to implement the inner reference current tracking control. The new sliding surface is proposed as (3.10):

S = k_{e} e + k_{d} \overset{•}{e} + k_{i} \int e

(3.10)

The voltage across the capacity is stabilized at 400V by feedback linearization control. The relationship between the input voltage of the active power filter and the designed switching function is shown as in (3.11), such that the voltages of a', b', c' obtained are constant [5]. The schematic diagram of the sliding mode controller is shown in Figure 3. The sliding mode control with a switch function is proposed as in (3.11):

Figure 3.

Schematic block diagram of the integral sliding mode control

u_{i'} = K_{i} u_{d c} sgn (S_{i}) K_{i} i s \frac{1}{3} o r \frac{2}{3}, i' = a', b', c', i = a, b, c

(3.11)

3.4 Calculation Method for the Reference Currents

The reference supply currents are produced by multiplying the amplitude signal I*_sp and the unit synchronized sinusoidal signals. However, in the actual grid the unit sinusoidal signals produced by a phase-locked loop (PLL) may appear as a wave distortion when the voltage contains harmonics. In this paper, a novel unit sinusoidal signal calculation method is presented.

The schematic block diagram of the novel unit synchronized sinusoidal signals calculation method is shown in Figure 4. The amplitude of the supply voltage is evaluated by synchronous frame transforming theory and the three-phase supply voltages are divided by the amplitude to calculate the unit supply voltage signals. Next, the PLL is applied to produce the unit synchronized sinusoidal signals (cos(wt),cos(wt − 2π / 3),cos(wt − 4π / 3). We multiply the amplitude and unit sinusoidal signals to produce the reference supply currents applied in the sliding mode controller.

Figure 4.

Schematic block diagram of the designed reference currents calculation method

4. Simulation Study

In this section, the active power filter system is implemented by a realistic model using the SimPowerSystem blockset of MATLAB. The nonlinear load consists of a three-phase universal bridge and a series RL branch. The components and parameters are listed in Table 2. In the actual power system, the supply voltage may contain harmonics. According to the electromagnetic compatibility (EMC) standards IEC-61000, the THD of the supply voltage should be under 5% in the low-voltage grid (≤1kV). The simulation study is implemented and a comparison is made between the unit sinusoidal signals' calculation by PLL and the designed method.

Table 2.
Components and parameters

Supply Voltage (Amplitude & Frequency) 220V 50HZ

Harmonic of Supply Voltage (Orders & Amplitude) 20^th
8

50^th
4

Inductance & Resistance in Nonlinear Load 1 2e-3H 5Ω

Inductance & Resistance in Nonlinear Load 2 2e-3H 20Ω

Inductance & Resistance in the Compensator 3.3e-4H 0.05Ω

DC side Capacitor 5e-3F

Reference Value of the DC Side Voltage 400V

Parameter of Feedback Linearization Control 1

Parameters of the Sliding Mode Controller 1, 9.1e-7, 6e-2

Supply Voltage (Amplitude & Frequency)	220V 50HZ
Harmonic of Supply Voltage (Orders & Amplitude)	20^th 8
50^th 4
Inductance & Resistance in Nonlinear Load 1	2e-3H 5Ω
Inductance & Resistance in Nonlinear Load 2	2e-3H 20Ω
Inductance & Resistance in the Compensator	3.3e-4H 0.05Ω
DC side Capacitor	5e-3F
Reference Value of the DC Side Voltage	400V
Parameter of Feedback Linearization Control	1
Parameters of the Sliding Mode Controller	1, 9.1e-7, 6e-2

Supposing the supply voltage contains 20^th and 50^th harmonics, the THD of supply voltage is 8.91% given harmonic analysis. As unit sinusoidal signals derived by the traditional sinusoidal signals calculation method appear distorted and cannot be used to calculate the supply reference currents, the novel method based on synchronous frame transforming theory is applied. The wave of the unit sinusoidal signals calculated by the designed method is shown in Figure 5.

Figure 5.

Wave of the unit sinusoidal signals calculated by the designed method

Obviously, in actual power systems the nonlinear loads change with any change of user. Thus, the simulation is carried out with mutative loads. An extra nonlinear load is connected into the system at simulation time t=0.2s and cut off at t=0.4s so as to analyse the robustness to the load disturbance.

The simulation comparisons among feedback linearization-controlled APF, feedback linearization sliding mode-controlled APF and the fuzzy parameter self-tuning feedback linearization sliding mode-controlled APF are presented in Table 3. The THD of the load currents when t=0.1s, 0.3s and 0.5s are, respectively, 21.87%, 20.51% and 21.87%. Feedback linearization indirect current-controlled APF with the hysteresis comparison controller, the sliding mode controller and the fuzzy parameter self-tuning controller are each formed with the same components and parameters. The simulation results reveal that the THD of the supply currents is reduced to 3.81%, 3.79% and 3.53% by feedback linearization indirect current-controlled APF with the hysteresis comparison controller. This is decreased to 3.18%, 3.46% and 3.26% after the designed sliding mode controller is applied to the system. The THD of supply current following compensation by the feedback linearization sliding mode-controlled APF with the designed fuzzy controller is 3.22%, 3.47% and 3.25%, respectively, and the overshoot of the DC side voltage is reduced from 42% to 38.4%, as shown in Figure 9 and Figure 10. The harmonic analysis of the supply current is shown in Figures 11–19.

Table 3.

Comparison of compensation effects

Time	Feedback linearization hysteresis comparison-controlled APF	Feedback linearization sliding mode-controlled APF	Feedback linearization sliding mode-controlled APF with fuzzy controller
0.1s	3.81%	3.18%	3.22%
0.3s	3.79%	3.46%	3.47%
0.5s	3.53%	3.26%	3.25%

Figure 6.

Amplitude of the reference supply current

Figure 7.

Enlarged view of the load current and supply current during A-phase

Figure 8.

Reference current tacking error during A-phase

Figure 9.

Voltage of the DC side capacity with feedback linearization sliding mode control

Figure 10.

Voltage of the DC side capacity with fuzzy feedback linearization sliding mode control

Figure 11.

t=0.1s, THD of the supply current compensated by feedback linearization-controlled APF

Figure 12.

t=0.3s, THD of the supply current compensated by feedback linearization-controlled APF

Figure 13.

t=0.5s, THD of the supply current compensated by feedback linearization-controlled APF

Figure 14.

t=0.1s, THD of the supply current compensated by feedback linearization SMC APF

Figure 15.

t=0.3s, THD of the supply current compensated by feedback linearization SMC APF

Figure 16.

t=0.5s, THD of the supply current compensated by feedback linearization SMC APF

Figure 17.

t=0.1s, THD of the supply current compensated by fuzzy feedback linearization SMC APF

Figure 18.

t=0.3s, THD of the supply current compensated by fuzzy feedback linearization SMC APF

Figure 19.

t=0.5s, THD of the supply current compensated by fuzzy feedback linearization SMC APF

The static and dynamic performances of the designed fuzzy parameter self-tuning feedback linearization sliding mode-controlled APF are described in Figures 6–8. Figure 6 shows the evaluated value of the amplitude signal I*_Sp used to calculate the reference supply currents. The amplitude signal is equal with the amplitude of load currents by a short-time adjustment. The waves of the load currents and the supply currents are shown in Figure 7. The reference current tracking error, which is shown in Figure 8, will become a little bigger when the load current increases. The DC side voltage, shown in Figure 10, is regulated by the feedback linearization control theory, which presents good dynamic and static characteristics. When load change happens, the voltage converges on the given value with a short-time adjustment.

5. Conclusion

In this paper, a fuzzy parameter self-tuning feedback linearization control has been applied to implement the DC side voltage regulation in an indirect current-controlled active power filter. A novel sliding mode controller has also been designed and applied in the reference current tracking control. In the actual power system, the supply voltage may contain harmonics, which limits the application of the APF. A novel unit synchronous sinusoidal signals calculation method based upon synchronous frame transforming theory is designed. The simulation results verify that the sinusoidal signals calculation method can produce ideal synchronous sinusoidal signals, even when THD of the supply voltage is high. The fuzzy parameter self-tuning feedback linearization control method can reduce the overshoot of the DC side voltage regulation and the designed sliding mode controller can reduce the THD of the supply currents dramatically. A modelling approach of APF based on feedback linearization theory can be applied to develop an adaptive sliding mode control or the intelligent control of an APF system.

Footnotes

6. Acknowledgments

This work is supported by the National Science Foundation of China under Grant No. 61374100 and the Fundamental Research Funds for the Central Universities under Grant No. 2612012B06714.

References

Saetieo

Devaraj

Torrey

. The design and implementation of a three-phase active power filter based on sliding mode control, IEEE Trans. on Industry Applications, 31(5), pp. 993–1000, 1995.

Haddad

Chandra

. A new control approach to three-phase active power filter for harmonics and reactive power compensation, IEEE Trans. on Power Systems, 13(1): 133–138, 1998.

Stanciu

Teodorescu

Florescu

Stoichescu

. Single-phase active power filter with improved sliding mode control, Proceedings of the 17th IEEE International Conference on Automation, Quality and Test, Robotics, pp. 15–19, 2010.

Fei

Wang

Juan

. A novel sliding mode control technique for indirect current controlled active power filter, Mathematical Problems in Engineering, Volumn 2012, Article ID 549782,18 pages,2012.

Mendalek

Haddad

. Modeling and nonlinear control of shunt active power filter in the synchronous reference frame, Proceedings of 9th International Conference on Harmonics and Quality of Power, 1, pp. 30–35, 2000.

Mendalek

Haddad

Fnaiech

Dessaint

. Nonlinear control technique to enhance dynamic performance of a shunt active power filter, IEEE Proc-Electr. Power Application, 150(4), pp. 373–379, 2003.

Matas

Vicuna

Miret

. Feedback linearization of a single-phase active power filter via sliding mode control, IEEE Trans. on Power Electronics, 23(1), pp. 116–125,2008.

Shyu

Yang

Chen

Lin

. Model reference adaptive control design for a shunt active power filter system, IEEE Trans. on Industrial Electronics, 55(1), pp. 73–78, 2008.

Ribeiro

Azevedo

Sousa

. A robust adaptive control strategy of active power filters for power-factor correction, harmonic compensation, and balancing of nonlinear loads, IEEE Trans. on Power Electronics, 27(2), pp. 718–730, 2012.

10.

Lee

. DC-bus voltage control of three-phase AC/DC PWM converters using feedback linearization, IEEE Trans. on Industry Applications, 36(3), pp. 826–833, 2000.

11.

Tan

Lai

Tse

Cheng

. Adaptive feedforward and feedback control schemes for sliding mode controlled power converters, IEEE Trans on Power Electronics, 21(1), pp. 182–192, 2006.

12.

Fei

Juan

. An adaptive fuzzy control approach for robust tracking of a MEMS gyroscope sensor, International Journal of Advanced Robotic Systems, 8(5), pp. 125–133, 2011.

13.

Carranza

Medina

Chang

. Real-time shunt active power filter compensation, IEEE Trans. on Power Delivery, 23(4), pp. 2623–2625, 2008.

14.

Santos

Jacobina

Dias

Rocha

. Single-phase to three-phase universal active power filter, IEEE Trans. on Power Delivery, 26(3), pp. 1361–1371, 2011.

15.

Rahmani

Mendalek

Haddad

. Experimental design of a nonlinear control technique for three-phase shunt active power filter, IEEE Trans. on Industrial Electronics, 57(10), pp. 3364–3375, 2010.

16.

Rahman

Haddad

Kanaan

. A comparative study of shunt hybrid and shunt active power filters for single-phase applications: simulation and experimental validation. Mathematics and Computers in Simulation, 71(4–6), pp. 345–359, 2006.

17.

Lee

. Fuzzy vector field orientation feedback control-based slip compensation for trajectory tracking control of a four track wheel skid-steered mobile robot, International Journal of Advanced Robotic Systems, 10(218), pp. 1–14, 2013.

18.

Tsengence

Georgios

. Shunt active power filter control using fuzzy logic controllers, 2011 IEEE International Symposium on Industrial Electronics, pp. 365–371, 2011.

19.

Colak

Bayindir

Kaplan

Tas

, DC bus voltage regulation of an active power filter using a fuzzy logic controller, 2010 Ninth International Conference on Machine Learning and Application, pp. 692–696, 2010.

e_c	e
NL	NM	NS	ZO	PS	PM	PL
NL	L/L	L/L	L/L	L/L	L/L	L/L	L/L
NM	S/L	L/L	L/S	L/S	L/S	L/L	S/L
NS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
ZO	S/L	S/L	S/L	L/S	S/L	S/L	S/L
PS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PM	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PL	S/L	L/L	L/S	L/S	L/S	L/L	S/L

e_c	e
NL	NM	NS	ZO	PS	PM	PL
NL	L/L	L/L	L/L	L/L	L/L	L/L	L/L
NM	S/L	L/L	L/S	L/S	L/S	L/L	S/L
NS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
ZO	S/L	S/L	S/L	L/S	S/L	S/L	S/L
PS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PM	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PL	S/L	L/L	L/S	L/S	L/S	L/L	S/L

e_c	e
NL	NM	NS	ZO	PS	PM	PL
NL	L/L	L/L	L/L	L/L	L/L	L/L	L/L
NM	S/L	L/L	L/S	L/S	L/S	L/L	S/L
NS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
ZO	S/L	S/L	S/L	L/S	S/L	S/L	S/L
PS	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PM	S/L	S/L	L/L	L/S	L/L	S/L	S/L
PL	S/L	L/L	L/S	L/S	L/S	L/L	S/L